Exact reflection amplitudes for the Rayleigh profile

Physica 116A (1982) 235-247 North-Holland

EXACT

REFLECTION

Publishing Co.

AMPLITUDES

FOR THE RAYLEIGH

PROFILE

John LEKNER Physics Department,

Victoria University of Wellington, New Zealand

Received 3 March 1982

We give an exact solution of the electromagnetic s and p wave equations for a medium where the reciprocal of the refractive index varies linearly with distance. This solution leads to analytical expressions for the reflection amplitudes rr and r,, which reduce at normal incidence to the result obtained by Lord Rayleigh. These results are compared at long wavelength with those obtained recently to second-order in the (interfacial thickness/wavelength) expansion.

1. Introduction

Lord Rayleigh’) in an 1880 paper entitled “On reflection of vibrations at the confines of two media between which the transition is gradual”, gave the solution to the one-dimensional wave equation for a particular transition between two media. In optical terms, this interface is one for which the reciprocal of the refractive index varies linearly with distance. The Rayleigh solution is for normal incidence only. In this note we shall show that the electromagnetic s and p waves for this dielectric profile are both expressible in terms of Bessel functions of imaginary argument. The reflection amplitudes rS and rP are thus determined analytically. There appears to be only one other profile for which analytic results are known for both the s and the p waves: the exponential profile studied by Abelbs2. We consider plane waves incident from medium 1 of dielectric constant cl onto an interface lying between zl and zz; some of the wave is reflected back into medium 1, some is transmitted into medium 2, of dielectric constant e2. For Rayleigh profile, E-“’ (2) is a linear function of z between z, and z2. The corresponding E(Z) is shown in fig. 1. Since E-“~ is linear in z, it will be useful to work in terms this function, which we will call q(z):

E-“2(z)= q(z)

= fj + (z - 2)

2,

(1)

where An = q2 - nl = e;“‘- E; “2, AZ = z2- zI, f = i(n, + 93, Z = f(z, + ~2). Let the plane of incidence-reflection-refraction be the zx plane. For the electromagnetic s-wave, the electric field is transverse to this plane: E = (0, E,,0). 0378437

ll82lOooO-owO BO2.75 @ 1982 North-Holland

JOHN LEKNER

236

c

0

-1

0

1

Fig. 1. Dielectric function E(Z) for the Rayleigh profile. Air, with cl = 1 is on the left, water with lz = (4/3)* is on the right. The scale is marked with values of Z/AZ. The origin is placed to satisfy (13); for this choice of origin, and the EI, ~2 values given above, ZI = -(4/7)Az, zz = (3/7)Az.

When the dielectric function is independent of x and y (the case considered here), the wave equation separates and E&r, x) = e’“E(z),where E(Z) satisfies3)

(2) with K = v/EI(o/c) sin 8, = v/a
E(z) + t, eiq2’,

(3)

where rS and t, are the reflection and transmission amplitudes for the s-wave, and qi = d/i(o/c) cos 8i. For the p-wave, B = (0, B,, O), B,(z, x) = eiK”B(z), and B(z) satisfies3) d2B 1 dcdB P-;drZ+(+K2)B=0, and has the asymptotic

(4) formr?)

eiqlz - r, e-iql* c B(z) +

J

2 t,

eiq2ze

(9

At normal incidence there is no physical difference between the s and p waves. The mathematical statement of this fact is that dE/dz (E being the solution of (2)) satisfies (4) when K = 0, i.e. B is proportional to dE/dz at normal incidence. The derivation of the Rayleigh result can now be outlined. From (1) and (2), on changing the independent variable from z to q, we have

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231

where

o AZ cAq ) (-_

2

+I_ 4

(7)

.

Eq. (6) has the solutions E, = q “2*“, so the complete incidence can be written as eik,z

+

r e-iklr,

2 s

solution

at normal

21,

c&‘~+” + /3~$‘~-“, z, s z s z2, 1 t eik2z, z 3 22.

