thesis-setup of a stable high-resolution laser system

Nils Nemitz

Setup of a stablehigh-resolution laser system

Diploma Thesis

March 2004

University of Stuttgart5th Institute of Physics

Chair of Professor Dr. Tilman Pfau

2

Nils Nemitz · Setup of a stable high-resolution laser system ·Diploma thesis · 5th Institute of Physics · Chair of ProfessorTilman Pfau · University of Stuttgart · First edition · March2004

Nils Nemitz · Aufbau eines stabilen hochauflosenden Lasersy-stems · Diplomarbeit · Funftes Physikalisches Institut · LehrstuhlProf. Dr. Tilman Pfau · Universitat Stuttgart · Erstabgabe ·Marz2004

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Scope of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . 9

2.1.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Zeeman-effect and Hyperfine Structure . . . . . . . . . 10

2.1.3 A Trap for Neutral Atoms . . . . . . . . . . . . . . . . 11

2.1.4 Equations and Limitations . . . . . . . . . . . . . . . . 12

2.1.5 Cooling Rubidium . . . . . . . . . . . . . . . . . . . . 13

2.1.6 Cooling Ytterbium . . . . . . . . . . . . . . . . . . . . 15

2.2 Cavities: Resonance and Transmission . . . . . . . . . . . . . . 16

2.2.1 Optical Resonators . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Confocal Cavities . . . . . . . . . . . . . . . . . . . . . 20

2.3 Lock Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Direct Locking to a Fringe . . . . . . . . . . . . . . . . 24

2.3.2 Traditional Hansch-Couillaud Lock . . . . . . . . . . . 24

2.3.3 Transmission Hansch-Couillaud Lock . . . . . . . . . . 28

2.3.3.1 Basics . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3.2 Theory . . . . . . . . . . . . . . . . . . . . . 28

2.4 Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 The Master Laser . . . . . . . . . . . . . . . . . . . . . 33

2.4.1.1 Extended Cavity Laser Diodes . . . . . . . . 33

2.4.1.2 Spectroscopy . . . . . . . . . . . . . . . . . . 34

4 Contents

2.4.2 Injection Locking . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 Modulated Diode Lasers . . . . . . . . . . . . . . . . . 40

2.4.3.1 Current-Dependency of Diode Laser Output . 40

2.4.3.2 Output of a Modulated Laser Diode . . . . . 42

2.4.3.3 Modulating an Injection-locked Laser . . . . . 42

2.4.3.4 Extending the Theoretical Model . . . . . . . 46

2.4.3.5 Vestigial Sideband Operation . . . . . . . . . 47

2.4.3.6 Sideband Injection . . . . . . . . . . . . . . . 52

3. Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Master Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Slave Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Modulation System . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Green Laser System . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Stabilization Cavity . . . . . . . . . . . . . . . . . . . . . . . . 62

3.7 Analyzer Assemblies . . . . . . . . . . . . . . . . . . . . . . . 63

4. System Characteristics . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 About Linewidths . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Stability of the Master Laser . . . . . . . . . . . . . . . . . . . 68

4.3 Noise Introduced by the Slave Laser . . . . . . . . . . . . . . . 69

4.4 Performance of Modulation Electronics . . . . . . . . . . . . . 71

4.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2 Tuning Range and Power . . . . . . . . . . . . . . . . . 74

4.5 Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Lock Signals - Cavity . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Lock Signals - Dye Laser . . . . . . . . . . . . . . . . . . . . . 85

4.8 The System as a Whole . . . . . . . . . . . . . . . . . . . . . 89

5. Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1 Ytterbium Spectroscopy . . . . . . . . . . . . . . . . . . . . . 91

5.2 Double-Locked Operation . . . . . . . . . . . . . . . . . . . . 93

Contents 5

6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Things to do . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Appendices 98

A. Schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.1 Radio-frequency Amplifier . . . . . . . . . . . . . . . . . . . . 99

A.2 VCO Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.3 Difference Amplifier . . . . . . . . . . . . . . . . . . . . . . . . 101

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

1. Introduction

1.1 Scope of this Thesis

This thesis deals with the construction of a laser system designed to providegreen laser light with a narrow linewidth and good long term wavelengthstability.

Special consideration has been given to keeping the whole system as simple aspossible and avoiding the use of expensive optical components while retainingthe advantage of not being limited to any specific frequency. A dye laser isused as a widely tuneable laser source. It is locked to an optical resonator toreach the required narrow linewidth.

The key idea in achieving long term stability was to stabilize the resonant fre-quency of this resonator using the reference provided by a laser diode lockedto a spectroscopy setup on rubidium. The free choice of operating wavelengthfor the dye laser is provided by introducing an adjustable frequency offsetto the reference laser by using a sideband-injected modulated slave laser asdescribed in [1].

1.2 Motivation

Since the Nobel Price in Physics for 2001 [2] was awarded to Eric Cornell,Wolfgang Ketterle and Carl Wieman “for the achievement of Bose-Einsteincondensation in dilute gases of alkali atoms, and for early fundamental studiesof the properties of the condensates”, the field dealing with matter in thisunusual state has been growing quickly.

Several BEC experiments are currently being undertaken at the University ofStuttgart. This thesis deals with one of them, which is designed to examinethe effects of mixing rubidium and ytterbium atoms at ultra-cold tempera-tures. Rubidium is one of the most commonly used elements for experimentslike this due to its level scheme, which is well suited for optical trapping andcooling while being easily accessible with inexpensive semiconductor diodelasers. Ytterbium was chosen because it has both fermionic and bosonic isoto-pes, which will increase the variety of experiments that can be done, while alsooffering a suitable level structure for laser cooling. Trapped ytterbium is alsobeing used in experiments investigating parity non-conservation (PNC) [3,4],permanent electric dipole moment (EDM) in elemental particles [5,6] and the

8 1. Introduction

possibility of creating a precise optical clock [7]. Bose-Einstein-Condensationwith ytterbium has also been achieved recently [8].

Ytterbium has two transition lines that are useful for laser cooling, but bothhave disadvantages. The transition at 398.3nm is quite broad with a linewidthof Γ = 2π · 28MHz [9] which makes laser stability unproblematic, but theachievable temperature is limited by the Doppler temperature TD = 672µK.For this reason, the laser system described here was designed to use the muchnarrower transition at 555.8nm, where Γ = 2π · 180kHz and TD = 4.4µK.To take full advantage of this, the laser will need a linewidth and stabilitycomparable to this natural linwdith. This is not easily achieved, since thelow saturation intensity makes fast high-resolution spectroscopy difficult.

1.3 Definitions

Some of the terms in the following chapters are used with specific meaningsthroughout this thesis. These will be clarified here.

red light The 780.233nm light from a laser diode locked to a spectroscopysystem which serves as a stable frequency reference as well as thatemitted by the slave laser diode.

red Components using the red light will be referred to as red, e.g. the “redlock” that stabilizes the cavity using the stable reference beam.

green Any components using or generating the green 555.80nm light usedfor trapping ytterbium, e.g. the green MOT.

reflectivity (R) The fraction of the incident intensity reflected by a mirroror other optical component.

transmission (T) The fraction of the incident intensity transmittedthrough a mirror or other optical component. For lossless systemsR + T = 1.

(electric) field reflectivity (r) The amplitude of the electric fields in thereflected beam from an optical component compared to the fields of theincident beam: Eref = r ·Ein. Phase shifts on reflection can be encodedby using complex numbers for the field reflectivity coefficient r. Relatesto the reflectivity as R = |r|2.

(electric) field transmission (t) Transmission equivalent of the field re-flectivity: Etrans = t · Ein, T = |t|2. Obviously r + t 6= 1 in general.

2. Theoretical Foundations

2.1 Laser Cooling and Trapping

Trapping and cooling atoms with lasers was first proposed by Ashkin, Hanschand Schawlow [10, 11] in the 1970s. The first working magneto-optical trap(MOT), that could both trap and cool atoms was reported in 1987 [12].

Since the laser system described here is being set up to operate a magneto-optical trap for ytterbium, this section will provide a basic outline of theoperation and requirements of such a trap.

2.1.1 Doppler Cooling

Since photons carry an impulse of pphoton = ~k, their absorption and emissionaffects atomic movement. If the corresponding probabilities can be properlycontrolled, this can be used to cool and trap atoms.

In a Doppler-cooling setup, several red-detuned laser beams are directed atthe atoms. An atom can absorp a photon from any of the beams, but theprobabilities depends on the amount of detuning. If the atom is moving itwill see the beams Doppler shifted according to

ωD = −~k~v (2.1)

for non-relativistic speeds. Here ~k is the wave vector of the laser beam and~v the velocity of the atom. If the atom moves towards the source of the laserbeam, the Doppler shift will reduce the detuning. The result of this is thatthe atom is more likely to absorp a photon travelling against its direction ofmovement, which will reduce its velocity.

Before this process can repeat itself, the atom has to return to its groundstate. This can happen by either stimulated or spontaneous emission of aphoton. Stimulated emission is most likely to result in a photon identical tothe originally absorped one. This would return the atom to its original velo-city and therefore would not contribute to the cooling process. Spontaneousemission, however, will occur in a random direction and while each individualphoton emitted will change the atom’s momentum by pphot = ~~k, the effectstend to cancel out, resulting in a vanishing net effect.

In this approximation, which treats the repeated absorption and reemissionas one continuous process, the total effects can be described as the so-called

10 2. Theoretical Foundations

spontaneous force [13, 14]~Fsp = ~~kγsc , (2.2)

where ~k is the wave vector of the laser and γsc is the scattering rate for theabsorption-spontaneous emission-cycle described above:

γsc = ηCGΓ

2

I/I0

1 + I/I0 + (2(δ + ωD)/Γ)2(2.3)

Here ηCG is the Clebsch-Gordan coefficient of the transition, I0 its saturationintensity and I and δ are the intensity and the detuning of the incident beam.Γ is the decay rate of the excited state and ωD is the Doppler shift describedabove.

Using a combination of beams from different directions results in a slowingforce for movement in any direction. This is called optical molasses and isusually done using a setup of six beams on three orthogonal axes. This cancool atoms, but since the force depends only on the velocity of the atoms andnot on their position, this alone is not enough to make a trap. The followingsections will describe how a position dependent component can be added byusing magnetic fields.

2.1.2 Zeeman-effect and Hyperfine Structure

Except for configurations with no magnetic dipole moment, atomic energylevels (determined by the atomic quantum numbers n, J and S) will split intoseveral components with different values for mS if a magnetic field is applied.These are separated by an energy difference

4EB = µB ·B · gJ ·mJ

= µB ·B ·mJ

(1 +

J(J + 1) + S(S + 1)− L(L + 1)

2J(J + 1)

)(2.4)

where gJ is the Lande g-factor (neglecting nuclear spin), which results fromthe mixing of orbital and spin angular momentum. A theoretical backgroundfor this can be found in many books dealing with the quantum mechanicaldescription of atoms. For a simple explanation of the involved factors see [15].

If sufficiently strong magnetic fields are used, this Zeeman splitting will havea much stronger effect than any hyperfine splitting that might be present.This will result in several groups of equidistant transition lines as shown infigure 2.1.2, some of which may be degenerate in what is called the “normal”Zeeman effect.

For weaker magnetic fields (usually ¿ 0.1T ) the hyperfine structure, if pre-sent, will still be the dominating feature and the Zeeman effect will causean additional splitting of the hyperfine levels. Since the relative alignmentof electronic ( ~J) and core (~I) angular momenta is the distinguishing factorbetween the hyperfine structure levels, the only free parameter when looking

2.1. Laser Cooling and Trapping 11

Fig. 2.1: Splitting of energy levels due to the Zeeman effect and itseffect in transition lines. Left: 3 spectral lines due to energeticallydegenerate transitions in the so-called normal Zeeman effect, Right:6 lines for anomalous Zeeman effect in a level scheme as found inrubidium. From [15], p.124.

at one particular line is the projection mF . For any F there will be 2F+1 pos-sibilities for the projection to the quantization axis defined by the externalfield. Equation 2.4 can be adapted to this case by using mF and the modifiedg-factors gF instead of mJ and gJ . Values for the relevant levels in 87Rb areshown in figure 2.3.

Since the splitting of the levels depends on the total angular momentum of thevarious states, transitions require specific changes in angular momentum. In asimple emission or absorption process, this has to be provided by the photon,which always carries a spin of 1. This gives rise to a set of selection rules, themost basic of which say that F and mF can change by a maximum value of 1for absorption or emission of a single photon. Using circularly polarized lightcorresponding to photons with a defined projected spin, it is possible to drivecertain transitions deliberately. A σ+ photon, with a clockwise polarizationrelative to the magnetic field, has a spin mphoton = +1. If it is absorbed,it will cause a change in angular momentum 4mF = +1. An absorbed σ−

photon will cause a 4mF = −1 transition.

2.1.3 A Trap for Neutral Atoms

By using a combination of Doppler effect and Zeeman splitting, it is possibleto create a magneto-optical trap that both holds the atoms in one place andcools them. In the following this shall be described for the one-dimensionalcase. With minor adaptions the same scheme works in 3D as well.


Two slightly red-detuned laser beams are sent into the trap region from bothsides, creating a Doppler cooling setup as described before. It can be turnedinto a trap with a spatially restoring force by applying an inhomogeneousmagnetic field with a zero value in the trap center, causing a Zeeman shiftthat increases with the distance to the trap center. For the simplest case,where the ground state has zero angular momentum and the excited statehas J = 1 (or Z = 1), this is shown in figure 2.2.

Fig. 2.2: Zeeman-split energy levels in a MOT. The laser is runningat a frequency of ωlaser, which has a detuning−δ0 from the unshiftedtransition frequency ω0 at z = 0. For atoms not in the center of thetrap the Zeeman shift reduces the detuning for the beam whichwill push the atom back towards the center to δ∗ while bringing theother beam even further from resonance. rMOT is the capture radiusat which the preferably absorbed beam passes through resonance.From this point outwards detuning for both beams will increase.

If the incident beams have the appropriate circular polarizations with regardto the z axis, the beam coming from the left can only drive the transitionsthat are close to resonance left of the origin and vice versa, resulting in arestoring force.

2.1.4 Equations and Limitations

Leaving details to the literature [14], some equations for the MOT shall begiven anyway to illustrate the limitations and requirements.

In the low intensity limit, where the influence of stimulated emission can beignored and assuming that all processes are so slow that many absorption-emission cycles smooth out the random walk of the individual processes, the


radiative force is given by

~F± = ±~~kγ

2· s0

1 + s0 + (2δ±/γ)2, (2.5)

where s0 = I/Is is the on-resonance saturation parameter, describing theintensity I of the incident light compared to the saturation intensity Is cha-racteristic for the transition, ~k is the light’s k-vector and γ = 1/τ is thetransition linewidth. The detuning δ± incorporates the effects of both Zee-man and Doppler shift:

δ± = δ ∓ ~k · ~v ± µ′B(~r)/~ (2.6)

Here δ is the detuning from the unshifted resonance, B(~r) is the magneticfield at the current position and the effective magnetic moment µ′ includesangular momenta and g-factors for both ground and excited state:

µ′ = (geme − ggmg)µB (2.7)

For small Doppler and Zeeman shifts this can be linearized to the form

~F = −β~v − κ~r , (2.8)

which is the well known form for a damped oscillator. In a magneto-opticaltrap the damping constant β turns out too be greater than the spring con-stant γ resulting in an overdamped oscillation.

This equation seems to imply that the atoms can be slowed to a completestop, but unfortunately there are limits to that. When the atoms get tooslow, the transition linewidth becomes large in comparison to the dopplershifts. This evens out the probabilities to absorb a photon from the “right”or “wrong” beam, lowering the effectiveness of the cooling mechanism untilthe fluctuations introduced by the random-walk nature of the absorption-emission-cycle lead to a lower temperature limit.

This is the Doppler temperature TD given by

kBTD =~γ2

(2.9)

This is directly proportional to the linewidth, which will become importantin the next section.

2.1.5 Cooling Rubidium

Although rubidium is not in the focus of this work, a look at the coolingsystem used in the experiment is interesting nonetheless. Besides providinga reference to compare the green system to, its laser system is used as thestable frequency reference in the setup for ytterbium.


Rubidium can be cooled on the 5S1/2(F=2)→ 5P3/2(F’=3) transition, thelevel scheme of which is given in figure 2.3. A complete theoretical descriptionwould take an inappropriate amount of space here, as the atoms will populate5 states with different projections of angular momentum in the lower stateand 7 in the excited state. And since the Lande-factors differ for both states,there will be almost no degeneracy except for the zero-field state in thetrap center. But since this mostly affects the spring constant in the dampedoscillator approximation described by equation 2.8, the atoms will still feeltrapping and cooling forces.

Fig. 2.3: Relevant energy levels in 87Rb together with transitionsused in the experiment. From [16,17]

A bigger problem is that the linewidth γ/2π ≈ 5.9MHz is not particularlysmall compared to the spacing of the hyperfine energy levels. Especially sincethe other transitions see Zeeman and Doppler shifts as well, this opens upa loss channel where atoms can be excited to the F’=2 state instead, fromwhere the selection rules allow them to fall back to the F=1 ground state. Dueto the strong effect of the nuclear spin on S-states, the hyperfine splitting inthe ground state is so big that the atoms cannot be excited by the MOT-laseranymore.

To reduce the effects of this loss, a “repumper” laser is used, driving5S1/2(F = 1) → 5P3/2(F

′ = 2). This will keep exciting the lost atoms untilthey fall back into the F=2 ground state.


2.1.6 Cooling Ytterbium

While the laser system, which is the main focus of this thesis, is not limitedto operation at a specific wavelength, its motivation is the cooling of ytter-bium atoms to low temperatures. Since this defines the desired stability andlinewidth, this section will give some data for ytterbium as far as it relatesto optical trapping and cooling.

The energy level scheme (see figure 2.4) of the most abundant ytterbiumisotope 174Yb is simpler than for rubidium due to the lack of nuclear spin. Infact, there are two transitions that can be used for a MOT, both without theneed for repumping and with a Zeeman splitting that works as illustrated infigure 2.2.

Fig. 2.4: Relevant energy levels in 174Yb together with transitionsused in the experiment.From [9,18]

The difficulties lie in the linewidths of the transitions. The blue 1S0 → 1P1

transition with γ =28.0MHz·2π is very broad, allowing for easy constructionof optical systems of adequate stability and good capture parameters at thecost of a high Doppler limited temperature of 672µK.

