A novel numerical technique for solving the one-dimensional Schroedinger equation using matrix approach-application to quantum well structures

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A Novel Numerical Technique for Solving the One- Dimensional Schroedinger Equation Using Matrix

Approach—Application to Quantum Well Structures

AJOY K. GHATAK, K. THYAGARAJAN, AND M. R. SHENOY

Abstract—In this paper we present a novel numerical technique which allows straightforward determination of bound state and quasi-bound state energy eigenvalues (and lifetimes of the latter) for arbitrary one- dimensional potentials. The method involves straightforward multipli- cation of 2 x 2 matrices and does not involve any iterations. The ap- plicability of the technique to the analysis of quantum well structures has also been shown. Since the Schroedinger equation for a spherically symmetric potential can be transformed to a one-dimensional equa- tion, all such problems can also be solved using the present method.

I. INTRODUCTION

R

ECENT advances in the fabrication of thin semicon- ductor layers have added a new dimension to both the application and perspective of the traditional discussion of the particle in a potential well. Fabrication of quantum well heterostructures and superlattices have allowed tail- oring of optical and electronic responses of the semicon- ductor materials, resulting in a considerable increase in research activity directed towards the development of novel electronic and optoelectronic devices [1]. In order to understand the physical properties of these heterostruc- ture devices, one has to solve the eigenvalue problem of a charged particle in the quantum well.

It is well known that exact analytical solutions to such problems are available only for simple potential structures such as the square well, parabolic well, etc., and even in these structures, in the presence of perturbations such as electric fields, etc., the problem in general cannot be solved exactly. The variational method has been widely used for studying quantum well structures in the presence of perturbations like electric field [2]; however, the method does not give the correct value of the wave function outside the well, and does not predict the lifetime of quasi-bound states. The energy level and wave function depend upon the choice of the trial function, which becomes very difficult to make for structures involving more than one quantum well, and also when the potential energy variation is graded. The Monte Carlo method used

Manuscript received July 10, 1987; revised March 23, 1988. This work was supported in part by the U.S. National Bureau of Standards and by the Department of Science and Technology of India.

The authors are with the Department of Physics, Indian Institute of Technology, New Delhi 110016, India.

IEEE Log Number 8821735.

by Singh [3], also solves the variational problem. And, as is well known, the variational method is not convenient for states other than the ground state.

In this paper we present a matrix method involving straightforward multiplication of 2 x 2 matrices, using which one can obtain bound state eigenvalues, quasi- bound state energy values, lifetimes of the quasi-bound states, and the corresponding eigenfunctions of arbitrarily graded, one-dimensional potential wells. The technique is based on a matrix method that was recently developed for the determination of the propagation characteristics of arbitrarily graded planar optical waveguides [4]. The applicability of the method is shown for the analysis of quantum-well structures with one or more wells. It is well known that in the presence of a static electric field the bound states become "leaky" and using the present technique one can determine the energy values, wave functions and lifetimes of such quasi-bound states. Since the Schroedinger equation for a spherically symmetric potential can be transformed to a one-dimensional equation, all such problems can also be solved by using the present method. We should mention here that matrix methods have earlier been used for solving the Schroedinger equation for multiple quantum well/barrier structures [5], [6];

however, these methods usually involve iterations and do not give the lifetime of quasi-bound states, and often re- quire the wave function to be "well-behaved" [5].

In Section II, we briefly discuss the matrix method and its applicability to the potential-well problem, using the equivalence of a refractive index "hump" and a potential

"well." In Section III, we demonstrate the usefulness of the method to obtain the bound and quasi-bound states of a single quantum-well, as well as to a double quantum- well structure in the presence of an applied electric field.

