Reflection and transmission of a plane TE-wave at a lossless nonlinear dielectric film

Physica D 158 (2001) 197–215

Reflection and transmission of a plane TE-wave at a lossless nonlinear dielectric film H.W. Schürmann a,1 , V.S. Serov b,2 , Yu.V. Shestopalov c,∗ b

a Department of Physics, University of Osnabrück, Barbarastrasse 7, D-49069 Osnabrück, Germany Department of Computational Mathematics and Cybernetics, Moscow State University, 119899 Moscow, Russia c Institute of Engineering Sciences, Physics and Mathematics, Karlstad University, S-651 88 Karlstad, Sweden

Received 29 January 2001; received in revised form 18 June 2001; accepted 20 June 2001 Communicated by F.H. Busse

Abstract A general analytical solution of the Helmholtz equation describing the scattering of a plane, monochromatic, TE-polarized wave with a film exhibiting a local Kerr-like nonlinearity is presented. The film is situated between two semi-infinite media. All media are assumed to be non-absorbing, non-magnetic isotropic and homogeneous. The results derived contain conditions for unbounded field intensities expressed in terms of the imaginary half-period of Weierstrass’ elliptic function ℘. The reflectivity R is calculated as a function of the film thickness and the transmitted intensity. The critical values of R are determined. The results are a generalization of the linear optics results. © 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65.Tg Keywords: Kerr-like nonlinearity; Film; Reflectivity; Weierstrass’ function

1. Introduction The reflection and transmission of electromagnetic waves at a single nonlinear homogeneous, isotropic, nonmagnetic layer situated between two linear homogeneous, semi-infinite media is of particular interest in linear optics [1]. In nonlinear optics, the Kerr-like nonlinear dielectric film has been the focus of a number of studies [2–7]. With respect to the nonlinear Fabry–Perot system, the present problem has been approached under special conditions by several authors: Marburger and Felber [8] simplified the analysis by imposing boundary conditions which suppose the nonlinear slab is separated from the linear media by perfect mirrors. Danikaert et al. [9] treated the steady-state response of a nonlinear Fabry–Perot resonator including nonlinear absorption and oblique incidence for transverse-electric and transverse-magnetic polarized fields. ∗ Corresponding author. Fax: +46-54-7001851. E-mail addresses: [email protected] (H.W. Schürmann), [email protected] (V.S. Serov), [email protected] (Yu.V. Shestopalov). 1 Fax: +49-0-541-969-2406. 2 Fax: +7-095-939-2596.

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 3 1 0 - 4

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Haeltermann et al. [10] and Vitrant et al. [11] presented a unified nonlinear theory for transverse effects of Fabry–Perot resonators simplifying numerical calculations and providing a good understanding of optical bistability. In Refs. [2–6], the solutions of the nonlinear Helmholtz equations have been given (partly for special cases) in terms of various Jacobian elliptic functions. The explicit form of these functions depends on the associated parameter regimes. As shown in Ref. [7] no classification of the solutions with respect to different parameter regimes is necessary, since the general solution can be presented in terms of Weierstrass’ elliptic functions ℘, containing the complete parameter dependence (cf. Ref. [7, Eqs. (29) and (32)]). In the following, we give a simplified version of this result. As is well known, elliptic functions exhibit poles in general. In Refs. [2–7] an analysis of the unbounded solutions (if the Kerr-constant is negative) is missing. In particular, no conditions (in terms of the parameters of the problem) for the existence of real, non-negative and bounded field intensities are given. The single general solution can be used straightforwardly to determine the critical values of the reflectivity R, covering all configurations uncountered and thus leading to a simplification of the discussion. The paper is organized as follows. In Sections 2–4, an exact analytical solution of the Helmholtz equation is presented and used for a singularity analysis. Section 5 briefly shows how the linear case emerges from the nonlinear case. In Section 6, the reflectivity R is investigated with respect to the existence of extrema following the lines indicated by Born and Wolf [1] in linear optics.

2. Reduction of the problem to a nonlinear ODE and to a quadrature Referring to Fig. 1, we consider the reflection and transmission of electromagnetic radiation at a nonlinear dielectric layer between two linear semi-infinite media. All the media are assumed to be homogeneous in x- and z-direction, isotropic and non-magnetic. The nonlinear material of the layer is assumed to be characterized by a dielectric function which is linear in the (time-averaged) local field intensity.

Fig. 1. Configuration considered in this article. A plane wave is incident to a nonlinear slab (sandwiched between two linear media) to be reflected and transmitted.

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A plane wave of frequency ω0 and intensity E02 , with electric vector E parallel to the z-axis (TE) is incident on the layer of thickness d. Since the geometry is independent of the z-coordinate and because of the supposed TE-polarization the fields are parallel to the z-axis (E = Ez ). Neglecting higher harmonics we look for solutions of Maxwell’s equations that satisfy the boundary conditions (Ez and ∂Ez /∂y must be continuous across the interfaces at y = 0 and y = d). Due to the requirement of the translational invariance in x-direction and partly satisfying the boundary conditions the fields tentatively are written as (ˆz denotes the unit vector in z-direction)  1 zˆ [E0 ei(px−q1 (y−d)−ω0 t) + Er ei(px+q1 (y−d)−ω0 t) + c.c.], y > d,    2 E(x, y, t) = zˆ 21 [E(y) ei(px−ϑ(y)−ω0 t) + c.c.], (1) 0 < y < d,    1 zˆ 2 [E3 ei(px+q3 y−ω0 t) + c.c.], y < 0, where E(y), p, q1 , q3 , and ϑ(y) are real and Er = |Er | exp(iδr ) and E3 = |E3 | exp(iδt ) are independent of y. Since a plane wave is assumed to exist for y < 0, q3 > 0 holds (cf. Eq. (30)). Modeling the nonlinearity by a Kerr-like dielectric function  y > d,   ε1 , 2 ε(y) = ε2 = εL + aE (y), 0 < y < d, (2)   ε3 , y<0 with real constants ε1 , εL , ε3 , a and inserting (1) and (2) into Maxwell’s equations we obtain the nonlinear Helmholtz equations valid in each of the three media (j = 1, 2, 3) ∂ 2 E˜ j (x, y) ∂ 2 E˜ j (x, y) + + k02 εj E˜ j (x, y) = 0, ∂x 2 ∂y 2

j = 1, 2, 3,

(3)

where k02 = ω02 /c2 and E˜ j (x, y) denotes the time-independent part of E(x, y, t), respectively, associated to the three media. Insertion of Eq. (1) into Eq. (3) yields for the semi-infinite media qj2 = k02 εj − p 2 ,

j = 1, 3.

