Introduction to Power Diode Lasers

8/8/2019 Introduction to Power Diode Lasers

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Introduction to Power Diode Lasers

Peter Unger

Department of Optoelectronics, University of Ulm,89069 Ulm, [email protected]

Abstract. An introduction to the physics, design, and fabrication of semiconduc-tor-diode lasers is presented with emphasis on high-power operation. Beginningwith a general section about fundamental aspects and elementary physics of theseoptoelectronic devices, topics like optical gain, quantum-well structures, optical

resonators, mirror coatings, optical waveguides, mode patterns, beam profiles, laserrate equations, device properties, high-power design, epitaxy, and process technol-ogy are discussed in more detail.

1 Fundamental Aspects of Diode Lasers

This section provides a basic understanding of the physical phenomena indiode lasers. A more comprehensive treatment can be found in books onsolid-state physics [1], semiconductor physics [2,3], optoelectronics [4,5,6,7],

and semiconductor-diode lasers [8,9,10,11,12].

1.1 Emission and Absorption in Semiconductors

Gas and solid-state lasers have electronic energy levels which are nearly assharp as the energy levels of isolated atoms. In semiconductors, these en-ergy levels are broadened into energy bands due to the overlap of the atomicorbitals. In an undoped semiconductor with no external excitation at a tem-perature ofT = 0 K, the uppermost energy band, called the conduction band,

is completely empty and the energy band below the conduction band, calledthe valence band, is completely filled with electrons. Conduction and valencebands are separated by an energy gap, which has a value of Eg = 0.52.5 eVfor semiconductor materials which power diode lasers are made of.

Two types of carriers contribute to electronic conduction, these are elec-trons in the conduction band and holes (missing electrons) in the valenceband. A free electron has a kinetic energy of E = p2/(2m0) where m0 =9.109 534 1031 kg is the free-electron rest mass and p the mechanicalmomentum. When treated as a quantum-mechanical particle, the momen-tum p = hk is proportional to the wavenumber k = 2/ with the re-

duced Planck constant h = h/(2) = 6.582 173 1016 eVs and the wave-length . Thus, for a free electron, the dependency of energy versus wavenum-ber is E(k) = (h2k2)/(2m0). In semiconductors, the electron energies in the

R. Diehl (Ed.): High-Power Diode Lasers, Topics Appl. Phys. 78, 154 (2000)c Springer-Verlag Berlin Heidelberg 2000


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2 Peter Unger

conduction band Ec(k) and in the valence band Ev(k) behave similarly forsmall wavenumbers k.

Ec(k) = Eg +

h2k2

2me , Ev(k) = h2k2

2mh . (1)

Figure 1 shows this behavior which is called the nearly-free carrier approx-imation. The interaction of the carriers with the solid-state lattice is takeninto account by the introduction of effective masses for the electrons meand for the holes mh which are in general different from the free-electronrest mass m0. Since the E(k) dependence in the valence band is a negativeparabolic curve, holes can be regarded as particles with positive charge.

Radiative band-to-band transitions are generation and recombination ofelectronhole pairs associated with absorption or emission of photons. Forthese transitions, conservation of energy E and momentum hk must be ful-filled. Due to the high value for the speed of light c = 2.997 925 1010 cm/s,the momentum of the photon hk = h/c = Eph/c for photon energies Ephin the 0.52.5eV range can be neglected in comparison to the momentumof the electronic carriers. Thus, a radiative transition between an electron inthe conduction band with energy E2(k2) and a hole in the valence band with

Fig. 1. Parabolic band structure E(k) for electrons in a direct semiconductor. Theconduction band is separated from the valence band by an energy gap Eg. A recom-

bination of an electron at E2(k2) in the conduction band and a hole at E1(k1) inthe valence band generates a photon with energy h. Since the momentum of thephoton hk is negligibly small, radiative electronic transitions between conductionand valence bands only occur at the same wavenumber k


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Introduction to Power Diode Lasers 3

energy E1(k1) under emission or absorption of a photon can only occur atthe same wavenumber k.

Eph = h = E2 E1 , k2 = k1 . (2)As shown in Fig. 1, these transitions can be illustrated by vertical ar-

rows with the length of the photon energy h pointing upwards for genera-tion and downwards for recombination of an electronhole pair. In thermalequilibrium, the carriers tend to occupy the states with lowest energy. Forelectrons, these are states at the minimum of the conduction band. On theordinate axis of a band diagram, the electron energy is plotted; therefore,the minimum energy of the positively charged holes is in the maximum ofthe valence band. The valence-band maximum and the conduction-band min-imum of direct semiconductors are both located at the point k = 0. In indirect

semiconductors like silicon and germanium, minimum and maximum have dif-ferent k-values; therefore, band-to-band recombinations can only occur withthe contribution of phonons or traps. These transitions are unsuitable forlaser activity, because the spatial density of phonons and traps is very low.Furthermore, they are mostly nonradiative and rather unlikely since morepartners are involved.

In thermal equilibrium at a temperature T, the probability whether astate with the energy level E is occupied by an electron is expressed by theFermi function f(E, T).

f(E, T) = 1

expEEFkBT

+ 1

. (3)

At T = 0 K, the Fermi function is a step function which has a valueof 1 (all electronic states filled) below the Fermi-level energy EF and a valueof 0 (all states empty) for higher energies. In undoped semiconductors, theFermi level is located in the middle between conduction and valence bandedges. For higher temperatures T, the Fermi function is smeared out in therange EF

2kBT, with kB = 8.617 347

105 eV/K being the Boltzmann

constant.For a fixed photon energy h, it is entirely correct to consider only two

discrete energy levels E1(k) and E2(k) since the transition can only occurat the same wavenumber k as illustrated in Fig. 1. Three types of radiativeband-to-band transitions can be found in semiconductors, which are sketchedin Fig. 2. The first process is called spontaneous emission, where a recom-bination of an electronhole pair leads to the emission of a photon. This isthe predominant process in Light-Emitting Diodes (LEDs). The emission ofthe photon is random in direction, phase, and time resulting in incoherent

radiation. Since this process depends on the existence of an electron at E2and a hole at E1 simultaneously, the transition rate for spontaneous emis-sion Rsp is proportional to the product of the electron density at E2 andthe hole density at E1. The electron density at the energy E2 is the product


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Fig. 2. Radiative band-to-band transitions in semiconductors

of the density of electronic states D(E2) and the probability that they areoccupied by electrons given by the Fermi function f(E2, T). The hole density

at the energy E1 is the density of electronic states D(E1) multiplied withthe probability of not being occupied by electrons [1 f(E1, T)]. So, thetransition rate per volume for spontaneous emission of photons with fixedenergy h = E2 E1 can be written asRsp = A D(E2)f(E2, T) D(E1) [1 f(E1, T)] , (4)

with A being the proportionality constant for spontaneous emission.Absorption, also called stimulated absorption, is the second process illus-

trated in Fig. 2. A photon is absorbed and an electronhole pair is gener-

ated. This is a three-particle process and the transition rate R12 thereforeis proportional to the product of three particle densities: first, the density ofnonoccupied states D(E2) [1 f(E2, T)] in the conduction band at the en-ergy E2, second, the density of states occupied by electrons D(E1) f(E1, T)in the valence band at E1, and third, the density of the photons (h) withenergy h = E2 E1.R12 = B12 (h) D(E1)f(E1, T) D(E2) [1 f(E2, T)] . (5)

B12 is a proportionality constant for stimulated absorption.

The third process is stimulated emission. A recombination of an electronhole pair is stimulated by a photon and a second photon is generated si-multaneously which has the same direction and phase as the first photon.This process can be used to amplify optical radiation, since the photons areemitted into the optical mode of the stimulating photon resulting in coher-ent radiation. Light sources based on this emission process are called lasers,which is an abbreviation of light amplification by stimulated emission of ra-diation. Analogous to the stimulated absorption (5), the transition rate R21for stimulated emission can be described as

R21 = B21 (h) D(E2)f(E2, T) D(E1) [1 f(E1, T)] , (6)with B21 being the proportionality constant for stimulated emission.


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When the semiconductor is in thermal equilibrium with the photons, noenergy is transferred from the semiconductor to the optical radiation field;thus, absorption and emission must be balanced:

R12 = R21 +Rsp . (7)Using (4), (5), and (6) for the rates Rsp, R12, and R21, respectively, gives

B21 (h) + A

B12 (h)=

f(E1, T) [1 f(E2, T)]f(E2, T) [1 f(E1, T)] , (8)

B21B12

+A

B12

1

(h)=

f(E1, T) f(E1, T) f(E2, T)f(E2, T) f(E1, T) f(E2, T)

=

1f(E2,T)

11f(E1,T) 1

. (9)

The spectral photon density (h) in thermal equilibrium does not dependon the specific function for the density of states D(E). Inserting the Fermifunction f(E, T) from (3) and the relation h = E2 E1 (2), gives

B21B12

+A

B12

1

(h)=

expE2EFkBT

exp

E1EFkBT

= exp h

kBT

, (10)

(h) =A

B12 exp

hkBT

B21

. (11)

The spectral energy density u() d at the frequency in a medium withrefractive index n for radiation in thermal equilibrium is given by Planckswell-known formula for blackbody radiation.

u() d =8hn33

c31

exp

hkBT

1 d . (12)

Dividing the energy density u() by the photon energy h yields the pho-ton density (h). Additionally, the relations = 2, h = 2h, and d(h) =d(h) = h d have been used.