E(r) =

(8)

where ki = d/a(w/c), r = rs = rp, and t = t, = t,. The continuity of E and dE/dz at z, and z2 gives four equations in the four unknowns r, t, (Y,p. The reflection amplitude is thus found to be (cf ref. 1, eq. (18)) Zik,z,

r=e

CL

2v- 1 (9)

2v(p2”+ l)-2iWdz c *rl W” - 1)’ where /.Lis the ratio of the refractive

indices,

(10) It is interesting

to compare (9) with the general second-order

rs= K_‘l-

result’)

(11)

2q,q,f2)+ 0(qA#,

where m

f’2 El

-

452

I dz Z(E- ~step),

(12)

-m

and the reflection amplitude is calculated 4) m I -cc

in the gauge where (see fig. 1 of ref.

dz(e - E,& = 0

(13)

(%tep= er for z < 0 and l2 for z > 0). From (1) and (12) we find 12s

2 log(e,/et)

( )I g

This function

El-E2

-&I*

is positive-definite,

as f2 must be for all monotonic

(14) profilesq.

238

JOHN LEKNER

The limiting value of I as IE, - •~1tends to zero is A&/12, in agreement with that for the linear profile (ref. 4, table 1.) To compare (9) with (11) and (14) we need first to use the same gauge (13). This implies zI =

-lZ9.!-AZ.

(1%

r)l+rlZ

From (9), using (15) and yx = 1 +x log y +0(x2), we find

(16) in agreement

with (11) and (14).

2. A general expression

for the reflection

amplitude

When the dielectric function is ll for z < zl, l2 for z > zZr and has some form in between which allows solution of the wave equation in terms of linearly independent functions A(z) and B(z), a general expression for the reflection amplitude may be written down in terms the values of A and B and of their derivatives at the end-points .zl and z2. We will here assume that E(Z) has no delta-function singularities (or worse), but in dealing with the p-wave later we will need to include the effect of delta functions at zI and z2. The general expression for the reflection amplitude follows from the continuity of E and dEldz at zl and z2. Since eiqlz+ rs eeiqlL, z < z, E(z) =

aA

+ PB(z),

z1 zz z < z2,

(17)

( t, eiqZZ, 2 2 Z2, we obtain four linear equations in the four unknowns rs, t,, CY,p. Solving for rs we find (writing A, for A(zJ, A{ for dA/dz at zl, etc.)

x q1q2(A1B2- B,A2) + iq,(A,B; - B,A;) + iq,(A;B* - B ;A3 - (A;B; - B ;A;) q,q2(A,B2-B1A2)+iql(A1B$B,AI)-iq2(A~B2-B;A3+(A;B5-B;A;)’

(18) This expression

2 +q2(z)+

applies to any wave equation which can be put in the form =0

(and for which q2 does not contain delta-function

(19) singularities or worse).

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We note that when total internal reflection occurs, (18) shows explicitly that rS lies on the unit circle. For example, when q2 is pure imaginary, (18) takes the form e2iq1z1(a+ ib)/(-a + ih), with a and b real (since A and B are solutions of a linear differential equation with real coefficients, they can be chosen to be real). When one solution of a linear homogeneous second-order differential equation is known, a second (linearly-independent) solution can be found by quadrature6). For example we can write B(z) = A(z) j” dz’/A’(z’). Let J: denote Jl; dz/A2(z). Then 2

A,B2 - B,Az = A,A,

II

, 2

,

A,B; - B,A; = A,IA,+

A,A;

A;B2-

+ A;A2

B ;A2 = -AZ/A,

I

(20)

2 ,

1

I

A;B;-B;A;=A;/A,-A;/A,+A;A;

2 .

1

This form of the coefficients explicitly demonstrates that the reflection amplitude for an arbitrary non-singular profile of extent AZ approaches the step profile value e2iq1zl (sl - q2)Nqr + q3 as AZ + 0.

3. The s-wave for the Rayleigh

profile

For arbitrary angle of incidence, (l), transforms to g+

[-(KS,‘+++

the s-wave equation (2) with E(Z) given by

=o.

(21)

Comparison with eq. (9.1.49) of ref. 7 shows that E is proportional to Y)“~ times a Bessel function of order v and imaginary argument +iK(Az/An)n. For analytical and numerical reasons, it is convenient to have a positive imaginary argument, which we accordingly take to be iKJAz/Anlq = ix. We have a choice of linearly independent pairs of Bessel functions of imaginary argument: we can take I,(x), I_,(x) with Wronskian equal to -2 sin TV/TX, or K,(x), I,(x) with Wronskian l/x (9.6.14 and 9.6.15). In this section we will use I,, I-,; the special case v = 0 ((o/c)lAz/Anl = 2 will be considered in section 6. The linearly independent functions A(z) and B(z) are accordingly A(z) = #‘IV(x),

B(z) = T$‘~I_~(x).

