Frequency doubling and sum frequency generation in semiconductor optical waveguide devices

Pr0g. Quanr. E/em. 1996, Vol. 20, No. 3 pp. 181-218 Copyright 0 1996 Elsevier Science Ltd

Pergamon

0079-6727(95)00006-2

Printed in Great Britain. All rights reserved. 0079-6127196 $32.00

FREQUENCY DOUBLING AND SUM FREQUENCY GENERATION IN SEMICONDUCTOR OPTICAL WAVEGUIDE DEVICES K. A. SHORE*, X. CHENt

and P. BLOOD?

*University of Wales, Bangor, School of Electronic Engineering and Computer Systems, Bangor LL57 IUT, Wales, U.K. and tuniversity of Wales, Cardiff, Department of Physics and Astronomy, PO Box 913, Cardiff CF2 3YB, Wales, U.K.

CONTENTS 1. Introduction 1.1. Background 1.2. Frequency doubling in semiconductors 1.3. Waveguide frequency doublers 1.4. Outline of review 2. Basic physical and mathematical formalism 2.1. Nonlinear polarisation and second harmonic generation 2.2. Counter propagation 2.3. Second order polarisation and waveguide geometry 2.4. Periodic layers--quasi phase matching 2.5. Cavity effects 2.6. Miller’s rule 2.7. Nonlinear refractive index 3. Interband transitions in Quantum Well materials 3.1. Second order susceptibilities 3.2. Calculation of x in asymmetric Quantum Well structures 3.3. Biased Quantum Well structures 4. Temporal response 4.1. Cross-phase modulated second harmonic generation 4.2. Sum frequency generation 5. Intersubband processes 5.1. Asymmetric Quantum Wells 5.2. Theoretical basis 5.3. Optimisation of Intersubband SHG 5.4. Conversion efficiency 6. Device applications 6.1. Optical correlator 6.2. Wavelength demultiplexing and parametric spectrometer 6.3. Sources 6.4. Diagnostics 7. Future prospects 7.1. Material systems 7.2. Devices and structures 8. Conclusion Acknowledgements References

181 181 182 183 184 185 185 185 186 189 189 190 192 193 193 193 197 197 197 198 201 201 201 204 205 206 206 207 207 210 211 211 213 215 215 215

1. INTRODUCTION 1.1. Background In his Bakerian lecture of 5 February 1857 Michael Faraday wrote: “At one time I hoped that I had altered one coloured ray into another by means of gold, . . . and though I have not confirmed that result as yet, still those I have obtained seem to me to present a useful experimental entrance into certain physical investigations respecting the nature and action of a ray of light. I do not pretend that they are of great value.. but they may save much trouble to any experimentalist inclined to pursue and extend this line of investigation”.(s2) Here, 181

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et al.

suggests Sir John Meurig Thomas(53’ in his introduction to the bicentennial edition of “Experimental Researches in Chemistry and Physics”, is an indication that Faraday was seeking to devise a frequency doubler. In the first article published in Progress in Quantum Electronics, Yariv and Pearson made reference to work reported by Faraday in 183 1 concerning parametric processes-albeit in mechanical systems.“@”As also noted by Yariv and Pearson, it was in 1961, following the invention of the laser, that Franken et al. reported the observation of second harmonic generation where the red light of a ruby laser to ultraviolet thus initiating the field of nonlinear optics.(55’It is apparent that second harmonic generation has a quite considerable history. The theoretical and experimental approaches to second harmonic generation, sum frequency generation and other nonlinear optical phenomena have been extensively treated in a number of textbooks and monographs(26, 69,‘19,‘59)and thus no attempt will be made here to provide an exhaustive overview of this burgeoning field. It has been argued that one major requirement in the field is a consistent notation, a treatment of this non-trivial issue has been given by Roberts.““) A significant development relevant to progress in this general area of activity has been the emergence of the semiconductor laser or laser diode as a highly versatile optical source with widespread applications. Considerable attention has been given to expanding the spectral range which can be accessed using semiconductor lasers. In particular a number of information processing applications have driven a requirement for efficient semiconductor laser-based sources of visible light. Progress is being made in the direct generation of visible light using wide band-gap semiconductor lasers but further research and development will be required to deliver devices with operating characteristics suitable for commercial applications. An alternative approach to visible light generation is the use of semiconductor laser sources in conjunction with a suitable nonlinear optical component for effecting frequency doubling or else sum frequency generation. In this context great success has been achieved using hybrid systems which combine the reliability of near-infra-red laser diodes with strong nonlinearities in glass materials such as lithium niobate. On the other hand, it is well known that semiconductor laser materials themselves possess strong nonlinear properties suitable for second harmonic generation. (‘03)Looking towards ease of device fabrication it would be generally preferable to utilise a single material system and device technology for the implementation of such laser-frequency doubler sources. The opportunity for monolithic integration of the two components which follows from the use of just one material system would also be expected to offer advantages in terms of practical utilisation and hence commercial exploitation of such devices. With these benefits in mind it is the aim of the present article to provide a detailed assessment of techniques and processes for effecting second harmonic and sum-frequency generation using semiconductor lasers as source of the fundamental frequency and semiconductor waveguide devices as the required frequency doubling elements. The context for this work is, in part, defined by early work on parametric oscillator processes in semiconductor waveguides.‘25,‘29) 1.2. Frequency doubling in semiconductors The nonlinear properties of the properties of semiconductor laser materials were revealed in experiments which were performed at the start of the era of nonlinear optics and soon after the invention of the semiconductor laser. Second harmonic generation via laser irradiation of the surface of a GaAs crystal was reported in 1963(49)andthe effect was observed in a number of other semiconductors. (31,Lo3) At the same time, second harmonic generation was observed in semiconductor lasers.(9s59.94,124)H owever, the optical power contained in the observed second harmonics was severely attenuated due to strong inter-band absorption within the semiconductor. The re-absorption of the generated second harmonic effectively

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limited the nonlinear interaction to a small region near the mirror facets of the laser.C’03) It is apparent therefore that internal second harmonic generation in such semiconductor laser geometries is not an efficient process. The early Fabry-Perot semiconductor lasers used in those experiments were highly multi-moded and hence the observed second harmonic output powers were obtained as a combination of second harmonic generation from the individual modes and via sum frequency generation using pairs of modes. Moreover the lasers were broad contact devices which would generally be expected exhibit multi-transverse mode or filamentary lasing action. As a consequence of these factors it is not possible to make an accurate deduction of the second order nonlinearities in such semiconductor lasers on the basis of the observed second-harmonic output powers. This issue has been addressed in more recent work by Furuse and Sakuma which has taken advantage of the developments in semiconductor laser design and operating characteristics. w Here use was made of devices with a controlled transverse and longitudinal mode structure to observe and calibrate second harmonic and sum frequency generation in an InGaAsP laser diode with an operating wavelength of 1.3 pm. Also Ogasawara et al. (‘O’)have observed second harmonic generation in a transverse-mode stabilised AlGaAs laser. It was shown by Furuse and Sakuma that second harmonic genration occurred when the laser was constrained to operate in a single longitudinal mode whilst sum frequency components became significant as the laser was taken into a regime of multi-longitudinal mode operation. Moreover the experiments demonstrated that the efficiency of the sum frequency generation was a factor of four times that of the second harmonic process. This observation is to be expected on the basis of basic theory of the nonlinear process(‘25)but is significant in revealing a feature of the spectral properties of semiconductor lasers namely the simultaneous oscillation of two or more longitudinal modes. Other observations had suggested that multimode laser action did not involve the co-existence of different longitudinal modes. Furuse and Sakuma@) suggested that spatial inhomogeneities of the population inversion, enhanced in InGaAsP lasers compared with GaAs lasers due to a longer operating wavelength and smaller diffusion length, gave rise to the coupling between co-existing longitudinal modes. Such an observation is of interest in connection with so-called FM locking operation of semiconductor lasers which again depends upon nonlinear couplings between co-existing longitudinal modes. The work of Furuse and Sakuma also indicated that the second order nonlinear susceptibility of the InGaAsP material was an order of magnitude greater than that of GaAs. However, despite the significant improvements in conversion efficiency obtained by the using improved semiconductor laser designs, the re-absorption of the shorter wavelength radiation remained as a basic limitation on the frequency conversion efficiency capabilities of the devices. The interaction length for the second harmonic and sum frequency generation is essentially determined by the absorption coefficient for the higher frequency component. Since that absorption coefficient is of order 10’ cm-’ it is clear that severe limitations are imposed upon achievable output powers of the higher frequency components. In order to pursue the objective of achieving all semiconductor source/frequency doublers alternative approaches must be sought. In the next section one such approach is presented. 1.3. Waveguide frequency doublers As indicated above, the main limitation on achieving significant second harmonic and sum frequency output power is the severe attenuation of the generated short wavelength radiation which limits the interaction length of the nonlinear process. A significant proposal for the resolution of this difficulty was made by Normandin, Stegeman and co-workers who showed theoretically that the use of counter-propagating guided waves could offer orders of magnitude improvement in second harmonic generation. W) Their work had been motivated by experiments on nonlinear mixing of guided waves in lithium biobate waveguides but their

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calculations also treated the case of GaAs waveguides. From the viewpoint of semiconductor waveguides the critical aspect of the proposal can be appreciated from Fig. 1. There it is shown that the nonlinear mixing of counter propagating guided waves gives rise to a radiation mode second harmonic signal. The second harmonic is thus extracted from the surface of the waveguide rather than the end-face of the structure and so the second harmonic signal does not need to propagate along the whole length of the waveguide. As such, the propagation distance of the second harmonic in the semiconductor is determined by the layer structure of the semiconductor waveguide. Typically such a distance would be of order l-10 pm rather than the 100-1000 pm propagation distance which would be typical for axial propagation in semiconductor optical waveguide devices such as semiconductor lasers. It is immediately apparent that this approach offers clear advantages due to the reduced re-absorption of the second harmonic which would thus be expected to occur. Progress in the experimental demonstration of semiconductor frequency doublers using counter propagating guided waves is detailed in Sections 2.2 to 2.4. The experimental verification of the opportunities for enhanced second harmonic generation via surface emission has motivated further attempts to enhance second harmonic generation by the use of appropriate resonator structures which, in general, may be utilised in both semiconductor and other nonlinear materials. Proposals have included the use of distributed output coupling using non-resonant first-order gratings;(‘48) vertical resonator These aspects are treated in greater detail structures’g”92’ and multilayer structures. (4’,43,44,96) below. 1.4. Outline of review Having established the context for the present review, a brief explanation is now given of the structure of the remainder of the paper. Section 2 aims to establish the basic physical and mathematical basis of present day efforts aimed at developing efficient semiconductor second harmonic generators. In Section 3 attention is given to processes which occur when the input wave has an optical energy close to that of the semiconductor bandgap. Specific features of the second harmonic generation process when the fundamental wave is in the form of a pulse are treated in Section 4. The design and properties of quantum well structures which permit

3 Pibre

I

Radiated bmoaic light I I AlGas buffer

Fig. 1. (a) Illustration of the nonlinear combination of counter-propagating fundamental beams in the guide and the surface emission of the sum frequency beam. (b) The conservation of the momentum along the guide which determines the direction of the radiation of the signal beam (After Ref. 142).