(h) d(h) =n3 (h)2

2 h3 c31

exphkBT 1

d(h) . (13)

In thermal equilibrium of the semiconductor material with the radiationfield, the spectral photon density described in (11) must be identical with


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the photon density of the blackbody radiation described by (13). Comparingthese equations gives

B12 = B21 = B , A =n3

2h3

c3(h)2 B . (14)

With the knowledge of these relations between the proportionality con-stants, the considerations can be extended to nonequilibrium conditions.When a p-n junction is forward biased, electrons and holes are injected intothe depletion region where they can either recombine or travel further to theother side of the junction and recombine there with the majority carriers. Inthe transition zone, where electrons and holes coexist, the carrier distribu-tion cannot be described by a single equilibrium Fermi function (3). Instead,separate quasi-Fermi functions are used for the electrons in the conduction

band fc(E, T) and for the holes in the valence band fv(E, T).

fc(E, T) =1

expEEFckBT

+ 1

, fv(E, T) =1

expEEFvkBT

+ 1

. (15)

The equations are identical to the equilibrium Fermi function but differ-ent Fermi-level energies EFc and EFv are employed for the carrier distri-butions in the conduction and valence bands, respectively. The nonequi-librium situation can be described by replacing f(E1, T) fv(E1, T) andf(E2, T) fc(E2, T).

To determine whether an optical wave with quantum energy h is ab-sorbed or amplified by stimulated emission, the quotient of the correspondingrates R12 and R21 is calculated.

R12R21 =

fv(E1, T) [1 fc(E2, T)]fc(E2, T) [1 fv(E1, T)] =

fv(E1, T) fv(E1, T) fc(E2, T)fc(E2, T) fv(E1, T) fc(E2, T)

=

1fc(E2,T)

11

fv(E1,T) 1 =

expE2EFckBT

expE1EFvkBT

= exp

h (EFc EFv)

kBT

. (16)

Again, the result does not depend on the specific density of states D(E).In thermal equilibrium EFc = EFv = EF, the exponent [h/(kBT)] is posi-tive, the exponential function is larger than 1, and therefore the absorptionrate R12 is always larger than the rate R21 of the stimulated emission. Laseroperation in semiconductors can only be achieved if the condition

EFc

EFv > h > Eg (17)

is fulfilled. In this status, which is called inversion, the exponential function issmaller than 1 and the rate of stimulated emission is larger than the absorp-tion rate. Laser operation requires a process called pumping which builds up


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and maintains a nonequilibrium carrier distribution in the semiconductor ma-terial. From (16) it can be deduced that a laser-active transition always showsabsorption in thermal equilibrium. Although pumping can also be providedby optical excitation of electronhole pairs, one main advantage of semicon-ductor lasers over other types of lasers is the fact that they can be easilypumped with electrical currents as a forward-biased semiconductor diode asshown in Fig. 3. For this reason, electrically pumped semiconductor lasersare called diode lasers.

All state-of-the-art semiconductor-diode lasers use forward-biased double-hetero p-i-n structures to achieve carrier inversion. In this type of structure,an undoped semiconductor layer with a direct band gap is sandwiched be-tween p-doped and n-doped material with a higher band gap. When the

junction is forward biased, the quasi-Fermi levels EFc and EFv in the intrin-

sic layer are located inside the conduction and valence bands as illustratedin Fig. 3. Thus, this region acts as a laser-active layer which amplifies opticalradiation by stimulated emission. Furthermore, the double heterostructurehas two additional advantages. First, the carriers (electrons and holes) areconfined between the double heterobarriers in the conduction and the valencebands and are therefore forced to recombine inside the intrinsic layer of di-rect semiconductor material. Second, this layer sequence works like an opticalwaveguide since for most semiconductor-material systems, the low-band-gaplayer in the middle of the structure has a higher refractive index.

Fig. 3. Forward-biased double-heterostructure p-i-n junction. Conduction Ec andvalence band edges Ev are plotted as solid lines. The Fermi-level energy EF, repre-sented by dashed lines, splits into quasi-Fermi levels EFc and EFv in the undopedtransition region, where holes and electrons coexist. In this region, inversion isachieved since the quasi-Fermi levels are inside the bands


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To assess which proportion of the carriers recombines by stimulated andspontaneous emission, the ratio ofR21 and Rsp is determined using (4), (6),and (14).

R21Rsp =

BA

(h) = 2h3c3

n3(h)2(h) . (18)

This equation shows that a high photon density (h) is necessary to suppressspontaneous emission. Since the term (h)2 is in the denominator of (18), ahigher value of the photon density (h) is required for lasers with higherphoton energy (h) to achieve the same suppression of spontaneous emission.To obtain a high photon density in semiconductor lasers, optical waveguidesare implemented to confine the photons in the laser-active region of the de-vice. Furthermore, an optical resonator, mostly a FabryPerot resonator, isused to increase the photon density in the resonator cavity. A semiconduc-tor laser can be regarded as an optical oscillator consisting of an opticallyamplifying medium and a resonator which provides optical feedback to theamplifier. Waveguides and resonators for high-power semiconductor lasers arediscussed in more detail in the next sections.

1.2 Basic Elements of Semiconductor-Diode Lasers

Several basic elements are necessary to realize a semiconductor-diode laser:

a medium providing optical gain by stimulated emission, an optical waveguide which confines the photons in the active region of the

device, a resonator creating optical feedback, and a lateral confinement of injected current, carriers, and photons which is

required for operation in a fundamental lateral mode.

The optical-gain medium consists of an active undoped layer of directsemiconductor material embedded between high-band-gap p- and n-doped

regions. When this p-i-n junction is forward biased, electrons and holes areinjected into the active region and optical gain by stimulated emission be-comes possible. Furthermore, the double heterobarriers confine the carriersin the active region. The active layer may consist of bulk material with atypical thickness of 100 nm or of one or more quantum wells having typicalthicknesses of 10 nm. Quantum-well structures are discussed in Sect. 1.6.

A dielectric optical waveguide consists of a core film with high refractiveindex embedded in cladding material with lower refractive index. Figure 4 il-lustrates the optical waveguide for a double-heterostructure laser. The active

film with band gap Eg, refractive index nf, and thickness d is sandwichedbetween cladding layers with band gap Eg,cl and refractive index ncl. If theindex step n = nfncl and the core thickness d of the waveguide are smallenough, only the fundamental mode with nearly Gaussian field distribution


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Fig. 4. Confinement of the electronic carriers (electrons and holes) and the electricfield (photons) using a double heterostructure in the vertical direction x of an edge-emitting diode laser. Plotted are the energy-band diagram E(x) with conductionand valence bands (top), the refractive-index profile n(x) of the dielectric waveguide(center), and the electric-field distribution E(x) of the fundamental optical modetraveling in the z direction (bottom)

can propagate in the waveguide. The optical wave traveling in the directionof the waveguide experiences an effective refractive index neff which is dif-ferent from the refractive indices of core and cladding (ncl neff nf).Figure 4 shows a structure where the same layer provides the confinement ofthe carriers and the optical wave. In quantum-well lasers, so-called separateconfinement structures are necessary, where the carriers are confined in quan-tum wells and the optical wave is confined in a separate dielectric-waveguidestructure.

For high-power diode lasers, FabryPerot resonators are used. Figure 5shows this type of resonator consisting of two mirrors with distance L around

a laser-active material having an optical waveguide with effective refractiveindex neff in a propagation direction normal to the mirror surfaces. The


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Fig. 5. A standing wave having m = 7 nodes in a FabryPerot resonator with acavity length L. The wave propagates in a waveguide with an effective refractiveindex neff. The distance between two nodes is 0/(2neff) with 0 being the vacuumwavelength

resonator provides feedback, when a standing wave develops between the

mirrors.

L = m0

2neff, m = 1, 2, 3, . . . . (19)

m is the number of nodes of the standing wave, the order number of thelongitudinal mode, and 0 is the vacuum wavelength. Lasers for optical com-munication systems have other types of optical resonators like DistributedFeedBack (DFB) or Distributed Bragg Reflector (DBR) resonators [13,14].

Up to this point, carrier and optical confinement have been discussedfor a direction which is perpendicular to the active-layer plane. This direc-tion is called the vertical direction. To obtain single-mode operation in bothtransversal directions, an additional lateral confinement is required. As dis-cussed in Sect. 1.5, three types of lateral-confinement mechanisms are possi-ble: current confinement leading to a gain-guided lateral waveguide, opticalconfinement providing index guiding, and buried heterostructures providingan additional carrier confinement.

1.3 Optical Gain and Threshold Condition

When passing through an absorbing material in the z direction, the inten-sity J of a planar optical wave exponentially decreases.

J(z) = J0 exp(z) , (20)with J0 being the initial intensity and the intensity-absorption coefficient.In laser-active semiconductor material, an amplification of the optical wave isachieved. In this case, the exponential increase in intensity can be describedby a negative value of which is referred to as optical gain g = . In anoptical waveguide, only a part of the intensity pattern of the optical mode

overlaps with the active region, which usually is located in the core of thewaveguide. One has to distinguish between the gain of the active materialitself, called the material gain g, and the significantly lower gain of the opticalmode, called the modal gain gmodal.


