1586 J. Opt. Soc. Am. B/Vol. 12, No. 9 /September 1995 Ding et al.

Cavity-enhaced and quasi-phase-matched opticalfrequency doublers in surface-emitting geometry

Yujie J. Ding

Department of Physics and Astronomy and Centers for Materials and Photochemical Sciences,Bowling Green State University, Bowling Green, Ohio 43403

Jacob B. Khurgin and Seung-Joon Lee

Department of Electrical and Computer Engineering,The Johns Hopkins University, Baltimore, Maryland 21218

Received November 28, 1994; revised manuscript received May 8, 1995

We show that we can efficiently achieve surface-emitting second-harmonic generation in vertical andhorizontal cavities. The fundamental beam is coupled into the waveguide, which consists of III–V orII–VI semiconductor multilayers or asymmetric quantum-well domain structures. The generated second-harmonic radiation propagates along the growth direction of these layers (which is normal to the propagationdirection of the fundamental beam). The quasi-phase matching is achieved when second-order susceptibilityis modulated along the growth direction in these structures. By the proper design of these structures, thefrequency doublers based on these structures together can cover the spectrum of 0.8–2.0 mm. If the pumppower density is sufficiently large, the conversion efficiency approaches saturation. The saturation powerper unit waveguide width is between ,0.9 and ,435 mWymm. At such a power density, 72% conversionefficiency can be achieved. In addition, the proposed frequency doublers are, in principle, broadband.

1. INTRODUCTION

More than 25 years ago, Bloembergen proposed a phase-quasi-matching scheme by spatially modulating second-order susceptibility in the parallel wave propagationconfiguration.1 This scheme has become essential forachieving second-harmonic (SH) generation (SHG) inmany kinds of second-order nonlinear material.2 Withthis concept, surface-emitting green light (0.53 mm) wasobtained3 by a frequency-doubling infrared laser beam(1.06 mm) in the waveguide based on periodically modu-lated second-order susceptibility by the use of alternatingAlxGa12xAs and AlyGa12yAs layers (x fi y, and the thick-ness of each layer is l2vy2n2v , where l2v is the SH wave-length in vacuum and n2v is the refractive index at the SHfrequency 2v). The maximum conversion efficiency ob-tained so far is 0.1%yW. Obviously, if the multilayers aresandwiched between quarter-wave stacks to form a verti-cal cavity for SH light, there is a huge buildup of the SHpower density, resulting in an efficient energy exchangebetween fundamental and SH waves. Recently such anenhancement was demonstrated in Ref. 4, in which a fac-tor of 240-fold increase in the conversion efficiency wasachieved. However, because the cavity layer consists ofonly one thin AlGaAs layer, a quasi-phase-matching con-dition was not established in Ref. 4, as shown below. Asa result, the conversion efficiency is still low (less than1%yW). In all the previous studies, the output powerwas proportional to the square of the input power. Tothe best of our knowledge, no one has ever observed orstudied saturation of the conversion efficiency in semi-conductor structures in the surface-emitting geometry.Therefore the conversion efficiency of SHG in semicon-

0740-3224/95/091586-09$06.00

ductor materials for generating visible light from infraredlight is still too low for any practical applications. Canwe combine the quasi-phase-matching concept demon-strated in Ref. 3 with the vertical cavity enhancementdemonstrated in Ref. 4 to increase the conversion effi-ciency to close to 100% at a modest input power densitybased on the multilayers?

Because each of the multilayers has different linearproperties, such as absorption and refraction, the wave-guide coupling, absorption, and interface reflections mayreduce the conversion efficiency. Is there an alterna-tive structure? It was demonstrated theoretically5 thatasymmetric quantum wells (QW’s) possess large second-order nonlinear optical susceptibility. Can we design astructure based on asymmetric QW’s to modulate thesecond-order susceptibility for achieving quasi-phasematching? Recently, following Ref. 5, Janz et al. grewan asymmetric coupled quantum-well (ACQW) domainstructure and used it to achieve surface-emitting SHG,6

although the efficiency was quite low. In Ref. 6, thethickness of each domain was l2vy2n2v to achieve quasi-phase matching.

There are several advantages of using asymmetricQW’s over multilayers. First, because the effective bandgap of asymmetric QW’s is larger than that of bulk GaAs,the band-to-band absorption of the SH radiation in asym-metric QW’s is lower. Second, because the thickness ofeach layer for each specific operating wavelength cannotbe exactly l2vy2n2v, there is reflection of SH radiationat the interfaces between AlxGa12xAs and AlyGa12yAslayers, which results in the loss of SH radiation in themultilayers. Such a loss does not exist in asymmetricQW’s. Third, the coupling efficiency of the fundamen-

1995 Optical Society of America

Ding et al. Vol. 12, No. 9 /September 1995 /J. Opt. Soc. Am. B 1587

tal light into the multilayer waveguide is lower becauseof the periodically modulated refractive index along thegrowth direction.

