New Basis Functions for Wave Equation

Scientia Iranica D (2016) 23(6), 2928{2933

Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering www.scientiairanica.com

New basis functions for wave equation

S. Khorasani

School of Electrical Engineering, Sharif University of Technology, Tehran, Iran.

Received 19 April 2016; received in revised form 22 June 2016; accepted 13 August 2016

KEYWORDS Abstract. The Di erential Transfer Matrix Method is extended to the complex plane, which allows dealing with singularities at turning points. The results for real-valued Physical optics; systems are simpli ed and a pair of basis functions are found. These bases are a bit less Quantum optics; accurate than WKB solutions but much easier to work with because of their algebraic form. Electromagnetic Furthermore, these bases exactly satisfy the initial conditions and may go over the turning optics; points without the divergent behavior of WKB solutions. The ndings of this paper allow Inhomogeneous explicit evaluation of eigenvalues of con ned modes with high precision, as demonstrated optical media. by few examples.

1. Introduction having the exact solution using the DTMM [4,5]: Many physical problems in optics and quantum me- Z x chanics are modeled using the equation [1-3]: fF(x)g=Texp [K(t)]dt fF(0)g=[Q0!x]fF(0)g: y00(x) + f(x)y(x) = 0; (1) 0 (4) where f(x) = g(x) + ih(x) and y(x) = u(x) + iv(x) are The transfer matrices [Q(p!q)] observe [4] the self- complex analytic functions. Eq. (1) may be written as: 1 projection [Qp!p] = [1], inversion [Qp!q] = [Qq!p] , d determinant jQ j = exp(trf[K ]g), and decom- fF(x)g = [K(x)]fF(x)g; (2) p!q p!q dx position [Qp!r] = [Qq!r][Qp!q] properties. One may omit the ordering operator T [5-7] to reach the explicit t 0 0 fF(x)g = fu(x) v(x) u (x) v (x)g ; but approximate solution: [0][1] Z [K(x)] = ; x [E(x)] [0] fF(x)g=exp [K(t)dt] fF(0)g= [Q0!x]fF(0)g; 0 (5) g(x) h(x) [E(x)] = ; h(x) g(x) while the trace trf[Q0!x]g [7] and the above rst three in which [0] and [1] are, respectively, the zero and iden- properties remain intact. We now de ne: tity matrices. Using the Di erential Transfer Matrix Z Method (DTMM), the DTMM-like system (Eq. (2)) is x [0] x[1] subject to the initial conditions: [M(x)] = [K(t)]dt = ; 0 [B(x)] [0] fF(0)gt = fRf(0) Jf(0) Rf 0(0) Jf 0(0)g ; (3) Z x G(x) H(x) [B(x)] = [E(t)]dt = : (6) *. E-mail address: [email protected] 0 H(x) G(x) S. Khorasani/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2928{2933 2929