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In Fig. 6, the material gain of GaAs at room temperature is plotted for dif-ferent carrier densities N. The maximum gain is observed at photon energieswhich are slightly higher than the band gap energy. Figure 7 illustrates theoptical-intensity pattern J(x) of the fundamental optical mode in a double-heterostructure edge-emitting laser having an active-layer thickness d. Therelation between modal gain gmodal and material gain g is expressed by defin-ing a confinement factor which depends on the overlap of the optical-modepattern with the gain region of the laser.

gmodal = g , =

+d/2d/2

J(x) dx

+

J(x) dx. (21)

In double heterostructures with active-layer thicknesses of 50300nm, theconfinement factor has values in the range 1070%. If the active layerconsists of a quantum well with a typical thickness around 10 nm, confinementfactors of a few percent are obtained.

For a mode traveling along the optical waveguide, the intensity-absorptioncoefficient is usually split into two parts, one describing the intrinsic modal

Fig. 6. Optical-gain spectrum g(0) of GaAs bulk material at carrier densitiesof N = 26 1018 cm3. For photon energies below the band-gap energy of GaAsof Eg = 1.42 eV, the material is transparent. Optical gain occurs for energies nearthe band gap. The maximum of the optical-gain curve shifts towards shorter wave-lengths for higher carrier densities N due to the band-filling effect. If the photonenergy is further increased, absorption takes place. The gain data have been calcu-lated using the model described in [15]


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Fig. 7. The confinement factor is defined by the overlap of the intensity pattern Jof the optical mode with the active region. The illustration shows the intensitydistribution J(x) in the vertical direction x of a fundamental optical mode withnearly Gaussian shape for an edge-emitting laser with an active-layer thickness d

absorption i and the other describing the modal gain gmodal = g whichdepends on the density of the injected carriers.

= i g . (22)The intrinsic modal absorption is caused by scattering of the optical mode atdefects or rough interfaces and by free-carrier absorption. Whereas scatteringis extremely low for semiconductor-diode lasers with good crystalline qual-ity, free-carrier absorption cannot be avoided since part of the optical-modepattern overlaps with the p- and n-doped cladding regions. When the modal

gain g is larger than the modal loss i, the propagating optical mode isamplified.

In a laser device, the optical waveguide is combined with a FabryPerotresonator having mirror reflectivities R1 and R2. Some optical intensity leavesthe cavity at these mirrors and contributes to the laser output beam. Asillustrated in Fig. 8, the intensity Jrt of the optical mode after a roundtripin the cavity is given by

Jrt = J0 R1R2 exp [2( g i)L] . (23)

Lasing occurs when the gain provided to the optical mode compensates theintrinsic absorption and the mirror losses for a roundtrip. The minimumgain g where the device starts lasing operation is called the threshold gain gth.


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Fig. 8. Intensity of an optical wave during a roundtrip in a FabryPerot resonatorwith cavity length L and mirror reflectivities R1 and R2

In this case, the intensity Jrt after a roundtrip in the cavity again has its initial

value J0.

Jrt = J0 ,

1 = R1R2 exp[2( gth i)L] , gth = i +

1

2Lln

1

R1R2

= i + mirror . (24)

At laser threshold, the modal gain gth is the sum of the two terms in (24),the intrinsic absorption i and the mirror losses mirror. The mirror lossesdepend on the cavity length L and the mirror reflectivities R1 and R2.

1.4 Edge- and Surface-Emitting Lasers

Figures 9 and 10 schematically show two basic implementations of semicon-ductor-diode lasers, the edge-emitting laser and the Vertical-Cavity Surface-Emitting Laser (VCSEL). The mirror facets of edge-emitting lasers are ob-tained by cleaving the wafer along crystal planes. The mirror reflectivities R1and R2 are approximately 30% if the facets are uncoated. Mirror coatings canbe applied to change these reflectivities and to passivate the surfaces. The

propagation direction of the optical mode in the resonator is in plane withthe substrate surface and is referred to as the axial direction. A planar op-tical waveguide and the laser-active region are formed by depositing a layersequence onto the substrate surface using a growth technique called epitaxywhere the deposited single-crystal layers are lattice-matched to the substrate.The growth direction which is perpendicular to the substrate surface is calledthe transverse or vertical direction. The lateral direction is in the substrateplane normal to the axial direction. The active region has a lateral width W, avertical height d given by the thickness of the epitaxially grown active layer,

and an axial length L which is identical to the cavity length. In Fig. 9, astripe laser is shown, where the width of the active layer is defined by the topohmic contact resulting in a gain-guided optical waveguide without carrierconfinement. The reference coordinate system used to describe edge-emitting


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Fig. 9. Schematic drawing of an edge-emitting laser with the coordinate system.The laser cavity in the axial direction z is located in the substrate plane havinga cavity length L in the range 3002000

m. The mirror facets are generated bycleaving the substrate along crystal planes. The direction perpendicular to thesubstrate plane is called the transverse or vertical direction x; the lateral direction yis in the substrate plane normal to the axial direction. The active region below thetop contact area has a lateral width W and a vertical height d

lasers has its x axis in the vertical, the y axis in the lateral, and the z axis inthe axial direction. Since edge-emitting lasers have typical cavity lengths L inthe range 3002000 m, the order number m of longitudinal optical modes isvery large (m = 100020 000), the spectral density of the longitudinal modesis very high, and a lot of possible modes can exist within the bandwidth ofthe spectral gain.

In vertical-cavity lasers, the optical propagation (axial) direction is normalto the substrate surface and the effective cavity length is very short (typi-cally 13 m) allowing the existence of only a single longitudinal mode withinthe spectral-gain range. To avoid extremely high mirror losses in the shortcavity, the reflectivity of the FabryPerot mirrors must be close to 100% ac-

cording to (24). This can be achieved by using Bragg reflectors which consistof typically 20 pairs of epitaxially grown GaAs-AlAs layers having alternat-ing high and low refractive index and a thickness of a quarter wavelength.Between the mirrors, a set of quantum wells is sandwiched, providing theoptical gain in the active region. In the case of strained InGaAs quantumwells, all semiconductor material including the substrate is transparent forlight in the wavelength range 0 = 8701100nm generated in these quantumwells. The choice of the appropriate mirror reflectivities allows the VCSELto be operated as a top or a bottom emitter. In the schematic illustration of

a VCSEL shown in Fig. 10, the electrical current is supplied through the p-and n-doped mirrors. The emitting lateral aperture normally has a circulargeometry with a diameter of a few microns allowing single-lateral-mode oper-


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Fig. 10. Schematic drawing of the Vertical-Cavity Surface-Emitting Laser (VC-SEL). The laser cavity of the top-emitting VCSEL is perpendicular to the substrateplane. The mirrors of the FabryPerot resonator consist of Bragg reflectors withreflectivities close to 100%. The total length of the device in the vertical directionis about 7

m and the effective cavity length is in the range 13

m

ation and a highly effective coupling to optical fibers. Additional advantagesare the ultra-low threshold current (< 1 mA), excellent dynamic properties,a high electrical-to-optical power-conversion efficiency, insensitivity to opti-cal feedback, and the absence of sudden device failures attributed to mirrordamage. Although the epitaxial growth of VCSEL structures is rather sophis-ticated, the remaining fabrication process is similar to the manufacturing ofLEDs which allows wafer-scale processing and on-wafer device testing. VC-SELs can be easily arranged in two-dimensional arrays and coupled to paralleloptical-fiber bundles. The single-mode output power of a VCSEL is in themW range. Higher power levels can be achieved by enlarging the diameterof the emitting aperture or by a densely packed arrangement. Since VCSELs

are rather novel devices, their full potential has not yet been exploited. Cer-tainly, there will be an increasing range of applications for VCSELs in thehigh-power regime. More comprehensive information on the properties andapplications of VCSELs can be found in [15,16,17].

1.5 Lateral Confinement

As already mentioned in Sect. 1.2, different implementations are utilized toachieve the lateral confinement of the current, the photons, and the carriers

in edge emitting lasers. As illustrated at the top of Fig. 11, current confine-ment is provided by a current aperture which is mostly realized by a dielectricisolator as shown for a stripe laser at the top of Fig. 12. In another imple-mentation, the current aperture is formed by ion implantation. Diode lasers


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Fig. 11. The three basic types of lateral confinement. Current confinement (top):the current is injected through an aperture. Optical confinement ( center): a stepin the effective refractive index neff builds up a dielectric lateral waveguide forthe optical mode. Carrier confinement (bottom): a double heterostructure barrierprevents the lateral diffusion of electrons and holes

which exclusively have current confinement are called gain-guided lasers. Onlythose modes are amplified, which are propagating under the stripe, since op-tical amplification only occurs in areas which are pumped by the electrical

current. Outside the stripe, an optical wave experiences high optical losses.Fundamental-lateral-mode operation can be obtained in these devices forsmall stripe widths. A single-mode stripe laser is rather easy to fabricatebut has some disadvantages. Compared to index-guided lasers, the thresholdcurrent is larger because the waveguide is rather lossy. Since the mode par-tially propagates in absorbing material, the phasefront of the mode is curved,leading to a significant astigmatism in the output beam.