In this paper we show that we can efficiently achieveresonantly enhanced SHG in the surface-emitting geome-try based on the semiconductor multilayer or asymmetricQW domain structures. We show that, in the surface-emitting geometry for the most of the structures and theoptimal structure dimensions, the conversion efficiencyundergoes saturation at the input power, which is com-mercially available. Although this saturation phenome-non was briefly discussed in Ref. 7, there is no detailedformulation of the phenomenon. In addition, we have ob-tained many new results in this paper. Furthermore, byincorporating the horizontal cavity for the fundamentalbeam, we can achieve saturation at a much lower inputpower.

There are several advantages of resonant surface-emitting SHG over collinear SHG. First, because thebirefringent effect is small for zinc-blende semiconductormaterials, this is probably the only geometry to achievequasi-phase matching in these materials. Second, be-cause the SH radiation emits in a direction normal tothe propagation direction of the fundamental wave, wedon’t need to separate SH light from fundamental light.Third, the frequency doublers based on surface-emittingSHG can be easily integrated with laser diodes to generatelight at a shorter wavelength. Fourth, we can achieveefficient conversion efficiency for a large range of pumpwavelengths. Fifth, a vertical cavity can be readily incor-porated into the structure during the growth to improvethe conversion efficiency by several orders of magnitude.Finally, a horizontal cavity can be incorporated into thestructure to improve the conversion efficiency further.

Because of these resonant enhancements, several non-linear optical devices based on second-order susceptibil-ity, such as correlators,8spectrometers,8 phase detectors,9

and blue lasers by the integration of laser diodes withfrequency doublers,3 can be implemented with signifi-cantly improved responsivities. Therefore, if our predic-tions are confirmed in the experiments, they may have adramatic impact on the applications of nonlinear optics.Furthermore, by the cascading of the second-order non-linearities (similar to the work of Ref. 10), another classof nonlinear optical devices based on effective third-ordernonlinearities, such as optical power limiters,11 opticalphase-conjugate mirrors,11 optical frequency shifters, di-rectional couplers,12 and self-phase modulators,13 can beimplemented.

This paper is organized as follows: In Section 2, weobtain the equations of motion for the surface-emittingSHG. In Section 3, we estimate the saturation powerdensities for two configurations, without and with thehorizontal cavity. In Section 4 we discuss several issues.Finally, in Section 5 we give our conclusions.

2. EQUATIONS OF MOTIONOur proposed structure, which is shown in Fig. 1, con-sists of a waveguide sandwiched between two multilayermirrors with high reflectivities R1,2 at the SH frequency2v. The waveguide is made from nonlinear materialwhose x s2d is modulated with the spatial period l2vyn2v

Fig. 1. Configuration for surface-emitting SHG. The middleplot is the band diagram of the large-scale ACQW domains.The bottom plot shows the sinusoidal component of the sec-ond-order susceptibility (solid curve) from the Fourier expan-sion of the periodically modulated second-order susceptibilityin the large-scale ACQW domains (dashed line) for achievingquasi-phase matching.

and the amplitude xs2d0 , where l2v is the SH wavelength

in vacuum and n2v is the linear refractive index at 2v.The SH radiation is generated from the surface of thesample with the propagation direction normal to the QWor the multilayer plane (see Fig. 1).

We follow exactly the same derivation procedure devel-oped in Ref. 11. Although, as one can see below, someof the introductory equations are essentially the sameas those in Ref. 11, we should still go through the en-tire derivation procedure. This is because Ref. 11 dealswith optical power limiting and optical phase conjugation,which are quite different from the SHG considered here.These two phenomena are based on effective third-ordernonlinearity. Furthermore, as shown below in the caseof the optical power limiting, the reflected power is limitedat a relatively high input power. In the same configura-tion, the surface-emitting SHG conversion efficiency ap-proaches the maximum value. Thus one device performstwo functions at different ports: power limiting in the re-flected beam and SHG in the surface-emitted beam.