(22)

JOHN LEKNER

240

Since x = KlAz/AnIn, and q is linear in z (eq. (1)) we have d/d.z = crK d/dx, where u = sgn(An/Az). The derivatives of A and B accordingly become, using (9.6.26),

2 =(+-

v)

2

q-1’21v(x)+ uKq 1’2I,_l(x),

g = (; -

v)

2

q-1’21_y(x) + uK~"~I,_,(x).

(23)

From (22) and (23) we can now evaluate expression (19) for r,. These are

AJ3i

-

BlA;

=

(+ -

v) 2

AIB2 - B IA2 = (; - V) 2

the coefficients

in the general

(;)"'

CJx,, x2) + aK(‘17,~)“~D,(x,, x3,

(z)“’

C,(x,, x2) - ~rK(7),772)“~D,(x2,x,),

A;B; - B;A; = (; - v)’ (2)2(n,nJ1’2C.(x,,

Arl +(f-V&UK

xz) + K2(n,“r12)“2E,(~,, x,)

I(q,)1’2 9

(24)

D v (Xl, XJ -

(3 )1’2 7)2

D (x2, x,) y

I '

where CvhxJ=

~"~~1~~-"~~2~-~-"~~1~~"~x2~,

MXlr

x2)

=

~"~~~~~1-"~~*~-~-"~~1~~"-1~~2~,

E,(x,,

x3

=

L-lag-.(x2)-

(25)

I,-u(x,)L,W

Note that lrsl is manifestly translationally invariant, since the values of 7, and xi depend only on the physical parameters E,, l2, w/c, AZ and angle of incidence, and not on the choice of origin. For a given set of physical parameters, the evaluation of rs requires the calculation of the four functions I,, I-,, IV_,, I,._, each at the two points x1 and x2. Ascending series, integral representations, and asymptotic forms for I, are given in sections 9.6 and 9.7 of ref. 7; some numerical aspects will be discussed in section 6. At normal incidence (K = 0) the arguments xl and x2 are both zero. We can obtain the normal incidence results by a limiting process, using I,(x)* (x/2)“/r(l+ V) (9.6.7). From this limiting form and the result r(l+

v)F(l-

Y) = -J!L sin ~7r’

(26)

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241

we find that as K + 0,

(27)

Using the limiting forms (27) we can verify that (18) reproduces incidence result (9).

the normal

4. The p-wave for the Rayleigh profile We saw in ref. 5 that the p-wave transformation B = v/(e/er) b, 2 ,/2d

E dz2

equation

(4), becomes,

on using the

-l/2

b =O.

]

(28)

Thus the Rayleigh profile, where E-“~ is linear in z, has the unique property that the E and b equations are the same, except at the end-points z, and z2 of the profile. There, because of the discontinuity in the slope of n = E-“~ the equation for b contains additional delta-function terms: S(z - ZJ - $ S(z - z3). As a consequence,

dbldz is discontinuous

at z1 and z2:

b’(zI+)-b’(zl-)=$zb(z,), bYzz+)-

b’(z,-)

=

-+-s

b(zJ.

(30)

The three analytic parts of b(z) are (cf. eq. (32) of ref. 5) eiqiz - r, ewiqlZ, z d z,, b(z) =

aA

+ BB(z),

z1 s z c z2,

(30

1 t, eiq2’, z 3 z2, where again A and B are equal to T)“~ times a Bessel function of imaginary argument. The continuity of b at z1 and at z2, and the discontinuity in dbldz as

JOHN LEKNER

242

given by (30), lead to _

rp

=

e2iwl

x qlq2(A1B2- BIA2) + iql(A1B2- B1& + iq2(A,B2- B,AJ - (A$,qlq2(AIB2 - B1A2) + iq,(A,&BI&) - iq2(A1B2 - B1A2) + (AI&-

BIA2) BP%) ’ (32)

where, for i = 1, 2 Ai =

A;--!-!-+,

jji = Bf-167)~~. 7);

I

(33)

AZ

Thus -rP has the same form as rs, with A’ being replaced by A, B’ by B. When I, and I-, are chosen as the basis, A and B are given by (22), and

( 12

A=- ; + B=-

v

;+, (

)

2

qp2r”(x) + aKq 1’2r”_l(x), (34) n-“*I_“(x) + UK?-)1’21,_“(x).