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second harmonic generation in the mid and far infra-red are examined in Section 5. Device applications of semiconductor frequency doublers are discussed in Section 6 whilst Section 7 seeks to identify areas where future developments can be expected. In the latter attention is given to both materials and devices. A short conclusion is given in Section 8. 2. BASIC

PHYSICAL

AND

MATHEMATICAL

FORMALISM

2.1. Nonlinear polarisation and second harmonic generation The first observations of optical second harmonic generation (SHG) were made by Franken et al. in 1961 .(55) In their experiment, a ruby laser beam (A = 0.94 pm) was focused on a quartz

crystal, the forward propagating beams passed through a prism and were detected on a photographic plate. A weak signal was detected at a wavelength of 0.47 pm-the second harmonic of the ruby laser fundamental beam. The demonstration of optical harmonic generation marked the birth of the nonlinear optics. The generation of harmonic wave results in nonlinear polarisation of optical medium. In general, the polarisation, P, is a complicated nonlinear function of electric field E. When E is sufficiently weak, the polarisation can be written(“9) as P(k, w) = x’(k, w) * E(k, o) + x*(k = ki + k,, w = wi + oj): E(ki, o,)E(k,, 0,) +

X3(/I

=

ki +

k, + k/yo = 0, + W, + o,):E(ki,

Oi)E(kjy mj)E(k,y 01) + . . .

(2.1)

where x’ is a linear susceptibility, x” is the nth order nonlinear susceptibility. The linear and nonlinear susceptibilities characterise the optical properties of the medium. Physically, x” is determined by the microscopic structure of the medium and can be properly evaluated only with a full quantum mechanical calculation. The second order susceptibility x2 in general defines sum frequency generation of two electromagnetic waves of different frequencies and leads to second harmonic generation when the two waves have the same frequency. As will be seen in Sections 3 and 5, the x2 can be greatly enhanced in semiconductor microstructures such as asymmetric quantum wells. By inserting the second order nonlinear polarisation x*EE into Maxwell’s equation, under the plane wave approximation, the harmonic conversion efficiency can be defined as(“9~‘59) 3’2o*d?P sin’(Ak1/2) po n' (Ak1/2)* A

(2.2)

where d is called the nonlinear optical coefficient, which is related to second susceptibility by x2 = 2d. This equation indicates a few key strategies which should be pursued in order to obtain efficient second harmonic generation: (1) seeking materials and structures having large nonlinear optical coefficients, (2) effecting an increase in the interaction length, 1, in the medium, (3) increasing the fundamental beam intensity, p, and (4) realising phase matching (k = 0) between the fundamental beam and the second harmonic. 2.2. Counter propagation

As indicated in the introduction second harmonic generation in waveguides has been studied intensively in recent years. The combination of a SHG waveguide with a laser diode source would make it possible to obtain efficient, compact and coherent blue light source for various applications. Most semiconductor materials have large nonlinear optical coefficient, for example, the nonlinear optical coefficient, d, of GaAs is about 200 times larger than the widely used KDP crystal. Furthermore, semiconductor materials susceptibilities may be dramatically increased via the use of synthetic semiconductor quantum well structures.‘~0’

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However, in many cases, the harmonic propagation loss is large if the photon energy exceeds the semiconductor bandgap. Secondly, semiconductors are optically isotropic and thus the birefringence effect obtained in anisotropic crystals cannot be used to obtain the phase-matching which is needed to achieve efficient harmonic conversion. In order to benefit from the large nonlinearity of semiconductor materials, it is essential to overcome both harmonic loss and refractive index dispersion. As can be understood from equation (2.2) a large interaction path is required to increase the conversion efficiency, but the optical path of harmonic wave in waveguide should be minimised to reduce the harmonic propagation loss. One way to compromise the interaction length and the harmonic loss is to achieve distributed out-coupling of SHG as proposed by Weissman and Hardy.“48) This scheme is based on using a first-order refractive index grating to achieve surface emission of the generated harmonic. This approach is suitable when moderate levels of harmonic loss are encountered. However, if the harmonic wave propagation is very lossy, as in the case of GaAs guides in the vicinity of 1 = I pm, counter-propagation schemes, in particular, appear to be more promising for these kinds of materials, The physical principle of counter propagation was first analysed by Bloembergen et a1.@” It was shown that two counter-propagating beams incident on the interface between a linear and nonlinear material in a total internal reflection geometry generate a sheet of nonlinear polarisation which in turn radiates two plane harmonic waves perpendicularly out of the surface and into the substrate. Later on, Stegeman and Normandin’99’ predicted and subsequently demonstrated the nonlinear mixing of oppositely propagating guided waves. The resultant field was coupled to radiation modes emitted in a direction perpendicular to the waveguide surface in the case of equal frequency fundamentals. The first generation of second harmonic using counter-propagation in a thin GaAs-based optical waveguide was demonstrated by Vakhshoori et al. In their geometry, the oppositely propagating beam was from the reflection of the exit surface. The second harmonic generation using counter-propagating scheme in a semiconductor waveguide is shown in Fig. 2(a). 2.3. Second order polarisation

and waveguide geometry

The number of independent elements of the second order susceptibility tensor x2 depends upon the crystal symmetry. GaAs-based crystals have a point group symmetry of 43 and only one independent component of x2 exists. The second harmonic polarisation can be written as a matrix form

OOOd,, 0 0 0 0 0

0 0

0 d,, 0

0 0 d,,

I

(2.3)

where d,4 = x2/2 is the second order optical coefficient. The crystallographic axes in the (lOO),(OlO) and (001) directions are denoted as i, j, k. The magnitude and the direction of harmonic polarisation are very much dependent on the polarisation of the input fundamental beam. This is important for counter-propagating semiconductor waveguides where the geometry can be chosen by growing the waveguide on the substrate with different orientations. Therefore the harmonic polarisation is very much dependent on the orientations of the waveguide with respect to the substrate. The waveguide axes are defined here as x, y, z: the x axis is taken to be normal to substrate surface and it is in this direction that the harmonic light is emitted; the y axis is in the direction of the input fundamental field; the z axis is the direction of propagation of the guided fundamental.

Semiconductor optical waveguide devices

GZAZ AIGaAz

(b)

Radiating region

Multilryer waveguide

Fig. 2.

187

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Fig. 2. (a) Experimental geometry with two input fibres. The A1,,Gao ,As/AI,,G~,~As waveguide layers on the AI,,Ga,,.,As buffer are shown. (After Ref. 100) (b) Electrically pumped, integrated visible harmonic InGaAs laser. (After Ref. 100) (c) Schematic side view of a vertically resonant surface-emitting SHG device. The hatched areas are mirrors, and the backward-propagating pump wave, rE,,,is provided by an endface reflection. A circulating second harmonic, Ez,,,is built up inside the cavity. (After Ref. 91)

In the case of (111) substrate, the x axis in the (111) direction, the z axis is chosen to be in the
Re[&(x)e”‘“’ - fl--)+ &(x, t)P’

+‘“)I

(2.4)

By combining equations (2.3) and (2.4), the harmonic polarisation is created in x-y plane of the waveguide axes. The polarisation, which contributes the surface emitted harmonic, in (111) geometry is given by (2.5) A similar analysis applies to (110) ojented substrate, the polarisation contributed to surface emitted harmonic is times 42 larger compared to a (111) substrate since the x component of the polarisation is smaller. Because radiated intensity is quadratically proportional to the polarisation, the conversion efficiency in (110) substrate is twice larger than that in (111). In a (100) substrate orientation, the TE or TM mode polarisation coincides with the crystal axes and mixed TE and TM mode interaction is required to generate the harmonic polarisation. The polarisation contributed to surface emitted harmonic becomes P,.(x, t) = - 4~,d,,E,,(x)E,(x)e’2’“‘cos(A/?z)

(2.6)

where Afi = /ITE- BTMis the difference between the propagation constants of the TE and TM modes. This equation indicates that the emitted harmonic is not exactly along the surface normal and the radiated intensity has a interference pattern along the fundamental propagating z direction. This effect has been observed in a GaAs ridge waveguide.(‘42) The dependence of harmonic polarisation on substrate orientation is important not only for the enhancement harmonic generation intensity but also for the particular device application. The (100) grown substrate is unsuitable for an integrated semiconductor frequency doubler since only TE or TM lasing occurs in normal laser diodes Efforts have been made to realise an integrated doubler on (11 l)(“*) and (211) structure. Such a structure is shown in Fig. 2(b). The (211) structure has the same polarisation efficiency as (110) but unlike (1 lo), the cleaved facets are available.~‘s”‘sz~

Semiconductor

2.4. Periodic layers-quasi

optical

waveguide

devices

189

phase matching

In a co-linear optically pumped SHG, the propagating speeds of the fundamental beam and the second harmonic in the medium are, in general, different and consequently the second harmonic generated at some planes is not in phase with that generated in the other plane. Two adjacent peaks of spatial interference pattern are separated by one coherence length. On the other hand, if it can be arranged that the difference in phase between the fundamental and the harmonic changes by 7r/2 over one coherence length then a quasi-phase matching condition is obtained. In a counter-propagating SHG regime, the fundamental and the harmonic propagate orthogonally and the quasi-phase matching condition is different from that of co-linear SHG. The counter-propagating fundamentals create nonlinear dipole sheets and the second harmonic which is radiated along the surface normal is the sum of the contributions of each dipole sheet. The harmonic intensity thus depends strongly on the phase distribution of the nonlinear polarisation. Since the fundamental beams have constant phases in the y-direction, i.e. the direction of confinement, the nonlinear polarisation will also be in phase. Actually, this phase distribution is determined by the harmonic propagation constant k = 2on,,,,/c, where w is the frequency of the fundamenta1.(9*~‘4”Therefore, if the separation of two polarisation sheets is L = i/2n,,, (the half wavelength of harmonic beam in the medium) then the two fields will interference destructively and cancel each other. In this way efforts can be made to design a periodic layer structure of semiconductor to realise quasi-phase-matching. A genera1 theoretical approach to the analysis of optical frequency conversion in nonlinear media with periodically modulated linear and nonlinear optical effects has been reported.“b3) Expressions are obtained for both difference frequency generation (DFG) and second harmonic generation. It is indicated that conversion frequencies of order 10% can be obtained from suitably designed AlGaAs DFG devices. 2.5. Cavity eflects As seen from equation (2.2) the second harmonic conversion efficiency is proportional to the fundamental beam intensity. When a non-linear medium is incorporated in a resonant cavity, the intensity of fundamental inside the cavity will exceed its value outside by (1 - r))‘, where r is the mirror reflectivity. Thus in a high Q cavity, the enhancement of conversion efficiency inside the cavity is very large. The resonance or optical feedback also extends the interaction length of fundamental beam with non-linear medium. On the other hand, by providing the resonant cavity for the harmonic Ashkin et al.(‘O)argued that when the harmonic beam begins to grow at less than the maximum rate it is reflected back and refocused by an optical resonator in such a phase that it can continue to interact with the fundamental power. In this way the effective length of interaction can be increased greatly, and the presence of harmonic electric field enhances the power radiated from the harmonic polarisation. The resonant cavity effects have been widely used in co-linearly optical pumped nonlinear crystal for second harmonic generation, frequency mixing and parametric oscillation. In a counter-propagating surface emitter, the limited vertical interaction length reduced the efficiency of second harmonic conversion although a semiconductor waveguide increases the circulating pumping power. The harmonic power is still too low for many applications. Recently, Lodenkamper et a1.‘9’)extended the concept of cavity effects in co-linear case to second harmonic surface emitter in semiconductor waveguide. A schematic of the cavity configuration is given in Fig. 2(c). Lodenkamper et al. developed and applied a simple theoretical model for doubly resonant SHG with orthogonally propagating fundamental and second harmonic waves.‘92)In the nondepleted-pump approximation, the on-resonance conversion efficiency of