Consider the equations that govern three coupledwaves11,14:

=2E1s2d 2nv

2

c2

≠2E1s2d

≠t2­

1c2

≠2

≠t2f x s2dE2s1dE2vg , (1a)

1588 J. Opt. Soc. Am. B/Vol. 12, No. 9 /September 1995 Ding et al.

=2E2v 2n2v

2

c2

≠2E2v

≠t22 m0s

≠E2v

≠t­

1c2

≠2

≠t2f x s2dE1E2g ,

(1b)

where s indicates the losses of the vertical cavity ex-pressed as the effective conductance; it can be relatedto the photon lifetime as s ­ eyt, where e is the di-electric constant at 2v. In Eqs. (1), nv is the linear re-fractive index of the material at frequency v. x s2d is thespecific element of the second-order susceptibility tensor;for mutlilayers, it is the nonzero element of the second-order susceptibility tensor of the bulk materials, i.e., forGaAsyAlGaAs, x s2d ­ x s2d

zxy . For asymmetric QW struc-tures, x s2d is the nonzero element of the second-order sus-ceptibility that is due to the asymmetry created along thegrowth direction, i.e., x s2d ­ x s2d

yxy , where y and z are inthe layer plane and x is the axis parallel to the growthdirection. In Eqs. (1) we have used the components ofthe electric fields that correspond to those based on thenonzero element of the x s2d tensor. In Fig. 1, we havechosen the polarizations of three fields to reflect this de-pendence in the asymmetric QW’s. The electric fields canbe expressed as products of the amplitudes’ slowly vary-ing in space or time (in general complex) and normalizedwaveguide or resonator modes (real):

E1s2d ­ A1s2dszdc1s2dsxdexpfis6bz 2 vtdg 1 c.c. , (2a)

E2v ­ A2vstdc2vsxdexps22ivtd 1 c.c. , (2b)

where b ­ svycdnv. Substituting Eqs. (2) into Eqs. (1),one obtains

2ibdA1s2d

dzc1s2dsxd ­ 7x s2dsxd

3v2

c2A2vA2s1d

pc2s1dsxdc2vsxd , (3a)

4ivn2v

2

c2

≠A2v

≠tc2vsxd 1 2ivm0sA2vc2vsxd

­ 24v2

c2 x s2dsxdA1A2c1sxdc2sxd . (3b)

Now, multiplying Eqs. (3) by cisxd and integrating overx, one obtains

dA1s2d

dz­ 6

iv

2nvcx

s2d0

deff1/2

A2vA2s1dp, (4a)

dA2v

dz1

A2v

2t­ i

v

n2v2

xs2d0

deff1/2

A1A2 , (4b)

where the effective SHG waveguide thickness is

deff ­ f x0s2dg2

, "Z `

2`

x s2dsxdc1sxdc2sxdc2vsxddx

#2

. (5)

The steady-state solution of Eq. (4b) is

A2v ­2ivt

n2v2

xs2d0

deff1/2 A1A2 . (6)

Substituting Eq. (6) into Eqs. (4), one obtains two coupledequations:

dA1

dz­ 2

2v2t

cnvn2v2

f xs2d0 g2

deffA1jA2j2, (7a)

dA2

dz­

2v2t

cnvn2v2

f xs2d0 g2

deffA2jA1j2. (7b)

In general, the fundamental amplitudes can be writ-ten as A1,2 ­ jA1,2jexpsif1,2d. We can readily show fromEqs. (7) that the phases f1,2 are constants. We assumethat they are 0. Introducing the power density, i.e., thepower per unit waveguide width, Pi ­ 2nv jAij

2yh0, whereh0 is the vacuum impedance, we can obtain the coupledequations for the power densities of the counterpropagat-ing beams at the fundamental frequencies:

dP1ydz ­ 2 kP1P2 , (8a)

dP2ydz ­ kP1P2 , (8b)

where the coupling coefficient is

k ­v2th0

cnv2n2v

2

f xs2d0 g2

deff

, (9)

with t ø 2n2vdycs1 2 R1R2d. Introducing the saturationpower density,

Ps ­ skLd21 ­l2nv

2n2vs1 2 R1R2d

8p2Lh0f xs2d0 g2

deff

d, (10)

where L is the waveguide length, and making a transfor-mation to normalized length z ­ zyL and power densityui ­ PiyPs in Eqs. (8), one obtains

du1ydz ­ 2 u1u2 (11a)

du2ydz ­ u1u2 . (11b)

We can see from the equations above that

u1 1 u2 ­ const. (12)

This is the modified Manley–Rowe relation for thesurface-emitting configuration. From Eq. (12), we canmake the following transformation:

u1 ­ a cos2 B, u2 ­ a sin2 B , (13)

where a is a constant and B is a function of z . Substi-tuting Eqs. (13) into Eqs. (11), we obtain

dBdz

­a cos B sin B

2. (14)

The solution of Eq. (14) can be written as

tan B ­ c1 expsazy2d . (15)

We assume that the beam at the fundamental frequencyenters the nonlinear material at the plane z ­ 0 and thatthe side z ­ 1 is coated by the dielectric layers, resultingin a high reflectivity R4 ø 1. The boundary condition atz ­ 1 is given by

u2s1d ­ R4u1s1d . (16)

From this condition, we can determine c1 to be

c1 ­p

R4 exps2ay2d . (17)