We see, on comparison with (23), that the factor i- v has been replaced by -(i + Y). Formulae corresponding to (24) may be obtained using this substitution, and thus need not be listed here. Similarly, the K + 0 limits may be obtained from (27) using this substitution, and lead to the normal incidence result (9). Thus the computation of rp for the Rayleigh profile runs in parallel to the computation of rsr and the values of the Bessel functions at xl and x2 serve both calculations.

5. Solution using I,, K, as basis For some applications, and in particular for the case v = 0 discussed in the next section, it is more convenient to use the basis A(Z) = n”21y(x),

For this basis a convenient 2 dB dz=

(35)

B(z) = q”‘Ky(x). form of the derivatives

is, using (9.6.26)

n-“21y(x) + aKn”21,+,(x),

(36) -“2K,(~) - aKn “*Ky+,(x).

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243

From (35) and (36) we find A,Bz - B,AI = (rl,q2)“%(x,,

x3,

A,B:-B,B~=(~+~)~(~)1’2L.(xl,x2)-uK(qlq2)’~2~~(xl,x2),

A;&-

Bl&

=

(;+

A;B;-B;A;=

4

2

($*z&,,

($+ v)*(2) -

(;+

9

h%-“*L”h,

2

x2)

+

uK(~,~~)~~*M~(x~,

x2) - K2(q1r12Y’*L”+,(X1,

UK { (3”‘Wh

~2) -

(37)

xl),

($)“*iWx2,

x3

x1)),

where LAX,, Mb,,

x2) = W,)Kh*)

- K,(x,Mx*),

~2) = &b,)Kv+1(~2)

(38)

+ K,h)L+,(x2).

The K +O limits of (37) are proportional to those given in (27), with the common factor sin VT/VT being replaced by V-I. The results for the p-wave are obtained as before by replacing A’ by A and B’ by B as defined by (33). These are

A(z) =

(-; +v) 2

B(z) = (-;+

V) 2

q-“*Iv(x) + wKvJ”~IV+,(X), (39)

r)-“*K,(X) - aKn”*K,+,(x).

Comparison with (36) shows that the p-wave reflection amplitude may be obtained by using the coefficients as given by (37), with the substitution ;+v-+-;+Y. 6: The case v = 0 The order V, defined by (7), is zero when (w/c)(Az/An( = i. For example, when E, = 1, e2 = (4/3)2 (as for the air-water interface of fig. 1) An = -f and v = 0 when (o/c)Az = i, i.e. when the wavelength in air is about fifty (16~, to be precise) times the extent of the interface. This illustrates the first reason for our detailed look at the case Y = 0: we can use it to check the accuracy of recent results4P5) giving r, and r, to second order in the parameter AZ/A. The result for rS has already been given in equations (11) to (13); the r, result is5) Q, - Q2

2iQlK2D

rp=-Q1f(~(QI+~2~5 + CQ12$21’

4 2 -+Q~(E,-~2) t :+DQ2

2 2K2L2->I* C

11

+ ‘X@Z)~,

(40)

244

JOHN LEKNER

where Qi = qi/ei, 1’ is given by (12), and (41) (42) For the Rayleigh profile (l), I2 is given by (14), and D/AZ = ; (An)2,

Comparison of the general second-order theory with the Rayleigh profile for u = 0 will be made below. A second reason for considering the case v = 0 in detail is that it provides a check on the analytic expressions for rS and rP in the two representations (using I,, I_, or I,,, K, as basis). In the I,, K, basis the Y = 0 case is obtained simply by setting I, = 0 in the formulae of section 5. In the I,, I_, basis, v = 0 must be obtained as a limiting case. From (9.6.46), (9.6.44) and (9.6.26) we find I”(x) = lo(x) - vK,,(x) + S(v2), I,-,(x) = I,(x) - v (K,(x) -b lo(x)) + WV’), L,(x)

