CHAPTER 15. LASER AND FIBER OPTICSThe laser is essentially an optical amplifier. The word laser is an acronym that

stands for light amplification by the stimulated emission of radiation.

15.1 Einstein’s theoryConsider atomic transition between two states, E1 (ground state) and E2 (excited

states). Stimulated absorption occurs whenever radiation containing photons of energy h= E2E1 is incident on the matter. The spontaneous emission takes place whenever atoms are in an excited state. No external radiation is required to initiate the emission. The emitted photon has an energy of h= E2E1 but is emitted in a random direction. By contrast, stimulated emission requires the presence of external radiation. When an incident photon of resonant energy h= E2E1 passes by an atom in excite state, it “stimulate” the atom to drop to the lower ground state. In the process, the atom releases a photon of the same energy, direction, phase, and polarization as that of the photon passing by. The net effect is two identical photons in the place of one, or an increase in the intensity of the incident “beam”. It is precisely this process of stimulated emission that makes possible the amplification of light in lasers.

Spontaneous emission (A21):

(15.1)

where N2 is the population of level E2, A21 is the radiative rate, and is the spontaneous radiative lifetime.

Stimulated emission (B21):

(15.2)

where () is the photon density as a function of frequency.

Absorption (B12):

(15.3)

Absorption is also a stimulated process since it depends on the strength of the photon field. In effect, stimulated absorption and stimulated emission are inverse processes. The A and B coefficients are constants characteristic of the atom.

Einstein has made the following assumptions:(a). Thermodynamic equilibrium at arbitrary temperature T exists between the radiation

field and atom. 84

Fig.15.1 Radiative processes that affect the number of atoms at energy E1 and E2.

(b). The radiation field () has the spectral distribution characteristic of a blackbody at temperature T.

(c). The atom population densities are distributed according to the Boltzman distribution at that temperature.

and

(d). Population densities are constant in time.

Finally, we have,

(15.4)

and (15.5)

Eq.(15.5) is valid for the case of nondegenerate energy states. By comparing Eqs.(15.2) and (15.3), if N2 is greater than N1 and a radiation field interacts with the atoms, stimulated emission exceeds absorption and photons will be added to the field. This leads to an increase in (), an amplification. This is the condition of population inversion since under normal equilibrium condition N2 is less than N1.

According to Eq.(15.4), when the frequency is higher, A21 becomes larger than B21. Since A21 is related to the spontaneous emission and does not contribute to the photon amplification, lasers of short wavelength radiation (UV or X-ray, for example) would be more difficult to build and operate.

For the successful operation of a laser, two ideas are important: stimulated emission and population inversion.

15.2 Essential Elements of a LaserThe laser device is an optical oscillator that emits an intense, highly collimated

beam of coherent radiation. The device consists of three basic elements: a pump, an amplifying medium, and an optical cavity or resonator.

The pump is an external energy source that produces a population inversion in the laser medium. Pumps can be optical, electrical, chemical, or thermal in nature, so long as they provide energy that can be coupled into the laser medium to excite the atoms and create the required population inversion.

Many lasers are named after the type of laser medium used, for example, helium-neon (He-Ne), carbon dioxide (CO2), and neodymium:yttrium aluminum garnet (Nd:YAG). The laser medium, which may be a gas, liquid, or solid, determines the wavelength of the laser radiation.

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Resonator is an optical “feedback device” that directs photons back and forth through the laser (amplifying) medium. In its basic form, it consists of a pair of carefully aligned plane or curved mirrors centered along the optical axis of the laser system. One of the mirrors is chosen with a reflectivity as close to 100% as possible. The other is selected with a reflectivity somewhat less than 100% to allow part of the internally reflecting beam to escape and become the useful laser output beam. The fundamental mode that appears in the output laser beam is the TEM00 mode. The transverse variation in the irradiance of this mode is Gaussian in shape, with peak irradiance at the center and exponentially decreasing irradiance toward the edges.