Substituting Eq. (17) into Eq. (15), we obtain

tan B ­p

R4 expfasz 2 1dy2g . (18)

Ding et al. Vol. 12, No. 9 /September 1995 /J. Opt. Soc. Am. B 1589

Substituting Eq. (18) into Eqs. (13), we obtain

u1 ­a

1 1 R4 expfasz 2 1dg,

u2 ­aR4 expfasz 2 1dg

1 1 R4 expfasz 2 1dg. (19)

If a ,, 1, u1 ­ ays1 1 R4d and u2 ­ aR4ys1 1 R4d, i.e., bothu1 and u2 are nondepleted at the low pump power density.If R1 ­ R2 ø 1, the total output power density is given by

P3

Psø 2

Z 1

0u1u2dz ­

2af1 2 exps2adgR4

s1 1 R4df1 1 R4 exps2adg. (20)

The conversion efficiency is determined to be

h ­P3

Psu0­

2af1 2 exps2adgR4

s1 1 R4df1 1 R4 exps2adgu0

. (21)

3. SATURATION OF CONVERSIONEFFICIENCYFirst let us consider a configuration in which there is amirror at z ­ 1, no mirror at z ­ 0, and an input at z ­ 0,i.e.,

u1s0d ­ u0 . (22)

From Eqs. (19) we obtain

u0 ­a

1 1 R4 exps2ad. (23)

Equation (21) can be reduced to

h ­2R4

1 1 R4f1 2 exps2adg . (24)

If the pump power density is low, i.e., u0 ,, 1, followingEq. (23), we obtain a ,, 1. Therefore we can simplifyEq. (23) to

u0 ø ays1 1 R4d (25)

Eliminating a in Eq. (24) by approximation (25), we ob-tain the conversion efficiency:

h ø 2R4u0 ­ 2R4P0yPs (26)

i.e., it is proportional to the pump power density. Forexample, if u0 ­ P0yPs ø 0.05 and R4 ø 99%, then h ø9.9%. This corresponds to the level of the pump powerdensities considered in Refs. 3, 4, 6, 8, and 9. On theother hand, if the pump power density is high, i.e., u0 ..

1, we obtain a .. 1 by following Eq. (23). Therefore, wecan simplify Eq. (23) to

u0 ø a (27)

Substituting approximation (27) into Eq. (21), we obtainthe conversion efficiency:

h ø2R4

1 1 R4f1 2 s1 1 R4dexps2u0dg ø

2R4

1 1 R4

. (28)

This corresponds to the saturation of the conversion effi-ciency. If R4 ø 99%, then h ø 99.5%. From Eqs. (23)

and (24) we plot the conversion efficiency as a function ofu0 ­ P0yPs in Fig. 2. One can see from Fig. 2 that theconversion efficiency is saturated if P0 .. Ps. Becauseof the large enhancement that is due to the presence ofthe vertical cavity, the saturation power density is greatlyreduced. Therefore the saturation of the conversion ef-ficiency can occur at a relatively low input power den-sity. The reason for the saturation’s occurring is that asthe input power density increases above the saturationpower density, the input pump power density, as wellas the reflected power density, decreases as the wavespropagate parallel to the z axis, i.e., the fundamentalbeams are depleted along their propagation directions (seeFig. 3). Because of such depletions, the power density ofthe reflected wave at z ­ 0 is close to 0. Therefore mostof the fundamental power is converted into SH power.

Fig. 2. Conversion efficiency versus the normalized pump powerdensity without the horizontal cavity.

Fig. 3. Normalized input and reflected power densities (P1yPsand P2yPs) versus the propagation distance zyL: line 1,P0yPs ø 0.05; curve 2, P0yPs ø 1; curve 3, P0yPs ø 5; inset;the cavity structure.

1590 J. Opt. Soc. Am. B/Vol. 12, No. 9 /September 1995 Ding et al.

Above the saturation power density, the conversion effi-ciency approaches 2R4ys1 1 R4d [see approximation (28)].Based on approximation (28), for P0 ­ 5 Ps and R4 ø 99%,h ø 98.9%. When P0 ­ Ps su0 ­ 1d, h ø 72.2%.

We now estimate the saturation power density based onEq. (10) for several realistic structures. If l2v ­ 0.9 mm(when III–V asymmetric QW’s or multilayers are used),deffyd ø 10, x

s2d0 ø 10210 myV , R1,2 ­ 99.9%, L ­ 10 mm,

and Ps ø 830 mWymm. For l2v ­ 0.49 mm (when II–VIasymmetric QW’s or multilayers are used), R1,2 ­ 99.9%,L ­ 10 mm, and Ps ø 110 mWymm. It is worth notingthat this structure can be used to generate blue light effi-ciently at a relatively low pump power. We can see that,although all these input power densities can be achievedby cw lasers, we may need to use pulsed lasers as ourpump sources to avoid the damage of the materials causedby the intense laser beams. Another important exampleis to frequency double 1.32-mm laser light: we can im-prove the conversion efficiency by replacing a single thinlayer of Al0.6Ga0.4As with alternating Al0.5Ga0.5AsyAlAslayers. Finally, the conversion of the light at 1.5 mm tothat at 0.75 mm is also important. We have listed all theoptimized structure parameters required for four differentinput wavelengths in Table 1.