For the film (j = 2), we get (by separating the result into real and imaginary parts)   dϑ(y) 2 d2 E(y) − E(y) + [k02 (εL + aE2 (y)) − p 2 ]E(y) = 0, dy dy 2

(4)

(5)

and E(y)

dϑ(y) dE(y) d2 ϑ(y) +2 = 0. dy dy dy 2

(6)

A first integral of Eq. (6) is E 2 (y)

dϑ(y) = c1 , dy

where c1 is a constant. Eliminating dϑ/dy in Eq. (5) by Eq. (7) and integrating Eq. (5) yields   c2 a dE(y) 2 + 2 1 + qL2 E 2 (y) + k02 E 4 (y) = c2 dy 2 E (y)

(7)

(8)

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with c2 being another constant of integration and qL2 = k02 εL − p 2 . Setting I˜(y) = E 2 (y), Eqs. (7) and (8) can be written as dϑ I˜(y) = c1 , dy 

dI˜(y) dy

(9)

2 = 2(−k02 a I˜3 (y) − 2qL2 I˜2 (y) + 2c2 I˜(y) − 2c12 ).

(10)

Thus, the determination of E(y) and ϑ(y) is reduced to the solution of Eq. (10) and a quadrature according to Eq. (9) y dy  ϑ(y) = ϑ(d) + c1 . (11) d I˜(y  ) The constants of integration c1 and c2 are determined in Section 3 by means of the boundary conditions.

3. Boundary conditions and associated relations The continuity of E and dE/dy at y = 0 implies E3 = E(0) eiϑ(0) ,

dE

= 0, dy y=0

(12) (13)

dϑ

= −q3 . dy y=0

(14)

At y = d the boundary conditions read E0 + Er = E(d) eiϑ(d) ,  2E0 e−iϑ(d) = E(d) 1 −

(15)

1 dϑ

q1 dy y=d

 +

i dE

. q1 dy y=d

(16)

Since E and dE/dy are continuous, so is dϑ/dy and thus Eq. (7) yields c1 = −q3 |E3 |2 .

(17)

Applying Eq. (8) to the interface y = 0, using Eq. (10) and the continuity of E 2 dϑ/dy at y = 0 the integration constant c2 is given by c2 = (q32 + qL2 )|E3 |2 + 21 ak20 |E3 |4 .

(18)

If the dielectric function ε(y) (see Eq. (2)) and the tangential component p of the wave-vector are given, all parameters in Eqs. (7) and (8) except the transmitted field intensity |E3 |2 are determined, so that a solution E 2 (y), ϑ(y) essentially depends on the parameter |E3 |2 only.

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From the physical point of view |E3 |2 is not given, but is related to the (physically) given incident intensity E02 and the parameters εj , p, d. Evaluating (dE/dy)|y=d and (dϑ/dy)|y=d (according to Eqs. (8) and (7), respectively, and using Eqs. (17) and (18)), aE20 is determined by Eq. (16) as  1 1 a aE20 = 2 (κ12 − κ22 )aE2 (d) − (aE2 (d))2 + (κ32 + κ22 + 2κ1 κ3 + |E3 |2 )a|E3 |2 , (19) 2 2 4κ1 2 2 2 = q 2 /k 2 = ε 2 2 2 where κ1,3 1,3 − ε1 sin ϕ1 and κ2 = qL /k0 = εL − ε1 sin ϕ1 . 1,3 0 The reflected intensity |Er |2 can be determined by evaluating the boundary conditions Eqs. (15) and (16). According to Eq. (15), |Er |2 = E02 + E 2 (d) − 2E0 E(d) cos ϑ(d) holds and this can be written as

|Er |2 = E02 −

κ3 |E3 |2 κ1

(20)

by expressing 2E0 cos ϑ(d) by means of Eq. (16) and using Eqs. (7) and (17). In terms of the reflectivity R and of the transmissivity T Eq. (20) reads R=

|Er |2 κ3 |E3 |2 = 1 − = 1 − T. E02 κ1 E02

(21)

Eq. (20) is consistent with energy conservation. As will be shown below, Eq. (19) is a generalization of the analogous formula of linear optics. To calculate the phase function ϑ(y) according to Eq. (11) the phase constant ϑ(d) must be determined. Eq. (16) leads to sin ϑ(d) = −

(dE/dy)|y=d . 2q1 E0

(22)

The phase shift δr of Er (phase shift on reflection) is determined by Eqs. (15) and (20) as sin δr = −

E(d)(dE/dy)|y=d

. 2q1 E02 1 − (κ3 |E3 |2 /κ1 E02 )

(23)

As is obvious from Eqs. (11) and (19)–(23), it is necessary to find a solution E 2 (y) of Eq. (10) in order to calculate reflectivity R, the phase shift on transmission δt = ϑ(0), and phase shift on reflection δr .

