chapter 4 stimulated emission ... -...

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課程編號：901 37500 - 01 科目名稱：光電導論授課教師：黃鼎偉

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3


υυ

υ υ

υ

Spontaneous emission:random directionE1 is emptyAs if e- is oscillating in freq. v

Stimulated emission:Two photons are

Incoming e- couples to the e- in E2

Basis for photon amplificationRe-absorbed? => population inversion

not in two level systems

Fig. 4.1

(in steady state: R12= R21 )

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υ

υ

υυ

(a) (b) (c) (d)

υυ

υ

Pumping(optical)

e.g. ruby laser: chromium ions Cr3+ in Al2O3 crystalFeedback: silvered mirror and partially silvered mirror

LASER: Light Amplification by Stimulated Emission of Radiation

Fig. 4.2

(Note: usu., more efficient in four level systems )

(See p.5)

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• Theodore Harold Maiman was born in 1927in Los Angeles, son of an electricalengineer. He studied engineering physics atColorado University, while repairingelectrical appliances to pay for college, andthen obtained a Ph.D. from Stanford.Theodore Maiman constructed this firstlaser in 1960 while working at HughesResearch Laboratories (T.H. Maiman,"Stimulated optical radiation in ruby lasers",Nature, 187, 493, 1960). There is a verticalchromium ion doped ruby rod in the centerof a helical xenon flash tube. The ruby rodhas mirrored ends. The xenon flash providesoptical pumping of the chromium ions inthe ruby rod. The output is a pulse of redlaser light. (Courtesy of HRL Laboratories,LLC, Malibu, California.)

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A useful LASER medium must have a higher efficiency ofstimulated emission compared with spontaneous emissionand absorption.

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Upward transition rate

Downward transition rate

( )νρ hNBR 11212 =

( )νρ hNBNAR 22122121 +=

B12 B21, A21 : Einstein Coefficients

( )hρ ν : photon energy density / freq.# photons / volume at hv

spontaneous stimulated

Thermal equilibrium: no change with time in the populations at E1 and E2

2112 RR = ( )

−−=

TkEE

NN

B

12

1

2 expBoltzmann statistics

( )

−

=

1exp

8

3

3

Tkhc

hh

B

eqν

νπνρ(only) in thermal equilibrium, by

Plank’s black body radiationdistribution law

=> 2112 BB = 3

3

21

21 8ch

BA νπ

= (ρ is much larger in laser operation)

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2112 BB = 3

3

21

21 8ch

BA νπ

=

( ) ( )21

21

221

221

21

21

)()(

AhB

NAhNB

sponRstimR νρνρ

== ( )3

38c hh

ρ νπ ν

=

Ratio of stimulated to spontaneous emission

Ratio of stimulated emission to absorption

1

2

12

21

)()(

NN

absorpRstimR

=

R21(stim) > R12 (absorp) =>

Stim. emission >> spon. emission => large photon concentrationachieved by optical cavity

Population inversion => depart from thermal equilibrium (negative abs. temp.)The laser principle is based on non-thermal equilibrium.

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Long haul communication suffers attenuation.⇒Regenerate signal by ptical- lectrical- ptical⇒Amplify optical signal directly => optical amplifier

EDFA: erbium (Er3+) doped fiber amplifierOther dopants: e.g. Nd3+

Splicing: fibers are fused together

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′

Optically pumped (usu. by LD)

E3: long-lived ~ 10ms

1240 (eV nm)=(nm)

E hvλ

⋅= ( )12 NNKGop −=

Optical gain:

K: dep. on pumping intensity

Fig. 4.3

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λ λ

λ

E’3 at 810 nm is less efficient.

Photodetector: monitor the pumping levelClosely spaced collection of several levels:

1525-1565 nm (40nm) used in WDM systemgain flattening to overcome non-uniform gain

Pumping at 1480 nm is also usedGain efficiency: 8-10 dB/mW e.g. 103 gain by a few mW

Fig. 4.4

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• Ali Javan and his associatesWilliam Bennett Jr. andDonald Herriott at Bell Labswere first to successfullydemonstrate a continuouswave (cw) helium-neonlaser operation (1960-1962).