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a doubly resonant vertical cavity surface emitter with a TE slab waveguide mode pumping is given by

(2.7)

where c = 32n2J2/Nfn2w~o CA,’(with N, the effective index of the fundamental waveguide mode, nzwthe index of the cavity at the second harmonic frequency, 1, the free-space wavelength of the fundamental and J is an overlap integral), L is the length of the device, W is the width of the fundamental mode, T, and T2ware the power transmissions of the coupling mirrors at fundamental and second harmonic, respectively, and similarly 6, and a2”,are the total cavity power losses. Their calculation shows that resonant devices can be several orders of magnitude more efficient than similar non resonant devices. Milliwatt visible outputs can be expected for tens of milliwatts of infra-red input in a doubly resonant structure of MNA (2-methyl-4-nitroaniline) for reasonable parameters. Lodenkamper et al. demonstrated a AlGaAs/AlAs surface emitting generator by embedding the waveguide core in a monolithic vertical resonant cavity.“‘) A 240-fold increase of conversion efficiency was obtained, which is quite close to the nominal value of 264. In their proof-of-principle device, the guided region is a non-quasi-phase matching layer and the efficiency is thus expected to increase significantly in quasi-phase matching active layers. By employing cavity effects, the conversion efficiencies of several percent per watt for SHG of green or blue light are possible in an optimised semiconductor resonant vertical cavity surface emitter.

2.6. Miller’s rule Calculations of second order susceptibilities of semiconductors have been performed using a sum rule approach by Scandolo and Basani. (“5,1’6)The method adopted facilitates the deduction of Miller constants for a range of semiconductors. The basic formalism, after incorporating a number of simplifying assumptions, provides a natural means for deducing Miller’s empirical rule which allows the expression of the second order susceptibility in terms of the first-order susceptibility. The general technique for the calculation of x2 (0, o) for semiconductors involves the computation of relevant matrix elements on the basis of a known band structure. Within such a theoretical framework strong peaks in the susceptibility would be associated with transition resonances. However, the general technique often fails to provide an accurate calculation of the static value of the second order susceptibility, x2 (0,O). Another failing in the general theory has been in the provision of a satisfactory theoretical basis for Miller’s rule which takes the form:

The coefficient, A,, is found to be a constant when the frequencies o and 20 are in the transparency region. On the basis of Miller’s rule A would be expected to be the same for different materials but this has not been found to be strictly correct. Deficiencies in the general theoretical approach can be attributed to two main factors. Firstly x2 is extremely sensitive to parameters such as matrix elements and energy differences. Secondly, a limited number of matrix elements is selected as opposed to effecting a sum on a complete basis set. It has been shown by Scandolo and Bassani (‘I’)that x2 is governed by a number of very stringent requirements which are expressed through seven sum rules written below. Such rules would

Semiconductor optical waveguide devices

be expected to be broken in any approximate

191

calculations. The sum rules take the form:

x2(0,0) = 2/7r Im &w’, o’)o’ do’ s

s s and

o” Re x2(0, w) do = 0

for

n = 0,2,4

o”Im&W,W)dw=O

for

n= 1,3

w~&,(w, O) do = - n/16N(e3/m3)(a3V/ax,ax,ax,> s

(2.9)

where N is the electron density; V(x) is the external potential experienced by the electrons and the average is performed in the ground state of the system The sum rules in conjunction with a simplified model for x2 (w, o) have heen shown to provide a justification of Miller’s rule and allows a calculation of Miller’s constant.“‘@ In turn the calculation is shown to provide values of the static limit x2 (0, 0) which are in agreement with experimental values. The main assumption made to facilitate this derivation is to give consideration to a single resonant frequency in which case it is found that: ~$~(o, 0) = cc,/(oO- w - iy) + a2/(oo - 20 - iy) + c13/(oo- 0 - iy)2 + a,/(o, + w - iy) + cl,/(oo + 20 - iy) + cr3/(w,+ 0 - iy)’

(2.10)

where the oscillator strengths C(iare found, via the above sum rules, to satisfy the relations: w,/4LX,+ WI_&+ LX3 = 0 0$16a, + &a, + 30&, = 0 0;/64u, + w& + 5o;c1, = c

(2.11)

From the above the oscillator strengths are given as u, = 64/9Clo,5;

u2 = - 22/9C/w:;

a, = 2/3Clo,4

(2.12)

where C is given in the form: C = - 1/16(e3/m3)(a3Vlax,~x,~x,),

(2.13)

For the single resonance model linear susceptibilities are written in the form: x:~(o) = N(e’lnt)lI(oi

- 0’)

(2.14)

Then the second order susceptibility can be expressed in terms of the linear susceptibilities as:

x$(0, 0) = - ~~i(2w)~(o)&,(o)(a’Vlax,~x,~x,)o/2e3NZ

(2.15)

Hence Miller’s constant can be obtained via the static limit: &(o,

0) = - x~i(o)3(a”vlax,ax,~x,)o/2e3N2

(2.16)

The calculation of the expectation value of the third derivative of the pseudopotential can be carried out in the knowledge that the contributions to the static limit originate from

192

K. A. Shore et al.

valence band electrons which respond to the pseudopotential. Moreover it has been shown that electron-electron interactions do not contribute to the sum rule constant and hence the unscreened pseudopotential is used. Excellent agreement is found between theoretical values for the Miller constant obtained using this approach and experimental values found for a number of materials.“‘@ 2.7. Nonlinear refractive index The generation of second harmonic radiation is accompanied by the depletion of the power carried by the fundamental wave. As a direct consequence the fundamental experiences an intensity-dependent phase shift which is associated with a large intensity-dependent refractive index. Such an effect can be interpreted as a x3 nonlinearity being generated via cascaded x2 nonlinearities.(48’ An alternative approach to this phenomenon has been suggested by Kador”‘) who utilises Kramers-Kronig relations to obtain a simple analytical expression for the phase shift in the limit of weak pump depletion. For this purpose the propagation of light through a distance L in the material is described in terms of a complex transmission function: Q(w) = 4(w) - i&(o)

(2.17)

whose real and imaginary parts are related to the refractive intensity-dependent absorption coefficient a (0) as follows:

index,

n(o),

and

fj(0.1) = n(o)oL/c S(0) = a(o)L/2

(2.19)

It is usually the case that Q(o) will be an analytical function whose real and imaginary parts are related through the Kramers-Kronig relations. Consideration is then given to the general expression for the second harmonic power generated in a x2 crystal (see equation (2.1) above) to write an effective amplitude coefficient for the fundamental wave: 6(O) = l/2CL2P,{sin(@/2)/@/2}2

(2.20)

In the above, C is a constant which depends on the second order coefficient and the dielectric constants of the crystal as well as on the cross section of the incident fundamental beam of power P,,,; 0 = (k, - 2kJL denotes the phase mismatch angle with k,and k, being the wave numbers for the fundamental and second harmonic, respectively. The expression given in equation (2.20) is valid for the case of small pump depletion i.e. when l/2 C L2 P, < 1. The attentuation due to pump depletion must be connected to a phase shift 4(O) of the fundamental wave in such a way that $(O) and the attenuation coefficient 6(o) form a Kramers-Kronig pair of real and imaginary parts of the analytical complex function a(o). By this means a closed form expression for $(O) is obtained: 4(O) = l/2CL2PW2/0(sin(0)/0

- 1)

(2.21)

The approach described here has been shown to compare favourably with numerical calculations using more complicated theories of cascaded second-order nonlinearities.“*’ It should be appreciated that the conclusions obtained here are generally valid so that in any situation where electromagnetic waves suffer attenuation there will be an associated phase shift applied to the transmitted wave. It is noted, inparticular, that provided that the conditions for the validity of the Kramers -Kronig relations are satisfied then the phase shift can be calculated from the attenuation coefficient without a detalied knowledge of the physical mechanism. The occurrence of the phase shift and the directly related

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193

intensity-dependent refractive index permits consideration of the application of this effect in the development of all-optical switching devices.