4. Exact solution To solve Eq. (10) let I (y) = 21 a I˜(y) + 13 κ22 , then Eq. (10) reads 

dI d(ik0 y)

2 = 4I 3 − g2 I − g3 = 4(I − I1 )(I − I2 )(I − I3 )

(24)

with 4 g2 = a|E3 |2 (a|E3 |2 + 2(κ22 + κ32 )) + κ24 , 3 1 g3 = − (3a|E3 |2 + 2κ22 )(3a|E3 |2 (κ22 + 3κ32 ) + 4κ24 ), 27

(25)

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1 I1 = − κ22 + 2 1 I3 = − κ22 + 2

 a |E3 |2 + 2  a |E3 |2 − 2

 1  2 κ2 + 2  1 κ22 + 2

2 a |E3 |2 + 2κ32 a|E3 |2 + 2 2 a |E3 |2 + 2κ32 a|E3 |2 + 2

κ22 , 3 κ22 . 3

I2 =

κ2 a |E3 |2 + 2 , 2 3 (26)

As is well known [12], the solution to Eq. (24) is given by Weierstrass’ function ℘ ℘ (ω¯ ± ik0 y; g2 , g3 ) = I (y) = 21 aE2 (y) + 13 κ22 . Since I (0) = 21 a|E3 |2 + 13 κ22 = I2 , ω¯ can be evaluated as an elliptic integral ∞ dI  . ω¯ = 4I 3 − g2 I − g3 I2 Thus, according to Eq. (27), the field intensity inside the layer is given by   κ22 2 2 ℘ (w¯ ± ik0 y; g2 , g3 ) − . E± (y) = a 3

(27)

(28)

(29)

The requirement that dϑ/dy must be real and E 2 (y) must be real, non-negative and bounded for 0 ≤ y ≤ d leads √ to necessary conditions for the parameters a, d, εj , p = k0 ε1 sin ϕ1 , E02 . According to Eqs. (7) and (17) κ32 ≥ 0

(30)

must hold since E 2 (y)(dϑ/dy) is proportional to the energy flux in y-direction, which is assumed to be positive (κ32 < 0 leads to a wave-guide problem). Further conditions can be derived from Eq. (29) by using properties of Weierstrass’ function ℘. Since Eq. (29) can be written as [13] E 2 (y) = |E3 |2 +

2 (I2 − I1 )(I2 − I3 ) , a ℘ (ik0 y; g2 ; g3 ) − I2

(31)

it is obvious, that E 2 (y) is real, because ℘ is real for pure imaginary arguments and real invariants g2 , g3 . The roots Ik are real, if the discriminant ∆ of [12] ∆ = 16(I2 − I3 )2 (I3 − I1 )2 (I1 − I2 )2

(32)

is non-negative. If ∆ < 0, I1 and I3 are complex conjugate. Thus, in any case, E 2 (y) is real and the remaining condition for non-negative and bounded solutions reads   κ22 2 ℘ (ω¯ ± ik0 y; g2 , g3 ) − 0≤ < +∞. (33) a 3 Discussion of (33) yields the following results (cf. Appendix A): If a > 0, all solutions represented by Eq. (29) are non-negative and bounded. and bounded for 0 ≤ y ≤ d if I2 > I3 which is equivalent to If a < 0 and ∆ > 0, then E 2 (y) is non-negative  2 κ3 > − 21 a|E3 |2 and κ22 ≥ − 21 a|E3 |2 + κ3 −2a|E3 |2 or 0 ≤ κ32 ≤ − 21 a|E3 |2 and κ22 > κ32 − a|E3 |2 (for E3 = 0 this is equivalent to κ22 > 0).

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Fig. 2. Dependence of the reflectivity R on (scaled) intensity aE20 and on the scaled thickness k0 d for κ12 = 0.2, κ22 = 0.5, and κ32 = 0.9. Paths in the trough denote reflectivities associated to Eq. (57), paths on the ridge belong to reflectivities with thicknesses according to Eq. (1). The dot denote a critical point (R = 0, h = 2.527, aE20 = −0.103).

 If I2 < I3 holds, which is equivalent to κ22 ≤ − 21 a|E3 |2 −κ3 −2a|E3 |2 or 0 ≤ κ32 < − 21 a|E3 |2 and − 21 a|E3 |2 +  κ32 −2a|E3 |2 ≤ κ22 < κ32 − a|E3 |2 (for E3 = 0 this is equivalent to κ22 < 0), then E 2 (y) is non-negative and bounded if 0 ≤ k0 y ≤ k0 d < |ω | (cf. Fig. 6), where ω denotes the imaginary half-period of ℘. E 2 (y) is non-negative and unbounded if k0 d ≥ |ω | (and I2 < I3 ). If a < 0 and ∆ < 0 holds, E 2 (y) is bounded if 0 ≤ k0 y ≤ k0 d < |ω2 |; E 2 (y) is unbounded if k0 d ≥ |ω2 |. If ∆ = 0, E 2 (y) is non-negative and bounded except if I1 = I3 < 0 (cf. Appendix A). To sum up unbounded field intensities are associated to parameters {k0 d, εj , p, a|E3 |2 } that satisfy  N |ω |, I2 < I3 , ∆ > 0, or       N|ω2 |, ∆ < 0, or k0 d ≥ (34)   N π    , I1 = I3 < 0 3I2 with N odd. For evaluation of R, δr , and δt these parameters must be excluded. The occurrence of unbounded field intensities is remarkable to a certain extent, since there is no counterpart in linear optics (cf. Section 5). A necessary condition for the existence of unbounded field intensities is a < 0. As a consequence, the dielectric function may vanish at some y inside the nonlinear layer. In other contexts, the occurrence of unbounded fields is well known [14–16]. With respect to the case in question it seems that the issue of an unbounded field is an artefact of our use of a real, local dielectric function according to Eq. (2). It may be that the singularity would be removed if absorption were introduced in the dielectric function ε(y). But the solution of Eq. (3) with a complex-valued function ε2 represents a non-trivial extension of the forgoing analysis which, to our knowledge, is not available in the literature. Thus it is not clear whether an unbounded field is an artefact due