The matrix exponentials here can be simpli ed as: removes singularR points since they are now located at zeros of x g(t)dt instead of g(x). In fact, the only 0 [1][0] singular expression now corresponds to [S(x)], which [Q0!x] = cosh[D(x)] [0][1] may also show the capability to be smoothed out. Hence, in real plane with v(0) = v0(0) = 0, the DTMM [0] x[D(x)]1 solution is given by: + sinh[D(x)] 1 ; (7) x [D(x)] [0] u(x) = C(x)u(0) + S(x)u0(0): (13) with sinh(.) and cosh(.) being de ned according to their Taylor expansions. Here, we have assumed that We obtain, after signi cant algebra [10], the new basis [D(x)] is a matrix root of [B(x)] in such a way that functions: x[B(x)] = [D(x)]2. If qij are elements of [Q ], 0!x 0!x s then the solution will be: Z x 2 11 12 13 0 C(x) = cos x k (t)dt; y(x) = q0!xu(0) + q0!xv(0) + q0!xu (0) 0 s Z + q14 v0(0) + i q21 u(0) + q22 v(0) x 0!x 0!x 0!x S(x) = xsinc x k2(t)dt; (14) 0 23 0 24 0 + q0!xu (0) + q0!xv (0) : (8) where g(x) = k2(x) and sinc(x) = 1 sin(x). In q x p quantum mechanics, we have k(x) = 2m [E V (x)] We also de ne [C(x)] = cosh[ xD(x)] = [C (x)] and ~2 ij where m is mass, ~ is reduced Planck's constant, E is [S(x)] = x[D(x)]1 sinh[D(x)] = [S (x)] to ultimately ij energy, and V (x) is the potential. In optical problems, obtain the function y(x) = u(x) + iv(x) as: p we have k(x) = c2 !2[(x) N] where c is the speed 0 of light in vacuum, ! is the angular frequency, (x) is u(x) =C11(x)u(0) + C12(x)v(0) + S11(x)u (0) the relative permittivity pro le of the dielectric, and N 0 is the normalized propagation constant. + S12(x)v (0); It is easy to verify that if g(x) = g(x), then 0 C(x) = C(x) and S(x) = S(x). These functions v(x) =C21(x)u(0) + C22(x)v(0) + S21(x)u (0) may be shown to exactly satisfy the initial conditions 0 C(0) = S0(0) = 1 and C0(0) = S(0) = 0. + S22(x)v (0); (9) These basis functions are not necessarily or- with the derivative y0(x) = u0(x) + iv0(x): thonormal, except for the trivial case of constant wave-function k(x). They furthermore do not span 0 0 u (x) =T11(x)u(0) + T12(x)v(0) + C11(x)u (0) a complete space; neither do xed k, sin(kx), and cos(kx). But, they may be always combined linearly 0 + C12(x)v (0); to satisfy any arbitrary initial or boundary conditions. In contrast, then, the well-known WKB bases are: v0(x) =T (x)u(0) + T (x)v(0) + C (x)u0(0) 21 22 21 Z 1 x 0 C~(x) = p cos k(t)dt ; + C22(x)v (0); (10) k(x) 0 where: Z 1 x 1 S~(x) = p sin k(t)dt : (15) [T(x)] = [B(x)][S(x)] = [T (x)]: (11) k(x) 0 x ij While WKB bases (Eq. (15)) clearly diverge at turning 2. Basis functions points with k(x) = 0, the newly introduced bases For real-valued equations having the following form, (Eq. (14)) do not. The reason is that at the turning 2 the DTMM [4,8] fails at zeros of g(x): points where k (x) changes sign, the relationships of these bases (Eq. (14)) remain algebraically continuous u00(x) + g(x)u(x) = 0: (12) and di erentiable. In contrast, WKB basis functions (Eq. (15)) are neither di erentiable norp continuous at While the generalized DTMM [5] and Airy functions [9] the turning points, because of the 1= k(x) prefactor. are able to alleviate some problems connected to these It should be pointed out that another pair of im- singularities, the method discussed in this paper easily proved basis functions using Airy functions also exist. 2930 S. Khorasani/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2928{2933

They are actually found from asymptotic expansions are actually Bloch waves. To verify this, we may write: near a turning point at x = , given by [11, p. 165]: s Z u(x + L; ) x+L 8 9 2 R " # 2 = exp i (x + L x) k (t)dt 1 < Z 3 = [ k(t)dt] 6 3 u(x; ) x C(x) = xp Ai k(t)dt ; k(x) : 2 x ; 0 s 1 Z L 8 9 = exp @i L k2(t)dtA : R " # 2 1 < Z 3 = 0 (22) [ k(t)dt] 6 3 S(x) = xp Bi k(t)dt ; (16) k(x) : 2 x ; Comparison with Eq. (19) reveals that the Bloch wave number is actually given by the dispersion relation: where the plus and minus signs are used across the s Z turning point, respectively, in the classically forbidden 1 L and allowed zones, and Ai(.) and Bi(.) are respectively = p k2(t)dt; (23) L the Airy's functions of the rst and second kinds. 0 While relatively accurate, these bases are too or: complicated for practical analytical purposes and have Z 1 L also to be exchanged across the singularities for smooth 2 = k2(t)dt: (24) solutions. Furthermore, it is impossible to obtain the L 0 analytical spectrum of con ned systems, even systems Doing the same procedure with the WKB bases leads as simple as harmonic oscillator, using the above to a di erent dispersion equation as: equation. Z 1 L = k(t)dt: (25) 3. Examples L 0 3.1. Periodic systems The dispersion equation using the new basis functions For periodic system with g(x) = g(x + L), one may (Eq. (24)) matches the long-wave length limit (alter- adapt DTMM to calculate the Bloch waves [3,5,8], natively known as homogenization) of the photonic which satisfy: crystals [12], while the WKB solution (Eq. (25)) does not. In that sense, the proposed basis functions provide u(x; ) = exp(ix)(x; ); higher accuracy for such class of problems. 3.2. Con ned bounded states (x; ) = (x + L; ); (17) Here, we demonstrate the power of new bases in analytical solution of a wide class of problems in optics where is the Bloch wavenumber. The DTMM and quantum mechanics. We suppose that an even solution of Eq. (17) takes the form: con ning potential or refractive index pro le is given, which is of the form: fF(x + L)g = [Qx!x+L]fF(x)g; (18) V (x) = Ujxj ; (26) while the periodic boundary conditions demand: where both and U are any positive real constants. u(x + L) = exp(iL)u(x); The turning points are therefore located at = (E=U)1= . The cases = 2[1; 11; 13; 14] and = u0(x + L) = exp(iL)u0(x): (19) 1[14] respectively correspond to the harmonic oscillator and quarkonium potentials. Hence, the wavenumber Finally, the simple result for the Bloch wave number function is de ned as: is: 2m k2(x) = [E V (x)]: (27) ~2 exp(iL) = eig[Qx!x+L]: (20) Noting the form of bases in Eq. (14), the eigenstates It is not dicult to see that the extended solutions may be found by the Wilson-Sommerfeld's quantiza- based on the functions in Eq. (14) as: tion [14] as: s Z ! s Z x 1 u(x; ) = exp i x k2(t)dt ; (21) 2 k2(x)dx = n + ; (28) 0 2 S. Khorasani/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2928{2933 2931 in which n is a non-negative integer denoting the state Also, for the case of harmonic oscillator with = 1 2 number. The =2 phase shift on the right-hand-side 2, and by taking 2 m = U, the spectrum according of the above has to be normally added because of the to the proposed bases is: re ection phase of wave-function at the turning point. r 32 1 In contrast, the WKB quantization will lead to E = ~ n + ; (36) the alternative form: n 32 2