3. INTERBAND

TRANSITIONS

IN QUANTUM

WELL

MATERIALS

3.1. Second order susceptibilities Nonlinear optical properties of quantum wells are of general interest both for device applications and also as a means of studying the electronic structure of mesoscopic media. It has been found, in particular, that asymmetric quantum wells (AQWs) exhibit very large second-order and indeed third-order susceptibilities. (2’-24~3c37~54~16’) In this section interest will be focused on enhanced nonlinearities associated with interband transitions in AQWs. In Section 5 it is shown that such structures are also of interest for intersubband nonlinearities responsive at mid-infra-red wavelengths. Calculations of the second-order susceptibility of an electric-field -biased single quantum well structure indicated that peak values of about 3 x 10-l’ m/V are expected.(‘*13@ The calculation included contributions from exciton and continuum states and concluded that the exciton states contributed about 30% of the total x2. Work by Khurgin on asymmetric quantum wells found a maximum x2 value to be approximately 1.8 x lo-l2 m/V.‘“**” That analysis omitted contributions from exciton states. It was pointed out by ShimiztP) that Wannier exciton states play a large role in second-order nonlinear optical phenomena and Shimizu estimated x2 values of 3 x 10 -9 m/V in unoptimised structures. Experimental work by Xie et al.(1s5-‘56) using multiple asymmetric quantum well structures found peak second order susceptibilities of order 8 x 10 - 9 m/V. That value is an order of magnitude larger than those calculated by Harshman and Wang. (64’It is noted that Scandolo et al.(1’4)have proposed step-like GaSb/InAsSb asymmetric quantum wells where it is expected that the second harmonic coefficient is 60 times that of bulk GaAs. In this context Atanasov et al.“” developed a detailed procedure for the calculation of the SHG susceptibility tensor in single and multiple asymmetric quantum wells. The work takes into account both excitonic and continuum states and includes the effects of non-homogeneity of charge distribution. The main features of this approach are given in the next section. The first measurements of second harmonic generation from a superlattice of asymmetric coupled GaAs quantum wells were reported by Janz et al. (74)In this work visible light was generated by using a fundamental at a wavelength of 1.319 pm. The approach exploited the use of quasi-phase matching for enhancing SHG in a reflection geometry.(7S’The orientation of the asymmetric quantum wells was periodically reversed in order to achieve quasi-phase matching at the required wavelength. The measured second harmonic susceptibility was found to be of order 1.5 x lo-” m/V and the second harmonic intensity generated in the superlattice structure was approximately 3.6 times that of a GaAs reference sample. 3.2. Calculation of x2 in asymmetric Quantum Well structures The formalism adopted by Atanasov’ “’ facilitated the calculation of all the non-zero components of x2 taking account of both the non-local response of the asymmetric quantum well to the electric field and the bulk-like inversion asymmetry of the material. A density matrix approach in the electric dipole approximation was adopted. The calculations were performed in an average field approximation which had been previously proposed by Agranovich.(4.5’ The method makes use of the fact that the radiation wavelength, 1, is large compared with the period, L, of a multiple asymmetric quantum well. This permits the use of unretarded Maxwell equations to perform self-consistent calculations to obtain the field distribution inside the well as well as to obtain the optical response. The internal electric field

194

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distribution is determined by the inhomogeneous charge distribution and the dielectric mismatch. The theoretical framework in Ref. 11 is established using three main steps. First of all, a density matrix formalism is adapted to take into account exciton and continuum states. The second order susceptibility tensor is then calculated in the electric dipole approximation using the local response approach. An averaged quadratic susceptibility tensor is then found taking into account non-local response. The starting point for the theory is a general expression for the local quadratic polarisation P(t, r) of a medium interacting with an external electric field E(t, r):

The integration over t, and t2 are performed in the infinite interval (-co, co) and the integration over r, and r, in the volume V of the sample. Here p, q, r label Cartesian coordinates and G,, is the quadratic response function whose time dependence is known as temporal dispersion. In the case of a homogeneous medium where spatial dispersion can be neglected and where attention is given to the interaction of two monochromatic waves of frequency oi and o2 it is possible to write: P,(R) = ErxP&I, %)-&(W,)MWJ

(3.2)

Here R = o, + o2 and Stakes into account possible permutations of indices. In the case of second harmonic generation o, = o2 and Y = l/2. Using the density matrix approach in the dipole approximation a general expression for x2 can be obtained. The space dependence of the Green function, G,, (r, rl, z2, r,, rJ, takes into account the nonlocal nature in the optical response whose main contribution arises from inhomogeneity of the medium under consideration. For quantum well materials the nonlocality influences the radiative and non-radiative polaritons the effect of which can be computed when account is taken of the dielectric mismatch and the charge distribution inside the quantum well. Taking account of the crystal symmetry properties of zinc-blende crystals it is found that for asymmetric quantum wells the constitutive equations (3.2) take the form P.r = ~,x.~.J,E:

(3.3)

P, = ~ox.&E:

(3.4)

Pz = ~,/2[X&z

+ G) +

Lz-@l

(3.5)

The requirement is then to obtain appropriate eigenstates and dipole matrix elements to evaluate the crossed component xXx:, the perpendicular component xzzz, and the in-plane component xz._. Atanasov et al. considered cases of electron-hole unbound pairs and bound exciton states. The formalism also was extended to treat the nonlocality of the interaction between the electromagnetic field and the charge distribution inside a single asymmetric quantum well. Results of those calculations are shown in Figs 3(a)-(d). Figure 3(a) shows the calculated crossed component of the second-order susceptibility spectrum, for a steplike AQW when the incident electromagnetic wave is near half of the fundamental bandgap. The effects of both direct (E,HH,, E,LH,, E,HH, etc) and indirect (EIHH2, E,LH,, E2HHI etc) excitations are apparent. Figure 3(b) shows the spectrum of x~:~(w) (the perpendicular component) for the same AQW considered in Fig. 3(a) in the half-band-gap region. This component is seen to possess both single and double resonances and therefore gives the largest contribution to the polarisation. The spectrum of the in-plane component xXx* (‘) for the same AQW is shown in Fig. 3(c). It is found that this component

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possesses only single resonances in the region of the fundamental band gap. It is found also that the peak values of the light-hole excitations are weaker than those of the heavy-hole excitations. The influence of nonlocal effects is made apparent in Fig. 3(d) where the crossed quadratic susceptibility line in the band-gap region calculated for a double AQW. The nonlocal interaction decreases the stronger E,HH, exciton peak and increases the light-hole exciton peaks. Atanasov et al. indicate that the use of sum rules, of the kind presented by Scandolo,(“5.“6) may provide one means for obtaining further improvements in their

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ho fmeV1 Fig. 3. (a) Half-band-gap second-order susceptibility spectrum I:!! for a steplike AQW composed of 1.5nm GaAs and 3.5 nm Ga,,Al,,4As in a barrier region of AIAs. The electron and hole confining potentials are shown in the insert with the lowest electron and heavy-hole levels and the corresponding envelope functions. The solid line indicates ( .r:‘:(w) ( and refers to the left scale and the dotted line shows the phase expressed in radians on the right scale. (After Ref. 11) (b) Spectrum of the I$: component for the same AQW as in Fig. 3.1 in the half-band-gap region. In the insert, we show the lowest light-hole levels and envelop functions. (After Ref. 1I) (c) Spectrum of the xl*/, component for the same AQW as in Fig. 3. I in a frequency region near the band gap (After Ref. 11). (d) Crossed quadratic susceptibility 1x:2:,(co)1 in the band-gap region calculated for the double AQW given in the insert and taking into account the internal nonhomogeneity of the well (solid line). The corresponding spectrum, without consideration of nonlocal effects, is shown by a dotted line. The AQW consists of Ga, IAlo5A~ barrier regions and two GaAs wells of width 2.2 nm and 2.8 nm and separated by a 2.2 nm barrier. Electron and heavy-hole eigenstates are shown in the insert. (After Ref. 11) 196

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197

formalism. Specifically identified is the need to satisfy gauge invariance conditions in scalar or vector potential representations. A particular requirement here is to ensure completeness of the relevant summations over transition states. 3.3. Biased Quantum Well structures Experimental work by Lue and Ma(93)aimed to test theoretical predictions of electric-field enhanced SHG by Ref. 133, 136 but no enhancement of the second order susceptibility was detected. On the other hand, Qu et al .(‘05,‘06)observed enhanced SHG in reflection mode using GaAs/ AlGaAs AQWs. The latter work indicated a good agreement between theory and experiment. Calculations of second order optical nonlinear susceptibilities responsive at wavelengths of order 1 pm in AQWs have been reported by Kuwatsuka and Ishikawa@3)who performed calculations for a range of quantum well widths and applied electric fields and examined the influence of Wannier excitonic states and non-excitonic states on x2. Their work showed that a x2 of 5 x lo-” m/V could be obtained in such structures. The relatively large value is attributed to the strong oscillator strength and large dipole of of Wannier excitons as well as to a strong resonance between the exciton and the pumping light.

4.

TEMPORAL

RESPONSE

Second harmonic and sum frequency generation (SFG) arising through the mixing of ultra-short optical pulses exhibit specific features which need to be taken into account in the interpretation of experimental results. In this section an overview is given of the theoretical framework which can be used to assess the influence of effects associated with higher-order nonlinearities, pulse shape and phase and group velocity mis-match in SHG and SFG. 4.1. Cross-phase modulated second harmonic generation The use of short high-power laser pulses to effect second harmonic generation can give rise to significant contributions from higher order nonlinearities. In particular third harmonic generation associated with x3 nonlinearities can introduce self-phase modulation (SPM) and cross phase modulation (XPM) of the pulses. These processes lead to spectral broadening and temporal pulse shape distortion and modulation which can, in turn, be used to achieve frequency and amplitude modulation as well as pulse compression. Materials such as ZnSe possess both x2 and xJ nonlinearities and measurements have indicated, for example, that the extent of spectral broadening of the second harmonic line was dependent on the intensity of the incident fundamental signal. c8)Analysis of the temporal behaviour of a laser pulse propagating in such a medium has been undertaken and compared with experimental results.@’ Consideration is given to an incident primary wave of the form: E, = 1/2{A(z, t)exp( - i(wt - k,z)) + c.c.} and a second-harmonic

(4.1)

written as: E2 = l/2(8(2,

t)exp( - i(2ot - k,z)) + c.c.]

(4.2)

where A(z, t) and B(z, t) are the pulse envelopes of the fundamental and second harmonic waves, respectively. Account is taken of both x2 and xs nonlinearities so that the nonlinear polarization is taken to be: Pn,

=

Gx2p

+

x8)

(4.3)

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198

First order wave equations for the pulse envelopes are obtained on the basis of a slowly-varying envelope approximation for non-phase matched second harmonic generation. The fundamental wave propagates according to the equation: &t/a2 + l/u,&4/dt = iyA JA I2

(4.4)

whilst the second harmonic pulse envelope is described by the equation:

aB/az + i/27,aB/at = i/3A2exp[i(2k, -

k,)~]

+

iyB(p12

+

2lA 1’) - aB

(4.5)