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Fig. 3. Dependence of the phase on reflection δr on a|E0 |2 and k0 d for κ12 = 0.16, κ22 = 0.5, and κ32 = 0.81.

to neglecting absorption. On the other hand, it should be noted that the form of ε(y) according to Eq. (2) is not an unphysical or artificial one since it gives rise to well-known phenomena in nonlinear optics. 3 From a mathematical point of view, the occurrence of unbounded solutions of Eq. (3) demands no explanation, since nonlinear ordinary differential equations of type (10) exhibit pole-type singularities. Nevertheless, a physical interpretation of a “nearly unbounded” field (for instance for a weakly absorbing medium) is not clear and, to our knowledge, not available in the literature. Subject to the above conditions for non-negative and bounded field intensities, the phase function ϑ(y) can be evaluated by using Eqs. (11), (17), (22) and (29) and some properties of ℘ [13]. The result is [7] ϑ(y) = ϑ(d) + iκ3

ik0 y

ik0 d

(℘ (u) − I2 ) du . ℘ (u) + I2 + κ22 /3 − κ32

(35)

The integration in Eq. (35) can be carried out straightforwardly yielding a lengthy expression that contains elliptic theta functions [7]. The phase shift on transmission δt is equal to ϑ(0) [7]. The phase  shift on reflection δr , determined −1 by Eq. (23), can be evaluated by using Eq. (29) and (dE/dy)|y=d =ik0 (aE(d)) 4℘ 3 (ω+ik ¯ ¯ 0 d)−g2 ℘ (ω+ik 0 d)−g3 . The result in [7]  −i 4℘ 3 (ω¯ + ik0 d) − g2 ℘ (ω¯ + ik0 d) − g3

sin δr = . (36) 2aE20 κ1 1 − (κ3 |E3 |2 /κ1 E02 ) Thus, reflectivity R, transmissivity T , phase shifts on reflection δr and on transmission δt have been determined in dependence on the parameters of the problem, in particular, on aE20 /εL . Evaluation of R, T , δr , δt is subject to the above constrains for non-negative and bounded field intensities. Plots of R, δr are shown in Figs. 2–6. It seems that, contrary to R, δr is not bistable. 3 Nevertheless, the consequence of an unbounded field intensity is ε(y) → −∞ according to Eq. (2). The physical meaning of this result is not clear.

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205

Fig. 4. Fundamental period parallelogram (FPP) of Weierstrass’ function ℘ for positive (rectangle) and negative (rhombus) discriminant ∆.

Fig. 5. Bistability of R for k0 d = 1. Parameters are as in Fig. 2. Arrows indicate switching.

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Fig. 6. Dependence of the phase on reflection δr on a|E0 |2 for k0 d = 2. Parameters are as in Fig. 2.

5. The linear case In order to show how the results of linear optics emerge from the analysis just presented the linear case a = 0 will be investigated in the following. Evaluation of Eq. (29) with a → 0 (I2 → I1 or I2 → I3 ) yields (κ22 = εL − ε1 sin2 ϕ1 )

|E3 |2 2 2 2 2 E (y) = [κ3 + (κ2 − κ3 ) cos κ22 k0 y], κ22 2

(37)

where the limits of ℘ if ∆ = 0 4 have been used. Eq. (19) reads in this case E02 =

|E3 |2 2 2 2 2 2 2 2 [(κ + κ κ ) − (κ − κ )(κ − κ ) cos κ22 k0 d] 1 3 2 2 1 2 3 4κ12 κ22

(38)

relating the intensity E02 of the incident field to the intensity |E3 |2 of the transmitted field. Since E 2 (0) = |E3 |2 , the reflectivity R and the transmissivity T can be evaluated straightforwardly by using Eqs. (21), (37) and (38). Hence

(κ22 − κ1 κ3 )2 − (κ22 − κ12 )(κ22 − κ32 ) cos2 κ22 k0 d

R= , (39) (κ22 + κ1 κ3 )2 − (κ22 − κ12 )(κ22 − κ32 ) cos2 κ22 k0 d T =

4κ1 κ3 κ22

. (κ22 + κ1 κ3 )2 − (κ22 − κ12 )(κ22 − κ32 ) cos2 κ22 k0 d

(40)

Turning to the phase function ϑ(y) it is suitable to start with Eq. (11). Inserting E 2 (y), given by Eq. (37), into Eq. (11) and using Eq. (17), ϑ(y) can be written as y dy 

. (41) ϑ(y) = ϑ(d) − q3 κ22 d κ 2 + (κ 2 − κ 2 ) cos2 κ 2 k y  3 2 3 2 0 4

Cf. Ref. [13, pp. 651–652].

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The phase ϑ(d) can be determined by Eqs. (16), (37) and (41) leading to

κ22 (κ22 − κ32 ) cos( κ22 k0 d) sin( κ22 k0 d)

. tan ϑ(d) = κ3 (κ22 + κ1 κ3 ) + κ1 (κ22 − κ32 ) cos2 κ22 k0 d Integration in Eq. (41), using Eq. (42), yields, after some algebra,

     (κ22 + κ1 κ3 ) tan κ22 k0 d   κ3 tan κ22 k0 y 

ϑ(y) = tan−1 − tan−1 .     2 κ2 (κ1 + κ3 ) κ22

207

(42)

(43)

Thus, the solution of the linear case is represented by Eqs. (37)–(40), (42) and (43). The phase shifts on transmission, δt , and on reflection, δr , are determined by Eqs. (12), (15) and (16), respectively. According to Eq. (43) δt is equal to q(0),

   (κ22 + κ1 κ3 ) tan κ22 k0 d 

. (44) δt = tan−1   κ22 (κ1 + κ3 ) According to Eqs. (15) and (16) δr is equal to tan−1 { tan ϑ(d)/(1 − (E0 /E(d) cos ϑ(d)))}, which can be evaluated to

    κ1 κ22 (κ22 − κ32 ) sin 2 κ22 k0 d

δr = tan−1 . (45)  2 2  κ1 κ3 − κ24 + (κ12 + κ22 )(κ22 − κ32 ) cos2 κ22 k0 d The foregoing results are consistent with the associated results of linear optics [17].