Consider a GaAsyAlAs ACQW domain structure. Asshown in Ref. 15, at l ø 20 mm, x

s2d0 ø 30 3 1029 myV .

Each domain consists of multiple ACQW’s, each of whichis composed of two GaAs QW’s with thicknesses of 60 and20 A and coupled by a 30-A-thick AlAs barrier. The po-larities of the adjacent domains are opposite each other,resulting in modulated second-order susceptibilities basedon the intersubband energy levels. We can use alternat-ing AlGaAsyAlAs layers as quarter-wave stacks for thevertical cavity. We use the reflectivities of the quarter-wave stacks of 99%. For nv ø 3.2 and n2v ø 3.3, Ps ø10 mWymm.

In the second configuration a horizontal cavity formedby two mirrors as a result of the coatings at the endsz ­ 0, 1 is used. To estimate the order of magnitude ofthe conversion efficiency, we use the following boundarycondition without considering the Fabry–Perot effect indetail:

u0 ­ u1s0d 2 R3u2s0d . (29)

From Eqs. (19), we obtain

u0 ­af1 2 R3R4 exps2adg

1 1 R4 exps2ad. (30)

Substituting Eq. (30) into Eq. (21), we obtain

h ­2R4

1 1 R4f1 2 exps2adgf1 2 R3R4 exps2adg . (31)

If the pump power density is high, i.e., P0 ­ u0Ps ..

Ps su0 .. 1d, we can simplify Eq. (30) to

u0 ø a (32)

The conversion efficiency is

h ø2R4

1 1 R4f1 2 s1 1 R4dexps2u0dg ø

2R4

1 1 R4

. (33)

This is exactly the same as approximation (28) for the con-figuration without the horizontal cavity. On the otherhand, if the pump power density is low, i.e., u0 ,, s1 2

R3R4d2, we can simplify Eq. (30) to

u0 ø as1 2 R3R4dys1 1 R4d . (34)

The conversion efficiency is

h ø2R4

s1 2 R3R4d2 u0 . (35)

Comparing approximation (35) with approximation (26),we can see that for the same input power density, we canincrease the SH power density by a factor of 1ys1 2 R3R4d2

by using the horizontal cavity. For example, if h ­ 9.9%and R3 ø R4 ø 99%, then u0 ø 2 3 1025, i.e., more than4 orders of magnitude lower than that without the hor-izontal cavity. Another example is that, without thehorizontal cavity, h ø 72% if P0yPs ø 1. With the cav-ity, R3 ø R4 ø 99% and h ø 72% if P0yPs ø 1.8 3 1023.Therefore the saturation power density is now Ps

0 ø 1.8 3

1023Ps. If l2v ø 0.9 mm and L ø 200 mm, the saturationpower density is Ps

0 ø 75 mWymm. If l2v ø 0.49 mm,the saturation power density is Ps

0 ø 10 mWymm. Ifl2v ø 0.65 mm, the saturation power density is Ps

0 ø345 mWymm. If l2v ø 0.75 mm, the saturation powerdensity is Ps

0 ø 435 mWymm (see Table 1). For theGaAsyAlAs ACQW domain structure, Ps

0 ø 0.9 mWymm.In Fig. 4 we plot the conversion efficiency as a functionof the input power density based on Eqs. (30) and (31).For comparison, we have also plotted the conversion effi-ciency versus the input power density without the hori-zontal cavity in Fig. 4. Therefore saturation occurs atmuch lower input power densities (see Fig. 4). Satura-tion of the conversion efficiency is due to the depletion

Table 1. Structure Parameters for Frequency Doubling the Laser Light in the Range of 0.98–1.8 mm

Parameter Value or Structure

Fundamental wavelength s mmd 0.98 1.3 1.5 1.8SH wavelength s mmd 0.49 0.65 0.75 0.9Doubling materials 53 ZnSeyZnS 53 Al0.5Ga0.5AsyAlAs 43 Al0.4Ga0.6AsyAlAs 43 GaAsyAl0.8Ga0.2AsQuarter-wave stacks ZnSeyZnS Al0.6Ga0.4AsyAlAs Al0.5Ga0.5AsyAlAs GaAsyAlAsReflectivity (%) 99.9 99 99 99.9Confinement layers ZnS Al0.6Ga0.4AsyAlAs Al0.5Ga0.5AsyAlAs GaAsyAlAsPs sWymmda 0.11 3.9 4.9 0.83

Ps0 smWymmdb 10 345 435 75

aPs, saturation power density without the horizontal cavity.bPs

0, saturation power density with the horizontal cavity.