II. THEORY

The bound-state energies of a one-dimensional finite potential well are given by the solutions of the time-in- dependent Schroedinger equation

(2)

which may be written in the form

(2) where V(x) represents the potential energy variation, m is the effective mass of the charged particle, and h = h/2ir, h being Planck's constant. E and ip respectively represent the energy eigenvalue and eigenfunction of the bound state. The solution of (2) for a simple square-well and a few other specific forms of V(x) is well known [7], [8]. We now recall the equivalence between the potential well problem and the optical waveguide problem; if we write

2 (3)

(4)

(5) V(x) = n\- n\x); E = n\ -

where n0 is a (suitably chosen) constant, and

= 2m/h2

then (2) takes the form

which is identical to that of the equation satisfied by TE modes in a planar waveguide described by the refractive index distribution n(x), with 0 representing the propa- gation constants of the various modes [9]. Since the boundary conditions on the wave function \p in a potential well are the same as those for TE modes in a dielectric waveguide, the matrix method described in [4] should be applicable to the potential well problem, and in our ex- amples below we have actually used (3)-(5) to solve the Schroedinger equation.

We now consider a potential energy variation V(x) as shown in Fig. l(a). Such a structure would have a finite number of bound states corresponding to Fmin < E < V.

Although all values of E > V are allowed, we will have a finite number of quasi-bound states corresponding to V

< E < V", similar to what we have in the alpha-decay problem. In order to obtain the eigenvalues corresponding to the bound and quasi-bound states, and the lifetimes of the latter, we introduce, by analogy with the waveguide problem [4], a region of potential energy which is less than or equal to Fmin as shown in Fig. l(b); we also choose the origin at the point where we introduce the potential.

Referring to Fig. l(b), the solution of the Schroedinger equation in each region may be written in the form

(6) where

A, = A2 = 0, A; = ki(d2 + ^3 +

i = 3, 4,

*, = \^(E- Vt)

*,--i);

1/2

(7) (8)

(b)

Fig. 1. (a) A general potential energy variation having a finite number of bound states, (b) In order to obtain the bound state and quasi-bound state energy eigenvalues for the potential energy variation shown in (a), we introduce a region of potential energy K, ( < Kmin) and obtain the solu- tions as d-, -» oo.

M,+ and w, represent the amplitudes of the waves propa- gating along the +x and — x directions, respectively.

Continuity of u and du/dx at each interface gives us [10]

u3

where

s -

l-

7; =

, - 1 6 ,

h = bt =

(9)

(10)

(11) / = 1,2, • • • , N — 1, N being the total number of re- gions. In the last region u^ would represent an exponen- tially amplifying solution and should vanish. Setting u^

= 0, we can calculate the amplitude of the wave function in any region in terms of wj1". The matrix relations (9)- (11) that we have obtained here are of identical form as those in [4]. By analogy, for the potential energy variation shown in Fig. l(b) if we calculate | u% /u\ |2 as a function of E, we would obtain Lorentzians corresponding to each bound state and quasi-bound state of the form (see the Appendix)

(E

- Erf + r

² ⁽¹²⁾

(3)

where E = E'b represents the position of the peak and 2F, the full width at half maximum (FWHM). As the distance d² is increased, E'^b would tend to the value E^b which represents the energy of the bound (or quasi-bound) state; the quantity F would tend to zero for bound states and to a nonzero value for quasi-bound states. The limiting values of h/2T would be the lifetime of the quasi-bound state (see Appendix).

Thus in order to obtain the energies and lifetimes of the various states of an arbitrarily graded potential well, we adopt the following procedure.

1) We first introduce a region of potential energy equal to or less than the minimum value inside the potential well, beyond the well.

2) We then calculate the ratio | « / /u\ |² as a function of E, where uf corresponds to theyth medium.

3) We fit a Lorentzian of the form given by (12) and obtain E'b and F.

4) We let the fictitious region move to oo and obtain the limiting values of E'b and F, which give the eigenvalue and lifetime of the bound/quasi-bound state. In practice, the fictitious region is moved sufficiently far for E'b and F to converge to the required accuracy.

We must mention here that an arbitrarily graded potential energy variation can be represented by a large number of steps and solved in a similar manner. However, if there is more than one well in the structure, we have to calculate the ratio | « / /w* |² inside each of the wells so that all energy states are obtained.