The green 1S0 → 3P1 transition on the other hand is very narrow at γ =182kHz · 2π, allowing for a very low Doppler limit on temperature of only4.4µK. The experiment described has been set up from the beginning to be


able to provide sufficiently slow and cold atom that can be captured even bya MOT on the green transition. Stabilizing a narrow-linewidth laser to thewavelength of this transition will be central topic in the following.

One additional feature of ytterbium is that it has seven stable isotopes thatdiffer in nuclear spin, some of them having bosonic and some fermionic cha-racter. The differences between the transition frequencies of the various iso-topes are on the order of GHz. This has the advantage that a specific speciescan be selected for an experiment simply by adjusting the laser frequenciesin the trapping and cooling system. A spectrum taken in our experiment isshown as figure 2.5 in the chapter on ytterbium spectroscopy.

0,5

1

1,5

2

2,5

3

3,5

-1000 -750 -500 -250 0 250 500 750 1000 1250 1500 1750

relative frequency [MHz]

flu

ore

sc

en

ce

sig

na

l

176

173F=5/2

174

171F=3/2 170

171F=1/2

172

173F=3/2

F=7/2

Fig. 2.5: Isotope shifts for the 399nm transition in ytterbium. Mea-sured fluorescence signal in arbitrary units over frequency relativeto the transition in 174Y b. Different isotopes and spin states arelabeled.

2.2 Cavities: Resonance and Transmission

Although there is no fundamental difference between light and other kindsof electro-magnetic waves, the fact that optical frequencies run in the rangeof 500THz makes it impossible to directly measure phase and frequency.Optical detectors will generally only measure the light’s intensity, which isproportional to the squared amplitude of the electric field in the EM wave.

2.2.1 Optical Resonators

Optical resonators or cavities are one commonly used way to overcome thisobstacle. In its most basic form, such a resonator consists of two parallel

2.2. Cavities: Resonance and Transmission 17

mirrors, bouncing the light back and forth between them as illustrated inFigure 2.6. In reality such a configuration is unstable even for perfectly ali-gned mirrors, since diffraction effects will always cause the beam to diverge.Using curved mirrors can compensate this, but in the interest of keeping thetheory simple for the beginning, diffraction and misalignment effects will beignored for now.

Fig. 2.6: Light in a resonator consisting of parallel mirrors. Drawnwith non-orthogonal incidence for clarity.

In order to understand how the beams transmitted or reflected by the cavitydepend on the mirror spacing and the wavelength, we will follow the lightfrom the point where it hits the first mirror (A). Here we have an electric fieldamplitude Ein, a part of which is transmitted into the resonator resulting inan initial electric field

EA0 = Ein · tA , (2.10)

where tA is the mirror’s electric field transmission. The light will now travelthrough the cavity, be partially reflected at mirror B and return to mirror Ato be reflected again. At the beginning of the next cycle the remaining electricfield amplitude will be EA1 = rA · rB ·EA0, with rA and rB being the electricfield reflectivities for the mirrors. It is important to note that in contrast tothe intensity reflectivities R, these need not be real and positive. In manysituations they will in fact be complex numbers, describing the phase jumpas well as the reflected amplitude:

r = |r| · e2π·i·4φ , (2.11)

where |r| is the reflected amplitude and 4φ is the phase jump that occursupon reflection. For the case of reflection on the interface to a medium witha higher index of refraction there will be a phase jump of 4φ = 2π [19],causing r to be real but negative. Reflection on the other side of the samesurface will not show a phase jump and therefore have a positive r.

To find the relative phase between the beam returning from the round tripand the beam just entering the cavity, it is easiest to keep the phase ofthe incoming beam constant and apply a phase propagation factor to thereflected beam:

EA(n+1) = rA · rB · e2πi 2dλ · EA(n) , (2.12)


where d is the mirror spacing and λ the wavelength of the light.

Summing over all EA(n)

EA =∞∑

n=0

Ein · tA ·(rA · rB · e2πi 2d

λ

)n

(2.13)

is possible by using the equality [20]

∞∑n=0

an =1

1− a(2.14)

(for |a| < 1), resulting in

EA = Ein · tA · 1

1−(rA · rB · e2πi 2d

λ

) . (2.15)

Now the fields for the reflected and transmitted beams can be found. For thetransmitted beam

Eout−B = EA · tB · e2πi dλ (2.16)

Eout−B = Ein · tA · tB · e2πi dλ


λ

) (2.17)

For the reflected beam the reflection of the incident beam has to be added tothe beam reflected one more time on mirror B and then leaving the resonatorthrough mirror A:

Eout−A = Ein · (−rA) + Ein · t2A · rB · e2πi 2dλ


λ

) (2.18)

The reflectivity for the beam that never enters the cavity is −rA, since ithappens on the other side of the same surface as the cavity internal reflectionsconsidered so far. This is obvious for the simple case of a reflection causedby a change in index of refraction, where the reflection remaining in the lowindex material will see a phase shift of π/2, but there will be no shift for thebeam on the high-index side.

But this relationship also holds for the general case. The phase jumps forreflection on both sides of the same surface always have to add up to π/2.This is equivalent to the additional minus sign in equation 2.18 and ensuresthat in resonance, the beam that was immediately reflected and the beamthat returns from the cavity have opposite signs and therefore tend to canceleach other out. If this was not the case, a resonant low-loss resonator wouldhave a total reflectivity greater than one, violating energy conservation asthe reflected beam would constantly have a higher intensity than the incidentbeam.


780.001 780.002 780.003 780.004wavelength

0.2

0.4

0.6

0.8

1

intensity

780.001 780.002 780.003 780.004wavelength

0.2

0.4

0.6

0.8

1

intensity

Fig. 2.7: Reflected (green) and transmitted (red) relative intensitiesfor a 100mm resonator depending on wavelength given in nm. Cal-culated as

(|Eout/Ein|2), using 2.18 and 2.17 for Eout. (top: R=0.95,

T=0.05; bottom: R=0.95, T=0.03)

It is also interesting to note that for any lossless configuration, the transmis-sion in resonance is always 100%, while the reflection falls to 0%.

For an assumed lossless cavity with a mirror spacing of 100mm and intensityreflectivities of 95% (RA = rA

2 = RB = rB2 = 0.95 ) the relative intensities

of the transmitted and reflected beams are plotted in the upper graph offigure 2.7. The reflectivities rA and rB are assumed to be real and negativehere. This eliminates any phase contribution from the reflections, as theproduct rA · rB in the denominator will be real and positive. For complexor mixed reflectivities this will not be the case, resulting in a shift of theresonance lines. However, as this does not affect the spacing of the lines, itwill be ignored here. In the simple case described initially, 2d

λneeds to be

an integer for resonance to occur. This corresponds to an equidistant set ofresonant frequencies

νres = n · c

2d, n ∈ N , (2.19)

where the frequency spacing is often referred to as the free spectral range(FSR) of the resonator:

4νFSR =c

2d(2.20)

Another important quantity is the FWHM linewidth of the resonance peakswhich for highly reflective mirrors in an otherwise perfect resonator can befound [21] to be

4ν 12−theor =

c

2d· 1− rA · rB

π√

rA · rB

(2.21)


in good approximation.

The ratio of free spectral range to linewidth is an indication of the opticalquality of the resonator and is commonly referred to as its finesse:

F =νFSR

4ν 12

(2.22)

For a perfect resonator the linewidth is given by equation 2.21. Therefore themaximum achievable finesse for a given set of mirrors is determined by

Fmax =νFSR

4ν 12−theor

=π√

rA · rB

1− rA · rB

(2.23)

For the values used in figure 2.7, Fmax ≈ 61.

In a real resonator, however, there will be additional losses. These will havetwo different effects. Any imperfection that reduces the amount of light re-turning to the first mirror after a round trip will show up as an additionalfactor (fint) in the rA · rB terms. This will result in a broadening of the re-sonance lines, reducing the finesse. Effects that reduce the amount of lightentering or leaving the resonator will lower the factors ta or tb in equations2.18 and 2.17. This will not reduce the width of the resonances, but insteadit will lower the signal amplitude.

Absorption in the silver mirrors used in the experiment is one factor thatwill cause this. Figure 2.7 illustrates this. Both diagrams plot transmittedand reflected intensities over the wavelength of the incident light. The up-per one is for lossless mirrors (R=0.95, T=0.05) and the lower one showsthe effect mirrors that absorp some of the incident light (R=0.95, T=0.03).The values used here correspond to mirrors coated with a 45nm silver layer.While the shape of the resonances remains unchanged, their amplitudes arereduced: The reflected intensity never drops below 15%, while the maximumtransmitted signal never reaches more than 40%.

The signal measured in the actual experiment show a mixture of these twoeffects. The resonance lines are wider than expected from the mirror reflecti-vities and the intensity of the transmitted beam is much lower than that ofthe incident beam, even in resonance.

2.2.2 Confocal Cavities

As mentioned before, a resonator with plane mirrors is unstable due to una-voidable diffraction effects. This will quickly cause the beam to diverge, whichcauses the beam to be clipped by mirrors or the resonator walls. These ad-ditional losses will result in a finesse that is lower than what the previouscalculations imply. Using suitably curved mirrors will prevent this by refo-cussing the beams upon each reflection.


The formalism of using Gaussian beams and matching the curvature of theirwavefronts to those of the mirrors is described in detail in almost every text-book on laser physics [21, 22] and shall not be explained here. Resonatorsconstructed in this way are called ‘stable’ and will virtually eliminate diffrac-tion losses as long as the mirrors are big compared to the spot sizes of thelaser beam.

For suitable mirror spacings even a beam that enters the cavity slightly off-axis or off-center will return to its point of entry after traversing the cavityseveral times [23].

Fig. 2.8: round trip in a confocal cavity for off-axis input

For the confocal cavity used in our experiment, the optical path is illustratedin Figure 2.8. In this configuration, where the focal points of the mirrorscoincide, a round trip for an off-axis beam consists of four passages throughthe cavity. Therefore the spacing between resonances will be

4νres =c

4d(2.24)

An equivalent, although less intuitive way to treat this is to find the resonantconditions for Hermite-Gaussian beams inside the cavity and then developthe incident beam in this basis. These calculations, which are done in detailin the book by Siegman [24], show that while the zeroth order Hermite-Gaussian modes still have resonances at νres = n · c

2d(n ∈ N) as in the case

of the parallel-mirror resonator, the higher transverse modes are offset byintegral multiples of 4ν = c

4d. Unless special care is taken to make sure the

incident beam matches the pure Gaussian mode, many transverse modes willbe excited. This results in a system of equidistant resonances equivalent toa parallel resonator of twice the mirror spacing, same as for the ray tracingapproach above.

In the following these additional complexities will be ignored, although theyare one of the limiting factors for the achievable linewidth in an imperfectlyaligned resonator as the transverse modes will not be completely degenerateanymore.

Another notable point is that since resonances occur twice as often, the freespectral range and therefore the finesse of a confocal resonator are half of


what they would be in a system with plane mirrors:

4νFSR-confocal =c

4d(2.25)

Using the picture of a roundtrip consisting of four passes through the cavity,it is possible to adapt the model of equations 2.17 and 2.18 to the newsituation. The field amplitude Eout−2 of the first transmitted beam (spot 2in figure 2.8) is very similar to that of the transmitted beam in the case ofparallel mirrors. Remembering equation 2.14, the terms in the denominatorcorrespond to the sum of contributions from all the complete roundtrips.Since both the distance travelled and the number of reflections are doubledcompared to the original situation, 2d

λin the phase propagation factor needs

to be replaced by 4dλ

and the factor rA ·rB by r2A ·r2

B. The latter will be referredto as the “roundtrip factor” f in later chapters. The terms appearing in theenumerator describe the changes in phase in amplitude and phase caused byentering the resonator through mirror A (tA), one additional passage through

the cavity (e2πi dλ ) in addition to the sum of complete roundtrips, and finally

leaving it through mirror B (tB). All of these are exactly the same as beforeand remain unchanged, resulting in the equation

Eout−2 = Ein · tA · tB · e2πi dλ

1−(r2A · r2

B · e2πi 4dλ

).

(2.26)

The electric field amplitude Eout−1 of the reflection from the cavity (spot 1in figure 2.8) is found by adding the beam that is reflected immediately bythe first mirror and that beam that returns from inside the cavity. The firstpart is still given by Ein · (−rA), while the second requires three reflectionsand four passes through the cavity now before it leaves the resonator again,in addition to the sum of complete roundtrips. This yields the equation

Eout−1 = Ein · (−rA) + Ein · t2A · rA · r2B · e2πi 4d

λ

1−(f · e2πi 4d

λ

) . (2.27)

Figure 2.9 illustrates the results of these equations. Since all sensors in ourexperiment generate signal proportional to the intensity instead of electricfield amplitude, the graphs plotted are all relative intensities found by using

Irel =

∣∣∣∣Eout

Ein

∣∣∣∣2

. (2.28)

Since there are now four possibilities for the light to leave the resonator, it isobvious that the transmitted intensity in each individual spot must be lowerthan for the parallel mirror setup. This is plotted here for spot 2, the outputsin spot 3 and spot 4 are similar with minimally lower intensities due to theadditional reflections that occur.

2.3. Lock Signal 23

It is important to note that in a confocal resonator, the reflected intensitywill never drop to zero, even for lossless mirrors. This is most easily under-stood in the picture of many transversal modes present in the resonator. Thecomplete pattern of resonance lines is actually the superposition of two setsof degenerate lines that are shifted against each other. Both sets can neverbe resonant at the same time, effectively halving the electric field amplitudethat can build up in the resonator compared to that in a cavity with par-allel mirrors. This also limits the field amplitude of the beam leaving theresonator to halve that of the incident beam, causing the cancellation to beincomplete. Even for lossless, highly reflecting mirrors, the remaining fieldamplitude will be one half of that in the incident beam, corresponding to a

relative intensity Irel =∣∣∣0.5Ein

Ein

∣∣∣2

= 0.25. The upper plot in figure 2.9 shows a

very similar value.

Introducing mirror losses has the same effect as described before. As long asthe reflectivities are kept the same, the width of the resonance lines remainsunchanged, but the signal amplitude decreases.

780.001 780.002 780.003 780.004wavelength

0.2

0.4

0.6

0.8

1

intensity

780.001 780.002 780.003 780.004wavelength

0.2

0.4

0.6

0.8

1

intensity

Fig. 2.9: Reflected (green) and transmitted (red) relative intensitiesfor a 100mm confocal resonator depending on wavelength given innm. (top: R=0.95, T=0.05; bottom: R=0.95, T=0.03)

2.3 Lock Signal

In our experiment, the cavity is locked to a stabilized diode laser, while ano-ther laser is locked to the cavity in turn. This chapter describes the methodused for locking and the reasons it was chosen.


2.3.1 Direct Locking to a Fringe

The simplest way to lock cavity or laser is to use the side of the transmissionfringe. Choosing a point where the intensity is about half of the maximumtransmitted intensity, the differences from this value can serve as error signalin a feedback loop. Unfortunately, the system will see any intensity fluctua-tions of the laser as errors and translate them into frequency fluctuations.Reducing this effect either requires active stabilization of the laser’s emittedintensity or increasing the slope of the resonance fringe by using a cavity ofhigher finesse. But doing so will increase the effect of the second shortcomingof this locking mechanism: Any disturbance that shifts the frequency by mo-re than one FWHM-linewidth of the resonance can push the system acrossthe maximum to the other edge of the fringe. Once it reaches the regionwhere the transmitted amplitude is lower than that at the original lockpoint,the error signal will take on the wrong sign, causing the system to becomeunlocked as illustrated in figure 2.10

Fig. 2.10: Stable and unstable regions for locking a laser to the sideof a resonator transmission fringe. A disturbance by as little as oneFWHM of the resonance can cause the laser to unlock, running offtowards the next resonance.

2.3.2 Traditional Hansch-Couillaud Lock

One way to reduce the influence of intensity fluctuations is to generate alocking signal that will not change its offset from zero for varying intensityfor at least one position, while at the same time offering a slope that can beused to create an error signal. This often takes the form shown in figure 2.11which is commonly called “dispersion shaped” due to its similarity to the

2.3. Lock Signal 25

structures found when plotting the index of refraction of a material overfrequency. It offers a good slope for stabilizing the system at the lockpointmarked in the diagram. Changes in intensity will ideally only scale the signal,leaving the frequency of the zero-crossing unaffected. Additionally, the systemwill return to the lockpoint for disturbances up to one half of the resonator’sfree spectral range.

Fig. 2.11: Stable and unstable regions for a dispersion shapedlocking signal.

A simple way to generate such a signal was first proposed by Hansch andCouillaud [25]. This uses the fact that similar to other systems that undergoforced oscillation, the phase of the light wave that builds up in the resona-tor will be lagging behind that of the incident beam if the frequency of thelatter is higher than that of the nearest cavity resonance, while for a drivingfrequency below the nearest resonance it will be advanced compared to it.Only at resonance and directly in the middle between two resonant frequen-cies will the phase difference be zero. Plotting the phase difference betweenthe incident beam and the resulting wave in the cavity over frequency wouldresult in a signal very similar to the one sketched in 2.11.

In order to generate an electronic signal from this, the Hansch-Couillaud se-tup uses a polarization filter inside the resonator. This might simply take theform of a glass plate inserted at Brewster’s angle to reduce transmission forthe vertical polarization as indicated in figure 2.12. The beam entering theresonator is linearly polarized at such an angle that it will have both verticaland horizontal components. A part of both polarizations is reflected back atthe first mirror, never entering the cavity. For the vertical polarization that isassumed to be blocked inside the resonator, the part that enters it is simplylost. The transmitted part of the other component, however, can excite oscil-lations as described in the previous section. For this horizontal polarization,the reflected beam also contains contributions from the light wave inside the


Fig. 2.12: Fictitious polarization components in a resonator witha polarizing filter. While the horizontal polarization (green) canpropagate normally inside the resonator, the part of the verticalpolarization that enters through the first mirror is reflected or ab-sorbed by the filter, preventing the build-up of a standing waveeven in resonance.

resonator,which will transfer some of the phase shifts mentioned above to theoutput beam.

Looking at the simplest case of a resonator with parallel mirrors or a confcocalcavity where only one transversal mode is excited, equation 2.18 describesthe reflected beam including the contribution from the resonator. For the twofictitious polarization components this yields

Ehor = Ein · (−rA) + Ein · t2A · rB · e2πi 2dλ


λ

) (2.29)

Ever = Ein · (−rA) (2.30)

Now the polarization that cannot propagate inside the resonator will serveas a phase reference for the other, orthogonal polarization. If there is nophase shift, then the resulting beam will still be linearly polarized, althoughthe axis of polarization will rotate slightly due to the different reflectivitiesfor the fictitious polarization components. In resonance the phase of thebeam returning from the cavity is π relative to the immediately reflectedone, resulting in a reduction of amplitude without changing the phase of theresulting beam:

Ehor = Ein · (−rA) + Ein · t2A · rB · 11− (rA · rB1)

(2.31)

Ever = Ein · (−rA) (2.32)

Outside of resonance, when the total optical path is not an integral numberof wavelengths, the phase factors e2πi 2d

λ will generally not be real, adding aphase shift to the horizontally polarized reflected beam that is not present inthe vertically polarized beam. This shift makes the total polarization of thereflected beam elliptical (see figure 2.13).