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He-Ne laser @632.8nm (red) from Ne atoms, He used to excite Ne

Ne: 1s22s22p6 or 2p6 He: 1s2

excited Ne: 2p55s1 Excited He: 1s12s1

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Using dc or RF high voltage:He atoms to become excited by collisions with drifting electrons

−− +→+ ee *HeHeExcited He*: 1s12s1 parallel spin, metastable,

not allowed to simply decay back to ground state,

** NeHeNeHe +→+Excited He collides with a Ne atom

=> population inversion of Ne

Fig. 4.5

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Fig. 4.6

16


µ

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Ne(2p55s1): 4 closely spaced energy levels

Ne(2p53p1): 10 closely spaced energy levels

Also energy level -- Ne(2p54p1)

Red: 632.8 nmGreen: 543 nm

IR: 3.39 µmprevented by freq. selective mirrors

Ne(2p53s1) is a metastable level,to ground state by colliding with the walls of laser tube=> narrow tube is better

Typ. He-Ne laserHe: Ne = 5:1 several torrs99.9% flat reflecting mirror, and 99% concave reflecting mirror, convergent lensdiameter: 0.5-1 mm divergence: 1 mrad few wattspolarized light by Brewster angle

θ ∆

θ

422 owλ

θπ

=( )

Gaussian beam

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Average kinetic energy of molecules: (3/2)kBT

Doppler effect:

−=

cvv xv101

+=

cvv xv102

movingaway

movingtoward

Doppler broadened linewidth: ∆v approx. v2-v1

∆v = 2-5GHz for many gas lasers, He-Ne laser ~0.02 Å

202/1)2ln(22

McTkvv B=∆Full width at half maximum

linewidth (FWHM)

Optical gain lineshape around λ0= c/v0

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(c)

δλ

λδλ

(b)

λλ

λ

(a)

λο

Fig. 4.8

axial (longitudinal) modes

Finite width due to NL of cavitiesacoustic and thermal fluctuations of Lnonideal end mirror (R<100%)

Typ. ~1Mhz for He-NeStabilized gas laser as low

as 1kHz

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Optical gain betweenFWHM points

δλm

(a) 5 modes

(b)4 modes

Number of laser modesdepends on how thecavity modes intersectthe optical gain curve.In this case we arelooking at modeswithin the linewidth∆λ1/2.

λ

Cavity modes

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Lm =

2λ

cavity modem: mode number

axial (or longitudinal) modesGaussian beam

Typ. freq. width of an individual spike in He-Ne laser is ~1MHz ( low ~1kHz)

Fig. 4.9

Example 4.5.1

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exp(-αx) α: absorption coefficient

exp(gx) g: optical gain coefficient

ph ph

ph ph

N NPP x N x cN t

δ δδδ δ δ

= = =n

g

( ) ( )( ) ( )νρ

νρνρhBNN

hBNhBNdt

dN ph

2112

211212

emissionphotonstimulatedofrateNet

−=−=

=

(stimulated – absorption)(considering directional wave

=> spon. emission is neglected)

(Fig. 4.10)

(optical gain coefficient)

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δ

δ

υ

υυο

∆υυ

υ

Fig. 4.10

lineshape function:spectral shape of the gain curve

( )ν

ννρ

∆≈ 0

0

hNh ph ( ) ( ) 21 0

0 2 1B hνν N N

c∆ν≈ −

ng

Radiation energy densityper unit freq. at hv0

General optical gain coefficient at v0

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RR

1==i

fop P

PG

Steady state conditions, no optical power loss in the round trip=> net round-trip optical gain Gop = 1.