III. APPLICATION TO QUANTUM WELL STRUCTURES

In order to show the applicability of the proposed technique to quantum well structures we first consider a conduction electron in a finite GaAs-Al^{0 38}Gao⁶²As single quantum well of depth Vo = 0.4 eV (which corresponds to an 85:15 splitting between the conduction and valance band) and of width 30 A with an effective mass m = 0.067m0. These parameters correspond to the calculations of Bastard et al. [2] and are chosen so that only a single bound state exists. For such a case, the eigenvalue equation is

V(x)

a tan aa = y where a and y are given by

2mE 2 _ 2m(V0 - E) H2 ' y ~ h2

(13)

(14) and la is the width of the potential well. The value of E

= 178.27 meV, obtained by using the matrix technique indeed satisfies (13).

In [4] we have shown that the matrix technique can be used to obtain with arbitrary accuracy the propagation constant in optical waveguides. Since, as described earlier, the quantum well problem is identical to the optical waveguide problem, energy eigenvalues of arbitrarily graded potential wells can also be obtained with any de- sired accuracy.

- a o o (a)

V(x)

(b)

Fig. 2. (a) A simple square potential well of depth Vo and width 2a. (b) The potential energy variation for an electron, corresponding to the well shown in (a) in the presence of an applied electric field.

The solution for a quantum well in the presence of an applied electric field is of considerable importance since the photoluminescence of quantum well structures is highly sensitive to the external electric field [11], [12].

We consider a quantum well in the presence of an applied electric field and calculate the field-induced variation of the bound-state energy and the intraband tunneling times out of the well. In the presence of an electric field 8 along the direction x the potential energy variation is given by

V(x) = Ko + eZx (15) where e represents the electric charge of the carrier. Fig.

2 shows the corresponding profile for an electron in a sim- ple square well of depth V^o. In the presence of an electric field there is no true bound state for a finite quantum well since the potential energy is negative and large in absolute value for large negative values of x. However, if the ap- plied field is not excessively large, the states will have a long lifetime and can therefore be considered as quasi- bound.

In the presence of an applied electric field we employ the matrix method by dividing the potential energy profile into a large number of layers (typically about 100) of ho- mogeneous regions [see Fig. 3(a)], and calculate the ratio I uplu\ |2 as a function off; herep corresponds to a layer inside the well. Numerical values of the well parameters 2a and V^o and the effective mass of the electron are the same as those used in the previous example. The origins of distance and of electrostatic potential are chosen at the centre of the well [see Fig. 3(a)]. Fig. 3(b) shows the variation of the energy shift AE with the applied electric field; the dashed curve corresponds to the variational calculations of [2] and the solid curves correspond to our calculations. We may point out here that the corresponding results for A£ using the variational and exact analyt-

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GHATAK el al.: NUMERICAL TECHNIQUE FOR SOLVING THE ID SCHROEDINGER EQUATION

V(x)

region

Ref. ( 2 ) Present method

• — - ^ ^

/ /

/

3 200

£ ( k V / c m )

— ' —

\ 400

,N

io12

108

IO4 1

(sec

(a) (b)

Fig. 3. (a) A single quantum well in the presence of an applied electric field and the corresponding staircase approximation for applying the matrix method, (b) Variation of the shift of energy eigenvalue and the inverse of lifetime of the quasi-bound state in a single quantum well with applied electric field.

x( A)

Fig. 4. The wave function corresponding to the quasi-bound state for applied electric fields of 100 and 400 kV/cm. Even for 8 = 100 kV/cm the wave function does show an oscillatory behavior at large distances from the center, but the amplitude of oscillation is very small.

ical technique show similar behavior in the case of an in- finite potential well [13].

Using our technique we have also estimated the lifetime of the quasi-bound state in the presence of an electric field and the results are also shown in Fig. 3(b). It can be seen that the lifetime is drastically reduced as the electric field is increased. Fig. 4 shows the wave function of the quasi- bound state for applied electric fields of 100 and 400 kV/cm. The oscillatory nature of the wave function in the latter case is obvious and is consistent with the extremely short lifetime of the state. The shift of the peak in the wave function inside the well to x > 0 is apparent for both cases.