2.3. Lock Signal 27

Fig. 2.13: Left: Phase shifted polarization components as the ba-sis of elliptical polarization. Right: A quarter-wave plate con-verts linear to circular polarization and vice versa. Both illustra-tions from Georgia State University’s “Hyperphysics” web page athttp://hyperphysics.phy-astr.gsu.edu/hbase/hph.html.

This can now be analyzed with a detector consisting of a quarter-wave plateand a polarizing beam splitter. The quarter-wave plate will convert circularpolarization to linear polarization and, for proper alignment of the opticalaxis, linear to circular polarization as shown in the right diagram of figu-re 2.13. Depending on the handedness of the circular component, the axisof the resulting linear polarization will be aligned at either +45°or -45°fromthe optical axis. If the beam splitter is properly oriented along these axes, itwill output all incident power to one port if the light before the wave-platehad a pure right-handed circular polarization, and to the other port if it wasleft-handed circular polarized. Any remaining linear polarization can be de-scribed as a superposition of equal parts left- and right-handed polarizationand will be evenly split up between the output ports of the beam splitter.The whole setup is shown in figure2.14

Fig. 2.14: Top view of the analyzer assembly for detection of circu-larly polarized light. The quarter wave plate mixes the horizontal(green) and vertical (red) components of the incoming light and thebeam splitter cube splits it into the new linear components, whichare detected by photodiodes.


Measuring the intensities with photodiodes and taking the difference bet-ween both channels will eliminate this contribution and result in a singledispersion-shaped signal. The laser can now be locked to the zero-crossingthat occurs at resonance, since any deviation will create an error signal thatnear resonance is approximately proportional to the deviation.

2.3.3 Transmission Hansch-Couillaud Lock

2.3.3.1 Basics

The system described above works very well in most cases, but it has severaldrawbacks for the application in our experiment.

Since the absorption losses for the silver coated mirror used are rather large,the amplitude of the light returning from the cavity is low compared to thatof the direct reflection. This degrades the signal-to-noise ratio considerably,since the noise caused by amplitude and polarization noise of the laser inconjunction with imperfect components in the analyzer is proportional tothe total intensity.

In a near-lossless resonator, the intensity of the reflected light will drop atresonance, as the beam reemerging from the cavity interferes destructivelywith the directly reflected one. As our setup uses a modulation technique(see section 3.4 to add sidebands to the electromagnetic wave entering thecavity, there will always be some components not at resonance, reducing theimpact of this beneficial effect.

Finally, imperfect mode-matching reduces the intensity that is actuallyavailable to excite oscillation of resonator-modes, without reducing the am-plitude of the initially reflected beam. This effectively lowers the transmissionof the first mirror, affecting the signal as indicated in figure 2.9. This furtherlowers the signal-to-noise ratio.

Since all of these effects are directly related to the reflection at the frontsurface of the resonator, the modification applied to the setup consists ofmoving the analyzer to the other side of the cavity and looking at the trans-mitted light. In the following pages the theory behind this “TransmissionHansch-Couillaud Lock” will be explained in detail.

2.3.3.2 Theory

Equation 2.26, which was only used to find the intensity of the transmittedbeam so far, also allows us to look at its phase.

Eout−2 = Ein · tA · tB · e2πi dλ

1−(f · e2πi 4d

λ

) (2.33)

2.3. Lock Signal 29

Here the factor for the mirror reflectivities have been replaced by theroundtrip factor f = r2

A · r2B, which will later be extended to include in-

tentional as well as accidential losses occuring during one roundtrip throughthe resonator.

By comparing the transmitted beam to an imaginary reference beam thatpasses through the cavity without undergoing any reflections, some of theterms in this equation can be dropped:

Erel =Eout−2

Ereference

=Eout−2

tA · tB · e2πifracdλ=

1

1−(f · e2πi 4d

λ

) (2.34)

The phase φrel = arg(Erel is plotted in figure 2.15.

780.001 780.002 780.003 780.004wavelength

-0.3

-0.2

-0.1

0.1

0.2

0.3

phase

780.0007 780.0008 780.0009wavelength

-0.3

-0.2

-0.1

0.1

0.2

0.3

phase

Fig. 2.15: Relative phase in units of π for the transmitted beamdepending on the wavelength (in nm) for a 100mm confocal cavity.Mirrors: R=0.95, T=0.03. The red graph is for the basic cavity,the green one for a cavity with an additional intensity loss of 8%per round trip. The top graph is plotted at the same scale as theprevious plots on cavity reflectivity and transmission, while thelower one is magnified to show the frequency range near a resonance.

In resonance, all the electric field contributions in the resonator have exactlythe same phase, so the emitted light has exactly the same phase as that of abeam passing through without being reflected. Therefore the relative phaseis zero, independent of f. Halfway between resonances the phase differencesbetween one round trip and the next are big and more or less cancel eachother out, resulting in a net phase of zero again.

The most interesting part is the area close to a resonance, as magnified in thelower plot. Here the circulating light only undergoes a minimal phase shift


between round trips, and the field amplitude effectively falls to zero beforethese can start to cancel each other out. For a high finesse, many reflectionswill occur before the intensity drops close to zero, resulting in an increasedcontribution from round-trips that have accumulated a larger phase shift.The extreme case for a low finesse would be a mirrorless cavity, resultingin zero phase shift for all wavelengths. So the slope of the phase aroundresonance depends strongly on the round trip factor f and therefore on the

cavity finesse F ≈ 12π√

f

1−f.

The effect is that in resonance the two modes with orthogonal linear polari-zations will have the same phase, resulting in linearly polarized light. Whenthe laser gets out of resonance, the phase slope is different for both polariza-tions. If the horizontal polarization is lagging behind the vertical one on thehigh frequency side of resonance, then it will be the other way around on thelow frequency side.

This will once again add a circular component to the polarization of the totalbeam. The only difference to the original Hansch-Couillaud lock is that theassumption of having a completely stable reference beam has to be dropped.In this setup, filtering one of the polarization components out completelywould remove the required phase reference. Instead, the polarization filterinside the cavity has to have an attenuation for one polarization that is highenough to spoil the cavity finesse for this component, while transmittingenough intensity to generate measurable circular polarization. Figure 2.16illustrates this.

Fig. 2.16: Transmission of two planes of polarization through thecavity. The Brewster’s angle plate reduces the transmission for thevertically polarized component (red) while letting the horizontallypolarized component (green) propagate freely. Although this redu-ces the cavity finesse for the red beam, there will still be sometransmission.

The analyzer itself is constructed in exactly the same way as described ear-lier (see figure 2.14). Giving a mathematical description of its operation isstraightforward if a little tedious. The retarder plate will delay the projectionof the wave projected to its slow axis by π

4relative to the projection to its

fast axis. This can be expressed as a multiplication by the imaginary uniti. Afterwards the partial beams must be recombined and then split into dif-

2.3. Lock Signal 31

ferent projections when the resulting beam reaches the beam splitter. Thisway the field amplitudes at its output ports can be found for varying am-plitudes and phases of the orthogonal polarization components leaving thecavity. The following equations give the results for a quarter wave plate withits fast axis rotated 45°from both the preferred plane of polarization for thefilter in the cavity and the orientation of the beam splitter:

EPD1/2 =

proj. on fast axis︷︸︸︷Eout−ver + Eout−hor√

2±

proj. on slow axis︷︸︸︷Eout−ver − Eout−hor√

2·ı

√2

(2.35)

EPD1/2 = Eout−ver · 1± ı

2+ Eout−hor · 1∓ ı

2(2.36)

This works out to the following equations, where equation 2.26 has been usedto find the complex field amplitudes Eout−ver and Eout−hor for the transmittedbeams:

EPD1 =

(1 + ı

2

)·(

Ein−h

−1 + fh · e 8dıπλ

+ı · Ein−v

1− f · fv · e 8dıπλ

)· tA · tB (2.37)

EPD2 =

(1 + ı

2

)·(

ı · Ein−h

1− fh · e 8dıπλ

+Ein−v

−1 + fv · fbrw · e 8dıπλ

)· tA · tB , (2.38)

where fv is the roundtrip factor for the vertical polarization and fh theroundtrip factor for the horizontal polarization component. If all other effectsare assumed to be identical for both polarizations, then fv = fh · fBrewster,where fBrewster is the reduction of electric field amplitude caused by theBrewster’s angle plate during one roundtrip. The horizontal polarization isassumed to pass through the plate unaffected.

The measured photodiode signals will now be proportional to∣∣EPD1/2

∣∣2:

UPD1 ∝∣∣∣∣

Ein−h

−1 + fh · e 8ıdπλ

+ıEin−v

1− fv · e 8ıdπλ

∣∣∣∣2

, (2.39)

UPD2 ∝∣∣∣∣

ıEin−h

1− fh · e 8ıdπλ

+Ein−v

−1 + fv · e 8ıdπλ

∣∣∣∣2

(2.40)

Taking the difference Udiff = UPD1 − UPD2 of these signals generates a di-spersion shaped signal. This is shown in figure 2.17 along with the signals forthe individual photodiodes.

More on the technical implementation of the system can be found in chap-ter 3, some data on the actual shape of the locking signals in section 4.6.

Having the equations for the theoretical shape of the error signal derived inthis section makes it easier to understand the signals observed in the experi-ment and helps tune the various variables to optimize system performance.Figure 2.18 is given as an example of this. In the experiment, the glass plate


780.001 780.002 780.003 780.004wavelength

0.01

0.02

0.03

0.04

0.05

0.06

PD signal

780.0007 780.0008 780.0009wavelength

0.01

0.02

0.03

0.04

0.05

0.06

PD signal

780.0007 780.0008 780.0009wavelength

-0.02

-0.01

0.01

0.02

diff. signal

Fig. 2.17: Photodiode signals relative to the incident intensity overlaser wavelength (top), magnified for a single resonance (middle).Dispersion shaped difference signal for the same frequency range(bottom). All in arbitrary units.

in the resonator introduces additional losses for the vertical polarization, re-sulting in a much lower maximum transmitted intensity for this contribution.It seems obvious that the amplitude of the error signal can be improved byadjusting the polarization of the beam fed into the cavity closer to verticalin order to compensate for these losses. Plotting the difference signal for dif-ferent input polarizations shows that this is not the case, however, and thatthe input polarization which generates the steepest slope at resonance, andtherefore the best lock signal, is always at an angle of 45°from the filter axis.This has been plotted in figure 2.18

2.4 Diode Lasers

Today’s widespread use of lasers is in no small part due to the developmentof semiconductor (or diode) lasers. High efficiency, small size and ease of useare only some of their advantages.

2.4. Diode Lasers 33

780.0007 780.0008 780.0009wavelength

-0.01

-0.005

0.005

0.01

diff. signal

Fig. 2.18: Error signal for different input polarization. Angles (fromred to blue): 0.05π, 0.10π, 0.15π, 0.20π, 0.25π. For higher angles upto 0.5π the signal returns to zero in the same way.

The experimental apparatus built in the course of my thesis also uses se-miconductor lasers for some of these properties. A diode laser locked to aRubidium vapor cell serves as a stable frequency reference and a second oneis operated as a modulated, injected slave laser and serves as an integral partof the mechanism that transfer this stability to the dye laser.

2.4.1 The Master Laser

For the injected modulation system to work properly, a stable single-modelight source is necessary. For our experiment it is provided by an extendedcavity laser diode that can be locked to a rubidium (Rb) vapor cell. The setupis basically the same as that described as Doppler free Dichroic Atomic VaporLaser Lock (DAVLL) in [26]. Further details on this specific implementationcan be found in [13].

2.4.1.1 Extended Cavity Laser Diodes

Normal laser diodes as used in CD-writers are optimized for maximum poweroutput in CW or pulsed operation. The spectral quality is rarely an import-ant factor for these applications, so these laser diodes will not usually operatesingle-mode, have linewidths of the order of 100MHz (see figure 2.19, takenfrom the datasheet for a Sharp GH0781RA2C diode again) and tuning is limi-ted by mode hops as described in chapter 2.4.3.1. One commonly used way tofix this is to vary the cavity losses depending on wavelength by incorporatinga grating into the resonator as shown in figure 2.20.

Since most diodes simply use the cleaved faces of the semiconductor crystalto work as resonator mirrors, the additional feedback of the grating has astrong impact on their behavior. The system described in [27] uses a holo-graphic grating providing 20% feedback into the laser with a 50mW diode toachieve stable single-mode operation. It has a tunable range of 8 GHz for sim-ply moving the grating with a piezoceramic actuator. A linewidth of about


Fig. 2.19: Emission spectrum for a free running laser di-ode. (Sharp GH0781RA2C at 25°C and 5mW output power,taken from the original datasheet available at http://sharp-world.com/products/device/lineup/opto/index.html

Fig. 2.20: Extended cavity laser diode created with a grating inLittrow configuration. The diffracted beam of first order is fed backinto the diode.

350kHz on a timescale of 200ms is reported without additional stabilizationtechniques. This requires current noise to be kept to a few µA.

2.4.1.2 Spectroscopy

In order to operate the reference laser in our experiment at a stable frequencyover a longer period as well as to decrease the linewidth even further, it isactively stabilized using a Doppler-free spectroscopy setup known as Doppler-free DAVLL (Dichroic Atomic Vapor Laser Lock).

This works by intersecting a probe beam with a counter-propagating pumpbeam inside a rubidium vapor cell (see figure 2.21). With the pump beamturned off, absorption will increase when the laser is tuned to a wavelengththat is close to a transition of the rubidium atoms, which occur at around780nm and 795nm as shown in 2.3.

At or slightly above room temperature the vapor pressure of rubidium is high


Fig. 2.21: Doppler-free spectroscopy setup for locking the masterlaser to a Rubidium transition. Drawn with exaggerated angle ofintersection inside the vapor cell.

enough to see a clear spectroscopy signal, but the hyperfine structure is com-pletely washed out by Doppler broadening, resulting in a single absorptionline of several hundred MHz linewidth. This is plotted as the red graph infigure 2.22.

If the pump beam is turned on now, it will also be absorbed by the rubidiumatoms, exciting them into a higher state. But since the lines are stronglyDoppler broadened, this will only affect the part of the atomic populationthat has the right velocity relative to the pump beam so that it appearsDoppler-shifted into resonance. Since pump and probe beam are counter-propagating, they will generally interact with different sets of atoms. Butif the laser frequency coincides with the unshifted resonance, the pump be-am will reduce the population of the ground state for the probe beam aswell. One mechanism for this is the saturation of the absorption line, wherethe absorption and stimulated emission processes caused by the laser lightdominate the spontaneous emission, leading to an nearly equal distributionof atoms in the ground and excited states, causing the gas to become moretransparent for the probe beam. Another is the optical pumping of atomsinto the other hyperfine ground state (F=1 in this case), where they do notinteract with any of the beams any more. The resulting dips in the absorptionspectrum are called Lamb dips. The one marked (c) in the diagram is causedby saturation/depletion of the 5S1/2(F = 2) → 5P1/2(F = 3) transition.

The dips (a) and (b) are called crossover-transitions. They occur when thefrequency of the incident light is exactly in the middle between the resonantfrequency of two transitions with the same ground state. That way bothbeams will interact with the same population of atoms again. These will seeone of the beams blue shifted to resonance with one transition and the otherbeam red shifted to another one.

The Lamb dip used for the stabilization of the master laser is that of the(2,3) crossover for the 5S1/2 → 5P3/2 transition, with a frequency exactlyin the middle between those of the 5S1/2(F = 2) → 5P3/2(F = 2) and the5S1/2(F = 2) → 5P3/2(F = 3) transitions.

There is another aspect of the laser locking mechanism that requires attenti-on. The problems of locking to a transmission or absorption line have already


Fig. 2.22: Observed spectrum for the D2 lines of 87Rb, with (black)and without (red) pump beam. Lamb dips: (a) (1,3) crossover tran-sition, (b) (2,3) crossover, (c) transition to F=3 state. (From [13])

been mentioned in chapter 2.3. For the spectroscopy this is overcome by split-ting the absorption signal into two parts that are shifted against each other.This gives a dispersion shaped locking signal in very much the same way asthat shown in figure 2.17 for the cavity lock.

This is achieved by winding a coil around the vapor cell and applying a ma-gnetic field in parallel to the beams. The linearly polarized light of the probebeam can then be treated as the superposition of the circularly polarizedcomponents σ+ and σ−. Each of these will drive a transition to a differentZeeman level, causing the absorption lines to be shifted towards each other.

Due to the high total angular momentum of the states involved, the lineswill not simply separate into two components, each of which is driven by onepolarization component. Instead, a whole system of lines will appear, one foreach of the transitions allowed by the selection rules. However, since σ+ lightwill always drive a transition into a higher projected spin state and σ− lightalways into a lower one, the lines will still be shifted, although there will alsobe additional broadening due to the unresolved additional lines.

The polarization components are independently detected by using the samepolarization analyzer setup used in the cavity lock, consisting of a λ/4 re-tarder plate and a polarizing beam splitter cube. The resulting photodiodesignals are subtracted, resulting in a wide dispersion shaped signal createdfrom the Doppler broadened absorption lines superimposed by a very similarsignal, which is much narrower and of opposite sign. This is created by theoffset Lamb dips and is used to lock the laser to the atomic transition. Acomplete spectrum taken with our system can be seen in figure 2.23.

The steepness of the slope can be optimized by choosing a suitable magneticfield to shift the Lamb dips with regard to each other. According to [13],


Fig. 2.23: Creating a dispersion shaped signal from two Zeemanshifted absorption spectra. From [13] (top: individual and differencesignals near the Lamb dip, bottom: full amplified signal and lockpoint)

detailed calculations shows a shift by 0.58 times the linewidth to yield theoptimum steepness for a lorentzian line shape.

Using the difference signal decouples the resulting signal from intensity fluc-tuations of the laser and the lock signal is also quite insensitive to stray ma-gnetic fields and temperature variations. For a more detailed analysis see [26].

The testing system reported there uses a feedback loop to control both thegrating of an extended-cavity laser and the driving current of the laser diode,working to counter slow (at up to 1kHz) and fast frequency changes respec-tively. It achieves an estimated linewidth of 350kHz when locked to the (2,3)crossover-transition of 85Rb and of 120kHz for the stronger (3,4) transition.