(45)

= I,(x) + v K,(x) +; IO(x)) + O(v’). (

Using (45) in (24) we find that each of the coefficients AlB2 - B1A2,. . . , goes to zero linearly with Y. On dividing by 2~ and then taking the limit v + 0 we regain the results obtained by setting v = 0 in the I,, K, basis. The same procedure may be carried out on the p-wave formulae. At normal incidence, the v = 0 s and p formulae reduce to r = exp{-ilp

- lJI(cL + 1)) 2!~~o~ Plv

in agreement with Rayleigh’s eq. (20). Finally, the third reason for examining v = 0 is for numerical convenience: there are polynomial approximations for I,,, I,, K,, and K, (9.81, 9.8.3, 9.8.5 and 9.8.7) and the ascending series (9.6.10 and 9.6.11) are easily programmed. Both of these methods provide sufficient accuracy for the comparison of the general second-order theory with the v = 0 case. The results are given in graphical form; we use the gauge (13) (leading to (15)) throughout, and the

EXACT REFLECTION

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FOR THE RAYLEIGH

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245

2

l

l

----I l

l

l

.

l

l

.

2

l

l

0

l

-1

l

i

I

Fig. 2. Reflection amplitude for the s-wave, plotted in the complex plane with Re(r,) horizontally, and lo6 x Im(r,) vertically. The arrow indicates increasing angle of incidence; the values shown are for 0, = 0°(5”)!W.

l

l* T

5

l

l

,

* - -5

Fig. 3. Fractional error in rs to second order, (r,o+ r,z)/r,(exact)- 1, in parts-per-million. complex plane. Angle of incidence increasing from 0” to 90” in steps of 5”, as in fig. 2.

0

in the

1

l l

. l

1

l l

l l

. .

l

II-5 Fig. 4. Reflection amplitude for the p-wave, with Re(r& plotted horizontally, and 10’ X Im(r& vertically. The arrow indicates direction of increasing angle of incidence; 6%= 0°(50)!900.

air-water optical dielectric constant values el = 1, e2 = (4/3)2. We see that the second-order theory has part-per-million accuracy for the s-wave, but only part-per-thousand accuracy for the p-wave near the Brewster angle. This is because the real part of rP passes through zero near O,, while the imaginary part is small (first-order in the small parameter qdz). The fractional error in effect becomes second-order instead of third-order near the Brewster angle.

JOHN LEKNER

246

Fig. 5. Fractional error in r, to second order, (IN+ r,l + r&/r&exact) - 1, in parts-per-thousand. Only the region near the Brewster angle (&I = 53.13”) is shown, from tI,= 51” to 5S”, in steps of 0.1”. The points just above and just below the real axis are at 01 = 53.1” and 53.2”, respectively.

Fig. 6. rp/rs in the complex plane, with angle of incidence increasing in the direction of the arrow from 0” to 90” in steps of 5”.

In fig. 6 we show the ratio rP/rS. The shift A@, in the Brewster angle, defined by’) Re(r,/rJ = 0 at angle of incidence OB+ A& = arctan

2 + dOa J El

is -2.953 millidegree. The predicted A& =

d$;;zi) +‘%@13.

(47) value (ref. 5., eq. (82)) is (4%

From (40), (12), (43) and (44) we find the value predicted by the second-order theory to be -2.951 millidegree, in agreement to better than one part in a thousand.

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References 1) J.W.S. Rayleigh, Proc. Lond. Math. Sot. XI (1880) 51; also Scientific Papers, vol. I p. 460, p. 460 (Cambridge, 1899), and Theory of Sound, Section 148 (Dover, 1945). 2) F. Abel&s, in: Ellipsometry in the Measurement of Surfaces and Thin Films, N.B.S. Misc. Pub]. No. 256 (1964), p. 41. 3) L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960) 668. 4) J. Lekner, Physica 112A (1982) 544. 5) J. Lekner, Physica A (to be published). 6) G. Birkhoff and G-C. Rota, Ordinary differential equations (Blaisdell, 1969), p. 35. 7) F.W.J. Olver, Handbook of Mathematical Functions, M. Abramowitz and I.A. Stegun, eds. (N.B.S. Appl. Math. Series No. 55 (1964)), chapter 9.