15.3 Simplified Description of Laser OperationThe four-step process is illustrated in Fig.15.3. In step 1, a large number of atoms

are excited from the ground state E0 to several excited states, collectively labeled E3. In step 2, these atoms decay in a short time (usually radiationless) to a special level (upper laser level) E2. This level has a long lifetime with a typical value of 10-3s, while most excited levels may have lifetimes in the order of 10-8s. Since level E2 is metastable, the population of this level easily piles up to a value much larger than that of level E1 (lower laser level), which is an ordinary level that decays to the ground states quite rapidly (step 4). The population N1 cannot build to a large value. The required population inversion for the light amplification is thus been built. When a photon of resonant energy h=E2E1

passes by any of the N2 atoms in the upper laser level, stimulated emission can occur (step 3).

In summary, the laser process depends on the followings: A population inversion Seed photons An optical cavity Laser beam through the output coupler mirror

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Fig.15.2 Basic elements of a laser. (a) Integral device with output beam. (b) Pump. (c) Resonator. (d) Laser medium.

15.4 Characteristics of Laser BeamThe light emitted by a laser is monochromatic. The emission determined by a

single pair of energy of an identical set of atoms levels is called fluorescence. The emitted light has a wavelength spread about a center wavelength 0, where 0=c/v0 and v0=(E2-E1)/h. is often called linewidth. When the linewidth is measured at the half maximum level of the lineshape plot, it is called the FWHM linewidth, that is, “full width at half maximum”. The FWHM linewidth for an ordinary discharge lamp is about 0.1nm, that for a cadmium low-pressure lamp is about 0.0013nm, while that for a He-Ne laser is only about 10-8nm.

The optical property that most distinguishes the laser from other light sources is coherence. The laser is regarded as the first truly coherent light source. Other light sources, such as the sun or a gas discharge lamp, are at best only partially coherent. Coherence is a measure of the degree of phase correlation that exists in the radiation field of a light source at different locations and different times. It is often described in term of a temporal coherence, which is a measure of the degree of monochromaticity of the light, and a spatial coherence, which is a measure of the uniformity of phase across the optical wavefront. In the stimulated emission, each photon added to the stimulating radiation has a phase, polarization, energy and direction identical to that of the amplified light wave in the laser cavity. Thus the laser light created and emitted is both temporally and spatially coherent.

For typical lasers, both the spatial coherence and temporal coherence of laser light are far superior to that for light from other sources. The coherence time tc of a laser is a measure of the average time interval over which one can continue to predict the correct phase of the laser beam at a given point in space. The coherence length Lc is related to the coherence time by equation Lc=ctc, where c is the speed of light. Thus the coherence length is the average length of light beam along which the phase of the wave remains unchanged. For the He-Ne laser, the typical coherence time is of the order of milliseconds

87

Fig.15.3 Operation mechanism for a general four-level laser.

(compared with about 10-11s for light from a sodium discharge lamp), and the coherence length for the same laser is thousands of kilometers (compared with fractions of a centimeter for the sodium lamp).

The astonishing degree of directionality of a laser is due to the geometrical design of the laser cavity and to the monochromatic and coherent nature of light generated in the cavity. The beam-spread angle is given by the relationship,

(15.6)

where is the wavelength of the laser beam and D is the diameter of the laser beam at its beam waist.

In Fig.15.4, the wavefronts that pass through the effective aperture or beam waist are plane waves, but the irradiance of the laser light is not uniform across the plane. For the lowest-order transverse mode, the TEM00 mode or the Gaussian beam, the irradiance of the beam decreases exponentially toward the edges of the beam in accordance with the Gaussian form , where y measures the transverse beam direction and D is the beam width at a given position along the beam.

Lasers are classified in many ways. Sometimes they are grouped according to the state of matter represented by the laser medium: gas, liquid, or solid. Sometimes they are classified according to how they are pumped: flashlamp, electrical discharge, chemical actions, and so on. Other classifications divide them according to the nature of their output [pulsed or continuous (cw)] and according to their spectral region of emission (infrared, visible, or ultraviolet).