2.4.2 Injection Locking

In normal laser operation, the noise of spontaneous emission serves as the seedfor the amplification process that finally results in coherent emission. Sendingan external signal into the resonator can have a huge influence on this process,under certain conditions causing the laser to lock on to the frequency off theinjected beam. A basic configuration to acheive this is shown in figure 2.24

Fig. 2.24: Basic setup for injection locking a laser diode. A smallpart of a stable injection (or seeding) beam is reflected into thelaser diode and causes it to emit an amplified beam at exactly thesame wavelength.

We use this to keep the slave laser system at the stable frequency obtainedfrom the spectroscopy. This section will give a basic theoretical backgroundfor the underlying processes.

Without any injected signal, the laser is assumed to run at a frequency ω0

with an output intensity I0. It can be shown that a second signal of frequencyω1 will have an intensity gain from input to output of approximately

gint = |g(ω)|2 ≈ γ2e

(ω1 − ω0)2, (2.41)

where γe is the energy decay rate of the laser cavity. For more details on thisand the equations in the following section, please see [24].

For injection far away from the free running frequency ω0 a small signal at thefrequency ω1 will appear. Bringing both frequencies closer together will resultin an increase of this signal, until it reaches an intensity which is comparableto the free running intensity I0. Since this intensity is limited by saturation,both signals will begin competing for available gain.

When ω1 is brought even closer to ω0, the gain available for the free runningsignal will continue to fall until it drops below threshold and lasing at thiswavelength will stop. Now all available power will be used in the amplificationof the injected signal, which will run at an intensity practically equal to thefree running value, possibly increased by the intensity of the injected light.This is illustrated in figure 2.25.

As soon as equilibrium is reached, the laser will continue to emit light exactlyat the frequency of the injection, as long as it stays in a certain range around


Fig. 2.25: Free running oscillation competing with the amplifiedinjection signal as a function of the frequency offset between freerunning frequency and injected beam. Intensities of oscillation atthe free running frequency ω0 and at the injected frequency ω1

plotted over ω1. From [24], p. 1133

ω0. This range, defined to be the area where the amplified output for theincident signal reaches the full free running intensity, is called the injectionlocking range. It is approximately given by

4ωlock ≈ 2γeE1

E0

=2ω0

Qe

√I1

I0

, (2.42)

where E0 is the amplitude for the beam emitted without any incident lightand E1 is the field amplitude of the injected beam. I0 and I1 are the corre-sponding intensities. Qe is the Q-factor for the cavity without gain medium,which is an alternative to γe in giving the cavity bandwidth.

The most common way to describe the behavior of an injection locked laseris the so called Adler equation [24]:

dφ(t)

dt+ ω1 − ω0 = −ω0

Qe

E1

E0

sin φ(t) = −ωm sin φ(t) , (2.43)

with ωm ≡ ω0

Qe

E1

E0≈ 4ωlock

2.

This has steady state solutions for the locking range above and reveals some-thing interesting about the phase of the emitted beam: Just like in a forcedoscillator, the resulting oscillation will always have the exact same frequencyas the driving force (here: the injected beam) in a steady state solution, butits relative phase will vary depending on how far away the system is fromresonance. Solving the equation above for the relative phase depending on


the frequency offset gives

φ(ω1) = arcsinω0 − ω1

ωm

, (2.44)

which is plotted in 2.26 and will play a role in the explanation of the observedbehaviour later.

-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5

freq.

offset

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

phase shift

Fig. 2.26: Phase shift (in units of π) between injected and emittedbeam for varying offsets from the free running frequency, given inunits of the locking range 4ωlock

2.4.3 Modulated Diode Lasers

2.4.3.1 Current-Dependency of Diode Laser Output

The output of a semiconductor laser can be conveniently modulated by chan-ging the driving current. This works up to frequencies of several GHz [28]and is used here to generate an offset between the frequency obtained fromthe spectroscopy and the one used for stabilizing the cavity length.

Varying the current on a free running laser diode will change both the inter-nal temperature and the amount of charge injected into the semiconductorjunction. This in turn influences the band gap and Fermi energy as well asresonator length and index of refraction. The final result is that for a givenexternal temperature there is a current range where the frequency of theemitted light increases continuously with increasing current.


Reaching the end of this continuous tuning range, the laser will “hop” to thenext resonator mode as the peak in the gain profile shifts faster than themode of the resonator. The various effects can best be explained by lookingat a diagram such as that given in figure 2.27

Fig. 2.27: Diagram showing several effects of varying the currenton a laser diode. From “laser diode application note 5” available athttp://www.ilxlightwave.com, for one of their laser diodes which isnot the same as used in the experiment.

The dashed diagonal line shows the shift of the gain maximum for the activemedium depending on the driving current, with a slope of 4λ1

4T1, which is

typically around 0.25 nmC . But even the very low finesse resonator formed

by the cleaved faces of the semiconductor crystal has resonance lines whichare much narrower than the gain curve. Therefore the wavelength of theemitted light will change more slowly than the dashed line would suggest,with a slope of 4λ2

4I2which is commonly around 10−3 nm

mA. This continues until

the gain maximum moves closer to the next cavity resonance at which pointthe laser will suddenly jump about 125GHz in frequency (about 0.3nm for a780nm or 850nm laser), corresponding to its cavity length of roughly 0.6mm.After this jump the laser wavelength will continue to shift at the slower rateagain, until the next mode jump happens.

Since these processes are controlled by the interdependent variables of junc-tion temperature and drive current, and are also influenced by variations inthe properties of the junction as well as the exact geometry of the resonator,the points where mode jumps occur are impossible to predict for an untested


laser diode and will vary wildly even between devices from the same batch.There are also hysteresis effects, where the oscillating mode depends on thehistory of current and temperature as well as their momentary values.

2.4.3.2 Output of a Modulated Laser Diode

The current-dependency of the laser diode output makes it possible to adda modulation by simply changing the driving current periodically.

If only the output intensity was changing proportional to the change in cur-rent, this would add a periodic envelope to the oscillation of the electric fieldin the laser beam. This can be expressed as the multiplication of two sinefunctions, one with the (optical) frequency of the laser and the other with themodulation frequency. Looking at the resulting spectrum would then showtwo sidebands offset from the free-running (fcar for carrier) frequency ofthe laser by the modulation frequency, the heights of which depend on thestrength of the amplitude modulation (AM).

However, changing the driving current will also directly affect the wavelengthof the emitted laser light. This will introduce an additional frequency modu-lation (FM) to the final signal, with the extra complication that the relationbetween frequency and current is highly non-linear due to the modehops andhysteresis effects.

While this greatly complicates a detailed analysis, some basic laws of Fouriertheory provide an insight into the form the spectrum of a current-modulatedlaser diode can have. Since optical frequencies (in the THz range) are muchhigher than typical modulation frequencies (up to several GHz), the resul-ting function for the electric field in the laser beam will be effectively periodicwith the modulation frequency. According to the sampling theorem, all fre-quency components in a truly periodic signal will be multiples of the signal’srepetition frequency and therefore the spectrum will consist of a series ofsidebands, offset from the laser’s free running frequencies by multiples of themodulation frequency.

Due to the presence of an amplitude modulated component and the spectrumof an ordinary frequency modulated signal, the expected spectrum wouldconsist of strong first order sidebands offset at the modulation frequency(fmod) and several higher order sidebands at integer multiples of fmod, withamplitudes falling off rapidly with increasing offset from fcar.

Because of the instability of the unstabilized laser diode, we have not beenable to measure a spectrum to confirm this.

2.4.3.3 Modulating an Injection-locked Laser

The setup described here uses a laser diode that is both modulated andinjection locked at the same time. The injection stabilizes the operation of the


laser diode, while the modulation makes it possible to create a new referenceoscillation at a variable offset from the frequency obtained from the rubidiumspectroscopy.

Modulating an injection-locked laser changes the situation described in theprevious section. As long as the modulation does not move the free runningfrequency more than half the locking range away from that of the stableinjection beam, the laser will continue to run with the exact same frequency.Only the intensity of the emission will change. This has a much simplerdependence on current, however, and can be treated as linear provided itstays above the threshold current and below the maximum allowed level.Figure 2.28 is taken from the datasheet1 of a Sharp GH0781RA2C laserdiode with a maximum CW power output of 120mW to demonstrate this.

Fig. 2.28: Power output for a Sharp GH0781RA2C semiconductorlaser diode depending on driving current given for different tempe-ratures.

Adding a sinusoidal modulating current to a constant driving current resultsin a basic amplitude modulated (AM) system. In this simplest case symme-tric sidebands will be added to the output signal, their separation from theoriginal line or “carrier frequency” given by the modulation frequency andtheir relative height depending on the amplitude of the modulation comparedto the static current.

The plots shown in figure 2.29 show the application of Fourier theory tothis situation. In the actual experiment the main oscillation occurs at opticalfrequencies in the THz range. The modulation happens at a frequency around1GHz. Because these frequencies are too different to result in legible plots,the exemplary treatment will be done for frequencies of fmod=1GHz andfcar=15GHz.

Modulation can be described in terms of multiplying the carrier wave by a

1 available at http://sharp-world.com/products/device/lineup/opto/index.html


temporally varying envelope as shown in the second diagram. The simplescase would be simple multiplication of two trigonometric functions [20]

cos fmod · t · cos fcar · t =1

2(cos (fcar − fmod) · t + cos (fcar + fmod) · t)

(2.45)resulting in a spectrum that consists of only two frequency components offsetfrom the original carrier frequency by the frequency of the modulation.

However, this mathematically simple case is unphysical in our experimentalsetup. The injected laser diode essentially serves as an amplifier for the si-gnal of the injected beam. The amplification will be high when the drivingcurrent is at its maximum, and it may drop to zero if the current falls belowthe threshold value for the laser diode. But at no time will there be a multi-plication of the original signal by a negative factor. In order to model this,an offset has to be added to the envelope function to keep it positive.

In the following, the laser diode to emit a beam oscillating between 20% and100% of maximum output power:

Pout(t) = (0.6 + 0.4 cos fmod · t) · cos fcar · t (2.46)

This can be reordered into a sum of trigonometric functions by using equa-tion 2.45.

Pout(t) = 0.6 cos fcar · t + 0.4 cos fmod · t · cos fcar · t= 0.6 cos fcar · t + 0.2(cos (fcar − fmod) · t + cos (fcar + fmod) · t)(2.47)

In this case there will still be a component remaining at the carrier frequen-cy in addition to the sidebands already found above. Their relative heightdepends on the depth of the modulation. It is obvious that for this kind ofamplitude modulation that sidebands can never have a height of more thanhalf the carrier, which will occur if the laser is modulated all the way frommaximum output to zero.

It is possible to measure these spectra directly by measuring the transmissionof a scanning optical resonator. The detector used for this will create a signalpropertional intensity of the transmitted light, and therefore to the squareof the amplitude calculated here. This leads to the expectation that thesidebands should have a maximum of 25% of the carrier intensity.

Nevertheless, the spectra taken in our experiment often show signals wherethe sidebands reach 70% of carrier intensity or even more. The intensities ofthe corresponding sidebands also tends to be highly unequal. The remainderof this chapter will introduce a model that attempts to explain these obser-vations. Since part of that model relies on Fourier theory, it will be helpful torederive the results found above in that context. Figure 2.29 helps illustratethis.

The fourier transform of the carrier signal, modelled by a cosine is a set oftwo delta functions with coefficients 0.5 (half the original signals amplitude


0.2 0.4 0.6 0.8 1t

-1

-0.5

0.5

1

0.2 0.4 0.6 0.8 1t

-1

-0.5

0.5

1

-15 -10 -5 5 10 15

-0.2

0.2

0.4

-15 -10 -5 5 10 15

-0.2

0.2

0.4

-15 -10 -5 5 10 15

-0.2

0.2

0.4

Fig. 2.29: Finding the frequency components for an amplitude mo-dulated signal. Relative amplitudes over time [in ns] or frequency[in GHz]. Top to bottom: carrier, envelope (red) and modulated si-gnal (green), transform of carrier, transform of envelope, transformof modulated signal


of 1) at the frequencies −fcar = −15GHz and +fcar = 15GHz. The envelopehas two similar components, at −fenv = −1GHz and fenv = 1GHz, with anamplitude of 0.2, half of the original amplitude of 0.4. In addition it has ano-ther delta function at zero frequency with a coefficient of 0.6 correspondingto the offset added to keep the envelope positive. These spectra are shown inthe third and forth diagrams.

The final spectrum AAM(f) can be found by calculating the Fourier transformof the product of carrier and envelope:

aAM(t) = aenv(t) · acar(t) →FT AAM(f) (2.48)

It is convenient to apply the convolution theorem (see [29]) to this problem. Itstates that a multiplication in the time domain is equivalent to a convolutionin the frequency domain. The transforms of carrier and envelope have alreadybeen described above to be

acar(t) →FT Acar(f) = 0.5 · δ(f + fcar) + 0.5 · δ(f − fcar) (2.49)

amod(t) →FT Amod(f) = 0.2 · δ(f + fcar) + 0.6 · δ(f) + 0.2 · δ(f − fcar) (2.50)

Using the convolution theorem

AAM(f) = Amod(f)⊗ Acar(f)

= 0.1δ(f + fcar + fmod + 0.3δ(f + fcar + 0.1δ(f + fcar − fmod

+ 0.1δ(f − fcar + fmod + 0.3δ(f − fcar + 0.1δ(f − fcar − fmod ,

as shown in the lowest graph of figure 2.29.

2.4.3.4 Extending the Theoretical Model

This section will offer several possible explanations for the spectra found inthe experiment, in particular the unexpected heigth of the sidebands andtheir asymmetry.

A way to generate a signal with very high sidebands would be to increasethe amplitude of the modulation envelope to values higher than its offset.In the experiment this would correspond to the diode current falling belowthreshold for part of the cycle. Any further reduction will have no effect, sothe envelope will remain fixed at zero for part of the cycle instead of goingnegative. This will increase sideband amplitude but also create additionalsidebands at multiples of the modulation frequency. Mathematically, fixingparts of the waveform at zero can be expressed as a multiplication by asquare wave. Since the transform of such a wave consists of a series of deltafunctions spaced at the modulation frequency, the convolution correspondingto the multiplication in the time-domain will impose additional sidebands onthe final spectrum.

Another way to account for the high sidebands is amplification by the activemedium in the laser diode. In this scenario, the laser is essentially seedingitself with the sidebands.


2.4.3.5 Vestigial Sideband Operation

A fascinating effect that occurs in the experiment is that if the offset currentis set just right, one of the sidebands will almost disappear, transferring itsintensity to the remaining one, which will reach an amplitude almost identicalto that of the carrier.

In radio transmission, signals with similar spectra are often used to reducethe power required for transmission. If only traces of one of the sidebandscreated by modulating the carrier wave remain, it is called vestigial. In thefollowing this term will be extended to the domain of optical frequencies.

Operating the laser with a vestigial sideband is very useful for the lockingmechanism due to the power transferred to the interesting sideband and isconsequently used in the normal operation of the system. This section willtry to explain what this effect is based on, even though the process is quitecomplex and not easily analyzed mathematically.

Looking at the time domain function corresponding to a spectrum consistingof two lines gives the first hints what kind of effects could cause the observedchange in the spectrum. The waveform (see figure 2.30) is that of the productof two cosines, one at the carrier and one at the modulation frequency. Thiscannot be reproduced directly, however, since it requires the envelope tobecome negative. An equivalent interpretation of the same function wouldbe to take the absolute value of the envelope and introduce a phase shift of πinto the carrier wave (initially as drawn in black) for half of the modulationcycle.

0.2 0.4 0.6 0.8 1t

-1

-0.5

0.5

1

Fig. 2.30: Time domain waveform for a two-line spectrum

This is not quite what happens in the experiment. For one thing, a big,abrupt phase jump like that would be hard to explain. Furthermore, as wasexplained before, this would cause the carrier to vanish instead of one of thesidebands, resulting in two emission lines separated by twice the modulationfrequency. It points in the right direction, though. One of our current theoriesis that the disappearance of the sideband is caused by the interplay betweenamplitude modulation and the phase modulation caused by the change infree running laser frequency at different driving currents. This creates a shiftin phase even for injected operation, since the relative phase of the resultingbeam depends on the frequency difference between free running and injectedfrequency as shown in figure 2.26.


When the current is tuned to the vicinity of a mode hop of the laser diode,a considerable phase jump occurs twice per modulation cycle as the lasercurrent is scanned back and forth across the mode hop. This creates theconditions necessary for vestigial sideband operation, but it is only the firstpart of the explanation.

Since the absolute values for the fourier coefficients of a real valued signal willalways be equal for positive and negative frequencies, an asymmetry in thespectrum can only be achieved by superimposing several components witha different rotation in the complex plane, which can be achieved by shiftingtheir phase. This is related to a technique for generating a single sidebandradio signal, named “phase shift method” (or “quadrature amplitude modu-lation”) [30].It works by adding a regular AM signal to a signal generated byshifting both carrier and signal by π

2, schematically shown in figure 2.31.

Fig. 2.31: Schematic for generating a single sideband signal by thephase-shift method. From [30]. In the top branch the modulationsignal is multiplied by the carrier signal directly, while in the lowerbranch both signals are shifted by π/2 before multiplication. Ad-ding both partial signals together creates a single-sideband signal.This is called single-sideband modulation (SSM) by the phase shiftmethod.

The easiest way to understand how energy can be shifted between sidebandsunder certain circumstances is to do a fourier transform on an example signaland try to find out which initial parameters will have what effect. Figure 2.32shows the construction of a suitable waveform.

It is made up of one cosine (blue) and one sine (red) function at a frequency of15GHz as carrier waves. A periodic phase shift is generated by changing therelative weight of the functions through multiplication with an envelope con-structed from a cosine at 1GHz. This generates a primarily phase-modulatedsignal printed purple in the lower diagram. There is also some amount of am-plitude modulation, but the advantage of this construction is the restrictionto trigonometric functions, resulting in a very simple spectrum. The resulting


0.5 1 1.5 2t

-1

-0.5

0.5

1

0.5 1 1.5 2t

-1

-0.5

0.5

1

Fig. 2.32: Construction of a signal for vestigial sideband operation(VSB) from basic trigonometric functions

functions for the red and the blue contribution are:

fblue(t) = cos (2π · fcar · t) · (0.5− 0.5 cos (2π · fmod · t)) (2.51)

fred(t) = sin (2π · fcar · t) · (0.5 + 0.5 cos (2π · fmod · t)) , (2.52)

where fcar is the carrier frequency and fmod is that for the modulation. Thecombined signal is now multiplied with another envelope function (black),which is slightly delayed to give a phase shift of φ with regard to the envelopefunctions. Adjusting this shift causes the rotation of the components in thecomplex plane according to the shift theorem [29].