Z which di ers with the exact spectrum E = ~ (n + 1 ) 1 p n 2 k(x)dx = n + : (29) 2 within a factor of 32=32 again corresponding to an error of 3.8%. While neither Eq. (28) nor Eq. (29) is generally exact, It is also instructive to check out the limiting the WKB quantization (Eq. (29)) cannot be analyt- case of ! 1 in Eq. (32). We rst note that the ically integrated except for = 1; 2. It is known wave-function should abruptly terminate at the turning that, only for the special case of = 2, however, the points because of the existence of the in nite potential WKB quantization leads to an exact result. Anyhow, wall at x = 1. Hence, the =2 phase shift on the plugging Eqs. (26) and (27) into Eq. (28) and noting right-hand-side of Eq. (28) would be unnecessary. Now, the even symmetry give: taking the limit here yields: 2~2 Z 2 2 2m 1 En = n ; (37) 4 [E Ux ]dx = 2 n + ; (30) 8m ~2 n 2 0 which is exactly the well-known spectrum of a con ned which after integration and some simpli cation leads massive particle in in nite potential well. to the algebraic equation: 3.3. Singular potentials As another example, we may also verify the existence 1+ 2 2 U E 2~2 1 n = n + : (31) of bounded solutions for the singular even con ning + 1 U 8m 2 potential of the form:

Hence, the nth energy eigenstate will be given by: V (x) = Ujxj ; (38)

2 where < 1 is any positive real constant less 2 2 +2 +2 ~ 1 2 1 En = (1 + )U n + : (32) than 1. The turning points are now located at = 8m 2 (U=E)1= . Using Eq. (28) results in: Therefore, the ground state energy according to Z 2 2 2 ~ 1 Eq. (32) will be: (En + Ux )dx = n + ; (39) 0 8m 2 2 2 +2 ~ 1 2 o ering the solution for the energy spectrum as: E0 = p (1 + )U : (33) 8 2m 2 1 2~2 2 1 2 E = 1 n + : (40) It is easy to verify that the expression for spectrum n 8U 2= m 2 Eq. (32) indeed agrees remarkably well with the known 1 2=3 Since 2 is negative, it would be better to rewrite behavior of spectrum for quarkonium En (n + 2 ) 1 the above as: and harmonic oscillator En n + 2 . For the case of quarkonium with = 1, we get 8 U 2= m 2 1 the following from Eq. (32): E = ; (41) n 2 2 2 ~ (1 ) 1 2 (n + 2 ) r 2 2 2 2 3 3 ~ U 1 1 2 E = n + ; (34) showing that E 1=(n + ) 2 . Therefore, the n 4m 2 n 2 ground state corresponding to the potential (Eq. (38)) while the WKB solution [14] is: is now given by: 2= 2 r 2 32 U m 92~2U 2 1 3 E = ; (42) E = 3 n + : (35) 0 2~2(1 ) n 32m 2 p which is divergent when ! 1. This justi es the These relations di er within a factor of 2= 3 9, corre- existence of a ground state bounded from below only sponding to an error of only 3.8%. when 0 < < 1. 2932 S. Khorasani/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2928{2933

Figure 1. Illustration of numerically calculated wave-functions (solid black) inside the singular quantum well (dashed) with = 1 . The ground and rst excited 2 eigenstates are leveled with horizontal lines (dot-dashed).