6. Critical reflectivities The investigation of the local extrema of R in the linear case is well known. 5 An analogous treatment in the nonlinear case is presented in the following. Setting h = k0 d, f = aE20 , and τ = a|E3 |2 and regarding f and h as the independent variables we investigate points {hc , fc } where R(h, f ) is critical [18], i.e., where Rh (hc , fc ) = Rf (hc , fc ) = 0

(46)

(subscripts h, f denoting partial derivatives). Eqs. (19) and (21) take the form R(h, f ) = 1 − F :=

 1  2 τ  2 2 2 2 2 2 1 (κ − κ )aE (h, τ ) − (aE (h, τ )) + κ + κ + 2κ κ + τ − f = 0, 1 3 1 2 3 2 2 2 4κ12

respectively.

5

κ3 τ (h, f ) , κ1 f

Cf. Ref. [1, pp. 63–64].

(47) (48)

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As is well known [18] the critical points can be discriminated by the Hessian determinant D 2 . D = Rhh Rff − Rhf

(49)

In particular, there are non-degenerate critical points (maxima if D > 0, Rhh < 0, minima if D > 0, Rhh > 0, saddle-points if D < 0) and degenerate critical points if D = 0. The latter points require further consideration. Rh and Rf can be evaluated straightforwardly, leading to Rh =

κ3 Fh , κ1 fFτ

Rf = −

(50)

κ3 f − τ F τ , κ 1 f 2 Fτ

(51)

According to Eqs. (50) and (51), the critical points are determined by   κ22 i 2 Fh = 2 κ1 − − 2℘ (ω¯ ± ih) ℘  (ω¯ ± ih) = 0, 3 2κ1 f = τ Fτ ,

(52) (53)

where the prime denotes differentiation with respect to ω¯ ± ih and Eq. (29) has been used. Since ℘  (u) = (4(℘ (u) − I1 )(℘ (u) − I2 )(℘ (u) − I3 ))1/2 Eq. (52) implies the necessary condition for critical points ℘ (ω¯ ± ihc ) = I1 ∨ I2 ∨ I3 ∨ I0 ,

(54)

where I0 = 21 κ12 − 16 κ22 . Eqs. (52) and (53) determine the critical values of τ and thus the critical values of the normalized (incident) intensity fc = aE20c . We first consider the case ℘ (ω¯ ± ihc ) = I2 , ∆ > 0. According to Eqs. (28) ω¯ is given by 6  ω if I2 > I1 > I3 and ∆ > 0,      ω + ω if I > I > I and ∆ > 0, 1 2 3 ω¯ = (55)   ω if I1 > I3 > I2 and ∆ > 0,     if ∆ < 0. ω2 Inversion of ℘ (ω¯ ± ihc ) = I2 (τ ) yields  ω + 2Mω + 2N ω      ω + ω + 2Mω + 2N ω ω¯ ± ihc =  ω + 2Mω + 2N ω     ω2 + 2Mω2 + N (ω2 + ω2 )

if I2 > I1 > I3 and ∆ > 0, if I1 > I2 > I3 and ∆ > 0, if I1 > I3 > I2 and ∆ > 0,

(56)

if ∆ < 0.

Combining Eqs. (55) and (56) the critical thickness is given by  NK((Imax (τ ) − Im (τ ))/Imax (τ ) − Imin (τ ))    , ∆ > 0, √   N |ω (τ )| = Imax (τ ) − Imin (τ ) hc (τ ) =  NK((1/2) + (3I2 (τ )/4|I1 (τ ) − I3 (τ )|)   , ∆<0 √  N |ω2 (τ )| = |I1 (τ ) − I3 (τ )|

(57)

6 Cf. Ref. [13, p. 633 (18.3.1)] in connection with the definition of ω , p. 630 (with different ω, 2 (y) can also be ¯ g2 , g3 this representation of E± i used in nonlinear waveguide theory, cf. Ref. [14]).

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209

with N even. The roots Ij are ordered according to Imin < Im ≤ Imax and K denotes the complete elliptic integral of the first kind. If ∆ = 0, ω is finite (I1 = I2 ) or infinite (I2 = I3 ). 7 Thus, there are no critical thickness hc if I2 = I3 (cf. Eqs. (A.7) and (A.8)). If ℘ (ω¯ ± ihc ) = I2 (τ ), Eq. (53) becomes an identity so that the critical incident intensities are given by fc (τ ) =

τ (κ1 + κ3 )2 . 4κ12

(58)

The Hessian determinant is zero so that only degenerate critical points exist in this case. According to Eq. (47) the critical reflectivities are  Rc =

κ1 − κ3 κ1 + κ 3

2 .