A phase shift like this might occur in the actual experiment due to the factthat gain of the laser diode is linked to the number of electrons injected intothe conduction band directly, while the emitted wavelength normally followsthe temperature-induced changes in resonator length. This is likely to lagbehind the modulation of the amplification factor due to the heat capacityof the diode. We do not currently have any data that might confirm or rulethis out, however.

Because all the components used in creating the function are trignonometricfunctions, the transform will consist of a set of delta functions:

S(f) = c1δ(f + fcarr + 2fmod) + c2δ(f + fcarr + 1fmod)

+c3δ(f + fcarr) + c4δ(f + fcarr − 1fmod)

+c5δ(f + fcarr − 2fmod) + c6δ(f − fcarr + 2fmod)

+c7δ(f − fcarr + 1fmod) + c8δ(f − fcarr)

+c9δ(f − fcarr − 1fmod) + c10δ(f − fcarr − 2fmod) (2.53)

Since this function is the convolution of all the individual transforms of thebase function, the frequencies for the delta peaks are combinations of thecarrier frequency and up to twice the modulation frequency. The fact thatboth the envelope and the phase variation have the same frequency leadsto a mixing of components with different phase shifts for the frequenciesf = ±fcarr and f = ±fcarr ± fmod.


Working through the calculations gives the coefficients c1 to c10 as shown intable 2.1.

coefficient valuec1

12(− cos φ− sin φ) + 1

2ı(− cos φ + sin φ)

c2 (+ cos φ− sin φ− 1) + ı(− cos φ− sin φ− 1)c3 (− cos φ + 2) + ı(− cos φ− 2)c4 (+ cos φ + sin φ− 1) + ı(− cos φ + sin φ− 1)c5

12(− cos φ + sin φ) + 1

2ı(− cos φ− sin φ)

c612(− cos φ + sin φ) + 1

2ı(+ cos φ + sin φ)

c7 (+ cos φ + sin φ− 1) + ı(+ cos φ− sin φ + 1)c8 (− cos φ + 2) + ı(+ cos φ + 2)c9 (+ cos φ− sin φ− 1) + ı(+ cos φ + sin φ + 1)c10

12(− cos φ− sin φ) + 1

2ı(+ cos φ− sin φ)

Tab. 2.1: Relative fourier coefficients for the test function, scaledby a factor of roughly 6.4.

As expected, the coefficients for the negative frequencies are the complexconjugate of the ones for the corresponding positive frequencies. This is agood check for the calculations, as this Hermitian quality is required in thetransform of a real valued function.

0.5 1 1.5 2phi

0.1

0.2

0.3

0.4

0.5

Fig. 2.33: Relative amplitudes of carrier (black) and first order si-debands (red and green) for a varying phase offset φ (in units of π)of the envelope as cescribed in the text.

Dropping any phase information, the absolute values describe the field am-plitudes of the detected lines. A plot of the positive frequency components isshown in figure 2.33.

For the parameters chosen here, a phase offset of π2

would generate a signalwith only one sideband present, which has the same amplitude as the carrier.In addition to that, two other bands appear, offset from the carrier by twicethe modulation frequency, with an amplitude that is independent of the en-velope phase. The carrier amplitude also undergoes some fluctuations as φis varied, but this is of little relevance to the experiment and may simply bedue to the way the phase modulated signal was modeled.


-4 -2 2 4f

0.2

0.4

0.6

0.8

1

relative Intensity

Fig. 2.34: Simulated spectrum for an envelope phase offset of 0.3πas given by the calculations in this chapter. Frequency measuredrelative to the carrier frequency fcar, with a modulation fmod = 1.

It would be interesting to analyze in more detail whether this model candescribe how the various properties of the diode interact under modulation.A particular point to investigate would be whether the measured signalscorrepond to a continuous phase modulation as modelled here or whether ahard jump in phase might be a better description. Such a jump might beoccur as the laser is scanned over a mode hop. This might even show anadditional lag relative to the modulation due to hysteresis effects.

A good place to start further investigation would be measurements of theeffects of varying the intensity injected into the modulated laser diode, asthis would change the shift between injected and emitted phase withoutaffecting the mode hops.

But since no specific experiments have been done yet to determine all this,the considerations in this chapter will be left standing as a theoretical modelthat might explain how the intensity can be shifted from one sideband tothe other. Its basic properties are in agreement with what is observed in theexperiment (see figures 2.34 and 2.35).


5

10

15

20

25

30

35

4 5 6 7 8 9 10

Uscan [V]

Usig

nal

[mV

]

Fig. 2.35: Spectrum measured in transmission through the cavityfor the injected, modulated slave laser. Signal repeats at the freespectral range of ≈750MHz. fmod = 543.33MHz, therefore the com-ponents highlighted actually belong to different repetitions. (Cur-ve averaged over five measurements and smoothed by a 5 pointrunning-mean)

2.4.3.6 Sideband Injection

Another effect that can occur in a modulated, injected system, either indi-vidually or in conjunction with the vestigial sideband operation describedabove, is injection into one of the sidebands.

This describes a situation where injection locked behavior of the slave laseroccurs although the injection is outside of the locking range. This happenswhen the free running frequency is adjusted so that one of the sidebandsalmost or fully coincides with the injected beam. The resulting spectrum willbe similar to the modulated spectrum of the free-running laser, but it willgain the stability characteristic for injected operation.

What happens in this case, is that the gain at the injection frequency istoo low for the injection to cause saturation for other frequencies, whichprevents normal locked operation. The laser will emit light primarily at thefree running frequency, with an additional small component correspondingto the injection as shown in figure 2.25. But as long as there is some amountof gain in the active medium, even if it is not enough to overcome the roundtrip losses, this component will be amplitude modulated. If the modulationfrequency is bigger than at least half the locking range, one of the resultingsidebands can be very close to the gain maximum of the laser diode and causeinjected operation although its amplitude is likely to be small. This can beunderstood by realizing that the sidebands will actually create photons forfrequencies offset from that of the carrier. With these the laser essentially


seeds itself.

Given enough modulation and injection power, it is even possible for thisprocess to repeat itself until finally the frequency reaches a value close enoughto the free-running frequency for locked operation to set in. This will appearas an injection into a higher-order sideband.

Commonly the only indication that this is happening is an asymmetry inthe spectrum, where the injection beam adds additional intensity to the si-debands on one side of the carrier. This also makes sideband injection acompeting process for the vestigial sideband operation described in the pre-vious chapter.

3. Construction

3.1 Overview

After the theoretical background has been given now, this chapter will de-scribe the actual setup of the laser system. Figure 3.1 shows a schematicoverview of the complete system.

Fig. 3.1: Schematic of the entire setup. Red and green arrows cor-respond to 780nm or 556nm laser beams, respectively. Blue arrowsindicate electronic signals.

In the following sections the individual parts will be described in detail.

3.2 Master Laser

The master laser is a self-built extended cavity laser (see section 2.4.1.1),using a Sharp GH0781RA2 diode with additional feedback provided by a 1800lines/mm holographic grating from Edmund Scientific (article no. NT43-775)in Littrow configuration. A Peltier element provides temperature control andstabilization. Both the current for the Peltier and the driving current for thelaser diode are controlled by an ITC102 laser diode controller available fromThorlabs. The grating angle can be changed by a piezoceramic actuator.

This combination of factors allows for single-mode operation in a wavelengthband several nm wide, with a continuous tuning range of 1-2 GHz by moving

56 3. Construction

the grating. Output power of the grating-stabilized diode laser is approxima-tely 45mW to 50mW at a driving current of up to 115mA.

A glass plate splits off part of the emitted laser beam for the spectroscopysetup. This uses the Doppler-free DAVLL system described in section 2.4.1.2and implemented as shown in figure 3.2.

Fig. 3.2: Master laser system with Doppler-free spectroscopy andsplit-off seeding beam for the slave laser.

The light emitted by the laser is split into three components by sending itthrough a glass plate with an antireflective coating on one side. The mainpart (A) passes through the plate and is used as the repumper for rubidiumin the main experiment. The weak reflection on the coated surface (B) isused to inject the slave laser, while the stronger reflection (C, at about 4%incident intensity) on the pure glass surface is used for spectroscopy.

A polarizing beam splitter with a rotatable λ/2-plate in front of it separatesthe light into pump and probe beams. Both are sent through polarizing foilto remove any residual circular polarization components. The vapor cell is asealed glass cuvette that contains some rubidium and is wound with appro-ximately 24 windings per cm operated at a current of 1A. At this current thesplitting of the Zeeman-levels that was explained in chapter 2.4.1.2 results ina good signal-to-noise ratio for the lock signal. No additional heating is requi-red, as the vapor pressure is high enough to generate a useable spectroscopysignal even at room temperature.

The output of the difference amplifier is used as error signal in a PI-loop thatdirectly drives the grating piezo. At the moment there is no feedback to thedrive current of the diode, which limits the bandwidth of the feedback loopto values below the resonant frequency of the piezo-grating-assembly, whichis in the low kHz range.

The injection beam (B) has an intensity of roughly 100µW . It is refocused by

3.3. Slave Laser 57

a f=300mm plano-convex lens and coupled into a single-mode fiber leadingto the slave laser assembly. The lens has a distance of approximately 25cmfrom the grating and 20cm from the fiber coupler. The fiber ensures a stableposition and mode for injection into the slave laser.

3.3 Slave Laser

The slave laser consists of another Sharp GH0781RA2 laser diode mounted ona Peltier element for temperature control. Both Peltier and diode current arecontrolled by a second controller. This way the free-running frequency can beadjusted until the injected beam is inside the locking range (see section 2.4.2).If mode-matching and alignment of the beams is good enough, the emittedbeam will have the same stability and linewidth as the master laser.

In order to achieve this, the beam from the fiber is focussed by an f=250mmlens and part of it is injected into the laser using the reflection from a glassplate with a 20nm silver coating protected by a 10nm MgF2-layer. Measure-ments show that this provides about 84% reflectivity and 10.5% transmission.A λ/2 retarder plate and polarizing foil on rotating mounts allow adjustmentsof polarization and intensity. The whole system is shown in figure 3.3

Fig. 3.3: Slave laser system with basic mode-matching. Lens1 ispositioned 17cm from the fiber coupler and 18cm from the laserdiode, lens2 at 25cm from the diode and 27cm from the cavity.

The high reflectivity of the coated glass plate reduces the available outputpower. But since the power available for the seeding beam is currently very

58 3. Construction

small, this is required to achieve stable injected operation. An alternative tothis might be the use of a lower power laser diode as the slave, which accordingto equation 2.42 would require less injected power. This would allow for ahigher transmission of the beam splitter, resulting in a higher overall outputpower. The drawback would be a higher sensitivity to feedback from thecavity which is also reduced by the splitter plate. Operating the currentlyused laser diode very close to threshold might have a similar effect.

In order to optimize injection performance, the following steps are takenperiodically:

• Rotate polarizing foil to maximize transmission of light from the free-running slave laser as measured before the light enters the fiber. Thisshould only be neccessary if the fiber has been moved.

• While measuring the intensity of the light transmitted through the fiberfrom the master laser, adjust focussing lens and mirror alignment onthe master laser side.

• Maximize intensity reaching the slave laser by adjusting the retarderplate without changing the polarizer orientation.

• Optimize the intensity of the slave laser’s light going backwards throughthe fiber by adjusting the fiber coupler and glass plate.

This creates sufficient mode matching between the beams in our case. Adap-ting the circular output of the fiber to the elliptical mode of the laser diodehas not proven necessary so far.

After adjusting the temperature to bring the emitted wavelength of the diodeto the desired range, the diode current is adjusted until the injected light isin the locking range, causing the emitted light of the slave laser to remainstable at the injection wavelength. This is most easily visible by looking at thetransmitted intensity of the cavity while scanning its length with the piezo.A slow change in the laser current will cause the observed lines to shift on theoscilloscope as the wavelength changes. Entering the locking range, there is aslight instability first, after which the line will stop moving for small changesin diode current. Finally, the signal will become unstable again, returning tothe slowly shifting behavior for a further change in current.

The effects are particularly visible when the diode is operated just barely overits threshold current of approximately 30mA. In this case even small opticalfeedback will make the system unstable, causing the signal on the oscilloscopeto jump while the laser does not operate single-mode, possibly completelywashing out into a flat, broad background. The switch to locked operation isvery obvious then, appearing as a sudden return of a clear, unmoving signalof noticeably higher amplitude.

Currently there is stable locked operation for an injected power at the laser of36µW (from 75µW exiting the fiber). The slave diode is operated at 48.7mA

3.4. Modulation System 59

and 16°C, yielding 11mW of output power, 1.05mW of which finally reachthe cavity. These numbers are already for modulated operation, however.

3.4 Modulation System

The modulation system serves to create sidebands in the slave laser’s output,providing a variable offset that allows stabilization of the cavity at almostany desired length instead of being limited to multiples of λmaster/4 (seechapter 2.2.2).

The system is set up as shown in figure 3.4.

Fig. 3.4: Schematic of the modulation system

A voltage controlled oscillator (Mini-Circuits ZOS-1025) is used to generate arange of modulation frequencies. This particular unit is capable of outputtingfrequencies between 550MHz and 1060MHz at a power of around 8dBm.

This signal is then amplified by a Mini-Circuits ERA-5 amplifier. Operatingthis at its power limits will introduce a certain degree of compression in thesignal. Since the short-term frequency stability of the VCO is in the lowkilohertz range, however, the resulting loss in spectral clarity is tolerable.

Now the maximum power of the modulation signal is between 15dBm and18dBm, depending on frequency. The modulation is added to the DC drivingcurrent immediately before the diode by a Mini-Circuits PBTC-3G bias-tee.

The high modulation power generates sidebands with an amplitude compara-ble to the carrier although no particular steps are taken to ensure impedancematching between the radio frequency components and the diode.

60 3. Construction

To keep the frequency generated by the VCO stable, its control current isgenerated by a self-built driver box that is designed to have a high degreeof thermal stability. Voltage noise is filtered down to below 1mV. The VCO-driver also has the capability to reduce the supply voltage from 12V down to6V in order to lower modulation power and sideband height. A schematic isgiven in A.2 and performance data can be found in section 4.4.1. The actualfrequency is monitored on an external counter connected to the secondaryoutput of the VCO.

In order to minimize losses, all the radio-frequency components are mountedon a board directly on top of the laser casing to keep connections short.

3.5 Green Laser System

The green laser beam is generated by a TekhnoScan Ametist-SF-07, a linearCW dye laser. It contains a birefringent (Lyot) filter, a thin etalon and athin absorbing film (TAF) mode selector, allowing for stable single modeoperation and a wide tuning range. The free-running linewidth without anyadditional stabilization is approximately 10MHz.

In normal operation the rough wavelength is set using the birefringent filterand then fine-tuned by adjusting the resonator length with a piezoelectricactuator.

The combination of etalon and TAF selector then introduces extra losses forundesired longitudinal modes, reducing their total gain to below the lasingthreshold if everything is set properly.

For electronic control of the laser wavelength, the birefringent filter and theetalon are left unchanged, while the TAF selector is adjusted automaticallyto keep the laser output at a local maximum. This is done by modulating itsposition at 2.5kHz and analyzing the change in laser output with a synchro-nous detector.

This way the laser continues to operate single mode while the resonator lengthis controlled by an external input 1. The tuneable range without mode jumpscan be up to several GHz. However, the modulation lock limits the bandwidthfor direct stabilization of the laser.

The laser is pumped by 2.5W of power from a Verdi V10 diode pumped solidstate laser internally frequency doubled to emit at 532nm wavelength. It usesRhodamin 110 (from Radiant Dyes) dissolved in ethylenglycol, absorbingaround 75% of the pump beam. Maximum output power at 556nm has beenslightly below 40mW, typical values are around 30mW.

1 Our laser is an early model and originally had both the external input and the stabi-lization circuit work on the resonator. It required a slight modification to the controlbox before it operated in the way described here. To my knowledge, this has beenfixed in newer models.

3.5. Green Laser System 61

Fig. 3.5: Setup of the green laser system. Distance from lens to laseris approximately 110cm, from lens to cavity 22cm.

The rest of the setup is laid out in figure 3.5. Glass plates serve to split offfractional beams for the dye laser’s TAF-lock and for the wavemeter usedto monitor the wavelength. Two polarizing beam splitter cubes are used incombination with rotatable λ/2-plates to control the relative intensities sentto experiment, cavity and an ytterbium spectroscopy cell. The latter is notcurrently in use, as the spectral resolution of the current setup is insufficientto stabilize the laser wavelength to it.

The polarization of the light used to lock the laser to the cavity can beadjusted with another λ/4-plate. The light enters the stabilization cavityslightly off-axis, resulting in a V-shaped beam configuration as shown in theschematic view (in figure 3.5). Besides reducing feedback into the laser, thisallows picking out one of the transmitted beams for the analyzer with a smallsilver coated mirror without blocking the red beam from the slave laser (seesection 3.3).

Although this is not implemented yet, the beam sent to the experiment issupposed to be switched by an acousto-optic-modulator in double pass con-figuration. This would also allow the frequency to be tuned by some tensof MHz, making it easier to measure a spectrum or to offset the frequencyif a desired value cannot be reached directly. In order to extract the retur-ning beam, it is sent through a quarter-wave plate twice on its way. Uponreturning to the beam splitter cube, the polarization will have changed fromhorizontal to vertical, causing the beam to pass straight through the cube

62 3. Construction

towards the main experiment.

Although this separation is not perfect, feedback is unlikely to be a problem,as the frequency off the returning light will be different by more than 100MHz.

3.6 Stabilization Cavity

The cavity is constructed from Invar alloy to reduce the effects of thermalexpansion. Its main part, the body, consists mainly of a cylinder 30mmin diameter with a longitudinal hole of 10mm. Both ends of the cylindercarry mirror assemblies spaced such that the mirrored surfaces are exactly100mm apart, corresponding to their radius of curvature. As described in sec-tion 2.2.2, this confocal arrangement provides degenerate transversal modesas well as stable operation for off-axis incidence.

Even more critical than the distance of the mirrors is their proper alignment.If one or both focal points do not lie on the axis connecting the mirrors, thelight will not return exactly to its point of entry. These errors will add up untilthe beam misses the mirror altogether, thereby introducing additional lossand lowering the finesse of the resonator. If the angle of misalignment is bigenough, the light will even be lost without overlapping at all. This will preventany interference effects, leading to a situation where the cavity transmission iscompletely described by the product of the mirrors’ transmission coefficients.