In the above: a describes linear absorption; p is proportional to x2 and y( = y’ + iy”) is proportional to x3. The real part of y contributes to the phase modulation of both the fundamental and the second harmonic whilst its imaginary part arises from induced absorption. The first term on the right hand side of equation (4.5) is the usual second harmonic generation process. In general A >> B and thus the self phase modulation (SPM) of the second harmonic (described by the term iyB I B I ‘) can usually be neglected relative to the cross phase modulation (XPM) contribution (governed by the term i2yBlA I’). On this basis, closed form expressions for the pulse envelopes A(z, t) and B(z, t) were obtained by Ho et ~1.‘~~) The general form for the second harmonic pulse envelope is rather a complicated function of the product of j and y but it can be used directly to analyse second harmonic generation due to XPM via the coupling of the second and third order nonlinearities. The pulse response is also dependent on group velocity dispersion (GVD) and phase velocity differences which are determined respectively by the group indices (n,, n2) and phase refractive indices (Q, nzp)of the fundamental and second harmonic waves. The relevant mismatches are characterised by parameters q = (n, - n&r and 5 = (nzp- nzp) 20/c where 7 is the pulse duration parameter. Experimental measurements (Ho 1989) of 1.06 pm picosecond pulse propagation through ZnSe indicate that the following parameter values are appropriate: n, = 2.58; nip = 2.484; n2 = 3.36; nzp = 2.697; x, = 10m9esu and y = lo-l2 m/V”. The formalism developed by Ho et al. permits a clear presentation of the influence of the role played by the different physical mechanisms which occur during pulse propagation. Fig. 4(a) shows four cases of interest. The case of simple SHG with no GVD (ye = 0), no XPM (y = 0) and with phase matching (5 = 0) is shown in Fig. 4(la) where the pulse duration of the SHG is approximately l/fi of the fundamental. In Fig. 4(a) the GVD is non-zero and cases zero and non-zero phase matching are shown. For 5 = 0 efficient SHG occurs throughout the nonlinear medium and form a longer pulse. When the phase velocity is not matched two peaks appear in the second harmonic at the entrance and exit surfaces of the nonlinear medium and correspond to the trailing and leading edges of the generated shg signal respectively. In this case SHG from the middle portion of the nonlinear medium is cancelled due to interference effects. The result of adding linear absorption to the previous case is shown in Fig. 4(c) - attenuation of the SHG is seen to occur. Inclusion of the XPM contribution to the SHG results in the pulse shown in Fig. 4(d) which matches very well with the experimentally observed response. The model has been utilised to identify temporal features which are similar to wave-breaking fringes and which are genral characteristics of both SPM and XPM. Further ramifications of the XPM process in SHG may be obtained from consideration of cross spatial modulation; pulse compression and THz optical modulation. 4.2. Sum frequency generation Autocorrelation measurements of the duration of ultra-short optical pulses may be performed by observing the second harmonic signal generated via the nonlinear mixing of a pulse and a delayed replica of the pulse. The intensity and half width of the pulse can then be deduced. When the fundamental pulse shape and width are known alternative correlation

Cb)

(a)

180

200

220

240

260

180

200

220

240

260

(C)

180

200

220

240

260 Time (psec)

Fig. 4. Temporal effects in second harmonic generation. (After Ref. 68)

techniques can be used to study the properties of the generated second harmonic. Thus, for example, a fundamental can be nonlinearly mixed with its own second harmonic and a cross-correlation measurement effected by observing the intensity at the sum frequency. Such experiments have been undertaken where a red pulse at a wavelength of 630 nm is mixed with its UV harmonic and the sum frequency intensity is generated at a wavelength of 210 nm in the deep UV. (“) An early analysis of the effect of group velocity mismatch on SHG autocorrelation measurements of ultra-short optical pulses was presented by WeineF4’) (1983). That approach, which was adopted by EdelsteitP in the analysis of their experiments, assumed that the 630 nm fundamental pulse and the 315 nm harmonic pulse were of identical functional form. An extension of the theory to describe the cross-correlation technique and also to take into account the group velocity mismatch between the harmonic and the sum frequency component was developed by Baronavski ct aj*(12,13) The approach followed by Baronavski et al .(“, 13)emphasises the requirement for a rigorous mathematical model in order to obtain a proper understanding of the effects of group velocity mismatch and crystal thickness. An accurate description of the sum frequency field is, in particular, seen as an important factor in the analysis of cross-correlation experiments. An important characteristic of the mathematical apparatus thus developed is that it facilitates the inference of the temporal width of the harmonic or sum frequency components. In the relevant experimental situations such pulse widths are typically of order a few tens offs and may not be amenable to direct measurement. Consideration is given to two pulses E, and E2 at frequencies ol and o2 respectively which enter a nonlinear medium and propagate in the z-direction with wave numbers k, and k,. It is assumed also that pulse E2 is delayed by a time z with respect to pulse E,. The pulse

K. A. Shore et al.

200

envelopes are defined by functions f;(z, t) and f2(z, t) such that E,(z, t) =f;(t - k,‘z) exp{jo,(t - k,‘z))

(4.6)

E2(z, t) = exp( - iw,rlfi(t - z - kz’z) expCjo,(t - kZ’z)}

(4.7)

where the ki.s are reciprocal group velocities defined as ki’ = w, dki/dw. The above pulses are combined in the nonlinear medium to produce a sum-frequency field E&z, t) at frequency w, = (w, + WJ prescribed, using corresponding envelope function s(z, r), wavenumber, k,, and reciprocal group velocity k,v’, in the form: E,(z, t) = s(z, t - k,‘z) expjw,(t - k,‘z)

(4.8)

The sum-frequency profile, s(z, t), is found from a slowly-varying envelope approximation using a first order equation of the form: &/8z = CE,(z, t)&(z, t) exp{ -j(w,t

- k,vz))

(4.9)

where C is a known constant. In this way Baronavski et al. obtained a widely applicable general expression for s(z, t). Consideration is given to the output sum-frequency after the fundamental pulses have propagated through a nonlinear medium of length, L. The response will depend, in general, upon the differences between the wave vectors and group velocities of the fundamental and the sum frequency waves. These differences are characterised by the following parameters: Ak = k,y- (k, + k,)

(4.10)

a = k,,’ - k,’

(4.11)

Ak’ = k,’ - kz’

(4.12)

The formal result obtained in this way is that: s(L,t) = D

do, dwZexp(jw,t)exp(

-jw,t)F,(w,

- 02)F2(wJT(x)

(4.13)

Here D is a constant; F, and Fz are the Fourier transforms of fi and fi, respectively and T(x) = exp( jx)sin(x)/x

(4.14)

x = (Ak + aw, + Ak’w,)LP

(4.15)

with

In order to facilitate computation of the sum-frequency wave some approximations are required to permit a simplification of the above general expression. So, for example, when the group velocity mis-match between the fundamental pulses is negligible (Ak’ = 0) the sum-frequency envelope is found to be of the form: s(L,t) = D exp( -jw2rs(t2fi(t

- z)*[l/aL sq(r/aL + 1/2)exp( -jAkr/a)]

(4.16)

where D is a known constant; * denotes the Fourier transform convolution and sq(x) = 1 for 1x1< l/2 and sq(x) = 0 otherwise. This expression represents a generalisation of the work of Weiner(‘47’to include the case of different fundamental pulse shapes. In treating the situation where a fundamental is mixed in a second nonlinear medium with its own second harmoniP the formalism of Baronavski et a1.(‘2’ demonstrates that, in contrast to previous analysis-which had assumed a simple sech’ (bt) dependence-the second harmonic pulse should be taken in the form: s(t) = sech*(at)*l/a L sq(t/aL + l/2)

(4.17)

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where a is a constant and CIthe group velocity mismatch arising in the medium where the second harmonic is generated. The foregoing analysis has been used to critically examine assumptions made in experimental work. It is shown (13) that the full-width half-maximum (FWHM) is fundamentally insensitive to the mismatch between the generated and incident optical fields. This provides confirmation of assumptions made in the earlier analysis. However, the asymptotic variation of the FWHM is also shown to be linearly dependent upon the mismatch between incident fields. The analysis was developed for the case of hyperbolic secant pulses but can be extended to other pulse shapes. An important practical consequence of the analysis is the identification of a constraint which is imposed upon crystal thickness used in narrow pulse measurement.

5. INTERSUBBAND

PROCESSES

5.1. Asymmetric Quantum Wells Quantum confinement of carriers in semiconductor quantum wells produces large oscillator strengths for transitions between the sub-bands of the quantum wells. The first observations of such large dipole transitions was reported in respect of conduction band transitions by West and Eglash.“49)Those transitions have resonant wavelengths which typically are of order 10 pm and are thus of interest in a range of applications but, in particular, are of considerable importance in the design of infra-red detectors. (“) Since second-order nonlinear coefficients are proportional to the square of oscillator strengths strong mid-infra-red nonlinear optical effects can be expected in suitably engineered quantum well structures. The structures can be considered to be quasi-molecules which can be optimised, via variation of the thickness of the quantum wells, for nonlinear optics at appropriate wavelengths. A basic requirement in this respect is the breaking of the inversion symmetry. The first approach to utilisation of intersubband processes for second harmonic generation by Fejer et al. used an applied electric field to break the inversion symmetry of a GaAs/AlGaAs multi-quantum well structure.‘“’ In this work values of the second harmonic coefficients of order 28 rim/V were obtained which is about 70 times that of the corresponding value for bulk GaAs. An alternative approach was advocated by Gurnick and DeTemple(62) who suggested the use of compositional gradients as a means for producing synthetic nonlinear semiconductors. Observations of both second harmonic generatiotP2) and nonlinear optical rectification”“’ in compositionally asymmetrical AlGaAs multiquantum wells have been reported by Rosencher et al. A typical structure is illustrated in Fig. 5(a). In their work on optical rectification (“‘) Rosencher and co-workers measured an electro-optic coefficient of 7.2 rim/V which is about three orders of magnitude greater than that of bulk GaAs. Their work revealed a second harmonic generation coefficient of 7 x lo-’ m/V at a wavelength of 10.6 pm which permitted experimental realisation using a continuous carbon dioxide laser. 5.2. Theoretical basis

represented an early attempt to develop a comprehensive Work by Khurgin(78~79*88’ theoretical basis for the evaluation of second order intersubband nonlinear susceptibilities of asymmetric quantum wells. The work utilised oscillator strength sum rules to obtain simple expressions for x2 where account was taken of higher subbands. Attention is given to two specific asymmetric quantum well configurations-a graded band gap quantum well (GBQW) and an asymmetric coupled quantum well (ACQW) structure. The theory permits the determination of the dependence of x2 on frequency, QW geometry,

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band-offsets and effective masses. A self-consistent solution of the Schrodinger equation is incorporated in order to ascertain the effect of screening which is expected to be substantial at donor concentrations up to lOI* cm-*. In regard to the frequency dependence of x2 it is shown by Khurgin that for the intersubband process a large nonlinear susceptibility of order lo-* m/V exists in asymmetric structures even far from resonance. This is in contrast to the case of interband processes in asymmetric quantum wells where x2 is found to have a sharply resonant character. This result supports the use of such structures for device applications in the 10 pm range. Due to the better charge separation in the ACQW this structure is found to have a larger x2 than that of the GBQW structure. It is also found that, in order to move towards device applications in the near infra-red, use should be made of materials with larger band-gap off-sets than those of GaAs/AlGaAs. It is estimated that a QW structure with a 1 eV conduction band off-set could provide a x2 of lo-” m/V at a wavelength of 1.5 pm. Possible material systems include GaInAs-GaAs and also indirect band-gap materials such as the Si-Ge superlattice. Material systems such as ZnSe/GaAs and ZnSe/Ge having large band offsets in the valence band may also be of interest. It is noted that the basic reason for 400 -

300

-

200

-

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-

0 E B :: ra

10

0

-10

Distance (nm) (a) -d_-B--w -T

AE

1 @I

-Bd2 AE

4-L

-

Fig. 5.