Losses: R1, R2,absorption (e.g. by impurities, free carriers)scattering (defects and inhomogenities)others

Fig. 4.11

( ) ( )1 2 exp 2 exp 2f iP PR R L Lγ = − g

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−

−

− −

Fig. 4.12

( ) ( )1 2 exp 2 exp 2f iP PR R L Lγ = − g

1 2

1 1ln2th L R R

γ

= +

g

( )2 121 0

thth

cN NB hν

∆ν− ≈

ng

Threshold optical gain

Threshold population inversion

( ) ( ) 21 00 2 1

B hνν N Nc∆ν

≈ −n

g

( )2 1 pumpingN N− ∝

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( )πφ 2trip-round m=∆Ef = Ei in Fig. 4.11 Phase condition for laser oscillations

( ) ( )2 2mk L m π≅nNeglect φ changesat mirrors

Lm m =

n2λApprox. laser

cavity modes

Ideally, infinitely wide mirrors plane waves are assumed

Practically, finite size mirrors Gaussian beams are the solutions

Off-axis modes can exist and replicate themselves=> transverse modes or transverse electric and magnetic (TEM) modesEach transverse mode with a given p,q has a set of longitudinal modes.

Usu. m is very large ~ 106 in gas lasers

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Wave fronts

Sphericalmirror

Optical cavityTEM00 TEM10

TEM01 TEM11

TEM00 TEM10

TEM01 TEM11

(b)

(c) (d)

Laser Modes (a) An off-axis transverse mode is able to self-replicate after one roundtrip. (b) Wavefronts in a self-replicating wave (c) Four low order transverse cavitymodes and their fields. (d) Intensity patterns in the modes of (c).

(a)


Transverse modes depend on: optical cavity dimensions, reflector sizes,..Cartesian (rectangular) or polar (circular) symmetry about the cavity axis

(Brewster angle)

highly desirable TEM00:lowest mode, radially symmetric, lowest divergence

Fig. 4.13

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34


(a) (b)

Fig. 4.14

degenerate dopingn > Nc, p>Nv

∆EF= eV > Eg

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No applied voltage, EFp = EFn

applied voltage V, ∆EF= eV

See Fig. 4.14(a) degenerate doping

Fig. 4.14(b) ∆EF= eV > Eg

Diminishes the built-in potential barrierSCL is no longer depletedIn SCL, more electrons in the conduction band at energies near Ec

than electrons in the valence band near Ev

Fig. 4.15(a) Population inversion between energiesnear Ec and those near Ev around the junction

Called inversion layer or active layerMore stimulated emission than absorption

-

> -g Fn Fp

Fn Fp

E hv E Ehv E E

< < Stimulated emission

Absorption

36


υ

−

≈−

(a) (b)

Injection pumping: pumping by the forward diode current

Fig. 4.15

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Fig. 4.16

Lm m =

n2λ

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λ

λ

λ

Threshold current Ith: optical gain in the medium can overcomethe photon losses from the cavity

Problems of homojunction LD:threshold current density Jth is too high for practical use~500 A/mm2 for GaAs, can only be operated at very low temperature

Jth can be reduced by orders of magnitude by using heterojunction LD

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Reduction of the threshold current by improvingA. Rate of stimulated emission

=> carrier confinementB. Efficiency of the optical cavity

=> photon confinement by waveguide

Both can be achieved by heterostructured devicesLEDs vs LDs: stimulated emission > spontaneous emission

Fig. 4.18p-GaAs and p-AlGaAs are degenerately doped

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∆n

(a)

(b)

∆

µ

(c)

(d)

Fig. 4.18

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Fig. 4.19

p-GaAs: 870-900 nmStripe contact: define current density

gain guided=> reduce Ith, better coupling to fiberW: few µm, Ith: tens mA

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Buried double heterostructureIndex guided: better lateral optical confinement (can be single mode)