(59)

This result and the linear limit of hc according to Eq. (57) (hc → N π/2 κ22 ) are consistent with the linear optics results. 8 In the literature, 9 a particular case of ℘ (ω¯ ± ihc ) = I2 was treated by considering I1 = I2 or I2 = I3 , leading to induced transparency. One obtains τ = κ32 − κ22 according to Eqs. (A.5) and (A.7). But induced transparency occurs for all values f and τ obeying Eq. (58), not only for τ = κ32 − κ22 . We consider next the case ℘ (ω¯ ± ihc ) = I1 . Inversion of this equation yields the conditions  ω + ω + 2Mω + 2N ω if I2 > I1 > I3 , ∆ > 0,       ω + 2Mω + 2N ω if I1 > I2 > I3 , ∆ > 0, ω¯ ± ihc = (60)  if I1 > I3 > I2 , ∆ > 0, ω + 2Mω + 2N ω     1   2 (ω2 − ω2 ) + 2Mω2 + N (ω2 + ω2 ) if ∆ < 0. Combining Eqs. (55) and (60) it is easy to see that no real hc exists if I3 > I2 or if ∆ < 0. The remaining conditions can be fulfilled if ∆ > 0 and I2 > I3 and hc = N |ω (τ )|

(61)

with N odd. Inserting I1 for ℘ (ω¯ ± ihc ) Eq. (53) reads −τ 3 + 8κ12 κ24 − 2τ 2 (κ22 + 2κ32 ) + 4τ κ12 (κ22 + 2κ32 )

= 2(τ 2 + 4κ12 κ22 ). 2 2 ((τ/2) + κ2 )2 + 2κ3 τ

(62)

Real solutions τ of Eq. (62) are τ1 = 0 if κ22 > 0, τ2 = 2κ1 (κ1 κ3 +κ22 )/(κ3 −κ1 ) if (κ12 −κ22 −2κ1 κ3 )/(κ3 −κ1 ) > 0, and τ3 = 7

2κ1 (κ1 κ3 − κ22 ) if κ12 + 2κ1 κ3 − κ22 > 0. κ1 + κ 3

Cf. Ref. [13, pp. 651–652]. Cf. Ref. [1, p. 63]. 9 Cf. Ref. [5, p. 5369]. 8

(63)

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Evaluation of Eq. (53) with τ1 and τ2 leads to f = 0, a result that can be disregarded from a physical point of view. With τ = τ3 Eq. (53) reads fc =

2κ3 (κ1 κ3 − κ22 ) κ1 + κ 3

(64)

The Hessian determinant D and Rhh are given by D=

(κ1 − κ3 )2 (κ22 − κ1 κ3 )2 4κ12

,

(65)

and Rhh =

(κ1 − κ3 )2 (κ12 + 2κ1 κ3 − κ22 )2 2κ12 (κ1 + κ3 )2

,

(66)

respectively. The critical (normalized) thickness associated to fc is (N odd)

NK((Imax (τ ) − Im (τ ))/Imax (τ ) − Imin (τ ))

 hc = N |ω (τ3 )| = √

Imax (τ ) − Imin (τ ) τ =τ3

(67)

with roots Ij according to I1 =

κ1 (3κ32 + κ22 ) − 2κ22 κ3 , 3(κ1 + κ3 )

I2 =

κ3 (3κ12 + κ22 ) − 2κ22 κ1 , 3(κ1 + κ3 )

I3 =

κ22 − 3κ1 κ3 . 3

(68)

Since no real hc exists for I3 > I2 , Eq. (67) must be evaluated subject to I3 < I1 < I2 or I3 < I2 < I1 . The constraint I2 > I3 (for hc being real) is equivalent to κ32 + 2κ1 κ3 − κ22 > 0.

(69)

This inequality must hold in addition to the constraint κ12 + 2κ1 κ3 − κ22 > 0

(70)

according to Eq. (63). If fc is not zero and if κ1 = κ3 Eqs. (65) and (66) indicate a minimum of R(f, h). Inserting τ3 and fc into Eq. (46) yields R = 0 (total transparency). If κ1 = κ3 only degenerate critical points aE20c = κ12 − κ22 , hc =

√ Nπ/ 2(3κ32 − κ22 ) exist, since D = 0 in this case (I1 = I2 holds so that |ω | = π/ 12I2 ). The present case (℘ = I1 ) can be summarized as follows. If the (normalized) thickness h and the (normalized) incident intensity f = aE20 = 0 are given by h = hc = N |ω [(κ1 /κ3 )fc ]|, N odd, and f = fc = 2κ3 (κ1 κ3 − κ22 )/(κ1 +κ3 ), respectively, then the reflectivity R has minima R = 0 if the conditions κ1 = κ3 , κ12 +2κ1 κ3 −κ22 > 0, and κ23 + 2κ1 κ3 − κ22 > 0 are fulfilled. Comparing the previous results with those of linear optics, the question arises whether additional (degenerate) critical reflectivities (Rh = Rf = 0) exist. Assuming ℘ (ω¯ ± ihc ; g2 , g3 ) = I3 in Eq. (54) the foregoing procedure yields conditions (analogous to Eq. (57) or Eq. (61)) that cannot be satisfied for real hc . Thus, Rh = 0 is impossible in this case. If ℘ (ω¯ ± ihc ; g2 , g3 ) = I0 , then τ = τ± = ±(κ12 − κ22 ) holds. τ = τ+ yields I0 = I2 and hence hc and fc are given by Eqs. (57) and (58), respectively. If τ = τ− (I0 = I2 ) there is no real hc (cf. Appendix B). Thus, no additional solutions of Eqs. (52) and (53) exist and it seems that the above results given by Eqs. (57), (58) and (67) (with R = 0) represent the appropriate generalization of the linear optics results.