Ding et al. Vol. 12, No. 9 /September 1995 /J. Opt. Soc. Am. B 1591

Fig. 4. Comparison between the conversion efficiencies versusthe normalized pump power densities with (solid curve, top) andwithout (solid line, bottom) the horizontal cavity.

of the pump power density along the propagation direc-tion (see Fig. 5). One can see that, even for P0yPs ,, 1,there is a strong depletion. However, without the hori-zontal cavity, if P0yPs ,, 1, the depletion is negligible (seeFig. 3). This is why the horizontal cavity can be used toimprove the conversion efficiency.

The enhancement factor for the surface-emittinggeometry that is due to the presence of the horizontal cav-ity is much larger than that for the collinear propagationconfiguration based on conventional second-order nonlin-ear materials. The reason is that, in the collinear con-figuration, if the depletion is negligible, the fundamentalpower density in the cavity is enhanced by ,1ys1 2 R3,4d2.However, the SH output power density is smaller thanthat in the cavity by a factor of ,1ys1 2 R3,4d. Thusthe conversion efficiency is increased by a factor of only,1ys1 2 R3,4d. This also illustrates one additional advan-tage of the surface-emitting geometry. In our geome-try, we can incorporate two cavities into the structureoptimized for the highest reflectivities possible at thefundamental and SH frequencies separately. Thus thesaturation power density is lower. However, in the col-linear propagation configuration, it is difficult to fabricatea single cavity with high reflectivities at both fundamen-tal and SH frequencies.

4. DISCUSSIONIt is worth noting that, to avoid the band-to-band ab-sorption of the SH light completely, the pump frequencyshould be fixed below half of the effective band gapof the materials. Thus GaAsyAlGaAs asymmetric QW’scan be ideally used for frequency doubling a laser beamfor the input wavelength in the 1.5–2.0-mm range. Forthe 1.3–1.5-mm range, AlxGa12xAsyAly Ga12yAs (x fi y,x fi 0, and y fi 0) multilayers seem to be the best mate-rial for SHG. For the 0.8–1.3-mm domain, ZnSeyZnS orZnSeyZnMnSe asymmetric QW’s or multilayers are idealmaterials for SHG.

There are other nonlinear effects that may competewith the SHG. Consider, for example, the GaAsyAlGaAsmultilayer structure. Even though the laser frequencyis below half of the effective band-gap frequency, the two-photon absorption can be important. Self-phase modu-lation and other nonlinear effects based on two-photonabsorption16 can be also important. All these effects mayreduce the conversion efficiency for SHG. We believe,however, that we can choose the proper experimental con-dition and design the structures in such a way that theSHG becomes dominant. We will consider these compet-ing effects in detail elsewhere.

Let us look into how the quasi-phase-matching condi-tion affects the conversion efficiency of SHG. To achievequasi-phase matching, we can grow a waveguide consist-ing of many units of asymmetric QW’s, each of which iscomposed of two domains of multiple asymmetric QW’sthat correspond to left and right symmetries (middleplot of Fig. 1); the thickness of each unit is l2vyn2v,where l2v is the SH wavelength in vacuum and n2v

is the refractive index at SH frequency. As a result,the second-order susceptibility x s2d changes sign fromone domain to the adjacent one5 (see Fig. 1). We canuse ACQW’s, stepped QW’s, or graded band-gap QW’s toachieve a large magnitude of the second-order susceptibil-ity. We can use multilayers that consist of alternatingAlxGa12xAsyAlyGa12yAs layers (the thickness of eachAlGaAs layer is l2vy2n2v), resulting in the spatially mod-ulated second-order susceptibility. As a result, for bothstructures there is a sinusoidal component of x s2d thatoscillates at the spatial period l2vyn2v. This oscillat-ing component cancels the spatial oscillation of the SHwave. Thus the quasi-phase matching along the x axisis established. This is reflected by the magnitude ofdeff. Because deff, which is given by Eq. (5), dependson the input wavelength and the spatial modulation ofthe second-order susceptibility, deff reaches a minimum

Fig. 5. Normalized input and reflected power densities (P1yPsand P2yPs) versus the propagation distance szyLd: curve 1,P0yPs ø 1.1 3 1024; curve 2, P0yPs ø 1.7 3 1023, curve 3,P0yPs ø 0.02; inset, the cavity structure.