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extractionernalopticalout

external IVP

ηηη ⋅=⋅= int, %100

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45


46


Output Spectrum dep. on

A. Optical ResonatorL: longitudinal mode separationW, H: lateral modes small W, H => single mode, TEM00divergence: smaller aperture => larger diffraction (e.g. H in Fig. 4.21)

B. Optical Gain Curve

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Fig. 4.21

48


λ

Red shift: temperature-induced gain shiftingdue to heating

Fig. 4.22

49


°°

°

Fig. 4.23

thIIP−

= 0slopeη

Slope efficiency

Conversion efficiencymay be as high as 30-40%

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λo(nm)

Mode hopping

20 30 40 50Case temperature (° C)

Single mode

776

778

780

782

784

786

788

20 30 40 50Case temperature (° C)

Single mode

20 30 40 50

Multimode

Case temperature (° C)

Peak wavelength vs. case temperature characteristics. (a) Mode hops in the outputspectrum of a single mode LD. (b) Restricted mode hops and none over the temperaturerange of interest (20 - 40 °C). (c) Output spectrum from a multimode LD.

(a) (b) (c)


Fig. 4.24

Temp. Refractive Index center wavelengthcavity length

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52


phsp

I n CnNedLW τ

= +

phph

ph CnNN

=τ

Rate of electron inject by current= Rate of spontaneous emission + Rate of stimulated emission

C: constant dep. on B21

Steady state:Rate of the coherent photon loss = Rate of stimulated emission

Nph: coherent photon encouragedby the optical cavity

τph: due to trans. thru end-faces, scattering, absorption

phth C

nτ1

=sp

thth

edLWnIτ

=At threshold Nph = 0

Smaller Ith: heterostructureand stripe geometry

Stimulated emission justbalanced by all loss

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∝

phthth NCn

edLWII

=− ( )th

phph JJ

edN −=

τ

( )( )( )R

Δt

NP

ph

−

= 1energyPhotonVolumeCavity

21

0

( ) ( )2

0

12

phth

hc W RP J J

eτ

λ

−= −

n

Above threshold J=I/WL =>

O/P opticalpower ∆t=nL/c

=> Laser diode equation

Fig. 4.25

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Short haul: LED larger ∆λsimpler to drive, economic, longer lifetimeusu. with multimode graded index fibers

Long haul: LDnarrow linewidth and high output power

A laser diode pigtailed to a fiber. Two of the leads are fora back-facet photodetector to allow the monitoring of thelaser output power.(Courtesy of Alcatel)

Comparison, see Fig. 4.26 and Table 4.1

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Fig. 4.26

rise time: light output 10% to 90%Short τr => wide bandwidths

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InP InP

∆λ < 0.01 nm in single frequency LD

One mode by suppressing unwanted mode thru design, ∆λ ~ 0.01~0.1nmLaser diode

57


(a) (b)

Λ

λ n Λ

λ n Λ

Fig. 4.27

λB: Bragg wavelengthq: diffraction order

2Bq λΛ=

n

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Λ

(a)λλ

λ(b) (c)

Fig. 4.28

( )2

12

Bm B m

Lλ

λ λ= ± +n

Right and left traveling waves are coupledset up a standing wave if they are coherently coupled(round-trip phase change =2π)

m=0 dominatesbecause relative threshold gain of higher modes is higherIdeally, perfectly symmetric (b) Practically, asymmetry induced on purpose (c)

L>>Λ λm~λB Commercially available 1.55µm ∆λ~0.1nm

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(a)

λ

λ

λ

(b)

Two lasers are pumped by different currentsOnly those waves that can exist as modes in both cavities are allowed

Fig. 4.29

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61


Fig. 4.30Ultra thin, typ. < 50nm, narrow bandgap semiconductor devicesLattice match is required

d << Dx, Dy: dOne dimensional PE well in x-direction and as if free in yz-plane

2*

22

2*

22

2*

22

888 ze

z

ye

y

ec Dm

nhDmnh

dmnhEE +++=

Step density of statesa large concentration of electrons (holes) can easily occur at E1 ( E’1 )population inversion occurs quickly without a large current