20

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203

(a)

Ea = 334 E2 = 228

Et = 92 meV

II-I (b)

Ea =

386

E2 = 270

E, = 151

meV

IO

f

E

1

E & .,o g g c -0 B :: v)

O.’

0.01

20

50

100

200

Pump power (mW) Fig. 5. (a) Band diagram and square of wavefunctions of compositionally asymmetrical quantum well. (After Ref. 112) (b) Asymmetric QW structures. (a) GBQW. (b) ACQW (After Ref. 78). (c) Conduction band energy diagrams of a single period of the AlInAs/GaInAs coupled quantum well nonlinear optical structures. Shown are the positions of the calculated energy subbands and the corresponding modulus squared of the wave functions. (a) The GaInAs wells have thicknesses of 64A and 28A and are separated by a 16A AlInAs barrier. b) The GaInAs wells have thicknesses of 42, 20, and lg.&, respectively, and are separated by 16a AlInAs barriers. (After Ref. 29) (d) Second harmonic generation signal as a function of pump power at a pump photon energy 122.2 meV for zero and + 4V bias at room temperature. Positive-bias polarity refers to the band diagram configuration in which the thin well is lowered below the thick well by the electric field. The insert shows the band diagram of the coupled-well structure. The energies of the bottom of the subbands and the corresponding I$[ * are displayed. (After Ref. 123)

204

K. A. Shore et al.

large optical susceptibilities in hetero structure materials is the small electron effective masses in the materials. Material choice to achieve either large linear or nonlinear susceptibilities should be largely made with a view to obtaining low effective masses. In practical utilisation of such susceptibilities concern may be felt for the effects of intersubband absorption on the efficiency of the SHG conversion process. A figure of merit can be defined in terms of the power density, Z,,z, of the fundamental frequency which is required to achieve the maximum (50%) SHG conversion efficiency. Khurgin calculated both this parameter and x2 as a function of the asymmetry of an ACQW structure. Typically Z1,* for ACQW structures is of order l&SO MW/ cm’. A particular feature of the work was the demonstration that the minimum of Z,,2and the maximum of x2 did not coincide (as functions of the asymmetry). It was also found that, whereas the value of x2 falls off quite quickly for greater asymmetry, the corresponding increase in Z,,, is relatively slow. Thus to achieve a higher overall conversion efficiency it is thus necessary to balance a decrease in x2 with a much stronger decrease in absorption. In this process the interaction length of the process is necessarily increased and thus the achievement of phase matching conditions is made more difficult. The role of absorption in determining the efficiency of phase modulation via the linear electro-optic effect and for performing optical rectification was also treated by Khurgin within the same theoretical framework. In the latter respect it is shown that observation of ultra-short electrical pulses via optical rectification of a short optical pulse should be free of interference caused by absorption provided the optical pulse length is significantly shorter than the intersubband relaxation time. Calculations of x2 were also undertaken for graded quantum wells by Gurnick and deTemple’62’and for biased quantum wells by Rosencher and Bois,(“O)Boucaud et al.,‘?” Tsang er ~1.“~~’and Ikonovic.“‘) Optimisation of compositionally asymmetric quantum wells responsive at a carbon dioxide laser wavelength has been discussed theoretically by Dave. (47)In this work a comparison was made between the predicted second order susceptibility of a two-step square well with an applied electric field and that of a structure where composition grading is used to obtain an equivalent asymmetry. Energy eigenvalues and envelope wave functions were obtained for an AlAs/AlGaAs quantum well structure by numerical solution of the time-independent Schrodinger equation in the effective mass approximation. The numerical technique utilised in this work having earlier been developed for optical waveguide analysis.‘46’It was shown that for relatively modest applied electric fields of order 80 kV/cm the compositionally graded structure exhibited a second order susceptibility which is two orders of magnitude greater than that of the square well with an equivalent applied electric field. 5.3. Optimisation of Zntersubband SHG The second harmonic generation process is enhanced in an asymmetric quantum well having three energy levels E,, E2 and E3 such that the energy difference, E,-E,, is near the pump wavelength. The use of intersubband transitions for second harmonic generation is, in the first place, motivated by the large dipole moments which are available in asymmetric quantum wells. In such structures the second order susceptibility can be approximated in the form: x2 = (e3/~o~)NM,2M,3M231/{(W- o,* - iZ)(20 - o,, - ir)} where N is the electron density; the M,,s are dipole matrix elements and the Z the broadening. The energy separations between the levels are then given as E2 - E, = q2, etc. It is apparent from the above expression that a resonant enhancement of the nonlinear coefficient can be obtained if it can be arranged that E2 - E, = E3 - E2. and o = E2 - E,. Use has been made of an applied electric field to effect Stark tuning of the energy levels to meet the

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resonance condition by Sirtori. (‘23)The work utilised AlInAs/GaInAs coupled quantum well structures as illustrated in Fig. 5(c). It was found that the Stark shifts of the l-+2 transition was in the opposite direction to that of the 243 transition whilst the magnitude of the shift was equal to the potential drop between the centres of the two wells. In consequence a strong peak was obtained in the value of the second harmonic coefficient as a function of the applied electric field. A peak second harmonic coefficient of 7.5 x lo-’ m/V is obtained when the bound states are made to be equally spaced. Such a value is about 200 times that of bulk GaAs. Figure 5(d) shows typical values for the second harmonic power generated using such a configuration. A comprehensive treatment of the design of coupled quantum well structures for electric field tunable nonlinear properties in the infrared is treated by Capasso et ~1.‘~~) Such structures offer opportunities for large optical nonlinearities of both second order and third order. The emphasis of investigations of intersubband processes has been in respect of transitions responsive at mid infra-red wavelengths and, in particular, near 10 pm. Interest is also evident for nonlinear processing at shorter wavelengths and notably at 2 pm where a number of semiconductor laser sources as well as diode-pumped solid-state lasers are available. Measurements have been performed of second harmonic generation in coupled InGaAs/AlAs coupled quantum well structures which exhibited transitions resonant at 2.1 pm and 4.1 pm. Using 4 pm radiation from a free electron laser a second harmonic coefficient of magnitude about 20 rim/V and a phase angle of about 60” relative to the GaAs substrate was found.(39,40) In addition, both theoretical and experimental work has been undertaken on far-infra-red second harmonic generation using modulation-doped half-parabolic quantum wells.“‘) In this work pump wavelengths in the 200-300 pm range using molecular gas and free electron lasers as sources. The experimental work indicated a surface nonlinear susceptibility of 1 f 0.75 x 1O-8 esu-’ cm3. Such a value for a non-resonant x2 is larger than that obtained for resonance-enhanced x2 found at mid-infra-red wavelengths.(24.“) A simple calculation, which modelled the heterojunction as a collection of non-interacting electrons in a triangular potential, predicted a x2 of 8 x 10 - ’ esu - ’ cm3 which agrees very well with the measured value. It was noted that at the highest intensities used in the experiments the intensity of the second harmonic depends sub-quadratically on the intensity of the fundamental. In this regime phenomena such as higher harmonic generation can be anticipated.(‘22’ 5.4. Conversion eficiency The large second harmonic coefficients which can be obtained using asymmetric quantum wells do not necessarily imply that efficient frequency doubling is obtained in practical devices. Indeed under small optical pump conditions the conversion efficency has been found to be limited to about lo- 7.(23) On the other hand, under strong pump conditions a saturation of the second harmonic generation in GaAS/AlGaAs quantum wells has been observed and a power conversion efficiency of order 1O-4 was obtained. For pump intensities greater than about 3 MW/cm’ the conversion efficiency becomes a nonlinear function of the pump intensity having a maximum value of about 0.03% which is obtained for a pump intensity of 16 MW/cm2. The saturation of the conversion efficiency is readily explained within a framework of a standard semiclassical theory of resonant second harmonic generation where it is found that the saturation is basically determined by population saturation of the relevant energy levels in the quantum well. It can be calculated on this basis that only one half of the electrons remain in the lowest energy state when the optical pump intensity is 10 MW/cm2. One corollary of this successful comparison between theory and experiment is the confirmation provided of estimated values carrier lifetimes used in the calculations. The decrease of the SHG efficiency at higher pump intensities arises because of the resonant nature of the nonlinearity which gives rise to an optimum conversion length which increases with pump intensity. The larger the pump intensity, the higher the conversion efficiency but the

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slower the growth of the second harmonic signal with distance. In consequence there is an optimum pump intensity for each propagation length. Typical optimal lengths are between 2 and 10 pm for pump intensities in the range 10 to 50 MW/ cm’.@*)Calculations suggest that can be obtained using intersubband transitions slightly shifted from the resonance condition. With a view to improving the efficiency of the conversion process involving intersubband transitions use has also been made of surface-emitting second harmonic generator by Chen et of. Use was made of a multipass structure to increase the interaction length of the SHG. At pump intensities far from saturation the conversion efficiency was found to be about 3.5 x 10m4which was indicated as being about 10 times better than comparable schemes using Brewster angle excitation. For higher pump powers conversion efficiencies of order lo-’ were reported using the multi-pass technique. Some debate has arisen in respect of selection rules which are applicable to intersubband transitions and hence to the geometry of the excitation process in the surface emitting configuration. It was argued on the one hand that the nonlinear emitting dipoles in the configuration cannot radiate vertically and an alternative explanation was preferred by Berger et al. that the observed SHG was due to thermal emission.“5’ A justification of the experimental configuration was outlined using group theoretic arguments.(38) Here it was argued that the transitions were active to both TE and TM polarisations and it was, in fact, the TE component of the second harmonic which can radiate out of the top surface. Evidence for the validity of the symmetry arguments was also indicated on the basis of a number of absorption measurements reported in the literature. Another approach to increasing the efficiency of the intersubband shg process has been proposed on the basis of using refractive index changes associated with intersubband transitions to achieve phase matching.“) The changes in refractive index associated with intersubband transitions may also be used as the basis for optical phase conjugation and optical Kerr effect responsive at mid-infra-red wavelengths.@‘)