The wavelength for several common lasers is:He-Ne (gas): 632.8 nmRuby (solid): 694.3 nmCO2 (gas): 10.6 mNd:YAG (solid): 1.064 mAr (gas): 488 nm or 514.5 nm

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Fig.15.4 External and internal laser beam for a given cavity. Diffraction or beam spread appears to be caused by an effective aperture of diameter D, located at the beam waist. The beam waist is determined by the design of the laser cavity and depends on the radii of curvature of the two mirrors and the distance between the mirrors.

15.5 Optical FibersAn optical fiber is a cylindrical dielectric waveguide made of low-loss materials

such as silica glass. It has a central core in which the light is guided, embedded in an outer cladding of slightly lower refractive index (Fig.15.5). Light rays incident on the core-cladding boundary at angles greater than the critical angle undergo total internal reflection and are guided through the core without refraction. Rays of greater inclination to the fiber axis lose part of their power into the cladding at each reflection and are not guided.

Applications: Light pipes: transporting of light, imaging, and communications. The advantages of fiber-optics communication systems over the conventional two-wire, coaxial cable or microwave waveguide systems are high information-carrying capacity, immunity to electromagnetic interference, and freedom from signal leakage.

15.6 Optics of Propagation:Suppose the fiber itself has a refractive index n1, the encasing medium (called

cladding) has a index n2, and the end faces are exposed to a medium of index n0 (Fig.15.6).

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Fig.15.5 An optical fiber.

Fig.15.6 (a) Propagation of light rays through an optical fiber. Ray B defines the maximum input cone of rays that satisfying total internal reflection at the walls of the fiber. (b) Propagation of a typical light ray through an optical fiber.

The critical angle c must satisfy,

(15.7)

At the input surface, the maximum half-angle m of a cone of rays must obey the Snell’s law,

(15.8)Since , it comes that the numerical aperture is,

(15.9)The skip distance between two successive reflections of a ray of light propagating in the fiber is given by,

(15.10)where d is the fiber diameter. Relating ’ to the entrance angle by Snell’s law, we can obtain,

(15.11)

Surface scratches or irregularities, as well as surface dust, moisture, or grease, become sources of loss that rapidly diminish light energy. The cladding is used to protect the optical quality of the fiber. Cladding materials need not to be highly transparent, but must be compatible with the fiber core in terms of expansion coefficients, for example. The cladding of a fiber actually increases the critical angle for internal reflection and reduces the numerical aperture of the fiber. The cladding of a fiber can also prevent the cross talk between fibers in communication applications.

The optic fiber cores, which are homogeneous in composition, characterized by a single index of refraction n1 are called step-index fibers. If the refractive index changes continuously from the core axis as a function of radius, the fibers are called graded-index fibers.

15.7 Allowed ModesNot every ray that enters an optical fiber within its acceptance cone can propagate

successfully through the fiber. Only certain ray directions or modes are allowed.

As shown in Fig.15.7, noticing that the ray represents plane waves moving up and down in the waveguide, it is evident that such waves overlap and interfere with one another. Points A and C lie on a common wavefront of such waves. If the net phase change that develops between A and C is some multiple of 2, then the interfering wavefront experience constructive interference and corresponding ray directions are allowed. The optical path difference is,

(15.12)The maximum number of allowed mode is when =c, the critical angle,

(15.13)

90

The above analysis is for slab waveguide, in the case of a cylindrical fiber, the maximum number of modes is,

(15.14)

The number of possible modes increases with the ratio d/. Larger diameter fibers are multimode fibers. If d/ is small enough to make mmax<2, the fiber allows only the axial mode to propagate. This is the monomode (or single mode) fiber. A careful analysis indicates that single-mode performance results when

(15.15)

15.8 AttenuationThe intensity of light propagating through a fiber attenuates due to extrinsic and

intrinsic losses. Among the extrinsic losses are inhomogeneities and geometric effects. Other extrinsic losses occur as light is coupled into and out of the fiber. At the fiber input there are losses due to the restrictions of numerical aperture, as well as losses due to inevitable reflections at the interface, the so-called Fresnel losses. Losses can include mismatch of coupled fiber ends, involving core diameter and lateral and angular alignment. Separation and numerical aperture incompatibility are also possible to cause losses.