The principle of index-guided lasers is illustrated in the center of Fig. 11.A lateral effective refractive-index step neff provides the waveguiding. De-

pending on this index step and the width W of the waveguide, single-lateral-mode operation can be obtained. A typical example of an index-guided laseris the ridge-waveguide laser shown in the center of Fig. 12. The index stepis formed by a step in the thickness of the upper cladding layer. Since the


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Fig. 12. Examples of laser structures with different lateral confinement. A stripelaser (top) only has a current confinement, the lateral waveguide is formed by gainguiding. The ridge-waveguide laser (center) has current confinement and opticalconfinement with index guiding. The buried-heterostructure laser (bottom) has cur-rent, photon, and carrier confinement

current is injected at the top of the ridge, current confinement is also im-plemented. The beam quality of ridge-waveguide lasers is very sensitive tothe width and the height of the ridge. Therefore, a precise and reproduciblecontrol of ridge dimensions is necessary during device fabrication.

All three types of lateral confinement are combined in a diode laser hav-ing a buried heterostructure which is schematically shown at the bottomof Fig. 12. The lateral heterostructure is formed by an epitaxial regrowthtechnique. This heterostructure provides index guiding and carrier confine-ment since the barriers prevent the lateral diffusion of electrons and holes asillustrated at the bottom of Fig. 11. The p-n-p structure in the vertical direc-tion works as a current-blocking layer providing carrier confinement. Buried-heterostructure lasers are mostly used in communication systems where anextremely low threshold current is required to allow good dynamic properties

and a low power consumption.


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18 Peter Unger

1.6 Quantum-Well Structures

In double heterostructures, the typical thickness of the active layer is d = 50300 nm, resulting in a confinement factor in the range of 1070%. The

density of electronic states D(E) increases with the square root of the energyat the band edge ( D(E) EEg ). If the thickness of the active layer isshrunk to values of 510nm, the electronic wave functions in this quantumwell show quantization in the vertical direction x resulting in discrete energylevels. In this case, the density of electronic states D(E) increases in stepswhich are located at the electronic energy levels of the quantum well. Thus,the density of states close to the lowest-energy level in a quantum well ismuch higher than the density of states at the band edge in bulk material. Thedensity of electronic carriers at a given energy is the product of the densityof states D(E) and the probability of being occupied by electons fc(E, T) orholes [1 fv(E, T)], which are exponentially decreasing functions (accordingto (15)). Thus, the carrier distribution for a quantum-well laser structure hasa higher maximum value and a smaller energetic width.

Due to the small active volume of a quantum-well laser, low thresholdcurrents can be obtained. Additionally, the material gain is higher and thespectral shift of the gain curve due to the band-filling effect is much smaller,because of the higher carrier density and its narrower energetic distribution.The main advantage of quantum-well structures, however, is the possibility

Fig. 13. Material gain spectrum g(0) of a single compressively strained 8 nm-thick Ga0.8In0.2As quantum well sandwiched in GaAs bulk material at carrier den-

sities N = 26 1018

cm3

. Due to the high density of states D(E) in quantumwells, the maximum of the gain curve shows nearly no shift in wavelength. At highercarrier densities, transitions to the second subband of the quantum well contributeto the gain. The data have been calculated using the model described in [15]


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to introduce compressive or tensile mechanical biaxial strain. In this way, theuseable wavelength range of a particular material system can be extended,e.g. the incorporation of In instead of Ga into a thin GaAs quantum well-layer results in a compressively strained quantum well and the accessiblewavelength now ranges from 870 nm for bulk GaAs into the long-wavelengthregion up to approximately 1100nm. This is illustrated by the spectral gain ofan 8 nm-thick Ga0.8In0.2As quantum well plotted in Fig. 13 in comparison tothe gain of GaAs bulk material shown in Fig. 6. The product of quantum-wellfilm thickness and strain must be below a critical value. Above this value,the film experiences relaxation, which is associated with a high number oftraps leading to nonradiative recombination. In compressively strained quan-tum wells, the light and heavy hole bands are split and the effective massof the holes in the valence band is reduced. The effective masses of elec-

trons and holes are now comparable (me mh) resulting in a more-efficientpopulation inversion in the quantum well. In the long-wavelength range, anykind of strain is beneficial due to the reduced inter-valence-band absorptionand Auger recombination. Especially for compressively strained quantum-well lasers, a significantly improved reliability has been observed. For thesereasons, strained quantum wells are the rule rather than the exception instate-of-the-art diode lasers.

Fig. 14. Different vertical structures for separate confinement of electronic carri-ers and the optical mode. Plotted are the band-gap energies Eg versus verticalposition x. In the upper diagram, a Multi-Quantum-Well Separate-Confinement

Heterostructure (MQW-SCH) with three quantum wells is sketched. In the lowerpart of the figure, a Single-Quantum-Well GRaded-INdex Separate-ConfinementHeterostructure (SQW-GRINSCH) is shown


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20 Peter Unger

Since quantum wells are very thin, the confinement of the optical modeis poor. This can be overcome by a Separate-Confinement Heterostructure(SCH) where the confinement of the optical mode is provided by a separatewaveguide structure. Two examples of such vertical structures are shownin Fig. 14. If the waveguide includes a graded refractive-index profile, thestructure is called a GRaded-INdex Separate-Confinement Heterostructure(GRINSCH).

2 Fabrication Technology

The multilayer structures of diode lasers are fabricated using epitaxial growthtechniques. In these processes, single-crystal lattice-matched layers with pre-

cisely controlled thickness, material composition, and doping profiles are de-posited onto GaAs and InP substrate wafers. The layer sequence consists ofa buffer layer, p- and n-doped cladding layers, a layer for the ohmic contact,and the active region, which may be simple bulk material or a sophisticatedstructure containing one or more quantum wells with separate optical con-finement. Figure 15 shows suitable IIIV semiconductor compounds whichcan be epitaxially grown on GaAs and InP substrates.

Fig. 15. Band-gap energy versus lattice parameter of IIIV semiconductor alloysused for high-power laser diodes. The binary compounds are represented by dots,ternary alloys are drawn as lines. Direct semiconductors are plotted as full linesand dots whereas for indirect material dotted lines and open dots are used. Indi-cated with arrows are ternary compounds growing lattice-matched on the commonsubstrate materials GaAs and InP. The data are derived from [18]


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The following list gives an overview of material systems commonly usedfor high-power diode lasers. Not included are IIVI and GaN-based semicon-ductors for blue and green light emitters [19] since their output power is stillrather low.

AlxGa1xAs grown on GaAs is the classical material for diode lasers. Sincethe radii of gallium and aluminum ions are nearly equal, AlxGa1xAs canbe grown lattice-matched for any composition x, and laser emission in thewavelength range 700870nm can be realized.

Using an InGaAs quantum well, the emission range of the AlGaAs/GaAssystem can be extended to longer wavelengths. Since an indium ion has alarger radius than a gallium ion, the lattice parameter of GaInAs is higherthan the lattice parameter of GaAs. Therefore, only thin GaInAs quantum-

well layers containing compressive mechanical strain can be grown keepinglattice-matching. The emission wavelength can be adjusted in the range8001100nm by varying the thickness and indium content of the strainedquantum well.

The same wavelength range can be covered using the strained GaInAsquantum well in combination with GaInAsP separate-confinement layersand Ga0.51In0.49P cladding layers. Like GaInAs-AlGaAs/GaAs, this systemis also lattice-matched to GaAs but is completely aluminum-free [20,21].

The visible-red short-wavelength range 600700nm can be accessed using(AlxGa1x)0.5In0.5P. Again, this material is lattice-matched to GaAs forany aluminum concentration x since the ion radii of aluminum and galliumare approximately equal.

GaxIn1xAsyP1y grows lattice-matched on InP substrates if the equa-tion x = 0.4 y + 0.067 y2 is fulfilled. The material has a direct band gapranging from Eg = 0.75eV for Ga0.47In0.53As to Eg = 1.35eV for InP.Lasers of this type are implemented in quartz-fiber communication sys-tems at the wavelengths 1.3 m and 1.55 m.

The fabrication sequence following the epitaxial growth is illustrated in

Fig. 16 for a ridge-waveguide laser. In the first lithographic step, the patternof the lateral waveguide is defined and then transferred into the underlyingsemiconductor material using an etching process. For a ridge-waveguide laser,the width and the depth of the waveguide have to be controlled precisely toensure a proper lateral-mode behavior of the device. This can be obtained us-ing epitaxially grown etch-stop layers and anisotropic dry-etching processes.A dielectric isolator is then deposited at the bottom and the sidewalls of theridge leaving uncovered only the highly doped layer for the ohmic contacton top of the ridge. For other types of lasers, the lateral waveguide can beproduced by techniques like ion implantation or a sophisticated epitaxial re-growth technique. A second lithographic step followed by metal evaporationand a lift-off process in an organic solvent is used to define the structure ofthe top ohmic contact. Some diode lasers take advantage of a thick gold layer


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22 Peter Unger

Fig. 16. Process sequence for the fabrication of ridge-waveguide diode lasers:(a) etching of the ridge, (b) deposition of top contact metallization, (c) electro-plating of a gold heat spreader, substrate thinning, evaporation of the back-sidecontact, and mirror cleaving

which has been electroplated atop the ridge waveguide using a patternedphotoresist as a mold for the galvanic deposition. The gold layer providesspreading of the heat generated in the laser leading to a significantly lowerthermal resistance when the laser is mounted junction-side up. In addition,this layer protects the ridge against damage during processing, testing, andpackaging and makes junction-side-down mounting more reliable.