6. DEVICE

APPLICATIONS

6.1. Optical cot-relator

Techniques for the measurement of short optical pulses are of interest for both fundamental studies and for practical applications. Optical correlators based upon second harmonic generation offer one means for performing such measurements. Internal generation of second harmonics in semiconductor lasers has been exploited for this purpose by Chen and Liuo3) but the poor efficiency of this effect in conventional edge-emitting lasers makes this approach unattractive for general application. Utiiisation of surface -emitted second harmonic generation via nonlinear mixing of counter propagating beams in a semiconductor waveguide structure has been shown by Vakshoori and Wang to provide a means for achieving an integrated semiconductor waveguide optical correlator.(14’) The operation of the device is based upon the nonlinear interaction between incident and reflected fundamental modes at the facets of the waveguide structure. The strength of the nonlinear interaction and hence the power of the generated second harmonic is determined by the overlap between the forward and backward traveiling pulses. Away from the waveguide facets the overlap is reduced and hence the genrated second harmonic is reduced. The width of the optical pulse can be deduced from the variation in surface-emitted second harmonic along the waveguide. To obtain a quantitative estimate of the effect a simple approximation can be obtained by considering the second harmonic generated by incident (i) and reflected (r) Gaussian pulses at the end facet of a semi-infinite waveguide. Using the

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facet as a reference it is possible to write: E oc exp[ - (z - ut/z,,)‘] and

e

cc exp[ - (z + ut/zJ2]

which give rise to a second harmonic pulse which, after time averaging, takes the form:

PP x ev[ - (zlWJ2)21 It is clear therefore that the full width at half maximum of the second harmonic is smaller by a factor of J2 than that of the fundamental pulse. In order to make an accurate comparison with experiments it is necessary to take into account reflections at both facets of the waveguide. Using this technique Vakshoori and Wang successfully calibrated pulses of a few ps duration. The work also indicated that the general approach could be applied to optical correlators fabricated in a range of materials so that accurate measurements of short pulses of light of different wavelengths could be anticipated using this method. Choice of materials for such optical correlators is, of course, driven by the need to identify a strong nonlinear coefficient. In semiconductors the strength of the second harmonic nonlinearity is dependent upon the crystal growth direction and, in particular, the use of (211) B GaAs substrates has been shown to result in enhanced SHG. Whitbread et al.(‘s3’ have utilised this effect in the construction of an optical auto-correlator which has the potential of resolving pulses of order 250 fs. 6.2. Wavelength demultiplexing and parametric spectrometer The availability of a surface-emitted signal from an integrated semiconductor second harmonic generator has also been shown to have applications in a wider communications context. The device can be operated as a demultiplexer to separate the wavelengths in a multichannel optical fibre communications system(32’ and also to define stable optical channels. The latter is of particular importance in proposed coherent optical communications systems. The use of surface emission to effect demultiplexing is illustrated in Fig. 6(a). Practical use has been made of sum frequency generation in multilayer waveguides to effect optical frequency measurements for multichannel networks. The resolution of such optical frequency difference measurements is claimed by Guy et al. to be better than 2 GHz.@~’ Wavelength tuning suitable for spatially addressable coherent detection has been demonstrated in active surface-emitting harmonic generators.(42) In this case the device is composed of a single-quantum-well semiconductor laser integrated monolithically with a multilayer nonlinear waveguide as shown in Fig. 6(b). Tunability is achieved via wavelength dependent resonant absorption loss. Variation of the loss can be achieved via current injection or reverse biasing a control section in a two-segment device. The achievable wavelength shift via current is expected to be of order 20 nm though reported experimental results were more modest. Using a combination of current injection and reverse biasing a total tuning range of order 6 nm has been achieved. 6.3. Sources In many applications of surface emitting second harmonic generators (SESHG) there is a requirement for coupling from optical fibres with typical core diameter of 8 pm. On the other hand the thickness of a GaAs/AlGaAs waveguide SESHG would normally be of order 1 pm. Such a mismatch of dimensions would give rise to a substantial input coupling loss. As a means for alleviating this problem use has been made of a novel waveguide geometry the so-called nonlinear antiresonant reflecting optical waveguide (NARROW) structure.(4”96)The NARROW structure provides a thick guiding core with high modal discrimination and thus permits large coupling efficiency in the fundamental mode. In the work of Dai et a1.(4’)a nonlinear waveguide core of thickness 2.6 pm was formed using AlGaAs material and, through the use of a 1.06 pm Nd: YAG laser source about 1 PW of a second harmonic signal

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in the blue-green was observed. This is an order of magnitude higher power than that obtained with a conventional SESHG. The improvement in performance is attributed to the increased coupling efficiency. Dai et a1.c4’)undertook a careful design of the NARROW structure taking into account measured x2 values versus Al fraction. The SHG conversion efficiency was calaculated in terms of a surface emission cross section A”’ defined by A”’ = P2,/P, P2where P,and P2 are the powers in the counter-propagating fundamental waves and Ph, is the total radiated second harmonic from a normalised emission area. To obtain an optimised design the surface emission cross section was calculated as a function of Al (a) k=m/c -O+ml

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Fig. 6. (a) Waveguide used as the spectrometer and the basic physics of the nonlinear interaction. (b) The dependence of the direction of the sum frequency signal on the propagation constant of the counterpropagating beams. (After Ref. 139) (b) Schematic view of WDM demultiplexer and switch. (After Ref. 97) (c) Variation of the cross section as a function of duty cycle and Al fraction between layers in the NARROW core. (After Ref. 41)(d) Intensity of SH light propagating toward the waveguide surface (thick solid curve) and the fundamental TE, mode (dotted curve) and the refractive-index profile (thin solid curve) for the NARROW with (i) 0.5 and (ii) 0.25 duty cycles. (After Ref. 41)

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fraction and a duty cycle of the NARROW core layers. The duty cycle being the ratio between the generating layer thickness and the layer-pair period normalised by the appropriate refractive indices. The variation of the cross section A”’as a function of these two parameters for SESHG emission at 0.532 pm is shown in Fig. 6(c) where the spacing layer has an Al fraction of 0.9. From that calculation a maximum cross section is a obtained for 30% Al fraction and 0.25 duty cycle. The optimum duty cycle is noted to be different to that normally expected from simple considerations of quasi-phase matching where a value of 0.5 would be expected. This difference is the consequence of taking account of the effects of absorption which is particularly important in the relatively thick NARROW structure where second harmonic light generated in layers near the substrate may experience a substantial absorption loss along its propagating path to the waveguide surface. Figure 6 illustrates how absorption affects the distribution of second harmonic through the layers of NARROWS with duty cycles of 0.5 and 0.25. It is found that for the 0.5 duty cycle the absorption of the low Al generating layer near the surface causes the harmonic intensity to decrease before being radiated out of the surface. In contrast, the effect of generating layer absorption is relatively small resulting ina larger net second harmonic intensity radiating from the waveguide. On the basis Dai et al. predict that a suitable NARROW structure for blue emission (at a wavelength of 0.46 pm) would be formed with Al fractions of 0.3 and 0.9 alternating on a duty cycle of 0.1. The generation of microwave and millimeter waves using nonlinear optical mixing has also been examined by Qu and Ruda.(‘05)The approach is to utilise difference frequency mixing in asymmetric quantum wells to obtain frequency tunable generation. Advantage is taken of the large second order susceptibility of the asymmetric structures together with the opportunities for tunability of optical pump sources to obtain a flexible microwave/millimeter source. The generation of mid-infra-red radiation is of considerable interest for a number of applications including pollution monitoring, laser radar, non-invasive medical diagnosis and intelligent process controls. One approach to this objective is via difference frequency generation using near-infra-red semiconductor laser sources in combination with intersubband processes in quantum well materials. (39)This work utilised semiconductor laser sources operating at wavelengths near 2 pm to address an InGaAs/AlAs quantum well structure with energy transitions near 9 pm and 2 pm. The relevant SHG coefficient was measured to be 65 times that of GaAs. Three wave mixing is an alternative approach with essentially the same operating principle.@4) 6.4. Diagnostics The availability of second harmonic radiation from semiconductor device sources is of interest in a number of diverse measurement and diagnostic contexts. In a very fundamental context it has been proposed that quantum non-demolition measurements be effected by second harmonic generation. 14s)Optical parametric amplification has also been proposed as a means for quantum noise reduction.(‘4) Measurement of basic optical properties such as coherent phase and frequency detection has been described using sum frequency generation.“@ The integrable parametric waveguide spectrometer developed by Vakhshoori et a1.(13g* MO, 143) is capable of resolving the modes of a semiconductor laser. Here sum-frequency generation of counter-propagating modes is used to identify the laser modes. When the modes contain more than one frequency component then the sum-frequency output will be surface-radiated at angles determined by the propagation constant mismatch. This approach allows the determination of absolute and relative frequencies and is thus of wider use in the context of frequency division multiplexing communications systems. The surface emission of

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second-harmonic radiation from counter propagating nonlinear mixing is also amenable for use in spectrometer and deformation sensors.(56) Use has been made of internally generated second harmonic radiation from a semiconductor laser source in absorption spectroscopy. (‘13)Due to the availability of a sensitive and low noise photomultiplier was available in the wavelength range 380-410 nm a few pica-watts of generated SHG radiation was sufficient to to identify absorption lines associated with potassium and aluminium. A requirement exists for sources of tunable ultra-violet (UV) radiation for detection of a number of molecules which fluoresce in the 200-300 nm spectral region. Reliable laser diodes with wavelengths less than 600 nm are not available and hence the UV spectral range cannot be accessed via second harmonic generation. With the availability of high-power semiconductor lasers interest has been shown in effecting frequency quadrupling or fourth harmonic generation (FHG) of laser diode light in order to generate the required UV radiation. The first efforts in this direction used a potassium niobate crystal to frequency double the 860 nm output of a mode-locked external cavity laser diode combined with a tapered waveguide GaAlAs amplifier. The resulting 430 nm radiation is subject to a further frequency doubling using a p-- BaB,O, (BBO) crystal to produce about 15 ,LLWof UV radiation.‘60’ The generation of second harmonic radiation in silicon devices is also useful as a diagnostic tool. Analysis of thin film interfaces such as oxide-nitride-oxide (ONO) films occurring in metal-oxide-semiconductor (MOS) gate insulators has also been affected via optical second harmonic generation.(‘46’The essence of this approach is that a second harmonic response only occurs at interfaces where the centrosymmetry of the silicon crystal structure is broken.(‘62) The approach has been used to study wet oxidation where the second harmonic intensity increases as the oxide thickness increases. (72)A study has also been made of annealing of silicon dioxide/silicon interfaces using hydrogen and nitrogen.@” Nonlinear optical processes, including second harmonic generation, are very generally used in precision measurements of optical frequencies. In such measurements it is important to be able to define the precision to which the ratio between the fundamental and second harmonic optical frequencies is known. Small deviations in the frequency ratio in nonlinear optical processes can arise, for example, due to finite linewidth of the fundamental and due to frequency-dependent phase matching conditions of the nonlinear response. Recent work by Wynands et ~1.~‘~~) has indicated an upper limit for frequency-dependent shifts of order lo-l4 with opportunities for improvement by another order of magnitude. Such precision would be of interest in the construction of optical clocks. Sharp optical resonances could serve as a primary standard which can be transfered into the radio-frequency regime by means of several nonlinear optical processes in suitable crystals.