Intrinsic losses are due to absorption, both by the core material and by residual impurities, and by Rayleigh scattering from microscopic inhomogeneities, dimensionally smaller than the optical wavelength.

Absorption losses over a length L of fiber can be described by the usual exponential law for light irradiance,

(15.16)where is an attenuation or absorption coefficient for the fiber, a function of wavelength. The defining equation for the absorption coefficient in decibels (db) is,

(15.17)

where P1 and P2 refer to power levels of the light at two different cross sections 1 and 2.

15.9 DistortionThe signal waveform can be distorted due to several mechanisms: modal distortion,

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Fig.15.7 Section of a slab waveguide showing a successfully propagating ray or one of the possible modes. The geometry is used to determine the condition for constructive interference.

material dispersion, and waveguide dispersion. They are in an order of decreasing severity.

Modal distortion occurs because propagating rays (fiber modes) travel at different times, broadening the square wave as shown in Fig.15.8. The shortest distance L from A to B is taken by the axial ray; the longest distance L’ from A to B is taken by the steepest propagating ray that reflects at the critical angle c.

The modal distortion can be expressed as a temporal pulse spread per unit length,

(15.18)

Modal distortion can be lessened by reducing the number of propagating modes. The best solution is to use a monomode fiber, with only one propagating mode. The next best is to use a graded index (GRIN) fiber. A GRIN fiber is produced with a refractive index that decreases gradually from the core axis as a function of radius. In general, the variation of refractive index with radius can be expressed as,

, (15.19)

where . The parameter is chosen to minimize modal distortion.

Even if modal distortion is absent, some pulse broadening still occurs because the refractive index is a function of wavelength. The monochromatic the light, the less the distortion due to material dispersion. Fig.15.9 illustrates material dispersion by showing the progress of two square pulses (initially coincident) in a fiber at wavelengths 1 and 2

with corresponding refractive indices n1 and n2. The material dispersion can be expressed as,

(15.20)

where M is a property of the core material and is the spectral width of the input wave. Pulse broadening due to material dispersion is much smaller than that due to modal distortion.

The waveguide dispersion is a geometrical effect that depends on waveguide parameters. It is a small effect that becomes important only when the other pulse-

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Fig.15.8 Schematic of modal distortion. Shown are the extreme paths: an axial ray and one propagating at the critical angle.

broadening effects have been essentially eliminated. The waveguide dispersion results from the dependence of the field distribution in the fiber on the ratio between the core radius and the wavelength. If this ratio is altered, the relative portions of optical power in the core and cladding are different, the group velocity of the mode is altered.

The variation of the index of refraction causes not only a spread in the group velocity of the component waves but also a variation in the zigzag pattern. Dispersion due to the spread of the index of refraction is called material dispersion. Dispersion due to the variation in the zigzag pattern is called waveguide dispersion.

References:1. A.H.Tunnacliffe and J.G.Hirst, Optics, The Association of British Dispensing

Opticians, 1996. 2. F.L.Pedrotti and L.S.Pedrotti, Introduction to Optics, Prentice Hall, New Jersey,

1993.3. D.Halliday, R.Resnick and J.Walker, Fundamentals of Physics, 5th edition, John

Willey & Sons, Inc., Singapore, 1997.

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Fig.15.9 A square wave input arrives at the fiber end at different times, depending on wavelength.

Fig.15.10 Schematic for waveguide dispersion. The zigzag pattern is changed.