After completion of the front-side processing, the wafer is thinned downto a thickness of around 100 m to allow proper cleaving of the laser mirrorsand to reduce the thermal resistance of junction-side-up-mounted devices.An ohmic contact is evaporated onto the back side and a baking processat high temperature provides alloying of the contact with the highly dopedsubstrate material. To create the laser mirrors, the wafer is cleaved alongcrystallographic planes into bars having typical lengths of 1 cm. The cavitylength L of the lasers is determined by the width of these bars. Figure 17shows scanning electron micrographs of a laser chip on a bar with uncoated

facets. After cleaving, the facets must be immediately coated to protect themirrors against corrosion and to modify the reflectivity of the facets. Devicetesting is performed on the bar level. Single devices are cleaved from the bars


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Fig. 17. Scanning electron micrographs of a (Al)GaInP/GaAs ridge-waveguidediode laser with a cavity length L = 500

m. A gold heat spreader is deposited atopthe ohmic contact. The picture at the top features a single diode laser on a bar, theimage at the bottom shows a closer view of the facet region ([25], c 1993 IEEE)


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24 Peter Unger

and then soldered into packages together with silicon p-i-n monitor diodes.For higher output powers, the chips have to be mounted junction-side downand submounts with low thermal conductivity have to be used to spread theheat. The packaged devices are again tested and often have to pass a burn-inprocedure before being shipped to the customer. An alternative approach usesmirrors created by a highly anisotropic dry-etching process. Mirror coatingand the device testing are performed on the full wafer. This process technol-ogy allows the monolithic integration of the devices and has the potential tomake laser production easier and cheaper [22,23,24].

3 Optical Waveguides and Resonators

3.1 Effective Refractive IndexWhen a light wave propagates from a dielectric medium with refractive in-dex nf to a medium with refractive index ncl < nf as illustrated in Fig. 18,refraction occurs at the interface according to Snells refraction law

nf sin = ncl sin , (25)

with being the angle of incidence. The refraction angle > of thetransmitted beam cannot be larger than 90 (sin = 1) corresponding to acritical angle of incidence crit given by the equation

sin crit =nclnf

. (26)

If the angle of incidence is equal to or larger than the critical angle crit,total reflection of the incoming wave occurs at the interface. The phase shiftof the reflected wave depends on the incidence angle and the polarizationof the wave. The incidence plane is defined as the plane that contains thewave vector k orientated in the propagation direction and the normal vectorto the interface plane. For Transverse Electric (TE) polarization, where the

Fig. 18. Illustration of Snells refraction law for a light ray coming from a mediumwith refractive index nf and being transmitted into an optically thinner mediumwith refractive index ncl < nf


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electrical field vector E is perpendicular to the incidence plane, the half phase-shift angle TE of the totally reflected wave is given by

tan TE =n2f sin2 n

2cl

nf cos . (27)

In the case of Transverse Magnetic (TM) polarization, where the electricalfield vector is in the incidence plane, the half phase-shift angle TM is givenby

tan TM =n2fn2cl

n2f sin

2 n2clnf cos

. (28)

A three-layer dielectric slab waveguide consists of a film with refractiveindex nf embedded between a cladding layer with refractive index ncl and asubstrate with refractive index ns. The refractive indices of the substrate andcladding layers are lower than the index of the film. As shown in Fig. 19, awave undergoes total reflection at both interfaces for large incidence angles when traveling through the film. In the absence of optical absorption in thematerial, the wave will be guided without energy losses in the two-dimensionalfilm. Making the assumption that the wave vector k = (kx, 0, kz) has onlycomponents in the propagation direction z and the transverse direction x,the wave is reflected back and forth between the interfaces in the x direction.

Such a wave can be described by the wave equation

E = E0 ex p[ i (t kxx kzz)] . (29)The electric field vector E points in the y direction for a TE-polarized waveand is located in the xz plane for TM polarization. The k vector of the wave

Fig. 19. Guided optical wave in a film with vertical thickness d being back andforth reflected at the interfaces to cladding layer and substrate. The wave vector

is split into a component kz in the propagation direction and a component kx in

the vertical direction. To allow total reflection, the angle of incidence must belarger than the critical angle crit defined by (26) at both interfaces


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26 Peter Unger

can be split into a component kz which describes the propagation along thewaveguide and a component kx:

kx = nf k0 cos , kz = nf k0 sin , k0 =2

0. (30)

The component kz is used to define the effective refractive index neff for thetraveling wave in the propagation direction

neff = nf sin , kz = neff k0 . (31)

In the x direction, however, a standing wave has to build up, otherwise de-structive interference will cause an extinction of the wave. In an asymmetricwaveguide, the refractive index of the layers above and below the film maydiffer, e.g. ncl

ns. In order to allow total reflection at both interfaces, sin

must be larger than ncl/nf and ns/nf, according to (26). On the other hand,sin is always equal to or smaller than 1.

ncl/nf ns/nf sin 1 ,ncl ns nf sin nf ,ncl ns neff nf .

(32)

This equation limits the range for the effective refractive index neff.To allow the formation of a standing wave in the x direction, the phase

path must be a multiple of 2 after a roundtrip in the vertical cavity withthickness d. The phase-shift angles 2cl and 2s from (27) or (28) after thetotal reflection at the cladding layer and the substrate, respectively, must betaken into account.

2 d nf k0 cos m 2 cl 2 s = m 2 . (33)m is the number of nodes of the standing wave and is defined as the or-der number of the transverse optical mode. The mode m = 0 is called thefundamental mode. For given geometry (d), refractive indices (nf, ns, ncl),

and wavelength 0, (33) determines the allowed values m for the incidenceangle depending of the mode number m.

3.2 Normalized Propagation Diagrams

In order to present calculated values for the effective refractive index nefffor all kinds of geometries, refractive indices, and wavelengths, normalizedparameters are usually defined.

V = k0

dn2f

n2s

, b =n2eff n2sn2f n2s

. (34)

V is called the normalized frequency since k0 = 2/0 = 2/c is pro-portional to the frequency of the wave. b is the normalized propagation


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parameter which is a measure for the effective refractive index neff. Asym-metric waveguides are taken into account by defining normalized asymmetryparameters aTE and aTM for transverse electric and magnetic waves:

aTE = n2s n

2cl

n2f n2s, aTM = n

4f

n4cln

2s n

2cl

n2f n2s. (35)

Using these normalized parameters, (33) can be transformed into

V

1 b = m + arctan

b

1 b

+ arctan

b + aTE

1 b

(36)

for TE-polarized waves. This dependency is plotted in Fig. 20 for the firstthree modes. There is always a solution to (36) for the fundamental mode

(m = 0) in a symmetric waveguide (a = 0). In asymmetric waveguides,propagation is not possible below a cut-off frequency V0 = arctanaTE for

fundamental modes. For higher-order modes, the cut-off frequency is Vm =V0 + m.

Fig. 20. Normalized propagation parameter b versus normalized frequency V at dif-ferent normalized asymmetry parameters a for TE modes in a three-layer dielectricwaveguide [26,27]

In semiconductor-diode lasers, fundamental-mode operation is desired intransverse and lateral directions. So, in both directions, the V parameter mustbe below the cut-off frequency V1 = for higher-order modes. According to(34), for a given wavelength 0 = 2/k0 the V parameter is proportional to

the product of the film thickness d and the refractive-index step

n2

f n2

s be-tween film and cladding layers. Since the waveguide in the transverse directionconsists of epitaxially grown semiconductor material, small layer thicknesses(below d = 1.5 m) can be accurately controlled. Therefore, a relatively high


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28 Peter Unger

refractive-index step is possible, which has the additional advantage of goodcarrier confinement since a high index step n = nf ncl is attributed to ahigh step in band-gap energy Eg = Eg,cl Eg (Fig. 4). For lateral indexguiding, the width of the waveguide is defined by the ohmic contact or theridge geometry (Fig. 9) which is patterned by a lithographic process allowingonly inferior dimensional control. On the other hand, a small lateral wave-guide will result in a high series resistance of the device, causing a lot of unde-sired ohmic heat generation. For these reasons, a relatively wide (W > 3 m)lateral waveguide is used together with a low effective-refractive-index step.To form a lateral-index waveguide, a stripe must be defined, where the effec-tive refractive index of the transverse waveguide under the stripe is higherthan outside it. This lateral control of the effective refractive index can beachieved in several ways:

by a lateral change in the asymmetry a = a(y), e.g. changing the thickness(ridge-waveguide laser) or the refractive index of one of the cladding layers,

by a lateral change in the film thickness d = d(y), or by a lateral change in the refractive index of the film nf = nf(y) as in a

buried-heterostructure laser (see Sect. 1.5).