As another example we consider a 60 A wide quantum well formed in GaAs-Al0 3Ga0 7As which corresponds to Vo = 0.2144 eV. (We have used the 60:40 splitting of AEg between the conduction and valence bands and the relation AEg = 1.425* - 0.90jt2 + l.lx3 [13].) Such a well supports two bound states in the absence of any ap- plied electric field with £, =64.71 meV and E2 = 212.72 meV. It may be noted that the n = 2 state is very close to the continuum of states which appears beyond E = 214.40 meV. Fig. 5 shows the variation of the shift in the n = 1 and n = 2 energy levels and the corresponding

2.0

6 ( k V / c m )

Fig. 5. Solid curves represent the variation of the shift in energy value of the n = 1 and n = 2 quasi-bound states corresponding to a single quan- tum well of width 60 A and depth 0.4 eV. The dashed curve is the corresponding variation of the inverse of lifetime of the n = 2 state. For the electric fields considered, the inverse of lifetime of the n = 1 state is almost zero. For S > 10 kV/cm the potential well supports only the n = 1 state.

lifetimes with the applied electric field. The figure clearly demonstrates the extremely short lifetime of the n = 2 state with applied electric fields as low as 10 kV/cm. The

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same electric field has almost no effect on the n = 1 state.

A quantum well structure possessing such a characteristic is indeed being studied as a potential infrared detector [14].

We may mention here that the method described by Kolbas and Holonyak [5] cannot be directly applied to the calculations of lifetimes of quasi-bound states. However, if one assumes the presence of only an outgoing wave solution in the region x < 0 [in Fig. l(b)], then one can obtain an eigenvalue equation for the quasi-bound states, the solution of which would give the corresponding energy values and lifetimes [15]. Even for a step potential well in the presence of an electric field the eigenvalue equation would involve Airy functions of complex argu- ments, the solution of which would become very cumber- some.

We have also carried out calculations for a double-well structure in which each well has the same parameters as before corresponding to Fig. 2 and the wells are separated by 50 A. Since the two wells are close to each other, the interaction between the wells splits the energy levels. Ta- ble I gives the splitting of the eigenvalues of the conduction electron in a double well for different values of the applied electric field. On applying the electric field the two wells become nonidentical. The calculations show that as the electric field increases, the wave function corresponding to the two eigenvalues becomes more and more confined to the individual wells, resulting in reduced interaction. Fig. 6 shows the electron wave function corresponding to the symmetric and antisymmetric states of the double well. As can be seen from the figure, an applied electric field breaks the symmetry of the eigenfunctions. For 8 = 40 kV/cm the amplitude of each quasi- bound state is about one order of magnitude larger in the well to which it is preferentially confined than in the other well.

Recently Bajema et al. [16] have reported measure- ments of the field dependence of n = 1 to n = 2 inter- subband transitions of an electron in the conduction band of a GaAs-Al0 3Gao.7As multiple quantum well structure using Raman spectroscopy. The wells are of thickness = 264 A, separated by 198 A thick barriers. The barrier width of 198 A leads essentially to uncoupled wells and hence the problem is equivalent to that of a single quantum well. Corresponding to the values of the parameters in [16], with Vo = 157 meV, we find that the energy dif- ference between the n = 1 and n = 2 states with 8 = 0 is 18.4 meV and the shift in this difference with the applied electric field is identical to the calculations of Ba- jema et al. [16]. We have also found that for 8 < 12 kV/cm, the states have long lifetimes and the variational technique used by Bajema et al. [16] gives sufficiently accurate results.

Although we have presented results only for a single and a double quantum well, the analysis can easily be applied for calculations in multiple quantum well and arbitrarily graded quantum well structures without any addi- tional complexity.