3.3.1. Numerical example 1 For the case of = 2 in the normalized atomic units where ~2=2m ! 1 and U ! 1, we obtain from Eq. (40): 2 1 3 E = n + : (43) n 2 2 Hence, the energy of the groundp and rst excited state 3 2 is, respectively,p E0 = 16= 1:17474 and 3 2 E1 = 16=9 0:56475. The exact values Figure 2. Comparison of un-normalized ground state wave-functions for = 1 within a subdomain of the range by numerical computation are E0 = 1:6534 and 2 spanned by turning points [; +]: numerically exact E1 = 0:43804; thus, the error of estimation (Eq. (43)) for the rst two states is about 28.9%. Figure 1 (solid black); newly proposed basis functions (dashed); simple WKB (dot dashed); improved WKB (dotted). All illustrates the numerically calculated wave-functions solutions diverge at in nity beyond the turning points, but and the eigenstates inside the quantum well. the improved WKB bases diverge at the turning points. Figure 2 compares the numerically exact solutions of the un-normalized wave-functions. Here, the ground and rst excited states are shown versus the References solution obtained by our proposed bases (Eq. (14)), 1. Yariv, A., An Introduction to Theory and Applications improvedR WKB bases (Eq. (15)), and simple WKB of Quantum Mechanics, Dover (2013). basis cos( x k(t)dt). The superior accuracy of the 0 2. Khorasani, S. and Mehrany, K. \Di erential transfer solution by Eq. (14), close to the expansion point, is matrix method for solution of one-dimensional linear clearly visible in this plot. The improved WKB bases non-homogeneous optical structures", J. Opt. Soc. are highly erroneous for bounded modes as can be seen Am. B, 20(1), pp. 91-96 (2003). here, while simple WKB solutions are still less accurate than our proposed basis functions (Eq. (14)). 3. Mehrany, K. and Khorasani, S. \Analytical solution of non-homogeneous auisotropic wave equation based on di erential transfer matrices", J. Op. A: Pure Appl. 4. Conclusions Opt., 4(6), pp. 524-635 (2002). In summary, we presented a new pair of bases for 4. Khorasani, S. and Adibi, A. \Analytical solution solving the wave equation. Through several explicit of linear ordinary di erential equations by di eren- examples, we have established the analytical power of tial transfer matrix method", Electron. J. Di . Eq., the proposed basis functions and their advantage over 2003(79), pp. 1-18 (2003). WKB basis solutions. Our new basis functions are 5. Khorasani, S. \Di erential transfer matrix solution not divergent close to singularities at turning point of generalized eigenvalue problems", Proc. Dynamic and they maintain remarkable accuracy compared to Systems & Appl., 6, pp. 213-222 (2012). the analytical solutions in the vicinity of expansion 6. Wilcox, R.M. \Exponential operators and parameter point. The basis functions also exactly satisfy the di erentiation in quantum physics", J. Math. Phys., initial conditions. 8, pp. 962-982 (1967). S. Khorasani/Scientia Iranica, Transactions D: Computer Science & ... 23 (2016) 2928{2933 2933

7. Khorasani, S. \Reply to comment on `analytical so- 13. Khorasani, S., Applied Quantum Mechanics (in Per- lution of nonhomogeneous anisotropic wave equation sian), Delarang, Tehran (2010). based on di erential transfer matrices'", J. Opt. A: Pure Appl. Opt., 5, pp. 434-435 (2003). 14. Sakurai, J.J., Modern Quantum Mechanics, Rev. Ed., 8. Khorasani, S. and Adibi, A. \New analytical approach Addison-Wesley (1993). for computation of band structure in one-dimensional periodic media", Opt. Comm., 216, pp. 439-451 (2003). Biography 9. Zariean, N., Sarra , P., Mehrany, K. and Rashidian, B. Sina Khorasani received the MSc and PhD degrees \Di erential-transfer-matrix based on airy's functions in analysis of planar optical structures with arbitrary in Electrical Engineering from Sharif University of index pro les", IEEE J. Quant. Electron., 44, pp. 324- Technology, Tehran, in 1997 and 2001, respectively, 330 (2008). where he is a Full Professor. He was with the School 10. Khorasani, S. and Karimi, F. \Basis functions for of Electrical and Computer Engineering at Georgia solution of non-homogeneous wave equation", Proc. Institute of Technology as a Postdoctoral (2002-2004) SPIE, 8619, 86192B (2013). and Research Fellow (2011-2012). He is now with 11. Schleich, W., Quantum Optics in Phase Space, Wiley- Ecole Polytechnique Federalede Lausanne (2016-2017) VCH, Berlin (2001). as a Visiting Professor. His active research areas in- 12. Chui, S.T. and Lin, Z.F. \Long-wavelength behavior clude quantum optics and photonics, and quantum of two-dimensional photonic crystals", Phys. Rev. E, electronics. Dr. Khorasani is a Senior Member of 78, 065601(R) (2008). IEEE.