To prevent this from happening, the mirrors are not fixed to the resonatorbody directly, but are glued to mirror holders instead, which are metal ringsthat can be shifted transversally. They are being held in place by lockingrings, as indicated in figure 3.6. These in turn are fixed to the cavity body bythree M3 screws in a triangular pattern, only one of which is visible in thecross-section diagram. The mirror holders have small cut-outs where they areclosest to the screws, allowing for a total range of movement of about 2mmin each direction.

One of the mirror assemblies is more complicated, as it includes a 3mm sliceof a piezoceramic tube bought from Piezomechanik. This is used to varythe length of the resonator. It is isolated from the metal rings by thin micaplates for safety as well as to prevent short-circuiting. The wire connectedto the inside electrode is brought to the outside of the cavity through asmall channel in the mirror holder. The whole construction has a resonancefrequency of several kHz and makes it possible to vary the resonator lengthby about 700nm with a maximum applied voltage of 500V.

The mirrors are silver coated with a thickness of 60nm and 70nm. This resultsin theoretical reflectivities of 97.7% and 98.0% respectively. Measured valuesare close to these at around 97.5%, leading to an expected finesse F ≈ 62.

Measuring the actual finesse by analyzing transmitted intensity while scan-ning the cavity length yields results closer to F = 10, however. A possiblereason for this reduction would be a mismatch of mirror distance and radius

3.7. Analyzer Assemblies 63

Fig. 3.6: Cross-section of the cavity, drawn to scale.

of curvature, lifting the degeneracy of the transversal modes. This would sme-ar out the resonance line in a way similar to what is observed. An estimatefor this is given in section 4.5.

One indication that this might be the case is that inserting the Brewster’sangle plate does not cause a significant change in finesse. If the broadeningwas due to losses caused by mirror imperfections or the beam clipping onsome feature of the resonator, then the additional losses introduced by theglass plate would increase the effect.

The Brewster plate is a piece of a thin glass microscope slide. It is mountedto a flattened section of a circular plug and can be inserted into a matchinghole drilled into the top of the cavity. This keeps the position stable andallows the angle to be adjusted to optimize the signal from the analyzers.

Finally the whole cavity assembly is fixed to a post to mount it on the opticaltable.

3.7 Analyzer Assemblies

The relative phase of the transmitted polarization components is turned intoa dispersion-shaped lock signal as described in section 2.3.

The actual analyzer assembly consists of a polarizing beam splitter cube andtwo BPW34 photodiodes mounted in a small metal box with a 1cm holeto let the light in. A piece of Edmund Optics quarter wave retarder foil ismounted in front of this opening. It serves to convert the circular polarizationcomponents in the light from the cavity into linearly polarized componentsthat can then be detected with the help of the cube. Figure 3.5 shows thegeneral layout. Theory shows that the optimum angle for the quarter waveretarder plate in front of the cube will always be 45° between the mainpolarization axes and the fast axis of the plate.

64 3. Construction

However, in reality the difference in optical paths between fast and slow axiswill never be exactly λ/2. This makes it possible (and sometimes necessary)to adjust the angle to correct imbalances in the light distribution to the twophotodiodes, which would otherwise degrade the lock signal. To be able todo so, a rotating mount is used. The retarder foil is fixed to a wedge shapedfrom a sheet of thin aluminum, creating an angle between the plane of thefoil and the direction of the incident beam that is designed to adjust thewavelength where the foil works as a true λ/4 retarder to the respective laserwavelength as suggested by the Edmund Optics catalog. The wedge is gluedto the rotating side of the mount, the box containing cube and photodiodesto its back.

In order to simplify optical alignment as well as to save space, the electronicsof the difference amplifier are in a second metal box, which is grounded toreduce interference effects.

The setup for the green analyzer was originally identical. But when opera-ting both lasers simultaneously, its difference signal was showing an offsetdepending on the intensity of the red light present, but ranging up to themagnitude of the green signal itself.

The reason for this is that although the paths of the red and green beamsare separated as much as possible, some scattered red light will reach thegreen detector. This can be assumed to mostly linearly polarized and shouldtherefore have no effect on the difference signal. But since the conversionbetween circular and linear polarization does not work equally well for allwavelengths, part of the incident stray light will emerge from the retarderplate in the form of linearly polarized component that is not aligned alongthe diagonal of the beam splitter cube and will therefore mainly be directedto one of the photodiodes. This creates an offset on the green difference signalthat depends on the intensity of the red laser (and on the red signal for thegreen laser), not only shifting the lockpoint for the feedback loop, but alsogenerating additional noise on the signal from amplitude fluctuations of theother laser system and should therefore be minimized. The effect is strongerfor the green detector since a higher intensity of red light is necessary to geta good signal despite the distribution of power over several sidebands.

To counteract the effect, the entire detector assembly for green light is at-tached to the rotating side of the rotatable mount so that the cube itselfcan be oriented at a 45° angle to the remaining polarization. Wedge and foilare mounted with a fixed angle with the fast axis of the tilted retarder foilrotated 45°from the vertical.

Taking full advantage of this reduction in the error signal will require precisealignment of the incident beam and the analyzer assembly so that the beamintersects the active plane inside the cube in the same point as the axis ofrotation. This has not been fully achieved yet, causing the partial beams tomove away from the photodiodes as the assembly is rotated.

Finally, both difference amplifiers contain a simple, passive low-pass filter

3.7. Analyzer Assemblies 65

that can be switched on or off. For the analyzer operating on the red lightthe cut-off frequency is designed to be 10Hz. This follows the reasoning thatwhile any drifts of the resonator length are likely to happen on a timescalemeasured in seconds, this sensor has shown the tendency to pick up strong50Hz noise from the fluorescent room lights.

The detector for the green beam only shows 50Hz noise to a lesser extent.Its filter serves to prevent resonant oscillation in the feedback loop of thedye laser. The cut-off frequency here is designed to be 1kHz, which will alsoreduce the possibility that the feedback will have unwanted effects on thelaser’s TAF-lock, which utilizes a modulation frequency of 2.5kHz.

4. System Characteristics

4.1 About Linewidths

The spectrum from any kind of oscillator always has some kind of peak,or line, at the main frequency of oscillation. Observation at a high enoughresolution will show that this is never just an infinitely narrow line, but thatit extends over a certain frequency range, where the width of the peak iscalled the linewidth. For lines with a simple shape it will often be givenas the FWHM, the Full Width at Half Maximum. But, as illustrated infigure 4.1, the shape can also be quite complex. In this case a simplifiedmathematical function is often fitted to the actual data in order to give ameaningful description of its width. In the following sections this fit function

Fig. 4.1: Exemplary line shape. From Minicircuits’ datasheet “VCOPhase noise”, available at http://www.minicircuits.com.

will commonly be a Gaussian due to its mathematical simplicity:

Pgauss(f) = e−( f−f0w )

2

, (4.1)

where f gives the center frequency and w describes the width of the line.Unless mentioned otherwise, all measured signals will be powers or intensities.At a frequency f +w the signal has fallen to 1/e of the maximum. When thiswidth is used in the following it will be explicitly referred to as the Gaussianwidth, to distinguish it from the FWHM. Since approximately e−0.6931 = 0.5,the Gaussian falls off to 0.5 its maximum value at a frequency offset 4f1/2 =0.8326w, corresponding to a FWHM of 1.665w.

Finally, for measurements where frequencies are being measured directly, butobtaining a proper spectrum would require calculating a histogram for the

68 4. System Characteristics

whole series, the linewidth can be found from the properties of the normaldistribution known from statistics (see [20], for example). Its density, whichin this case is essentially the amplitude at a specific frequency, is

anormal(f) =1

σ√

2π· e− (f−f0)2

2σ2 , (4.2)

where σ is the root-mean-square (RMS) value of the distribution:

σ2 =n∑

i=1

(xi − µ)2

n. (4.3)

Here xi is the ith data point, n is the number of points and µ the averagevalue. This is easy to calculate and comparing equations 4.1 and 4.2 clearlyshows the simple connection between Gaussian width and σ:

wgauss2 = 2σ2 ⇔ wgauss =

√2σ (4.4)

So for a true Gaussian line, the conversions between the three different mea-sures of linewidth work out to:

wgauss = 0.601wFWHM =√

2 · σ (4.5)

wFWHM = 1.665wgauss = 2.355 · σ (4.6)

σ = 1√2· wgauss = 0.425 · wFWHM (4.7)

However, it should be perfectly clear that these conversions will only be arough estimate for non-Gaussian line shapes.

4.2 Stability of the Master Laser

There are currently no measurements that give an indication of the linewidthof the master laser. It would be possible to lock the second extended cavitylaser present in the experiment to the transition used by the repumper. Over-laying both beams and analyzing the beat signal with a spectrum analyzerlinked to a fast photodiode would give a good indication of the linewidth.This will be done in the near future.

Until then, it is safe to assume that the linewidth of the extended cavitylaser is comparable to that of other, similar systems. A commercially availa-ble design is the Toptica DL100 laser, which has a short-time linewidth of≈1MHz1. Another system, which is self-built like ours, is described in [31]and achieves linewidths around 3MHz. This number is for observation witha scanning Fabry-Perot resonator of high finesse. Contributions to this li-newidth are spontaneous emission (≈ 100kHz) as well as noise (partly shotnoise) on the laser current. The feedback loop working on the grating piezo

1 from the datasheet available at http://www.toptica.com

4.3. Noise Introduced by the Slave Laser 69

will suppress most of the effects of cavity vibrations which would otherwiseincrease the linewidths even further.

As mentioned before, one way to reach even narrower lines (down to 350kHzaccording to [27]) requires the driving current to have an extremely highpassive stability with fluctuations of not more than a few µA. Since the spec-troscopy setup is already in place, our feedback loop will probably be adaptedto include a second branch which controls the diode current. In [26] Petelskiet al. report linewidths down to 120kHz with a setup that is extremely si-milar to the one used in our experiment. This is on the order of the naturallinewidth for the green transition line in ytterbium which is ≈180kHz andwill be sufficient if it can be transferred to the green laser system. Broa-dening beyond the transition linewidth will reduce the effectiveness of thegreen MOT, reducing the number of atoms that can be caught [32], but notpreventing its operation altogether.

4.3 Noise Introduced by the Slave Laser

If temperature and current for the slave laser diode are set properly to ensurestable injection locking, frequency fluctuations will be virtually non-existentcompared to free-running operation. In fact, the basic theory for injectedoperation as outlined in section 2.4.2 requires the laser to operate at exactlythe injected frequency for any steady state situation.

In reality, variations in current and temperature will cause the free runningfrequency to shift against that of the injected light, resulting in phase fluc-tuations of the emitted beam as shown in figure 2.26. To reach an estimateof the magnitude for this broadening a worst case scenario is presented inthe following paragraphs.

An injection system on the verge of instability might oscillate throughoutthe whole locking range, resulting in an output phase shift varying between−π and π. In terms of signal theory this can be described as phase modulati-on [30]. Looking at the spectrum, this will cause sidebands at the modulationfrequency, with a height that depends on the maximum phase variation. Ifthe time scale for current and temperature variations is is wide enough, thensidebands will smear out to form a broad background which will have littleeffect on the actual line shape. The worst case here would be a modulationat a single frequency that is just small enough to not be resolved individual-ly, but big enough to broaden the line perceptibly. For a single frequencymodulation the phase variation can be described as

φ(t) = β sin ωm t , (4.8)

where ωm is the modulation frequency and β is the so-called modulationindex. For the maximum modulation described above β = π. The amplitudeof the nth-order sideband is then given by the Bessel function Jn(β), which isplotted in figure 4.2 for J0(x), J1(x) and J2(x). A maximum in the height of


the first sideband will occur near a modulation index of β = 2. This roughlycorresponds to oscillations through about 90% of the locking range. For anyhigher modulation more power will be found in higher order sidebands, whichwill complicate the line shape, but have only a small additional effect on thelinewidth. For this reason the further calculations will be done for β = 2,even though this is technically not the worst possible case.

Since the coefficients found from the Bessel functions describe the electricfield amplitude, the relative intensity for the carrier and the sidebands atmaximum modulation will be

Ic = J20 (β) = 0.22 (4.9)

Isb = J21 (β) = 0.32 . (4.10)

1 2 3 4 5x

-0.4

-0.2

0.2

0.4

0.6

0.8

1

J

Fig. 4.2: Bessel functions J0(x) (in green), J1(x) (in red) and J2(x)(light blue).

Assuming an originally Gaussian line shape with an amplitude and width of1 for simplicity, the sidebands will yield a maximum linewidth if they areoffset at a distance of 1. This is plotted in figure 4.3. For higher offsets itwould be possible to observe the sidebands separately from the carrier signal,returning the effective width to its original value. For this construction theFWHM nearly doubles from 0.832 for the unmodified Gaussian line to 1.641for the maximally broadened one.

Generalizing this, injection phase noise can lead to a doubling of the effectivelinewidth, but only if the injected laser is just marginally stable and all thechanges in free running frequency happen at a frequency corresponding closeto the original linewidth.

In reality, only current fluctuations are likely to occur at frequencies of se-veral hundred kHz and these can be strongly reduced by adding sufficientfiltering to the current supply if necessary. This would take the form of alowpass in the DC branch and a highpass with a cutoff frequency slightlybelow the modulation frequency in the AC branch of the bias-tee as shownin figure 4.4. All the thermal effects caused by external influences are likelyto occur on a frequency scale of up to several Hz, creating no noticeable

4.4. Performance of Modulation Electronics 71

-3 -2 -1 1 2 3x

0.2

0.4

0.6

0.8

1

y

-3 -2 -1 1 2 3x

0.2

0.4

0.6

0.8

1

y

Fig. 4.3: Top: Basic Gaussian line shape for a (Gaussian) widthof 1, corresponding to a FWHM width of 0.832. Bottom: Effect ofimposing sidebands on this line with an offset equal to the Gaus-sian width and with amplitudes for carrier (green) and first-ordersidebands (red) given by the respective Bessel functions for a mo-dulation index of 2.

offset for the sidebands. Effects of the modulation, like heating or shifts infree running frequency due to the changing current will always be periodicwith the modulation frequency and therefore conincide with the sidebandscreated by amplitude modulation or their higher harmonics. Furthermore,the influence can always be reduced by increasing the injection power or de-creasing the diode current to extend the locking range, leading to reducedphase fluctuations.

Therefore it can be concluded that phase-modulation broadening in the in-jected laser can be neglected, at least for the sake of this evaluation.

4.4 Performance of Modulation Electronics

4.4.1 Stability

The linewidth of the VCO is also insignificant compared to that of the lasersused. The datasheet2 for the ZOS-1025 gives a power of -92dBc (decibel rela-

2 available at http://www.minicircuits.com


Fig. 4.4: Filtering the laser diode current to reduce line broadeningby injection phase shifts.

tive to carrier) at an offset of 10kHz from the main frequency of oscillation.Once again assuming a Gaussian line shape, this is equivalent to a Gaussianwidth of 2.1kHz or a FWHM of 3.5kHz.

The line is slightly broadened by the amplifier used. For 12V supply voltage,the VCO provides 8dBm (6.4mW at 50Ω impedance) of power. The ERA-5amplifier has a gain near 20dB over the frequency range used. It can onlyprovide 17.2dBm (about 67mW at 50Ω) of output power, leading to strongcompression effects. This is illustrated in figure 4.5, with the compressioneffects simplified to a hard clipping at maximum output power. The resultingFWHM is now more than doubled to 7.4kHz. This is still narrow compared tothe laser linewidth and even the width of the green ytterbium transition. Thecompression effects can also be reduced or even eliminated by reducing thesupply voltage of the VCO and thereby its output power. This has a slightcost in the power of the modulation signal, since there is no hard cutoff atthe specified maximum output but rather an increasing nonlinearity as thegain drops with increasing input power.

-6 -4 -2 2 4 6f

0.2

0.4

0.6

0.8

1

amplitude

Fig. 4.5: Simplified shape of the spectrum for the modulation signalbefore (red) and after (green) compression effects of the amplifier.Drawn for a gaussian line shape and hard cutoff at maximum outputpower. Amplitudes relative to maximum, frequency in kHz relativeto the main oscillation frequency.

An effect that has a much bigger impact on the stability of the laser system

4.4. Performance of Modulation Electronics 73

are changes in the VCO’s oscillation frequency on longer time scales. Thefactors driving these changes are changes in temperature as well as controland supply voltages. Minicircuits’ documentation gives some parameters forthese variations, shown in table 4.1

parameter value meaningfrequency pulling 51kHz (ptp) frequency changes induced by feedback

on the output port. For feedback with apower reduction of 12dB.

frequency pushing 1MHz/V change of frequency with supply voltagetuning sensitivity 30MHz/V change of frequency with tuning voltagetemperature sens. <200kHz/K change of frequency with temperature

Tab. 4.1: Important factors on frequency stability for a VCO andtheir typical values for the ZOS-1025.

In order to keep these changes small, i.e. in the 10kHz range, the whole systemwill need to operate in a steady state to keep feedback constant and minimizefrequency pulling. If this turns out to be problematic, some isolation betweenVCO and amplifier will reduce feedback power. This might simply take theform of a resistive stage using up the excess power generated by the VCO toensure that the feedback is reduced as well.

Requiring 10kHz stability for each contribution means that supply voltagewill need to be kept stable to 10mV, tuning voltage to 0.33mV and tempe-rature to 0.05K. These are achievable values, provided some kind of tempe-rature stabilization is added and the signal cables are kept short to reducethe amount of EMF noise picked up.

Some measurements have been performed with the current setup which isstill suffering from long cables, low-pass filters only in the VCO driver boxand not even passive temperature stabilization. For this a webcam was set upto watch the display of the frequency counter linked to the auxiliary outputof the VCO. The pictures taken at regular intervals were then evaluated. Thisallowed measurement without thermal disturbance by human presence.

Data for a measurement with 730 data points at intervals of 1s is shownin figure 4.6. The frequency counter has an averaging time of about 0.5s atthis resolution. Excursions of more than 10kHz are rather rare, indicating astability on this time scale that is more than sufficient. The RMS value forthis series was σ = 3.04kHz, equivalent to a full width at half maximum of7.2kHz if the line is Gaussian (wgauss = 4.3kHz)

Another series of data points has been taken at intervals of 90 seconds. Thisplotted in figure 4.7. While the short term stability is similar to the one foundon a 1-second time scale, there are also long term drifts. These are probablytemperature effects corresponding to the building cooling off over the courseof the night. The measurement was started at around 9:45pm and the jumpat around 8:00am was caused by opening the door to the laboratory.