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7. Conclusion There are some connections between the foregoing analysis and the literature. In particular, it seems appropriate to compare the above results with those contained in Refs. [2–5]. 1. The ansatz (3) for the field intensity E 2 and the solution procedure of the nonlinear Helmholtz equation (2) as well as the formulation of the boundary conditions are essentially the same as in Refs. [2–5]. 2. The representation of the field intensity and phase shift (if considered at all in Refs. [2–5]) is different and equivalent if the roots (26) (Iν in Refs. [2–4] and Wν in Ref. [5]) are real and if the conditions (34) are not satisfied. In this sense (29) or (30) is more general than the corresponding representation in Refs. [2–5]. The evaluation procedure in the present paper is different from those in Refs. [2–5]. In Refs. [2–4], a consistency check of Eqs. (2.44) and (2.45) is used. In Ref. [5], the roots W1 and W2 for a parametric evaluation, and the root must be chosen appropriately according to the values of parameters (cf. Section 5 of Ref. [5]). As shown above, the complete parameter dependence is contained in the invariants g2 and g3 of Weierstrass’ function according to (25). Evaluation runs with a free parameter τ = a|E3 |2 subject to the constraints (cf. Appendix A) which are expressed by the roots with the help of (26). Containing the same free parameter a|E3 |2 the constrain W2 ≤ W ≤ W1 given in Section 4 of Ref. [5] is a subcase of the general constraints in Appendix A. In general, the roots W1 and W2 may be complex (if a > 0, as assumed in Ref. [5], and this is impossible, as we have shown above). Thus, evaluation is simplified by using a compact expression (29) for E 2 (y). 3. Another advantage of Eq. (29) is related to a concise determination of the local extrema of R considered as a function of two variables a|E0 |2 and k0 d. In Sections 7 and 8 of Ref. [5] induced transparency and induced resonance scattering are discussed. The results are consistent with the corresponding treatment in Section 6. Eqs. (86) and (87) in Ref. [5] correspond to Eqs. (58) and (59) of this paper; however, as pointed out, special cases (I1 = I2 and I2 = I3 ) are considered in Ref. [5]. With respect to the induced resonance scattering the results of Eqs. (92), (93) and (94) in Ref. [5] are consistent with Eqs. (57), (59) and (58), respectively (since a > 0 was assumed in Ref. [5] the case ∆ < 0 cannot occur in Eq. (57)). Eqs. (92)–(94) in Ref. [5] are derived using the Jacobian function sn2 which, as was argued above, does not represent the field intensity for all possible parameter values. 4. The case ℘ (ω¯ ± ihc ) = I1 (Eqs. (60)–(70)) has not been treated in Ref. [5]. 5. The problem of unbounded solutions and associated conditions has not been addressed in Refs. [2–5]. Summing up, we think that representing the solution to the Helmholtz equation (3) in terms of Weierstrass’ elliptic function instead of the Jacobian elliptic function has definite advantages.

Acknowledgements This work was part of the Research-in-Pairs Program at the Mathematisches Forschungsinstitut Oberwolfach, Germany. We gratefully acknowledge the support of the Volkswagenstiftung.

Appendix A. Discussion of condition (33) Investigating condition (33) it is appropriate to refer to the well-known properties of ℘ [12] and to consider the cases a > 0 and a < 0 separately.

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℘ (u; g2 , g3 ) is a doubly periodic function (periods 2ω, 2ω ) of a complex variable u whose only singularity in the finite plane is a double pole at u = 0 and, according to the periodicity, at u = 2Mω + 2N ω , with integer M, N. Thus the study of ℘ conveniently can be reduced to the consideration of its behavior in the FPP. If ∆ > 0 the FPP is a rectangle, if ∆ < 0 the FPP is a rhombus (cf. Fig. 4). If ∆ = 0, the FPP degenerates (ω → +∞, or/and ω → i∞) (see footnote 5). We first consider the case a > 0. In this case ∆ < 0 can be left out of consideration, since I1 , I3 are real according to Eqs. (26) and (30). Thus, ∆ > 0 holds and ω¯ can be equal to ω or ω + ω or ω . Hence we get 10 I2 ≤ ℘ (ω + ω ± ik0 y; g2 , g3 ) ≤ I1

if I1 > I2 > I3 (ω¯ = ω + ω ),

I1 ≤ ℘ (ω ± ik0 y; g2 , g3 ) ≤ I2 ,

if I2 > I1 > I3 (ω¯ = ω),

−∞ ≤

℘ (ω

± ik0 y; g2 , g3 ) ≤ I2

if I1 > I3 > I2 (ω¯ =

(A.1)

ω ).

It is obvious (cf. Eq. (26)) that the case I3 > I2 is impossible, since a > 0, κ32 ≥ 0. Using Eq. (29) the remaining inequalities (Eq. (A.1)) can be written as   κ22 2 2 2 |E3 | ≤ E (y) ≤ I1 − if I1 > I2 > I3 , a 3   (A.2) κ22 2 2 2 I1 − ≤ E (y) ≤ |E3 | if I2 > I1 > I3 . a 3 Since (2/a)(I1 − 13 κ22 ) ≥ 0, conditions (A.2) imply that 0 ≤ E 2 (y) < +∞. Thus, if a > 0, we have obtained the result, that all solutions given by Eq. (29) are real, non-negative and bounded. The only constraint is κ32 = ε3 − ε1 sin2 ϕ1 ≥ 0, which is (physically) meaningful. If a < 0 and ∆ > 0, instead of Eqs. (A.2) we get   κ22 2 2 2 |E3 | ≥ E (y) ≥ I1 − if I1 > I2 > I3 , a 3   κ2 2 (A.3) I1 − 2 ≥ E 2 (y) ≥ |E3 |2 , if I2 > I1 > I3 , a 3 +∞ ≥ E 2 (y) ≥ |E3 |2

if I1 > I3 > I2 .

Since I1 − 13 κ22 < 0, if a < 0, the inequalities in Eq. (A.3) imply the following statements: If a < 0 and ∆ > 0, E 2 (y) is non-negative and bounded for all y, if I2 > I3 , irrespective whether I2 > I1 or I2 < I1 . If I3 > I2 , then E 2 (y) is non-negative and bounded, if 0 < k0 d < |ω | (cf. Fig. 6), where ω is the imaginary half-period of ℘. E 2 (y) is non-negative and unbounded if k0 d ≥ |ω | and I3 > I2 . If a < 0 and ∆ < 0, ω¯ is equal to ω2 (cf. Fig. 6) and thus −∞ ≤ ℘ (ω2 ± ik0 y) ≤ I2 . Hence we obtain +∞ ≥ E 2 (y) ≥ |E3 |2 .