1592 J. Opt. Soc. Am. B/Vol. 12, No. 9 /September 1995 Ding et al.

at this quasi-phase-matching condition. We assumethat the oscillating waves are strongly confined by thewaveguide (i.e., the amplitudes of the oscillating wavesoutside the waveguide are negligible). For the quasi-phase-matched structures, x s2dsxd can be expanded inFourier series. The first nonzero oscillating componentof x s2d can be expressed as x

s2d0 cosfs2pn2vyl2vdxg. For

the SH standing wave, c2v ­ c2v0 cosfs2pn2vyl2vdxg. In

this case, from Eq. (5) we obtain

dspmdeff ø 2d . (36)

On the other hand, if there is no quasi-phase matching,e.g., the nonlinear material consists of only a single thicklayer f x s2dsxd ­ x

s2d0 g, we obtain

dsnpmdeff ø

"2pn2vdyl2v

sins2pn2vdyl2vd

# 2d2

. (37)

Obviously the minimal value of dsnpmdeff occurs when

sins2pn2vdyl2vd ­ 1, i.e., 2pn2vdyl2v ­ s2j 1 1dpy2,where j is an integer. In this case, the ratio betweend

spmdeff and d

snpmdeff is

dsnpmdeff

dspmdeff

øp2n2v

2d2

l2v2

. (38)

For l2v ø 0.9 mm, d ø 0.96 mm, n2v ø 3.5, dsnpmdeff yd

spmdeff ø

138; i.e., the effective waveguide thickness for the non-quasi-phase-matched structure is 2 orders of magnitudelarger than that for the quasi-phase-matched one. Thisillustrates the importance of the quasi-phase match-ing achieved in multilayers or asymmetric QW domainstructures. Because the saturation power density is pro-portional to the effective waveguide thickness based onEq. (10), we conclude that, because of the quasi-phasematching, we can reduce the saturation power densityby at least 2 orders of magnitude. For example, in thepresence of the horizontal cavity and to generate lightat l2v ø 0.9 mm, the saturation power density increasesfrom 75 mWymm when the alternating GaAsyAl0.8Ga0.2Aslayers are used to 10 Wymm when a single layer of GaAsis used.

Because the saturation power density is independentof the width of the waveguide [see Eq. (10)], the satu-ration power is proportional to the waveguide width.The smaller the width, the lower the saturation power.Therefore we can fabricate each individual frequency dou-bler that has the smallest width possible within presenttechnology. Therefore it is feasible to integrate a largenumber of the frequency doublers into an array on onewafer by optical lithography and chemical etching tech-niques (see Fig. 6).

One of the advantages of the surface-emitting geome-try is that we can generate efficient SH radiation for alarge input wavelength range or for a large output wave-length range. We assume that the pump power densityis the same as the saturation power density sP0 ­ Psd atthe center SH wavelength l0, where l0yn0 is the periodof the spatially modulated component of the second-ordersusceptibility of the multilayers or asymmetric QW do-main structures (n0 is the reflective index at l0). At the

generated SH wavelength sl2vd, the saturation power den-sity is Pssl2vd ø 20Pssl0d. The corresponding conversionefficiency decreases from 72% to 10%. We assume thatthis decrease is primarily due to the increase of the effec-tive thickness when the SH wavelength is tuned from l0

to l2v . Because c1,2sxd vary as functions of x much moreslowly than coss2pn2vl2v

21xd, we can assume that c1,2sxdare more or less constant across the waveguide. It canbe shown that in such a case, based on Eq. (5),

deff sl2vd ­ deff sl0d

(2psl2v

21 2 l021ddn2v

sinf2psl2v21 2 l0

21ddn2vg

) 2

. (39)

Therefore 2psl2v21 2 l0

21ddn2v ø 62.54. It turnsout that for n2v ø 3.5, d ø 1 mm, and l0 ø 0.9 mm,Dl ø 1890 A. If the input light is monochromatic, wecan then generate the SH radiation within such a largeoutput wavelength range. On the other hand, if thebandwidth of the SH radiation is narrow, we can generateefficient SH radiation for the input wavelength within thebandwidth of 3780 A. Such a large usable bandwidthis possible only for surface-emitting geometry. This isbecause the waveguide thickness is many orders of mag-nitude shorter than the interaction length for the parallelpropagation configuration. It is important to notethough that, because of the Fabry–Perot effect on the SHwave, we can efficiently generate SH radiation only fordiscrete wavelengths within the usable bandwidth. Theother factor that limits the usable bandwidth is the de-pendence of the reflectivities of the quarter-wave stackson the wavelength. For the quarter-wave stacks withextremely high reflectivities, the reflectivities decreaserapidly as the wavelength moves away from the centerwavelength. However, a large usable bandwidth limitedby the quasi-phase-matching condition discussed abovecan still be achieved for quarter-wave stacks with rela-tively lower reflectivities. When designing the structurefor the specific applications, we need to take into consid-eration all the factors above.