548,715

548,720

548,725

548,730

548,735

548,740

548,745

548,750

00:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:00 09:00 10:00 11:00 12:00

time [mm:ss]

freq

uen

cy [

MH

z]

Fig. 4.6: VCO oscillation frequency over time. 730 data points takenat 1s intervals.

Since these drifts cover more than 100 kHz in a time of one hour even withoutadditional disturbances by human presence, they will need to be reducedbefore a continuous stable operation at low linewidth is possible. If passivestabilization proves to be insufficient some kind of active temperature controlwill be implemented. An alternative would be setting up an automated wayto measure the current frequency and generate a feedback signal based onthat. But for the moment, readjusting the oscillation frequency based on thedisplay of the frequency counter will be sufficient.

4.4.2 Tuning Range and Power

Data has also been taken on the tuning range and power output of the mo-dulation electronics. This was measured after the amplifier with a spectrumanalyzer. The results are shown in figure 4.8.

By varying the tuning voltage from 0V to 16V, frequencies between544.20MHz and 1062.92MHz can be reached. This extends slightly furtherthan the range specified in the datasheet (685MHz to 1025MHz) but at thecost of reduced output power even at maximum amplifier gain. Being able totune the frequency over almost a full octave has advantages when it comesto locking the green laser to a specific wavelength, provided the cavity hasbeen designed to match the tuning range. If the free spectral range is equalto the minimum VCO frequency, then increasing the modulation frequencyup to twice the free spectral range while keeping the laser injected will mo-ve the adjacent sideband all the way from one fringe corresponding to theinjected wavelength to the next. This way the cavity can be locked to anydesired length except for a small range where the locking signal generatedfrom the sideband is obscured by the injection or by another sideband of high

4.5. Optical Spectra 75

548,50

548,55

548,60

548,65

548,70

548,75

548,80

21:00 22:00 23:00 00:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:00 09:00 10:00

time [hh:mm]

freq

uen

cy [

MH

z]

Fig. 4.7: VCO oscillation frequency over time. 494 data points takenat 90s intervals.

intensity.

Characterizing the final parts of the system, bias-tee and laser diode, hasturned out to be difficult. The main unknown factor is the noise added byfrequency pulling in the VCO due to feedback caused by poor impedancematching. While the RF components are all designed to work with 50Ω im-pedance, laser diodes can have impedances as low as 2Ω. This would leadto massive feedback, some of which might reach the VCO even through theamplifier. It also decreases modulation efficiency, but the observed sidebandsare still high enough for the cavity lock to work.

4.5 Optical Spectra

When scanning the resonator length with the piezo it functions as a sort ofoptical spectrum analyzer, since for any given length certain wavelengths oflight will be in resonance as described in section 2.2.2. The problem is thatif the resonance condition is met for one particular frequency, then it is alsomet for all frequencies that differ from it by an integer multiple of the freespectral range fFSR. A similar thing happens when scanning the cavity. If acertain wavelength λ is resonant at a certain cavity length, then it will beresonant again when the cavity is extended so that the optical path is exactlyone wavelength longer. For the confocal setup used this occurs for a change inlength of λ/4. Since for each of the two lasers the variations in wavelength arevery small compared to its absolute value, the spectrum taken when scanningthe cavity over λ/4 will repeat itself if the scanning range is increased. Theresult of these two effects is that a spectrum taken in this way will containall the spectral components present, but unless the spacing is known with an


15

16

17

18

19

20

500 550 600 650 700 750 800 850 900 950 1000 1050

frequency[MHz]

P[d

Bm

]

Pout[dBm]

Fig. 4.8: Power output in dBm over oscillation frequency in MHzfor the combination of VCO and amplifier over the full tuning range

accuracy of at least one spectral range, then it will be difficult or impossibleto reconstruct the real spectrum. Increasing the scanning range will not help.

5

10

15

20

25

30

35

40

4 5 6 7 8 9 10

piezo control voltage [V]

sig

na

l [m

V]

48.7mA

2

8

7

4

3

9

10

5 / 61

Fig. 4.9: Transmission spectrum for 780nm laser diode with side-bands at 543.33MHz modulation frequency. Signal averaged overfive sweeps to reduce noise.

Figure 4.9 shows a set of data that was obtained in this way. It was mea-sured with the modulation system set to a frequency of 543.33MHz and theslave laser operating at 47.8mA. Its temperature is at 16.9°C (14.9kΩ NTCresistance). The light transmitted through the cavity was measured with anunamplified photodiode. This gives a good sensitivity even for weak signals,but the response is strongly nonlinear, levelling off to a maximum outputsignal of around 0.6V. It also does not provide a low impedance output,increasing the noise picked up from electro-magnetic interference. But by


averaging over five sweeps (for this and all other data taken in this way) itis possible to extract a spectrum with a tolerable signal-to-noise ratio.

The plot clearly shows the main laser line (c), which will be called the carrierin the following, as well as contributions from several orders of sidebands (a,b and d, e). It also clearly repeats (1 and 2) after the control voltage has beenincreased by about 2.5V. The actual piezo voltage is generated from this byan amplifier with a fixed gain of 50 and a maximum output voltage of 500V.In order to analyze the signal in more detail, the various components havebeen fitted with Gaussians in figure 4.10. All lines can be well fitted withGaussians of width 0.18V. Their positions are given in table 4.2.

5

10

15

20

25

30

35

40

4 5 6 7 8 9 10


sig

na

l [m

V]

48.7mA

fit 2

fit 1

Fig. 4.10: Transmission spectrum with manual fits. The red andgreen parts mark different repetitions of the signal.

line position line position1 4.50V 6 7.03V2 5.15V 7 7.68V3 5.80V 8 8.41V4 6.45V 9 9.21V5 7.10V 10 9.90V

Tab. 4.2: Positions of the fitted Gaussian lines in figure 4.10, givenby means of the piezo control voltage. Numbering is from the leftto the right as plotted in the diagram.

Although neighboring lines should be spaced at constant distances, they ap-pear closer together for high control voltages. This can be easily explainedsince the length change of the piezo is reduced at high voltages. Calcula-ting the free spectral range from equation 2.25 gives a value of 749MHz fora 100mm confocal cavity. Therefore a modulation frequency of 543MHz willplace the sidebands closer to the next “echo” of the carrier than to the carrieritself.


Without this folding of the spectrum into the free spectral range of the reso-nator, the signal would look more like indicated in figure 4.11. But at leastfor higher order sidebands it is still difficult to tell if any particular sidebandbelongs to the high or low frequency side of the spectrum while they are allintermixed. For this reason a more detailed analysis of the signal will be doneat a different modulation frequency.

Figure 4.12 shows the spectrum measured for a modulation frequency of829.54MHz. The slave laser is operating at a current of 48.0mA and withthe temperature still at 16.2°C. Since the frequency offset of the sidebandsis just slightly greater than the frequency difference to the next repetitionof the carrier, the various lines combine into a single shape. This revealstwo additional lines between the groups centered around the repetitions ofthe carrier. Since these also shift when the injected frequency is varied, theymust be products of the modulated diode and should therefore also be lo-cated at integer multiples of the modulation frequency from the carrier. Toconfirm this, Gaussian lines have been fitted to the spectrum in figure 4.13.A Gaussian width of 0.16V for each of the lines seems to give the best fit.

Looking at the distances between adjacent lines shows that they are veryevenly spaced. Distances vary between 0.25V and 0.29V piezo control voltagefor voltages of 4V to 9V. In the range between 9V and 10V the spectrum ap-pears stretched out again. But there is an irregularity at the position markedwith red vertical lines in the diagram. The line spacing here is approxima-tely 0.36V, too far away from either the normal value or a multiple of it tobe likely to be a random fluctuation. Since it appears at a control voltagenear 7V, nonlinear effects in the piezo are also unlikely to be the cause. Abreak like that is expected to appear between repetitions of the signal un-less the ratio of modulation frequency to free spectral range happens to bea simple fraction. Now the repetitions of the signal can be isolated and theactual spectrum can be reconstructed from the folded data. This is plottedin figure 4.14.

The plot shows this to be highly asymmetric. The amplitude of the sidebandsdrops close to zero for the third order sidebands. The right side, however,shows strong forth and fifth order lines as well. It is quite likely that freqeun-cy of the master laser coincides with one of these, possibly the fifth ordersideband. The observed increased intensity can then be explained as directamplification of the injected light. Unfortunately our setup does not current-ly allow sending the injection beam into the cavity in a way that would makeit possible to compare its wavelength to those of the various sidebands, sothis cannot be shown conclusively.

The reason for the asymmetry of the spectrum with regard to the injectedfrequency might be suppression of one the sidebands by the mechanismsdescribed in the chapter on vestigial sideband operation (section 2.4.3.5) orsimply by lack of gain for its frequency. In the latter model, which is analternative to the one offered before, the modulation effects do not directlycreate the observed spectrum but instead transfer intensity from one sideband


5

10

15

20

25

30

35

40

2 3 4 5 6 7 8 9 10


sig

na

l [m

V]

48.7mA

fit 1

Fig. 4.11: Transmission spectrum for modulation at 543MHz again,with fit corresponding to the spectrum without folding caused bythe limited free spectral range.

5

10

15

20

25

30

35

4 5 6 7 8 9 10


sig

na

l [m

V]

Fig. 4.12: Transmission spectrum for modulation at 829.54MHz.Slave laser operating at 48.0mA and 16.2°C (14.9kΩ). Spectrumaveraged over five sweeps of the cavity.


5

10

15

20

25

30

35

4 5 6 7 8 9 10


sig

na

l [m

V]

!

Fig. 4.13: Spectrum for modulation at 829.54MHz, with fitted Gaus-sian lines. Green vertical lines mark the regular spacing of adjacentlines, while the red ones mark the different spacing between repetiti-ons of the spectrum. The transmission lines highlighted in red showthe spacing as it would appear without folding into the cavity’s freespectral range.

0

5

10

15

20

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

frequency / fmod

am

pli

tud

e [

mV

]

frequency of

injected light

Fig. 4.14: Reconstructed spectrum for modulation at 829.54MHz.Frequency is given in units of the modulation frequency relative tothe likely frequency of the injected light (marked in red).


to the neighboring ones. Each of the laser oscillations started in this way willhave its own gain or loss, depending on the gain in the laser medium and theproperties of the laser resonator at this frequency.

This is illustrated in figure 4.14: The heights of the various sidebands mea-sured fit a fictitious gain envelope (plotted in gray) centered at an offsetof approximately 4.2GHz from the injection frequency. This is in agreementwith observations on the behavior of the spectrum when the diode currentis changed: For low diode currents and a well adjusted injection setup, theindividual lines do not change their positions on the oscilloscope, but theiramplitudes change as the envelope is gradually shifted. It is important tonote here that this envelope does not correspond to the gain of the activemedium in the diode, as this would result in a much wider shape. Instead itis given by the shape of the resonance line of the diode’s resonator.

A good way to gain more insight into what happens inside the modulatedslave diode might therefore be to change the diode’s resonator, either byadding an external mirror or adding a coating. This should have a strongeffect if the “gain-per-sideband” model is correct but should change little ifthe modulation creates the whole spectrum directly.

For now we will assume that the injection in this example was not into the“carrier”, but in fact into a fifth order sideband. The main laser emissionthen occurs at a frequency more than 4GHz offset from that of the masterlaser. It also shows several sidebands at amplitudes given by an approximatelyGaussian envelope (indicated in grey in figure 4.14) created by the interactionof the laser diode’s gain profile and the intensity redistribution caused by themodulation.

In order to optimize the system’s stability, the laser should be injected intoa first order sideband so the cavity can be locked to the line with the highestamplitude. Locking the cavity to a sideband that is not adjacent to the oneused for the injection will multiply the effect of frequency fluctuations in themodulation system.

Finally, the data taken in these measurements can be used to characterizethe cavity. The spectrum for 829.54MHz modulation frequency will be usedfor this, with the data beyond 9.5V control voltage excluded due to theincreased piezo non-linearities. The average spacing between adjacent lines(ignoring the irregular gap) is 0.269V (±0.02V) and the average spacingbetween repetitions of the same line is 2.519V (±0.28V). In a confocal cavitythis corresponds to a change in length of one quarter wavelength. This allowsthe contraction coefficient of the resonator to be calculated as

ccontr ≡ 4laxial

4Uctrl

=λ/4

4Urep

=780.233nm

4 · 2.519V= −72.27nm/V (±8.61nm/V )

(4.11)Since the actual voltage at the piezo is 50 times higher than the controlvoltage, the contraction coefficient for the piezo tube is

cpiezo =cctrl

50= −1.45nm/V , (4.12)


which comes close to the values of 16µm contraction for a 36mm tube at1000V given in the datasheet3. This would mean an average contraction co-efficient of -1.33nm/V for the 3mm slice used. This was to be expected, sincethe contraction decreases as the voltage approaches its allowed maximum.

The change in resonant wavelength relates to the change in cavity length as

4λres

λ0

=4l

l0. (4.13)

Since all changes are small compared to the initial values, λ0 and l0 can betreated as constants. Linearizing the relation that way and plugging in thecontraction coefficient found above yields

4λres =λ0

l0· cctrl · 4Uctrl (4.14)

and therefore 4λres

4Uctrl

= −0.564pm/V (±0.007pm/V ) . (4.15)

This will later be used in the analysis of the error signals of the cavity feed-back loop.

Based on its design, the cavity has a free spectral range of 749.5MHz. TheFSR can also be found from these measurements by comparing the distancebetween adjacent sidebands to the distance between repetitions of the sameline:

fFSR =2.519V

2.519V + 0.269V· fmod = 749.5MHz (4.16)

This exact match is quite surprising considering the non-linearities and largeerror margins, but it shows that the calculations are generally correct.

The finesse of the resonator is another value that is conveniently found thisway. It is given by

F =fFSR

fFWHM

≈ 2.519V

1.665 · 0.16V= 9.5 , (4.17)

where the width of the transmission line has been converted from its Gaussianvalue to FWHM. This low value is surprising since the theory predicts afinesse of around F = 50 for the mirror reflectivities used. An explanationmight be deviations from the confocal setup of the cavity. The resonantfrequencies for higher transverse modes in a resonator with spherical mirrorsare offset from the lowest order resonance by

4fnm = (n + m)arccos

√g1 · g2

π︸︷︷︸phase shift factor

· c

2L, (4.18)

3 available at http://www.piezomechanik.de, the dimensions of the piezo tube beforecutting were 1mm×10mm×36mm.


where n and m are the parameters describing a Gaussian-Hermite transversalmode and g1 = g2 = 1− l

Rare the “g-factors” describing the relation of radius

of curvature (R) and spacing (l) for the mirrors (see [24]). For a perfectlyconfocal resonator g1/2 = 0, and the phase shift factor will be exactly 0.5,leading to the pattern of degenerate lines at half the original spacing asdescribed in section 2.2.2. But if the radius of curvature is just one millimetergreater than the resonator length, the factor will drop to approximately 0.497.Such a difference is well possible since the radius of the mirrors is not specifiedto much better than 1%. This will lift the degeneracy and the transversalmodes with (m + n = 2) will be offset from the next longitudinal mode,which it would normally be degenerate with, by 0.64% of the free spectralrange. Transversal modes with (m + n ≈ 15) would be shifted by 10%.Creating a broadened line shape that accounts for the low cavity finesse of9.5 would require transversal modes of decreasing amplitudes up to around(m + n = 30). These are likely to be excited in our case since the off-axisoperation of the resonator is based on exciting a mixture of transversal modes.

The lowest order mode, after being spread out some more by the effect of theplano-concave substrate of the cavity mirror, has an angle of divergence ofslightly below 0.2°(mode-matching calculations done according to [24]). Forhigher transverse modes, the mode width will approximately scale as

wn ≈√

n · w0 (4.19)

The laser beams enter the cavity at angles near 2°. To construct a beamlike that from the resonator modes will require contributions from modeswith n ≥ 100. Inversely, these modes will be excited by the incident beam,making broadening by an imperfect cavity length more than likely. In orderto reduce this effect the cavity length needs to be carefully matched to themirror radius of curvature. This requires adding a way to adjust the lengthof the assembled cavity.

The most important values found in this section have been summarized intable 4.3.

property symbol value errorlength l0 100 mm 0.2 mm

free spectral range fFSR 749.5 MHz 1.5 MHzfinesse F 9.5 0.6

contraction for control voltage ccontr -72.27 nm/V 8.61 nm/V

change in resonant wavelength 4fres

4Ucontrol-0.564 pm/V 0.007 pm/V

change in resonant frequency 4λres

4Ucontrol297.5 MHz/V 3.3 MHz/V

contraction for piezo voltage cpiezo -1.45 nm/V 0.17 nm/V

Tab. 4.3: Summary of cavity properties for the 780.233nm laser


-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

-200 -150 -100 -50 0 50 100 150 200

cavity length offset

red lock signal

-3,0

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

green lock signal

Fig. 4.15: Dispersion shaped locking signals for the red and greenlaser systems. Centered and normalized, with cavity length recon-structed from green signal, relative to the position of it center re-sonance line.

4.6 Lock Signals - Cavity

Stabilizing the lasers is the heart of the whole system. With both lasersoperating, the optical setup was adjusted to maximize the locking signalsand minimize the influences they have on each other. Then, the lasers wereadjusted to ensure stable injected operation of the red system and single modeoperation of the green dye laser. Subsequently the output of the differenceamplifiers with filters turned off was recorded with a digital oscilloscope asthe cavity was scanned over several free spectral ranges. The resulting datais shown in figure 4.15. The amplitudes have been normalized and the changein cavity length has been reconstructed from the distances between peaks inthe green signal. The original peak-to-peak amplitudes were 2.08V for thered and 5.64V for the green signal.

Since the distance between repetitions of the signal can safely be assumedto correspond to the free spectral range of 749.5MHz for the respective wa-velength, it is easy to find the width of the slopes. This is around 52MHzfor the red and 48MHz for the green signal, probably due to different mode-matching for the beams. Expressing this in terms of changes in resonatorlength yields 13.5nm for red and 8.9nm for green.

The red dispersion-shaped signal is used to lock the cavity to the laser. Toadjust the locking system and select the desired lock point, the cavity isscanned by adding a triangle wave to the output of the loop amplifier. Thenthe input offset is adjusted to center the signal so that the lock will operate atthe zero-crossing to minimize the effect of intensity fluctuations. The desiredresonance is selected with the output offset. Now the scan is turned off and

4.7. Lock Signals - Dye Laser 85

the loop is activated. Finally the low-pass filter in the difference amplifier isturned on to reduce the effects of line noise and background light.