3.3 Optical Near- and Far-Field Patterns

The amplitude of the electric-field vector E(x,y,z) in dielectric material withgiven refractive-index distribution n(x,y,z) for sinusoidal time dependenciesat a fixed angular frequency can be derived using the Helmholtz equa-tion [28]

2

x2+

2

y2+

2

z2

E(x,y,z) + k20n

2(x,y,z) E(x,y,z) = 0 . (37)

In a film waveguide, the refractive-index profile n(x,y,z) = n(x) only has adependency in the x direction as shown in Fig. 19. For a TE-polarized wave

propagating in the z direction, the electric field vector E is oriented in they direction. Such a wave can be described by

Ey(x, z) = Ey(x) exp(i neffk0z) . (38)Inserting (38), the Helmholtz equation (37) can be written as

2Ey(x)x2

+

k20n2(x) k20n2eff

Ey(x) = 0 , (39)which is a one-dimensional differential equation yielding the eigenvalue neff

and the y-polarized vertical electric-field distribution Ey(x).The field distribution and the effective refractive index of a one-dimen-sional dielectric waveguide can be approximated using the effective-indexmethod, where the lateral effective-index variation neff(y) is determined by


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solving (39) for the different lateral positions y. This effective-index distribu-tion is then again inserted into a one-dimensional Helmholtz equation similarto (39) for a lateral waveguide, this time for a TM-polarized wave (magnetic-field vector orientated in the

x direction) and yields the effective index

of the one-dimensional waveguide and the field distribution in the lateraldirection. This method is a rather good approximation for waveguides ofedge-emitting diode lasers, which are thin in the vertical and broad in thelateral direction. As illustrated in Fig. 21 for a ridge-waveguide laser, the fielddistribution E(x, y) at the laser facet is called the near-field profile.

When the emitted light leaves the laser resonator and propagates fromthe waveguide into free space, the beam is broadened by diffraction. Thediffracted pattern some distance away from the facet is called the far field.The transition occurs at a distance roughly W2/0, where W is the width

of the near-field pattern. The far-field intensity J(x, y) can be deducedfrom the FresnelKirchhoff diffraction integral [10]. For the direction x, thefar-field intensity distribution is given by

J(x) cos2 x

+

E(x) exp(i k0 sin xx) dx

2

. (40)

Replacing x by y in (40) yields the same equation for the y direction. Thefactor cos x in front of the integral is called the Huygens obliquity factor,

which accounts for the fact that the intensity from a surface tilted by anangle is reduced by cos2 . For very large off-axis angles, slight correctionsmust be added to the Huygens factor, which also depends on the polarization

Fig. 21. The transition from the optical near-field profile (x, y) at the laser facetto the far-field intensity pattern in free space J(x, y) is described by diffractiontheory


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30 Peter Unger

Fig. 22. Output characteristic and horizontal far-field patterns for a ridge-waveguide laser with a ridge width W = 4

m and a cavity length L = 500

m [29].Single-mode behavior with perfectly Gaussian far-field patterns is obtained

Fig. 23. Output characteristic and horizontal far-field patterns for a ridge-waveguide laser with a ridge width W = 7 m and a cavity length L = 500 m. Theridge width of this laser is close to the limit, where higher-order modes can developin the waveguide. When the laser is operated at higher power levels, the lateraleffective refractive-index step is increased due to the higher temperature under theridge allowing the propagation of a second lateral mode associated with a kink inthe light-output characteristic

of the beam. For small angles, the far-field pattern is the Fourier transform ofthe near-field distribution. Thus, a narrow emitting aperture leads to a wideangular profile in the far field and vice versa. If the near-field function E(x, y)has a constant phase front at the facet, the far-field profile, being the Fourier


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transform ofE(x, y), is always a symmetric function even if the emitted nearfield shows an asymmetric amplitude distribution. Waveguides with real indexguiding have planar wavefronts in the propagation direction. In gain-guidedlasers, the near field has a curved wave front originating from the opticalabsorption in the lateral cladding layers where no current is injected (seeSect. 1.5).

Figures 22 and 23 show lateral far-field patterns with increasing out-put powers of two ridge-waveguide lasers having different ridge widths W.Whereas for the small ridge width (Fig. 22, W = 4 m), nice Gaussian pro-files are observed, a second-order mode develops for higher output powers inthe laser with larger ridge width (Fig. 23, W = 7 m).

3.4 FabryPerot Resonator

An electromagnetic wave traveling in the z direction in a waveguide with anabsorption coefficient and an effective refractive index neff can be describedusing the wave equation

E= E0 exp

2

ex p[ i (t k0neffz)] . (41)

The effective refractive index neff describing the phase evolution and theabsorption coefficient describing the amplitude evolution can be combinedin a complex propagation number .

E= E0 exp(z) exp (i t) , (42)

=

2+ i k0neff , kz = k0neff =

2

0neff . (43)

In a FabryPerot resonator, multiple reflections occur at the mirrors whichhave transmission coefficients t1, t2 and reflection coefficients r1, r2 for theamplitude of the wave. As illustrated in Fig. 24, the transmitted wave Etcan be expressed as a superposition of the contributions from these multiplereflections.

Et = t1t2 Ei exp(L)

m=0

[(r1r2)m

exp(2mL)] . (44)

This geometric series can be transformed into

EtEi =

t1t2 exp(L)1 r1r2 exp(2L) , (45)

which is called the Airy function for the FabryPerot resonator. At laserthreshold, the denominator of (45) becomes zero, resulting in a transmitted

wave having infinitely large amplitude.

1 = r1r2 exp(2L) = r1r2 exp(L) expi 4

0neffL

. (46)


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32 Peter Unger

Fig. 24. Multiple reflections of a planar electromagnetic wave inside a FabryPerotresonator with a cavity length L. The mirrors have amplitude transmissions t1, t2and amplitude reflectivities r1, r2

This equation can be split into two parts, yielding conditions for the phase

and the amplitude.1 = r1r2 exp(L) , (47)

1 = exp

i 4

0neffL

. (48)

The amplitude condition in (47) can be written as

1 = r1r2 exp(L) ,0 = ln(r1r2) L , = 1

Lln(r1r2) . (49)

Replacing the absorption coefficient by i gth according to (22) for thethreshold gain gth and substituting the amplitude reflection factors r1, r2 bythe intensity reflectivities R1 = r21 and R2 = r

22, the equation can be further

transformed into the laser-threshold condition known from (24).

gth = i 1L

ln(r1r2)

= i +1

L ln 1

r1r2

= i +1

2Lln

1

R1R2

. (50)


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Equation (48) yields the phase condition known from (19).

4

0neff L = m 2 , m =

2Lneff0

,

L = m0

2neff, m = 1, 2, 3, . . . . (51)

As shown in Sect. 3.1, the effective refractive index neff of an opticalwaveguide is a function of the wavelength 0 even if there is no dispersionin the semiconductor material itself. The spectral separation between twoneighboring modes can be derived by building the total differential of

m 0 = 2L neff . (52)

m 0 + 0 m = 2L neff ,

2Lneff0

0 + 0 m = 2Lneff0

0 . (53)

According to (52), a decrease of m by 1 (m m = 1) switches to thenext-higher resonant value for 0. The wavelength separation FP 0between two optical modes is called the free spectral range of the FabryPerotresonator and results in

2Lneff0

FP 0 2L neff0

FP ,

2L neff FP 20 2L neff0 0 FP ,

FP 20

2L

neff neff0 0 20

2ngr,effL, (54)

which depends on the mode dispersion neff/0.

ngr,eff neff neff0

0 neff + neff

(55)

is the group effective index. vgr = c/ngr,eff is the group velocity of the opticalmode. The group effective index is typically 2030% larger than the effectiverefractive index depending on the specific photon energy relative to band-gapenergy.

3.5 Diode-Laser Spectrum

Figure 25 shows the spectrum of the longitudinal FabryPerot modes to-gether with the modal gain at the laser threshold. When the peak of the

modal gain at the wavelength p is equal to the threshold gain necessary toovercome the intrinsic absorption and the mirror losses (according to (50)),the diode laser starts to operate at the mode which is in the closest spectral


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34 Peter Unger

Fig. 25. Modal gain spectrum and FabryPerot modes of a diode laser at threshold.

The modal gain g(0) has its maximum value gth at a peak wavelength p. Thespectral distance FP of the FabryPerot modes is called the free spectral range

vicinity to the peak wavelength p. As will be shown in Sect. 4.1, the modalgain is clamped above threshold at the threshold gain value gth. Thereforeonly the mode closest to the peak gain is amplified, whereas for the otherlongitudinal modes, the losses are higher than the modal gain. This is shownin the spectrum of a diode laser above threshold in Fig. 26 where a sup-pression of the side modes of 31 dB is achieved. The emission wavelength of

a diode laser changes when the temperature of the device is varied. Sincethe refractive index increases with temperature, the wavelength of a longitu-dinal optical mode at a given order number m increases according to (52).Additionally, the length of the laser cavity increases with temperature dueto thermal expansion of the material, leading to a higher emission wave-length. The wavelength shift with temperature caused by these two effectsis in the range 0/ T = 0.060.2nm/K. A stronger effect is the shift ofthe spectral-gain curve which is mainly determined by the decrease of theband-gap energy with temperature. The shift of the peak-gain wavelength p

with temperature T is approximately p/ T = 0.33 nm/K. Both effectscan be seen in Fig. 27 showing the variation of the emission wavelength of alaser with temperature.