TABLE I

SPLITTING IN THE ENERGY VALUES OF A CONDUCTION ELECTRON IN A DOUBLE QUANTUM WELL STRUCTURE IN THE PRESENCE OF A STATIC ELECTRIC FIELD 6"

Electric Field

( k V / c m ) (meV)

0 20 40 100

9.01 18.38 33.28 80.67

"The various parameters used are: well depth = 0.4 eV (corresponding to an 85:15 splitting of a GaAs

— Al0 38Gao.62As quantum well), well width = 30 A, barrier width = 50 A, effective mass = 0.067m0.

Fig. 6. The wave function corresponding to the symmetric and antisymmetric states of a double quantum well in the presence of an electric field S = 40 kV/cm. Each well is of width 30 A and depth 0.4 eV, and the separation between them is 50 A.

IV. SUMMARY

We have presented a novel technique for solving the one-dimensional Schroedinger equation using a matrix approach, and have demonstrated the applicability of the method in the study of quantum well structures. In partic- ular, we have considered single and double quantum well structures and have studied their characteristics in the presence of an applied electric field. The proposed method involves only straightforward multiplication of 2 x 2 matrices and is simple to implement on a computer. The present numerical technique yields the energy eigenvalues, the wave function, and can even predict the lifetime of quasi-bound states making it a valuable tool in the analysis of multiple quantum well structures.

APPENDIX

We first consider a general potential energy variation as shown in Fig. 7(a). The solution of the Schroedinger

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GHATAK et al.: NUMERICAL TECHNIQUE FOR SOLVING THE ID SCHROEDINGER EQUATION

V(x)

Vo

0 x,

(b)

Fig. 7. (a) A general potential well described by V(x), and (b) the cor- responding potential energy variation for applying the matrix method.

It is obvious that for E satisfying (A2), B must vanish and therefore the oscillatory solution in region III will have a very small amplitude. These are the quasi-bound states of the leaky structure shown in Fig. 7(b).

We assume that at t = 0 the wave function of the par- ticle in the potential well shown in Fig. 7(b) is given by

\j/b(x). We may write \pb in terms of the complete set of functions given by (A3). Thus

t = 0) =

= j

X(E) +(E, x) dE. (A6) It may be noted that Hx) has dimensions of (energy • length)"'/2, while \pb(x) has dimensions of (length)~'^.

Thus x(E) has dimension of (energy)~'/2.

Now since the wave function at t = 0 corresponds to

\^b(x) and since \[/(x) shows a behavior similar to \l/b(x) only for E — Eb, x(E) will be sharply peaked around E

= Eh. Thus equation is given by

\l/b(x) = Ab<i

where

= Ah

7

x < Xi

X > X] (Al) 2m(V0- E)

Here 4>(x) is the well-behaved solution in the region 0 <

x < xx and for the sake of simplicity we have assumed V(x) = oo at x = 0 so that 4>(x) is that solution which vanishes at x = 0; the subscript b stands for bound state.

Applying the boundary condition at x = X\ we obtain 1

dx (A2)

which represents the transcendental equation determining the descrete energy eigenvalues. We next consider the profile shown in Fig. 7(b). The solutions in the various regions may be written as

Hx) = A<j)(x); 0 < x < x,

= Be>{x-xx) + ce-^x-xl); x{ < x < x2

= D+eia(x~X2) D_e-ia(x-x2). x X2

(A3) where now all values of E are allowed and one can deter- mine the coefficients B, C, and D± in terms of A. The normalization condition

!/-*(£", x) t(E, x)dx = 8(E - £ ' ) (A4) gives [17]

1/2

(A5)

HE, x) = — Ab

(A7) Using a method similar to that given in [17], [18] we obtain

(A8) Now using the boundary conditions at x = x{ and x = x2, we obtain relations between D+ and A. Making a Taylor expansion around E = Eb, we can show that

where

A (E

2ay (a2 + y2

2aT a

- Erf + r

²

4>(xi) .-27(«-,n ) ab

(A9)

(A10)

(All)

, _ 1 dB

b ~ AdE EL = Eh-

: = E^b

(a22-

(A12) Equation (A9) is of the same form as (12). Equation (A9) shows that there is a shift in the energy eigenvalue due to the presence of the newly introduced region III, and as x2

-> oo, E'b -» Eb as it should.