(A.4)

This inequality implies the statement: If a < 0 and ∆ < 0 then E 2 (y) is non-negative and bounded if 0 < k0 d < |ω2 | (cf. Fig. 4), where ω2 is the imaginary half-period of ℘. E 2 (y) is non-negative and unbounded if ∆ < 0 and k0 d ≥ |ω2 |. To complete the discussion of the solution (29), we consider the case ∆ = 0. According to Eq. (32) at least two of the roots Ik are equal, iff ∆ = 0. 10

Cf. Ref. [15, p. 167].

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213

1. If I1 = I2 , Eqs. (26) yield a|E3 |2 = κ32 − κ22

and

a|E3 |2 = 0

κ22 ≥ 0.

κ22 − 3κ32 ≤ 0,

(A.5)

or and

(A.6)

2. If I2 = I3 , Eqs. (26) yield a|E3 |2 = κ32 − κ22

and

κ22 − 3κ32 ≥ 0,

a|E3 |2 = 0

κ22 ≤ 0

(A.7)

or and

(A.8)

must hold. 3. If I1 = I3 , a|E3 |2 is given by

a|E3 |2± = −2(κ22 + 2κ32 ± 2 κ32 (κ32 + κ22 )) if κ32 + κ22 ≥ 0.

(A.9)

4. If I1 = I2 = I3 = 0, we get a|E3 |2 = 0

and

κ22 = 0,

(A.10)

or a|E3 |2 = −2κ32

and

κ22 = 3κ32 .

(A.11)

Evaluation of E 2 (y) according to Eq. (31) yields E 2 (y) = const. if (A.5) or (A.6) or (A.7) or (A.8) or (A.10) or (A.11) holds. If a|E3 |2 is given by Eq. (A.9) E 2 (y) can be determined by Eq. (31), where ℘ degenerates to a sinh−2 or a sin−2 dependence. 11 If I1 = I3 > 0, E 2 (y) is bounded for all y, if I1 = I3 < 0, and E 2 (y) is singular if y satisfies 3I2 k0 y = π, 3π, 5π, . . . . ¯ ± ihc ) = I0 Appendix B. Discussion of ℘(ω If ℘ (ω¯ ± ihc ) = I0 , Eq. (53) yields τ0± = ±(κ12 − κ22 ).

(B.1)

If τ = τ0+ Eqs. (26) imply I0 = I2 . Thus, these solutions are special cases of ℘ (ω¯ ± ihc ) = I2 investigated above. Hence ℘ (ω¯ ± ihc ) = I0 must be inverted subject to τ = κ22 − κ12 . According to Eq. (A.1) I0 must obey one of the conditions I3 < I1 ≤ I0 ≤ I2 ,

I3 < I2 ≤ I0 ≤ I1 ,

I0 ≤ I2

(B.2)

if ∆ > 0. If ∆ < 0, I0 ≤ I2 11

Cf. Ref. [13, pp. 651–652].

(B.3)

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must hold. Eqs. (B.2) and (B.3) are not consistent with τ = κ22 − κ12 . This can be seen as follows: If τ = κ22 − κ12 the roots Ij are √ κ12 5 2 A I1,3 = − κ2 ± , 4 12 4 I2 = − I0 =

κ12 5 + κ22 , 2 6

κ12 κ2 − 2 2 6

(B.4) (B.5) (B.6)

with A = (κ12 − 3κ22 )2 + 8κ32 (κ22 − κ12 ).

(B.7)

Considering Eq. (B.3) I0 ≤ I2 reads κ22 − κ12 ≥ 0

(B.8)

in contradiction to A < 0, κ32 ≥ 0. Considering the next case A > 0, (∆ > 0), the first inequality (B.2) implies √ 0 ≤ A ≤ κ12 + κ22 ≤ 5κ22 − 3κ12 .

(B.9)

Necessary for (B.9) is κ22 ≥ κ12 .

(B.10)

Evaluation of (B.9) yields (κ12 − κ22 )(κ22 + κ32 ) ≥ 0,

(B.11)

and hence, with (B.10), κ22 ≤ −κ32 .

(B.12)

(B.10) and (B.12) are not consistent, since κ12 > 0, κ32 ≥ 0. The second inequality (B.2) implies √ 3κ12 − 5κ22 − A < 0,

(B.13)

and κ22 ≤ κ12 ,

(B.14)

and κ12 + κ22 −

√

A ≤ 0.

(B.15)

It is appropriate to discriminate the cases 3κ12 − 5κ22 ≥ 0 and 3κ12 − 5κ22 < 0. Assuming 3κ12 − 5κ22 ≥ 0, (B.13) leads to (κ12 − κ22 )(κ12 + κ32 − 2κ22 ) < 0,

(B.16)

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and thus, by (B.14), to κ12 + κ32 − 2κ22 < 0.

(B.17)

If we assume further κ12 + κ22 ≥ 0, (B.15) yields (κ12 − κ22 )(κ22 + κ32 ) ≤ 0,

(B.18)

and hence, with (B.14), κ22 ≤ −κ32 ,

(B.19)

contrary to κ12 > 0. If κ12 + κ22 ≤ 0 holds, this is in contradiction with κ32 ≥ 0. If 3κ12 − 5κ22 < 0, then κ22 > 0 and inequality (B.15) yields (κ22 + κ32 )(κ22 − κ12 ) ≥ 0,

(B.20)

hence, with (B.14), κ22 ≤ −κ32 ,

(B.21)

contrary to κ22 > 0. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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