3.6 Mirror Coatings

Dielectric mirror coatings are deposited on laser facets for two main reasons.Mirror coatings passivate and protect the extremely sensitive surfaces of thefacets. Corrosion which results in device degradation and sudden failurescan be reduced or even completely eliminated by a suitable coating process.On the other hand, the reflectivity of the mirrors can be changed, allowingthe entire output power of the laser to be emitted through the front facet.Furthermore, by using a very low reflection at the output facet, the optical


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Fig. 26. Emission spectrum of an AlGaAs/GaAs single-mode diode laser with oxideaperture providing lateral current confinement and index guiding of the opticalmode [30]. Since the cavity length is rather small (L = 300

m), the lateral modescan be resolved by the spectrometer. At an operating current I = 3 Ith, a singlelongitudinal mode at a wavelength 842.1 nm dominates and a side-mode suppressionof 31 dB is obtained

Fig. 27. The peak-emission wavelength of a diode laser increases with tempera-ture. The device in this example shows a shift of 0.2 nm/K due to the change inrefractive index and due to thermal expansion of the material. Since the peak gainshifts by approximately 0.33 nm/K, longitudinal mode hops to the next-lower-ordernumber m occur with increasing temperature


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36 Peter Unger

density inside the cavity can be reduced, resulting in less filamentation ofthe lateral optical mode and an increase in the output power level wheredestruction of the facet occurs.

Figure 28 shows a typical facet coating for a high-power diode laser. Di-rectly on both facet surfaces, Al2O3 layers are deposited. This material hasa refractive index of approximately n = 1.65 and is well known for its goodadhesion and passivation properties. The reflectivity R of the front facet de-pends on the layer thickness d and its refractive index n.

R(n, d) =(1 neff)2 cos2 (nk0d) +

neffn n

2sin2 (nk0d)

(1 + neff)2 cos2 (nk0d) +

neffn + n

2sin2 (nk0d)

, (56)

with k0 = 2/0 and neff being the effective refractive index for the optical

wave traveling in the waveguide between the laser mirrors. This is a periodicfunction with a periodicity in thickness of 0/(2n) as illustrated on the left-hand side of Fig. 29. The minimum and maximum value Rmin and Rmax ofthe reflectivities are

Rmin =

neff n2

2(neff + n2)

2 , Rmax =(1 neff)2(1 + neff)2

. (57)

Rmax is the natural reflectivity for an uncoated facet. The minimum reflec-tivity Rmin is achieved at a layer thickness d = 0/(4n). If the coating has arefractive index n =

neff, antireflection can be obtained.

(56) and (57) are only valid for a dielectric film on bulk material with arefractive index neff at normal incidence of the planar electromagnetic wave.If the wave is confined in a dielectric waveguide, significant modifications tothese equations have to be considered [31,32].

Coatings for the back mirror having a high reflectivity consist of Braggstacks. These are pairs of layers with high and low refractive index. Thethickness of each layer is 0/(4n). The constructive interference in the stacksprovides a higher reflectivity which increases with the number of layer pairs

Fig. 28. Typical mirror coating for high-power edge-emitting lasers. The front facetis coated with a single layer of Al2O3 to reduce the reflectivity whereas on the backfacet, a Bragg-mirror stack consisting of two pairs of Al2O3/Si layers is used toobtain a high reflectivity


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and the difference in the refractive indices of the two materials. Standardcoating materials for the Bragg reflector are Al2O3 with a refractive indexn = 1.65 and Si with a refractive index in the range n = 3.54. The calculatedspectral reflectivity of such a back mirror is plotted on the right-hand sideof Fig. 29. With two layer pairs, a reflectivity of above 90% is obtainedfor a rather broad spectral range. The refractive index of Si strongly varieswith wavelength and, for short wavelengths (0 < 700nm), the absorptionbecomes significant, leading to facet heating. For the short-wavelength range,the low-absorbing TiO2 (n = 2.45) in combination with SiO2 (n = 1.45) is agood alternative.

Fig. 29. Reflectivity R of the front and the back mirrors of a 980 nm laser havingfacet coatings as illustrated in Fig. 28. For the front mirror, an Al2O3 layer witha refractive index n = 1.65 is used. In the diagram on the left-hand side, thereflectivity versus Al2O3 layer thickness is plotted. At a thickness of 200 nm, 10%front-mirror reflectivity is achieved. A Bragg reflector with two pairs of Al2O3/Silayers is used to increase the back-mirror reflectivity above 90%. The reflectivity

versus wavelength is plotted in the right-hand-side diagram. The data have beencalculated using the transfer-matrix model [33]

4 Rate Equations and High-Power Operation

4.1 Rate Equations for Electronic Carriers and Photons

A phenomenological approach to describe the behavior of diode lasers during

operation is the set of the rate equations. These are two coupled equationsexpressing balances for electronic carriers and photons. The rate equationsprovide relations between the external laser parameters extracted from themeasured device characteristics and the internal physical effects.


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38 Peter Unger

The generation and recombination balance for the electronic carriers inthe active region of a laser can be written as

dN

dt=R

gen

Rnr

Rsp

Rstim . (58)

A change dN/dt of the carrier density N in the active region is attributed tothe difference between the carrier-generation rateRgen and the recombinationrate (Rnr + Rsp + Rstim). The generation rate for carriers is the number ofelectrons per time and volume which are injected into the active region. Thenumber of the electrons per time is the electric current I divided by theelementary charge q = 1.602 189 1019 C of an electron. The volume of theactive region is given by the product of the active-film thickness d and thecurrent-injection area (LW). With the current density j being the current I

divided by the injection area (LW), the carrier-generation rate can be writtenas

Rgen = ijq d

, (59)

where the internal efficiency i is the fraction of the current which generatescarriers in the active region. The introduction of i < 1 takes into accountthat part (1 i) of the current does not enter the active region.

For the recombination rate of carriers, several processes must be consid-ered, the nonradiative recombination

Rnr, spontaneous emission

Rsp, and

stimulated emission Rstim. For a spontaneous-recombination process, thepresence of an electron and a hole is required. Therefore, the spontaneous-recombination rate Rsp is proportional to the product of the electron andhole densities. In undoped active regions, charge neutrality requires that bothdensities are equal; thus, the spontaneous-recombination rate is proportionalto N2.

Rsp = B N2 . (60)B is called the bimolecular recombination coefficient, and it has a valuearound B 1010 cm3/s for most IIIV semiconductor materials.

In semiconductors, there are two major nonradiative recombination mech-anisms. For the first mechanism, the recombination at point defects, the rateis proportional to the carrier density. The second mechanism is the Augerrecombination, where the photon energy from a spontaneous recombinationprocess is transferred to an electron in the form of kinetic energy (photo ef-fect). Since this mechanism depends on the presence of three carriers, therate is proportional to N3.

Rnr = A N + C N3

. (61)In the absence of photons in the device, the carrier-recombination rate Rstimof the stimulated emission is negligible. In this case, the device behaves like


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a conventional LED. After switching off the external current, the generationof carriers is terminated and the carrier density in the device decays.

dN

dt=R

nr

Rsp =

A N

B N2

C N3 . (62)

To simplify the description of this decay, an exponential law with a carrierlifetime is used.

N = N0 exp

t

, (63)

with N0 being the initial carrier density. This exponential function is thesolution of the differential equation

dN

dt = N

. (64)

Compared to (62), this is an improper description of the nonradiative andspontaneous recombination in the device.

As shown in Sect. 1.3, the density of photons Nph exponentially in-creases when light is traveling in a medium with optical gain originated fromstimulated-emission processes.

Nph(z) = Nph(0) exp[ g(N, 0)z] , (65)

with the material gain g(N, 0) depending on the carrier density N andthe vacuum wavelength 0. This exponential function is the solution to thedifferential equation

dNphdz

= g(N, 0) Nph (66)

describing the photon-generation rate per unit length, which can be convertedinto the photon-generation rate per unit time by

dNph

dt=

dNph

dz

dz

dt= g(N, 0) Nph vgr . (67)

vgr is the group velocity of the photons in the active material. In the caseof a dielectric waveguide, vgr = c/ngr,eff is given by the group effective in-dex ngr,eff neff (neff/0) 0 as defined in (55). The photons which aregenerated in the active material are carrier losses by stimulated emission

Rstim = vgr g(N, 0) Nph . (68)Now the rate equation for the carriers is complete and can be written as

dNdt

= ijq d

N vgr g(N, 0) Nph . (69)


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40 Peter Unger

It is common to approximate the dependency of the material gain g(N, 0)as a linear function of the carrier density N.

g(N) = ad (N

Ntr) , ad =g

N, (70)

with ad being the differential gain coefficient and Ntr the transparency carrierdensity. Ntr is the carrier density, where the material losses of the activemedium are compensated by the optical gain, resulting in material which isoptically transparent for light with vacuum wavelength 0.