Now the probability of finding the particle in the poten-

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tial well at any time t is given by P(t) =

Using

" Do

and (A8), we obtain P(t) =

(A13)

(A14)

(A15) Thus h/2T would represent the tunneling time of the par- ticle. Therefore, for known analytical forms of (j>(x) one can obtain analytical expressions for E'b and T. For ex- ample, for a step potential well (in the region 0 < x <

Xi) with potential energy Vo

<j>(x) = sin ax (A16) which gives

r =

_{( a}₂ ^-2y(X2-x\) ^(A17)

which is consistent with the corresponding expressions for the propagation constant and the leakage loss coefficient obtained for a step index planar optical waveguide [4], [17].

REFERENCES

[1] See, e.g., 1EEEJ. Quantum Electron., Special Issue on Semiconduc- tor Quantum Wells and Superlattices: Physics and Applications, vol.

QE-22, Sept. 1986.

[2] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, "Variational calculations on a quantum well in an electric field," Phys. Rev. B, vol. 28, pp. 3241-3245, 1983.

[3] J. Singh, "A new method for solving the ground-state problem in arbitrary quantum wells: Application to electron-hole quasi-bound levels in quantum wells under high electric field," Appl. Phys. Lett., vol. 48, pp. 434-436, 1986.

[4] A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical anal- ysis of planar optical waveguides using matrix approach," J. Light- wave Technol., vol. LT-5, pp. 660-667, 1987.

[5] R. M. Kolbas and N. Holonyak, Jr., "Man-made quantum wells: A new perspective on the finite square-well problem," Amer. J. Phys., vol. 52, pp. 431-437, 1984.

[6] B. Ricco and M. Ya. Abzel, "Physics of resonant tunneling: The one- dimensional double-barrier case," Phys. Rev. B, vol. 29, pp. 1970- 1981, 1984.

[7] L. I. Schiff, Quantum Mechanics, 2nd ed. New York: McGraw- Hill, 1968.

[8] A. K. Ghatak and S. Lokanathan, Quantum Mechanics: Theory and Applications, 3rd ed. New Delhi, India: Macmillan India Ltd., 1984.

[9] M. J. Adams, An Introduction to Optical Waveguides. Chichester, England: Wiley, 1981.

[10] We are assuming continuity of the wave function and its derivative (rather than the derivative divided by the effective mass) across the boundary which is reasonably accurate in most of the cases.

[11] E. E. Mendez, G. Bastard, L. L. Chang, L. Esaki, H. Morkoc, and R. Fisher, "Effect of an electric field on the luminescence of GaAs quantum wells," Phys. Rev. B., vol. 26, pp. 7101-7104, 1982.

[12] J. A. Kash, E. E. Mendez, and H. Morkoc, "Effective field induced decrease of photoluminescence lifetime in GaAs quantum wells,"

Appl. Phys. Lett., vol. 46, pp. 173-175, 1985.

[13] D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W.

Wiegmann, T. H. Wood and C. A. Burrus, "Effective field dependence of optical absorption near the band-gap of quantum well struc- tures," Phys. Rev. B, vol. 32, p. 1043, 1985.

[14] D. D. Coon, R. P. G. Karunasiri, and L. Z. Liu, "Narrowband in- frared detection in multiquantum well structures," Appl. Phys. Lett., vol. 47, p. 289, 1985.

[15] T. B. Bahder, C. A. Morrison, and J. D. Bruno, "Resonant level lifetime in GaAs/AlGaAs double barrier structures," Appl. Phys.

Lett., vol. 51, pp. 1089-1090, 1987.

[16] K. Bajema, R. Merlin, F. Y. Juang, S. C. Hong, J. Singh, and P. K.

Bhattacharya, "Stark effect in GaAs-Al,Ga,_,As quantum wells:

Light scattering by intersubband transitions," Phys. Rev. B, vol. 36, pp. 1300-1302, 1987.

[17] A. K. Ghatak, "Leaky modes in optical waveguides," Opt. Quantum Electron., vol. 17, pp. 311-321, 1985.