7.

FUTURE

PROSPECTS

7.1. Material systems The emphasis of the present article has been on the use of structures and devices, fabricated in the GaAs/AlGaAs quantum well system, for frequency doubling and mixing. Implementation of these processes using other materials is evidence of the wide interest in In the GaAs/AlGaAs system the exploitation of second order nonlinearities. (27~28) consideration has been given to opportunities for optimising second-order nonlinearities associated with quantum size effect (QSE) subband states formed by asymmetric conduction band potentials.@” The use of synthetic semiconductor structures based on these principles PQE.?O+C

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is predicted to offer a 10 to 100 fold increase in nonlinear coefficients compared with bulk materials. Attention has been given to the opportunities for enhancement of the nonlinearity using increased quantum confinement in quantum dots. (6) The approach is to utilise lateral confinement for manipulating asymmetric electron wavefunctions with a view to maximising the dipole moment elements which determine the strength of the nonlinearity. Fabrication of such laterally asymmetric quantum dots is proposed via the use of a modulation-doped heterostructure or a quantum well structure. Control of the lateral asymmetry is effected using a gate electrode thus providing a device with tunable nonlinear coefficients and is suitable for quasi-phase matching in waveguide structures. Enhanced nonlinearities using both interband and intraband transitions in low dimensional semiconductors have been proposed by Shimuzu et al. (‘21)The approach is suggested as being generally applicable to quantum well, quantum wire or quantum dot structures. It is necessary to create an asymmetry in the structures and the application of an electric field is seen as a practical means for this purpose. It is argued that large nonlinearities arise in this case due to resonance enhancement based on the discreteness and sharpness of the quantum levels in low-dimensional semiconductors. As such the effects would be anticipated to arise in a range of low-dimensional materials including organic materials.@2) The approach is suggested as a means for down conversion of light from semiconductor lasers to intersubband frequencies. Modulation of the applied electric field would give rise to intensity modulation of the down-converted light with modulation frequencies up to tens of GHz being feasible. Work on materials in the GaSbInAsSb system has suggested that significant improvements can be achieved in second order susceptibilities responsive at the optical communications wavelength of 1.5 pm. (‘14)The basic approach taken here is to obtain a double resonance in the denominator of the susceptibility. Working in the same material system a mechanism for normal incidence second harmonic generation has been proposed by Xie et a1.“55v’56)It was noted that emphasis has been given to SHG associated with intersubband transitions near the spherically symmetric r point where selection rules permit only one non-zero component of the second-oder susceptibility. Experiments using Brewster angle coupling would then achieve a relatively weak coupling to the intersubband optical interaction. Normal incidence, surface-emitting SHG can be obtained with an L valley transition in AlSb/GaSb/GaAlSb/AlSb quantum wells. The existence of L-valley transition matrix elements in the plane of the quantum well is assured when the quantum well growth direction is tilted with respect to the principal axes of the L-valley constant energy ellipsoids.(15’.15’) Optically pumped up-conversion lasers produce output at wavelengths between violet and red using infra-red pump wavelengths from semiconductor lasers. Such upconversion lasers are commonly fabricated in materials such as BaYYbF and YLiF. An obvious interest exists in the integration of up-conversion lasers with the semiconductor laser optical pump. Progress has been reported on the heteroepitaxial growth of BaYYbF, on GaAs using a CaF or LiF buffer layer to form the required optical waveguide. (‘O)Both (100) and (111) GaAs substrates were used for this purpose. The resulting structure was addressed by a 647 nm laser and shown to generate UV and visible radiation at 360 nm, 450-480 nm and 50&550 nm. It is recognised that III-V nitride materials offer great promise for optical devices over the visible and UV wavelength range. Aluminium nitride has received particular attention because of its piezo-electric properties but the non-centro-symmetric structure of the material makes it a clear candidate for second order nonlinearity. Early work”*, 19.57*90, 12’)confirmed the generation of second harmonics but failed to access the full nonlinearity expected from theory.@) The use of a waveguide geometry where phase-matching conditions can be achieved by tuning the fundamental wavelength has lead to second harmonic generation efficiencies

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of 4 x 10 - * which is comparable to that achieved using lithium niobate and poled dye-polymer waveguides. The development of light sources for optical storage and information processing has driven an activity in the development of blue light emitters using II-VI materials such as ZnSe. Wide gap II-VI semiconductors possess strong second order susceptibilities amenable to efficient second harmonic generation of near-infra-red radiation. Waveguide devices in the ZnSe/ZnTe/ZnSe/GaAs system have been grown by Wagner et ~1.~‘~~) and second harmonic generation has been observed using end-fire coupling. Phase matching was achieved by coupling a TE fundamental mode with a TM second harmonic mode. Work on CdTe has reported green light generation following picosecond pulse pulse excitation at a wavelength of 1064 nm.(‘04’In this case the SHG process makes a noticeable contribution to the generation of a free-carrier plasma in this material which is transparent at the fundamental but absorbing in the region of the frequency doubled light. It is suggested that CdTe may have a role as a relatively cheap and robust frequency doubling material which may be used in association with Nd: YAG lasers e.g. to perform a infra-red to visible up-conversion as a means for detecting the location of near-infra-red beams. As already indicated, it has been advances in III-V semiconductor growth technology which has facilitated many of the developments indicated above. There has been a parallel development in material processing capabilities in other materials and, in particular, epitaxial growth in the Si-Ge material system has matured to be capable of providing good interface properties in quantum structures of Si and SiGe grown on Si substrates. Such structures have been used for intersubband related second harmonic generation using Si/SiGe asymmetric quantum wells.(“‘) The material system offers two particular contrasting features as compared with III-V quantum wells. In the first place the inversion symmetry of Si precludes second harmonic generation from bulk material. Secondly, most of the band offset occurs in the valence band and use is thus made of p-doped valence band quantum well transitions. In this case the fundamental does not necessarily have to be polarised along the growth direction: several non-vanishing off-diagonal elements of the second order susceptibility have been predicted for the SiGe hole system. “‘*) Experiments undertaken using a Q-switched carbon dioxide laser indicated a nonlinear susceptibility of 5 x 10 - * m/V and a quadratic dependence of second harmonic power on incident power was demonstrated up to 75 W and giving second harmonic output powers up to 10 mw.(“‘) 7.2. Devices and structures Multilayer waveguide structures offer opportunities for performing a range of optical and harmonic generation and mixing functions.“@ Work by Rikken”“” has indicated the opportunity for achieving wavelength-uncritical SHG in a four layer waveguide where phase matching is achieved through modal dispersion. The use of modal dispersion for effecting phase matching is a well-established approachO’ and has been applied to several material The efficiency of the harmonic systems including organic crystals(65’and poled polymers. (130’ generation process is proportional to the spatial overlap of the fundamental and second harmonic waves.“**’Application of this concept in multilayer guides to achieve a large modal Specific examples of the approach have overlap has also been explored in earlier work. (H)*73,*9’ been adduced for four layer waveguide structures in Si,NdSiO, and poled polymers.“08’ This work has been extended to suggest that SHG bandwidths of order 10 nm to 100 nm can be achieved within reasonable fabricational tolerances for four layer waveguides in these materials. Such structures should be amenable to operation with semiconductor lasers. Quasi-phase-matching (QPM) layers formed with GaAs/AlAs multilayers are an essential element of semiconductor SHG devices reported by a number of workers’74.75. ‘37*‘3*’ and discussed above in Section 2.4. The GaAs/AlAs stack forms a QPM layer because GaAs and AlAs have different nonlinear coefficients. (‘02)Practical limitations exist to using such QPM

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layers to enhance the performance of semiconductor SH generators. In particular due to strong absorption of GaAs at the relevant wavelengths thickening the QPM layer cannot be used to help amplify the second harmonic. The SHG conversion efficiency can be improved, however, by inceasing the power of the fundamental. To this end Nakagawa et ~1.‘~”have used a high Q vertical cavity semiconductor structure to demonstrate SHG at 490 nm. Fig. 7(a) shows the theoretical prediction of the effect of absorption on the distribution of second harmonic power. Corresponding experimental results are shown in Fig. 7(b). The vertical cavity was formed by a GaAs/AlAs distributed Bragg reflector (DBR) bottom mirror and a Ti02/Si02 DBR top mirror. The latter is highly transmissive at the second 0)

(a) 3

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Fundamental power (mW) Fig. 7. (a) Distribution of the SH power calculated for the fundamental wavelength of 980 nm in (i) conventional half-wavelength QPM layers and (ii) QPM layers taking into account the absorption of SH power. The rectangular curves show the distribution of nonlinear coefficient, in which the higher and lower levels indicate GaAs and AlAs, respectively. (After Ref. 95) (b) SH power emitted from the vertical cavity as a function of input fundamental power. The SH wavelength is 492 nm. (After Ref. 95)

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harmonic wavelength. Due to the crystal symmetry, the second harmonic field is collinearly generated from the fundamental resonating vertically only if the structure is grown on a substrate whose orientation is other than (100). In the work of Nakagawa et al. the growth was performed on a (31 l)B GaAs substrate. Using this structure an efficiency of 1.4 x 10e4 % /W was m e a sured for the second harmonic generation. It is indicated that optimisation of two features of the device offer opportunities for significant improvements in the conversion efficiency. The requirements in this respect are to (i) achieve a better confinement of the fundamental and (ii) to achieve better control of the growth of the QPM layers. It is suggested that with such an optimised device an efficiency up to about 15% /W could be expected. Additional functionality can also be sought from second harmonic generators. Early work has pointed to the opportunities for achieving passive and active modelocking via intracavity optical frequency mixing. (‘26)More recently all-optical transistor action has been reported by Lefort and Barthelemy (U albeit using a KTP nonlinear crystal. Electroluminescence diodes fabricated in organic materials are of interest because of their potential as low-voltage sources for display applications. “. 2,13’.13’)Work has been reported on red to green upconversion using an organic electroluminescent diode combined with photoresponsive amorphous silicon carbide. (“) This arrangement provides photocontrol of the charge injection which is the basis of the light emission. The light transducer is seen to have applications in IR-visible image conversion and a laser diode driven electroluminescence displays. 8. CONCLUSION

An overview has been given of the opportunities for the second order nonlinearities in semiconductor structures. Considerable progress has been made in improving the performance of semiconductor devices which utilise these nonlinearities. Several challenges remain to be met in order to achieve the full potential of such devices. The widespread commercial and scientific applications of harmonic mixing in semiconductors should ensure a continued effort aimed at overcoming those challenges. AcknoM,/edgemenrs-This

work has been undertaken within EPSRC project GR/H30687.

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