The electronic carriers are confined in a layer with vertical thickness d.The photons are confined by a separate optical waveguide. The volume oc-cupied by photons is therefore larger. The confinement factor as definedin (21) can be regarded as a carrierphoton overlap factor, being the volume

occupied by carriers divided by the volume occupied by photons. Since thevolume for the photons is larger by the factor 1/ than the volume for thecarriers, the rates per unit volume R are smaller by the factor for thephotons generated by spontaneous and stimulated emission.

dNphdt

= Rstim + spRsp Nphph

, (71)

where sp < 1 is the spontaneous-emission factor, which accounts for thefact that in contrast to the stimulated emission only the fraction sp of the

spontaneously generated photons are emitted into the lasing mode. sp is ap-proximately the reciprocal of the number of optical modes in the bandwidthof the spontaneous emission and therefore strongly depends on the cavitylength and the geometry of the optical waveguide. While spontaneous andstimulated emission are photon-generation processes, the photon losses aretaken into account by introducing a lifetime ph for the photons in the lasercavity in analogy to (64). The differential equation (67) can be used to de-scribe the exponential decay of the photon density with time in the absenceof any photon-generation terms (Rstim = Rsp = 0).

dNphdt

= Nphph

= dNphdz

dzdt

= (i + mirror) Nph vgr . (72)

The two loss mechanisms for the photons are the intrinsic absorption i andthe mirror loss mirror. The mirror loss of a FabryPerot resonator has beendetermined in Sect. 1.3 where (24) provides a condition for the laser threshold.

i + mirror = i +1

2Lln

1

R1R2

= gth . (73)

A comparison of (73) and (72) yields a measure for the photon lifetime ph.

1

ph= vgr

i +

1

2Lln

1

R1R2

= vgr gth . (74)


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Inserting (68) into (71) yields the complete rate equation for the photons

dNphdt

= vgr g(N, 0) Nph + spRsp Nphph

. (75)

Both rate equations, (69) and (75), are strongly coupled by the stimulatedemission process, which is a loss mechanism for the carriers and a generationprocess for the photons. The rate equations can be utilized to describe thestatic and dynamic behavior of diode lasers and LEDs.

4.2 Electrical and Optical Characteristics of Power Diode Lasers

For all further considerations, the contribution of the spontaneous emissionto the photon density is neglected (sp = 0), which is a good approximationfor power diode lasers since these devices usually have large mode volumesresulting in rather low spontaneous-emission factors sp.

When a diode laser is operating in steady state, carrier and photon den-sities do not change,

dN

dt= 0 ,

dNphdt

= 0 . (76)

So, the rate equations for the steady-state operation of diode lasers can bewritten as

0 =ijq d

N vgr g(N, 0) Nph , (77)

0 = vgr g(N, 0) Nph Nphph

. (78)

Rearranging (78) yields

Nph

vgr g(N, 0) 1

ph

= 0 , (79)

and two solutions can be found to fulfill this equation.

Nph = 0 , (80)

which is valid below laser threshold, where no photons are emitted with theexception of spontaneously generated photons, and

vgr g(N, 0) =1

ph, (81)

which describes the situation above threshold. In this case, the gain is con-

stant and according to (74) has a value g(N, 0) = gth. This behavior is calledgain clamping. The carrier density N = Nth is also constant above threshold,since the gain g(N, 0) is a function of N.


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Fig. 30. Carrier and photon density versus injected current density of a diode laseras described by the rate equations. Below laser threshold, the device works like anLED with undoped recombination zone. Above threshold, the optical output powershows a linear increase with the injected current. The carrier density and thus theoptical gain have a constant value. This behavior is called gain clamping

According to (72), the loss rate for photons through the laser mirrors is

dNphdt

= mirror vgr Nph . (89)

The photons leaving the laser through the mirrors form the laser output.The output power P is the energy per time leaving the laser facets.

P = mirror vgr Nph h

LW d

. (90)

The photon density Nph above threshold is given by (86). Replacing thecurrent densities j and jth by the currents I and Ith divided by the injectionarea (LW) and inserting this equation into (90) yields

P = imirror

gth

h

q(I Ith) . (91)

Now the threshold condition for the modal gain ( gth) (see (24) and (73))

gth = i + mirror = i +1

2Lln

1R1R2

(92)


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44 Peter Unger

can be utilized to complete (91).

P = imirror

i + mirror

h

q(I Ith) = d h

q(I Ith) . (93)

Above threshold current, the output power P linearly increases with cur-rent I. The differential efficiency nd is defined as

d =q

h

dP

dI=

(dP)/(h)

(dI)/q. (94)

d is the differential increase in photons per time (dP)/(h) emitted fromthe diode laser divided by the differential increase in injected electrons pertime (dI)/q above laser threshold. (dP)/(dI) is the slope efficiency in W/Afrom the linear part of the laser characteristic above threshold.

In Fig. 31, the output-power characteristics of a real broad-area high-power diode laser are shown together with the currentvoltage characteristic.The device shows an electrical-to-optical conversion efficiency of 63%. Dur-ing operation of a high-power diode laser, 3550% of the input power isdissipated as ohmic heat in the device. To avoid high device temperatures,a proper mounting is necessary, usually by using sophisticated junction-side-down soldering processes and water-cooled heat sinks.

Fig. 31. Characteristics of a 950 nm InGaAs/AlGaAs broad-area diode laser withan active stripe width W = 100

m and a cavity length L = 500

m [34]. The opticaloutput characteristic (continuous line) exhibits a threshold current Ith = 135 mAcorresponding to a threshold current density jth = 270 A/cm

2 and a differentialefficiency d = 0.82. The currentvoltage characteristic (dashed line) has a kinkvoltage 1.34 V which corresponds well to the photon energy Eph = 1.305 eV for

photons with a wavelength = 950 nm. From the slope of the characteristic, adifferential resistance 163 m can be derived. The electrical-to-optical conversionefficiency (dotted line) exceeds its maximum of 63% at an output power of 1 W.Even at an output power of 2.7 W, the conversion efficiency is above 55%


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Internal laser parameters like the internal efficiency i and the intrinsicabsorption i can be determined by plotting the inverse differential efficien-cies nd versus the cavity lengths L. From (93), the equation

1d

= 1i

+ ii

2ln

1R1R2

L (95)can be derived. From the intersection of the linear fit with the 1/d axisextrapolated for L = 0, 1/i can be determined. Knowing the values for thefacet reflectivities R1 and R2, i can be calculated using the slope of thefitted line.

Figure 32 shows an example of this type of plot for broad-area lasers withuncoated facets. Usually a rather simple device-fabrication process is em-

ployed and no device mounting is necessary to perform this characterizationunder pulsed conditions. Beside measurements of the threshold-current den-sity, the characteristic temperature, the vertical far-field distribution, and theemission wavelength, such examinations have to be routinely performed onepitaxial laser material to control and verify the quality of the diode lasers.

A very important parameter for high-power lasers is the shift of thethreshold-current density jth with temperature T [35]. This behavior canphenomenologically be described by the equation

jth

exp TT0 , (96)

Fig. 32. A linear fit of a plot of the inverse differential efficiency 1 /d versus the cav-ity length L gives access to the internal efficiency ni and the intrinsic absorption i.

Since the facets of the devices are uncoated, facet reflectivities R1 = R2 = 0.29 havebeen used for the calculations according to (95)


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46 Peter Unger

where the characteristic temperature T0 is a measure of the temperaturestability of the device. Figure 33 shows examples of two diode lasers withdifferent characteristic temperatures.

Fig. 33. Light-output characteristics of two single-lateral-mode semiconductor-diode lasers having different characteristic temperatures T0

4.3 Design Considerations for High-Power Operation

When operating a diode laser at high output power, there are two main prob-lems manufacturers and customers have to struggle with. The first problem isthe distortion of the beam profile at high output levels. Devices with single-mode behavior in vertical and horizontal directions are available up to outputpower levels of approximately 200 mW. Above this power, the device showsbeam distortions in the lateral direction. This phenomenon is called filamen-tation, which implies that there are hot regions inside the cavity where the

refractive index is increased leading to parasitic optical waveguides, which de-stroy the lateral-mode profile. Especially in broad-area devices, filamentationis very pronounced. The main effect which causes filamentation, however, hasa different origin. When the local optical intensity in a device is very high,the carrier density is reduced in this area. This behavior is called spatialhole burning. Due to the decrease in the local carrier density, the gain is alsoreduced and the refractive index is increased [36]. Taking into account thelateral-carrier diffusion and additional contributions from thermal effects, fil-amentation is a highly dynamic process which occurs on a picosecond time

scale.Another very important effect is Catastrophic Optical Mirror Damage(COMD). An example of this phenomenon is shown in Fig. 34. When a deviceis properly cooled, the output power limitation is caused by a destruction of


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the facet. A scanning electron micrograph of a facet after COMD is shown inFig. 35. The dynamics of this mirror-degradation mechanism is described indetail in [37] and [38]. With the help of suitable facet coatings, the COMDeffect can be drastically reduced or even completely eliminated [37].

Fig. 34. Light-output power versus operating current for two single-lateral-modediode lasers. The output power of the device on the left-hand side is limited by ther-mal rollover, whereas the device on the right-hand side is destroyed by Catastrophic

Optical Mirror Damage (COMD)

The properties of high-power diode lasers with regard to device reliabilityand beam filamentation can be improved by lowering the optical density inthe device. This measure will result in

reduced spatial hole burning and thus better beam quality, more-reliable device operation since regions whi

Documents

Introduction to Power Diode Lasers