AdvantagesIth is markedly reduced (eg. 0.5-1mA in SQW, 10-50mA in DH laser)linewidth is substantially narrower because e- (h+) near E1 (E’1)

62


g

(a) (b) (c)

2*

22

2*

22

2*

22

888 ze

z

ye

y

ec Dm

nhDmnh

dmnhEE +++=

Fig. 4.30

63


′

υ ′

′

Fig. 4.31

64


A 1550 nm MQW-DFB InGaAsPlaser diode pigtail-coupled to afiber. (Courtesy of Alcatel.)

Fig. 4.32

Larger volume

65


66


67


See Fig. 4.33Dielectric mirrors λ

21

2211 =+ dd nn

(DBR structure)

High reflectance end mirrors are needed due to short cavity length20-30 or so layers to obtain the required reflectance (99%)

Active layer: very thin <0.1µm, likely MQWe.g. 980 nm InGaAs in GaAs substrate

Circular cross-sectionHeight: several microns => single mode longitudinally (maybe not laterally)Spectral width 0.5nm

68


λ nλ n

Fig. 4.33

λ21

2211 =+ dd nn

69


An 850 nm VCSEL diode.(Courtesy of Honeywell.)

SEM (scanning electron microscope) of the first low-threshold VCSELsdeveloped at Bell Laboratories in 1989. The largest device area is 5 µmin diameter (Courtesy of Dr. Axel Scherer Caltech.)

Microlaser: cavity dimension in the microns rangeMatrix emitter: arrayed microlaser, broad area surface emittiner laser sourse

optical inter connect, optical computinghigher optical power than conventional laser, few watts

70


Fig. 4.34

Amplified by induced stimulated emissionPumped to achieve optical gain

(i.e. population inversion)Random spontaneous emission

=> noise => filtered outTyp. ~20dB, dep. on the AR coating

Under threshold currentAmplified by stim. EmissionMultiple reflectionHigher gain, less stable

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A technique of reproducing three dimensional (3D) optical image of an objectby using a highly coherent radiation from a laser source

Fig. 4.35

Ecat: both amplitude and phase variations representing the cat’s surfaceReflected Ecat interferes with Eref at photographic plateInterference pattern depends on the magnitude and phase variations in Ecat.

Hologram: recorded interference patter in the photographic film

Diffracted beam: virtual image (Bragg condition d sinθ = mλ )

Thru beam: real image

72


(b)(a)Fig. 4.35

73


( ) ( ) tjr eyxUyxE ω,,ref = ( ) ( ) tjeyxUyxE ω,,cat =

( ) ( )( )**22catref, UUUUUUEEyxI rrr ++=+=+=

( ) ****, UUUUUUUUyxI rrrr +++=

( ) ( )****, UUUUUUUUUyxIUU rrrrrrt +++=∝

( ) ( ) ( ) *2***, UUUUUUUUUUyxIUU rrrrrrrt +++=∝

( ) ( )yxcUyxbUaU t ,, *++∝

Pattern on the hologram

Illuminating the hologram with reference beam

i.e.

a = Ur(UU*+UrUr*) constant: through beambU: scaled version of U: virtual imagecU*: complex conjugate of U: real image, conjugate image

Eye can not detect phase shift. Observer always sees the positive image.

Holography is a method of wavefront reconstruction.

74


Dennis Gabor (1900 - 1979), inventor of holography, is standing next to his holographic portrait.Professor Gabor was a Hungarian born British physicist who published his holography inventionin Nature in 1948 while he as at Thomson-Houston Co. Ltd, at a time when coherent light fromlasers was not yet available. He was subsequently a professor of applied electron physics atImperial College, University of London. (From M.D.E.C. Photo Lab, Courtesy AIP Emilio SegrèVisual Archives, AIP.)

chapter 4 stimulated emission ... -...

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