Fig. 6. Side-view structure of the one-dimensional array of theoptical frequency doublers. The plane is (a) parallel to and (b)normal to the propagation direction of the input beam.

Ding et al. Vol. 12, No. 9 /September 1995 /J. Opt. Soc. Am. B 1593

Because we use the horizontal and the vertical cavitiesto increase the conversion efficiency, our frequency dou-blers may be sensitive to temperature change. The highreflectivities of the vertical mirrors may be sensitive tothe change. To estimate the temperature sensitivity ofthe reflectivities (and therefore the conversion efficiency),we consider the ZnSeyZnS multilayer structure. The re-flectivities based on the ZnSeyZnS quarter-wave stackscan be expressed as R1,2 ø 1 2 4fn2vsZnSdyn2vsZnSedg2N ,where N .. 1 is the number of pairs of the ZnSeyZnSlayers. From Eq. (10), we obtain

dPs

dT­ 2NPs

"1

n2vsZnSddn2vsZnSd

dT

21

n2vsZnSeddn2vsZnSed

dT

#, (40)

where T is the temperature of the material. Followingthe Moss rule17 and from Ref. 18, we obtain

dPs

dT­ 21.9 3 1025NPs . (41)

Assuming that N ­ 25, we obtain Ps21dPsydT ø

24.8 3 1024 K21. Consider the nondepleted SHG. Thecorresponding conversion efficiency is determined byapproximation (26). Using Eq. (41), we can determinethat

Dh ø 4.8 3 1024hDT . (42)

Assuming that h ø 9.9% and DT ø 610 K, we obtainDh ø 60.047%.

Most importantly, the length of the horizontal cavity ismore sensitive to the temperature change. If aT is thecoefficient of the linear thermal expansion for ZnSe, wecan estimate the change of the waveguide length to beDL ø aT LDT . For L ø 200 mm, aT ø 7.8 3 1026 K21,and DT ø 610 K, we obtain DL ø 156 A. The waveguidelength must be Lm ø sly2nvdm (where m is an integer) toachieve the resonance condition in the horizontal cavity.If the device is relatively insensitive to the temperaturechange, the following condition must be satisfied:

jDLj ,, Lm11 2 Lm ­ ly2nv , (43)

i.e., the change of the waveguide length that is due tothe temperature change should not destroy the resonancecondition for the fundamental wavelength in the hori-zontal cavity. For the ZnSeyZnS structure, we obtainly2nv ø 1960 A. Therefore the condition determined byEq. (43) is satisfied. We conclude that the frequency dou-bler is relatively insensitive to the temperature change ifits structure is properly designed. Now we estimate thewavelength sensitivity. With the resonance conditionabove, we can estimate the difference between the adja-cent wavelengths to be Dl ­ l2y2nvL. For the ZnSeyZnSstructure, Dl ø 9.6 A. Therefore the linewidth of thepump laser should be much narrower than Dl. If thelinewidth is larger than Dl, the SH radiation consists ofmany sharp peaks in the wavelength domain.

5. CONCLUSIONSIn conclusion, we have shown that we can efficientlyachieve surface-emitting SHG in vertical and horizon-

tal cavities. A fundamental beam is coupled into thewaveguide consisting of II–VI or III–V semiconductormultilayers or asymmetric QW domains. The SH ra-diation is generated and propagates along the growthdirection of the epitaxial layers (which is normal to thepropagation direction of the fundamental beam). Thequasi-phase matching is achieved by the modulation ofsecond-order susceptibility along the growth direction inthese structures. If these structures are properly de-signed, these frequency doublers together can cover the0.8–2-mm spectrum. We have investigated the satu-ration of the conversion efficiency. If the pump powerdensity is sufficiently large, the conversion efficiency issaturated. We have introduced a characteristic pumppower density, the saturation power density, whichis a measure of the degree of saturation. At such apump power density, 72% conversion efficiency can beachieved. The saturation power density is between 0.9and 435 mWymm for the optimized structures consid-ered in this paper. In addition, the proposed frequencydoublers are, in principle, broadband. These frequencydoublers can be integrated into a one-dimensional array.

Surface-emitting SHG has potential applications in theefficient generation of blue light from laser diodes outsidethe laser diode cavity or by intracavity,3 and monolithicminiature devices, such as correlators,8 spectrometers,8

and phase detectors.9 Furthermore, it can be used toachieve optical power limiting,11 optical frequency shift-ing, optical phase conjugation,11 directional coupling,13

and self-phase modulation12 based on cascaded second-order nonlinearities similar to those of Ref. 10.

ACKNOWLEDGMENTSWe are thankful to A. G. Cui for the graphics help. Thiswork is supported by the U.S. Air Force Office of ScientificResearch and the National Science Foundation.

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