It is possible to get an indication for the quality of the lock by looking atthe error signal while the lock is operating. For the red lock this was doneusing the signal shown above, with an amplitude of 4V peak-to-peak at thedifference amplifier. After activating the feedback loop the fluctuations ofthe error signal are almost completely confined to the range of ±6mV. Usingthe width of the slope found above and assuming it to be approximatelylinear, this corresponds to laser frequency noise of ±78kHz or uncompensatedcavity length fluctuations of ±0.020nm. These numbers describe how well thecavity length is adjusted to keep the red beam in resonance. They are partlycaused by disturbances that happen too quickly to be compensated for, butalso partly by electronic or optical noise picked up by the system. For thelow remaining fluctuations observed here, the latter are probably even thedominating factor. When measured at the monitor output of the feedbackloop, the observed noise was around 50mV independent of whether the lockwas operating or not.

The low pass filter has been included based on the assumption that in theabsence of vibration any errors in cavity length will be thermally induceddrifts on a time scale of several seconds. In this case the filter will reducehigh-frequency noise caused by instabilities of the master-slave system orelectromagnetice interference while letting the biggest part of the actual (low-frequency) error signal through.

4.7 Lock Signals - Dye Laser

After the cavity has been locked to the red laser, the dye laser is in turnlocked to the cavity. The residual unfiltered noise found here is ±30mV for a4.4V peak-to-peak signal with the 1kHz low-pass filter active to keep the pie-zo actuator in the dye laser below resonance. This is equivalent to a changein resonator length of 0.061nm or in laser frequency of ±330kHz at frequen-cies below the bandwidth limit imposed by the photodiode amplifiers, whichshould operate well at up to 50-100kHz.

Since these numbers are considerably higher than those for the cavity lock,they represent the laser linewidth. Although no digital data has currently be-en taken, assuming a Gaussian width of 330kHz (*FIXME: this was 660kHZbefore, consequences on final linewidth?) will be a conservative estimate.While this is too wide to observe the linewidth of the 555.8nm line in yt-terbium, it will be sufficient for the operation of a MOT if the detuning isincreased slightly.

While some fine-tuning is still possible, a dramatic reduction in linewidthwill require a change in the setup used. This might take the form of puttingan acousto-optical modulator into the optical path between laser and cavi-ty/experiment to be able to increase the bandwidth of the lock beyond the


limit of 2.5kHZ imposed by the dye laser (see section 3.5). The new band-width limit for the feedback loop is then given by the time it takes for theAOM to translate a different driving frequency into a different optical outputfrequency. This has an unavoidable delay on the order of 1µs caused by thetravel time of the sound waves in the crystal medium. If an even higher band-width is required, a setup as described in [33], consisting of a combinationof electro-optic and acousto-optical modulators, can be used. This achieveslinewidths of around 20kHz or with considerably higher effort even in thesub-kHz range [34].

The dye laser also contains a second piezo actuator with a light-weight mir-ror. Operating this at a frequency above 2.5kHz to avoid disturbing theinternal locking system might make it possible to increase the bandwidth ofthe feedback loop without external components.

To obtain an estimate for the required bandwidth, the error signal of anearly version of the locking system was recorded on various timescales upto the maximum bandwidth (45MHz) of the oscilloscope used. This datawas then Fourier-transformed to obtain a spectrum for the fluctuations. Theresults after averaging the spectral data over 25 series, compensating foraliasing effects and merging the partial data from different frequency rangesare shown in figures 4.16 to 4.19. Even with the lock operating, 35% of the

0

5000

10000

15000

20000

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

frequency [Hz]

no

ise

po

we

r

5ms

50ms

Fig. 4.16: Noise power spectrum for the error signal of the dye laserlocked to the cavity. Plot shows data from measurements at 5msand 50ms sampling rate with power given in arbitrary units. Lowfrequency range shown.

total noise power is located in the low frequency band up to 1kHz. So thereis considerable room for improvement simply by tuning the loop behavior.Increasing the bandwidth to 10kHz will only cause a marginal decrease inlinewidth, as the 1kHz to 10kHz band only contains 14% of the noise power,with 1.5% present in the inaccessible range around the internal modulationfrequency of 2.5kHz.

4.7. Lock Signals - Dye Laser 87

0

25

50

75

100

125

150

2 4 6 8 10 12 14 16 18 20

frequency [kHz]

no

ise

po

we

r

5ms

500us

1527

Fig. 4.17: Noise power in arbitrary units over frequency. Interme-diate frequency range shown. Data taken with sampling rates of5ms and 0.5ms.

For a significant reduction the maximum operating frequency of the lockneeds to be pushed up to include a bump that appears in the spectrumcentered around 30kHz. The band from 10kHz to 60kHz contains 48% of thetotal noise power. Higher frequencies account for the remaining 3%.


0,1

1

10

100

1000

10000

0 10 20 30 40 50 60 70 80

frequency [kHz]

no

ise

po

we

r500us

Fig. 4.18: Noise power in arbitrary units over frequency. Logarith-mic scaling used for clarity, high frequency range shown.

0,001

0,01

0,1

1

10

100

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2

frequency [MHz]

no

ise

po

we

r

500us

50us

Fig. 4.19: Noise power in arbitrary units over frequency up to thehighest measured frequencies. Logarithmic scaling for clarity, fullmeasured spectrum plotted.

4.8. The System as a Whole 89

4.8 The System as a Whole

Now that estimates for the various subsystems have been found, their effecton the final stability and linewidth of the full system can be discussed.

The convolution of one Gaussian line with a second one will always result inanother Gaussian line [29]:

e−( xwa

)2

⊗ e−

(x

wb

)2

≡∫ ∞

−∞e−( y

wa)2

· e−(

x−ywb

)2

dy = A · e−x2

wa2+wb2 , (4.20)

where A is an amplitude factor that has no effect on the line shape. If theinitial distribution were normalized to have an area of 1, the final Gaussianwould be as well. This property is used in statistics, but shall not concernus here. The important result is that when a Gaussian line of width wa

undergoes broadening of width wb, the resulting line will have a Gaussianwidth of

√wa

2 + wb2. This is assuming that the broadening mechanisms are

completely independent of each other.

For finding the linewidth of the system in its current state, the widths fromthe previous chapters have been gathered in table 4.4. Of course these do not

component widthlocked master laser 1MHz

injection phase noise (small)modulation system 4.3kHz

cavity lock 156kHzdye laser lock 330kHz

Tab. 4.4: Summary of Gaussian linewidths for the individual com-ponents.

quite fulfill the requirement of independent subsystems. Especially the noiseon the cavity simply echoes part of the noise on the red laser system. But aslong as this does not force the system into oscillation, it will actually causethe resonator to filter out some of the laser noise as it will not follow all of itsfluctuations. Additionally, the 1Hz-lowpass used in the cavity feedback loopwill also serve to decouple the cavity length from high-frequency noise in thered laser system. So the linewidth found here will be an upper limit.

There is also an effect caused by the different wavelengths in the resonator.The total system linewidth up to the contribution from the cavity lock is1208kHz. By comparing this to the free spectral range, this can be translatedinto a Gaussian width for the resonator length fluctuations:

wlen =wfreq

fFSR

· λred

4= 0.31nm (4.21)

Translating this back into a frequency for the green laser yields

wgreen =4wlen

λgreen

· fFSR =λred

λgreen

· wfreq = 1696kHz (4.22)


The last factor will be the lock of the dye laser. This broadens the line to aGaussian width of 1728kHz.

The main contribution is still from the master laser. If all other numbers arecorrect, then a reduction of its linewidth to 120kHz will cause the linewidthafter the total system linewidth to fall to 431kHz, so this will be one optionto consider. Another option is to optimize the compromise between lockingbandwidth and noise reduction provided by the low-pass filter in the cavitylock. If the linewidth of the system can be measured directly, for example byDoppler-free spectroscopy on the ytterbium line, then adjusting the cutofffrequency might prove a valid alternative to modifying the master laser.

5. Operation

The first complete version of the system is operational. Both feedback loopswork well and stay in lock without strong external disturbances. There issome cross-talk between the red and the green systems, causing an offset ofthe locking signals depending on the intensity of the other laser. It is mostlikely caused by stray light in the resonator as there is no notable changewhen the laser crosses a resonance.

So far only basic tests have been run with the system, the results of whichare presented in this chapter.

5.1 Ytterbium Spectroscopy

Locking the dye laser to the resonator and controlling the length of thatwith the output offset of the feedback loop, the ytterbium spectroscopy cellshows fluorescence at different wavelengths for seven different transition lines,corresponding to isotope shifts and hyperfine structure effects for six of theseven stable isotopes. The only isotope that does not show an effect in ourmeasurement is 168Y b with a natural abundance of only 0.13%.

A spectrum for the different transition frequencies is shown in figure 5.1. Thewavelengths for which fluorescence was observed in our spectroscopy setupare correlated with the data from that spectrum in table 5.1.

wavelength strength isotope abundances555.7990 nm medium 171 (F=3/2) / 173 (F=3/2) 14.28% / 16.13%555.8000 nm medium 170 / 173 (F=5/2) 3.04% / 16.13%555.8015 nm medium 172 21.83%555.8030 nm strong 174 31.83%555.8045 nm medium 176 12.76%555.8060 nm weak 171 (F=1/2) 14.28%555.8065 nm medium 173 (F=7/2) 16.13%

Tab. 5.1: Observed wavelengths for the 555.8nm line in differentspecies of ytterbium atoms. Wavelength error is±0.001nm. Isotopesand spin states are also given for each line. Abundances are for thewhole isotope, not the particular spin state.

Due to the eye’s high sensitivity for the green range of the spectrum, thefluorescence is easy to see but difficult to pick up with regular photodiodes.

92 5. Operation

Fig. 5.1: Spectrum for the 1S0 → 3P1 transition of ytterbi-um, showing shifts based on isotope and nuclear spin. Measuredby R. Maruyama at the University of Washington, available athttp://www.phys.washington.edu/ reinam/

Converting the experiment to use photomultipliers to detect the green lightis planned, but has yet to be implemented.

After locating the interesting wavelengths, the green laser beam was fed intothe main chamber and aimed at the MOT operating on the blue transition.For the strongest line at 555.803nm there was a clear effect even at a lowpower of 260µW . This took the form of a small green spot in the center of theblue light emitted by the MOT itself. Measuring the intensity of the greenfluorescence was possible by filtering out part of the blue light with a simpleyellow filter foil.

Keeping the dye laser locked to the cavity and scanning the piezo voltage,spectra for the strongest line at 555.803nm were taken. One set of raw datais plotted in figure 5.2. Evaluating the data shows a linewidth of wgauss =45MHz. In order to get some data on the dominating broadening mechanisms,both the intensity of the green light and the current in the MOT coils werevaried. Lowering the intensity reduces saturation effects which lead to so-called power broadening. Reducing the MOT current will lower the potentialgradients in the trap, affecting the equivalent of the spring constant. This canreduce the kinetic energy of the trapped atoms, leading to reduced Dopplerbroadening.

Both parameters have an influence on the observed linewidth, as shown infigures 5.3 to 5.6. In addition, there will also be Zeeman-broadening caused bythe shift of the ground state of the transition in the inhomogenous magneticfield of the trap. Also, the absorption-emission cycle driven by the blue laser

5.2. Double-Locked Operation 93

0,025

0,030

0,035

0,040

0,045

0,050

0,055

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

time [s]

sig

na

l [V

]

6,4

6,6

6,8

7,0

7,2

7,4

7,6

pie

zo

co

ntr

ol

[V]

signal

piezo control

Fig. 5.2: Raw data taken for the spectrum of the 1S0 → 3P1 transi-tion in the most common ytterbium isotope 174Y b. 1400µW greenlight, MOT current 250A, averaged over 8 sweeps.

will effectively limit the “lifetime” of the ground state, resulting in additionalbroadening.

In the current state of development 30MHz is the lowest achievable linewidth.Any further reduction in intensity causes the signal to be swamped out bythe intensity fluctuations of the blue laser system. Lowering the current inthe coils will cause the trap to become unstable.

5.2 Double-Locked Operation

Operation with both locks turned on has also been demonstrated. In orderto set the green laser to a specific frequency the following procedure has beenthe most successful so far:

• With both locks and filters turned off, the resonator length is scannedwhile watching the locking signals using the input monitor connectionsof the loops.

• The wavelength of the dye laser is adjusted to the appropriate rangeaccording to the wavemeter using the Lyot-filter and selector etalon inthe laser. Care needs to be taken to ensure that the resonator controlsallow for a wide tuning range without mode jumps or the onset ofmulti-mode operation.

• Input gain on the loops is adjusted to bring the amplitudes of thelocking signals to standard values (currently 10V peak-to-peak areused) to allow for reproducible lock operation.

94 5. Operation

0,025

0,030

0,035

0,040

0,045

0,050

0,055

-150 -100 -50 0 50 100 150

frequency [MHz]

sig

na

l [V

]

fit

data

250A, 1.40mW

width=45MHz

Fig. 5.3: Photodiode signal corresponding to intensity of the fluo-rescence caused by the green beam in the ytterbium MOT, plottedover frequency relative to the line center. Linewidth of the manualfit is 45MHz. Spectrum taken at 250A MOT current and 1400µWgreen light, averaged over 8 sweeps.

0,026

0,028

0,030

0,032

0,034

0,036

0,038

-150 -100 -50 0 50 100 150

frequency [MHz]

sig

na

l [V

]

fit

signal

250A, 0.22mW

width=35MHz


5.2. Double-Locked Operation 95

0,030

0,035

0,040

0,045

0,050

0,055

-100 -50 0 50 100 150 200

frequency [MHz]

sig

na

l [V

]fit

signal

143A, 1.45mW

width=40MHz


0,025

0,030

0,035

0,040

0,045

-150 -100 -50 0 50 100 150

frequency [MHz]

sig

na

l [V

]

fit

signal

143A, 0.19mW

width=30MHz


96 5. Operation

• The output offset for the green lock is adjusted manually until the desi-red frequency is reached, for example until green fluorescence appears.This frequency needs to be held manually in the following.

• Now the modulation system is adjusted, until one of the green cavityresonances coincides with one of the red ones. This will require adaptingthe slave laser current to keep the injection stable. Scanning the cavityover several free spectral ranges is required here, because when lookingat a single one, bringing one of the sidebands into coincidence with thegreen resonance might place it to close to one of the repetitions of thecarrier for the lock to work properly.

• By lowering the amplitude of the scan and adjusting the output offsetof the cavity-locking loop, the cavity length is brought close to the pointof coincidence between red and green resonance.

• Finally the filters are activated and the locks turned on.

This way it was possible to keep the dye laser at the frequency of the greenytterbium line for a time of several minutes. After that the fluorescence slowlydisappeared, indicating that the reason was a slow drift rather than thesystem becoming unlocked or jumping to the next resonance. In addition tothis very strong drift, the feedback loops were also showing much strongernoise than in previous tests. This was probably caused by operating the blueMOT, and its trapping coils in particular. Especially the sensitive photodiodeamplifiers will pick up any field fluctuations as noise. By providing bettershielding and higher signal levels this effect will need to be corrected beforestable operation can be achieved.

6. Conclusion

6.1 Summary

It has been shown that the combination of a modulated slave laser and aresonator described here can be used to transfer the stability of an existinglaser system to another that operates at a different frequency.

While long-term stability and linewidth still need to be optimized before thesystem can fulfill its intended function, the idea appears to be valid. Undernormal circumstances stable locked operation is possible with a linewidthlikely to be below 2MHz and drifts of less than 1.5MHz over several hours1.

All of this uses very few complicated electronic components and currentlyno optical element that is more complicated than a polarizing beam split-ter cube. Specifically, no expensive optical insulators or AOMs with a widetuneable range are required while still providing access to essentially anydesired frequency the dye laser system can generate.

6.2 Things to do

While basic operation has already been achieved for the complete system,there are still some missing parts and obvious improvements.

• To use the system in actual experiments, some form of computer controlwill need to be implemented. This will most likely consist of the AOMalready shown in figure 3.5 for fast switching in combination with amechanical shutter to ensure complete blocking of the beam.

• The long term stability of the VCO can very likely be increased bykeeping it at constant temperature. As a first step some kind of pas-sive insulation will be added to reduce the effect of drafts. Adding anexternal modifier input to the control box would be advantageous forfine-tuning as well as for taking spectra.

• In addition to this a procedure to ensure injection into a low ordersideband is required to avoid multiplication of the errors introduced bythe modulation system.

1 For VCO frequency drifts of 250kHz multiplied by injection into a fifth sideband asfound in the measurements

98 6. Conclusion

• A metal casing will be constructed to enclose cavity and analyzers. Thiswould greatly reduce the room light picked up by the sensors, decreasethe effects of temperature fluctuations on the resonator length and offera degree of shielding against external electromagnetic fields.

• Since one limiting factor for the achievable linewidth is the masterlaser, this will need some work to improve the performance of the lock.A first step would be to include direct current modulation to increasethe bandwidth of the feedback loop.

• Alternatively the cutoff frequency of the low-pass filter in the cavitylock can be adjusted to filter out as much of the fast fluctuations aspossible.

• For all measures intended to reduce the linewidth, a way to measurethe current width will be required. This will either be implemented inthe form of a traditional doppler-free spectroscopy setup or by doinga simple form of absorption/fluorescence spectroscopy on the slow andfocused atom beam after the Zeeman slower as used in [8]

• The next version of the cavity will be fitted with mirrors with a thinnersilver coating, giving vastly better transmission at the cost of a minorreduction in reflectivity.

• To improve the slope of the lock signals at resonance and thereforereduce the influence of noise on the lock, the resonator finesse will beincreased by making its base length adjustable.

• More improvement is possible here by using a longer resonator to reducethe free spectral range and thereby the linewidth for constant finesse.A second effect of a longer resonator with a free spectral range close to500MHz would be better matching of the frequency range that can becovered by the modulation system and the range of sideband positionsthat are not blocked by repetitions of the carrier line.

Appendix A

Schematics

Since many of the electronic circuits used are self-designed, the followingchapter will provide schematics for the most important ones.

A.1 Radio-frequency Amplifier

Fig. A.1: Schematic of the circuit based on the Minicircuits Era-5RF amplifier.

100 Appendix A. Schematics

A.2 VCO Driver

Fig. A.2: Schematic of the driver box for the VCO. Provides bothpower and a thermally stable control voltage.

101

A.3 Difference Amplifier

Fig. A.3: Schematic for the difference amplifiers used in the circularpolarization analyzers.

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thesis-setup of a stable high-resolution laser system

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