[18] A. K. Ghatak, "Electromagnetics of integrated optical waveguides,"

J. Inst. Electron. Telecom. Eng. (India), Special Issue on Optoelec- tronics and Optical Communications, vol. 32, pp. 159-170, 1986.

Ajoy K. Ghatak was born in Lucknow, India, on November 9, 1939. He received the M.Sc. degree in physics from Delhi University, Delhi, India, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1959 and 1963, respectively.

In 1962, he worked for five months at General Atomics, San Diego, CA, and from 1963 to 1964, he was with Brookhaven National Laboratory, Upton, NY. From 1964 to 1966, he was with the Department of Physics, Delhi University, and since 1966, he has been at the Indian Institute of Technology, New Delhi, where he is currently a Professor with the De- partment of Physics. He has held visiting appointments at the Department of Electrical Engineering, Drexel University, Philadelphia, PA; in the De- partments of Applied Mathematics and Engineering Physics, Institute of Advanced Studies, Australian National University; University of New South Wales, Sydney; the National University of Singapore; and the Institut fur Hochfrequenztechnik und Quantenelectronik, Universitat Karlsruhe, West Germany. His current interests are in the theory of optical waveguides and propagation of electromagnetic waves. He has published over 130 research papers and several books including Contemporary Optics (coauthored with K. Thyagarajan, New York: Plenum), Inhomogeneous Optical Waveguides (coauthored with M. S. Sodha, New York: Plenum), Self Focusing of Laser Beams (coauthored with M. S. Sodha and V. K. Tripathi, New Delhi, In- dia: Tata McGraw-Hill), Optics (New Delhi: Tata McGraw-Hill), and La- sers: Theory and Applications (coauthored with K. Thyagarajan, New York:

Plenum). Inhomogeneous Optical Waveguides has been translated into Russian and Chinese, and Optics has been translated into Chinese.

Dr. Ghatak was the recipient of the 1979 Bhatnagar Award, instituted by CSIR, India.

K. Thyagarajan was born in Vishakapatanam, India, on March 29, 1952. He received the B.Sc.

and M.Sc. degrees from the University of Delhi, India, and the Ph.D. degree from the Indian In- stitute of Technology, Delhi, India, in 1971, 1973, and 1976, respectively.

In 1976, he joined the Department of Physics, Indian Institute of Technology, Delhi, where he is currently an Assistant Professor. During 1977-

* """ 1978, he was with the Ecole Normale Superieure, Paris, France, and Central Research Laboratories (LCR), Thomson-CSF, Orsay, France. During 1983-1984, he was on sab-

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GHATAK et al.\ NUMERICAL TECHNIQUE FOR SOLVING THE ID SCHROEDINGER EQUATION

batical leave at LCR, Thomson-CSF, Orsay. His current interests are in the field of fiber and integrated optics, fiber optics devices, fiber optics sensors and applied optics. He has published over 50 research papers, and is also the coauthor (with A. K. Ghatak) of two books: Contemporary Op- tics (New York: Plenum, 1978, reprinted by Macmillan India, 1984) and Lasers: Theory and Applications (New York: Plenum, 1981, reprinted by Macmillan India, 1984) and one review "Graded index optical wave- guides: A review," in Progress in Optics (E. Wolf, Ed.), vol. 18, 1980.

Dr. Thyagaragan is currently an awardee of the Indian National Science Academy (INSA) Research Fellowship. He is also on the Editorial Board of the Journal of Institution of Electronics and Telecommunication Engi- neers (India).

M. R. Shenoy was born in Mangalpady, India, in 1957. He received the B.Sc. and M.Sc. degrees from Mysore University, India, and the Ph.D. degree from the Indian Institute of Technology, Delhi, India, in 1977, 1979, and 1987, respectively.

From 1979 to 1982 he was a teacher with the Department of physics, R. J. College, Bombay, India. Since July 1982 he has been with the De- partment of Physics, Indian Institute of Technol- ogy, Delhi, where he is currently a Senior Scien- tific Officer. His research interests are in the field of fiber and integrated optics.