An optical study of the interfacial region of a surface-aligned nematic liquid crystal

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An optical study of the interfacial region of a surface-aligned nematic liquid crystal

J.P. Nicholson

To cite this version:

J.P. Nicholson. An optical study of the interfacial region of a surface-aligned nematic liquid crystal.

Journal de Physique, 1987, 48 (1), pp.131-139. �10.1051/jphys:01987004801013100�. �jpa-00210415�

An optical study ^{of the} interfacial region

of ^a surface-aligned ^nematic liquid crystal

J. P. Nicholson

Department ^of Physics, University ^of Strathclyde, John Anderson Bldg., ¹⁰⁷ Rottenrow, Glasgow ^G4 ONG,

U.K.

(Reçu ^{le 30} juin 1986, accepté ^{le 23} septembre 1986)

Résumé.

Le coefficient de réflexion optique ^{de la zone de} séparation ^{entre la} phase nématique ^{du 7CB et une}

frontière solide a été mesuré en fonction de l’angle d’incidence et de la température. Il existe une déviation considérable de la réflexion de Fresnel pour le rayon extraordinaire, mais le coefficient est semblable à celui de Fresnel pour le rayon ordinaire. La modélisation des données expérimentales exige ^{une zone de} séparation ^d’une

densité ~ 1 1/2 % plus grande ^{et un} paramètre d’ordre 10-20 % plus grand que celui ^{en masse.} L’épaisseur ^{de la} région interfaciale est ~ 850 A. On propose donc qu’il ^{existe une couche} quasi smectique ^le long de la surface.

Abstract.

The optical reflectivity from the interfacial region ^{of a} nematic/solid wall boundary has been measured

as a function of incident angle ^and temperature for 7CB. Considerable deviation from Fresnel reflectivity ^{occurs for}

the E-ray whereas reflectivities for the 0-ray ^{are similar to} Fresnel. To fit the experimental ^{data an} interfacial region having ^a density 2014 ^{1 1/2 %} greater ^{than bulk as well as an} increase of order parameter by ^{10-20 % is} required. ^The skindepth ^{of the} region obtained is

850 A. It is postulated ^{that a} quasi-smectic layer ^{exists near to the} boundary

wall.

Classification

Physics ^Abstracts

61.30

42.80

68.00

1. Introduction.

The alignment ^{of a nematic} liquid crystal (LC) ^{into a} single ^domain by ^the action of the enclosing ^{cell walls}

has been known since the earliest experiments [1]

where simple unidirectional rubbing ^{was found to} produce the desired results. A recent review of the various chemical and physical ^surface treatments has been given by Cognard [2]. ^{The effect} of the pre- treated boundary however is more complex ^than merely providing ^a uniform detector direction. In the region ^of

the nematic/boundary interface, the nematic molecules

experience short range forces due to the solid surface itself as well as the longer range intermolecular forces within the nematic which produce ^{such bulk} properties

as long-range order, anisotropic elasticity ^and viscosity,

etc. This is liable to produce localised variations of nematic parameters (such ^{as order} parameter, density, etc.) ^{in the} interfacial region from their bulk values.

Theoretical models [4-6] describing this interfacial

region have been mainly ^{confined to} temperatures

around the nematic-isotropic transition where the Land-

au de Gennes expansion [3] of the free energy in ^term of the order parameter, S, is valid. Their results derive

S ( z ) ^{as a} function of distance, ^{z from the} boundary ; S ( z ) ^is expected ^to decay from its value at the

boundary ^{to the bulk value over} distances of order 50- 500 A. A more ambitious approach ^{to the} problem using ^{mean field} theory ^to obtain the free energy in the presence of ^a boundary ^{has been} developed ^{for the case}

of homeotropic alignment by Telo da Gama [7] ^{with a}

further paper ^on the parallel alignment ^case expected.

Experimental measurements in the interface have been mainly ^{confined to the} isotropic phase [8-9] by obtaining ^the optical phase ^{shift between} light polarized parallel ^and perpendicular ^to the director after passing through ^{the cell} (parallel alignment). Measurements,

other than this paper, ^{of the} L.C./boundary ^interface

with the L.C. in the nematic state have been performed by ^{Mada and} Kobayashi [10], by effectively measuring

the optical transmittance of the interface, ^and by

Salamon [11] using fluorescence emission from a dye

loaded nematic. Both report ^a different order par- ameter at the surface than in the bulk but make no

estimates of the skindepth ^{of the} interfacial region.

Mada and Kobyashi’s work, ⁱⁿ fact, ^is seriously ^marred by ^lack of any consideration of the effect of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004801013100

132

skindepth, ^{d, on} their results. Their expression ^{for the} optical transmission is derived using ^the simple ^Fresnel expression ^{for the} amplitude reflectivity ^{at normal}

incidence : r

(n1 - no ) / ( nj ⁺ no ) [12] where n1 ^is

the refractive index of the L.C. at the boundary ^with

the substrate and no that of the substrate. In fact the

reflectivity ^{of a} thin interfacial layer ^{is a} complicated

function of no, nl, n2 (the bulk refractive index) ^{and d} (see ^below, Appendix, ^{and Refs.} [17-19]). ^In this paper the optical reflectivity of the interface is measured ^{as a} function of angle of incidence (not merely ^{for normal}

incidence as in Ref. [10]) ; this enables a calculation of the skindepth, d, ^{and some} indication of the profile, S ( z ) , ^{in the} interfacial region.

2. Experimental.

The cell containing ^{the L.C.} (which ^was heptyl- cyanobiphenyl ^or 7CB) ^{and the} optical arrangement

are shown in figure 1. The L.C. is contained between the hypotenuse ^{faces of two} right-angled prisms ^made

of F2 flint glass. The refractive index of F2 glass ( n =1.62 ) falls between the two refractive indices of

Fig. 1(a).

Experimental arrangement ^for measurement of

optical reflection from glass/nematic interface. The L.C. cell is wedge-shaped ^to enable the reflected beam from the lower interface to be rejected.

Fig. l(b).

Magnified ^view of upper glass/nematic ^interface showing nomenclature used. Media (0), (1) ^and (2) ^{are F2} glass, L.C. interfacial region and L.C. bulk respectively.

7CB (1.65-1.68 ^and 1.51-1.52) ^thus ensuring ^{a reason-}

able reflectivity for either polarization direction. The interface is illuminated by light ^{from a} He-Ne laser at 6 328 A which may be polarized ^{as shown in} figure 1,

either with its electric field parallel ^{to the} plane ^of incidence, Ell (transverse magnetic ^{or TM} case) ^or perpendicular ^{to the} plane ^of incidence, El (transverse

electric ^or TE case). The reflected light ^{from the first}

F2 Glass/L.C. interface is detected and measured by ^a

Hamamatsu S1227 type photodiode ^followed by ^a

stabilized D.C. amplifier. ^The reflection off the lower surface emerges ^{at a} slightly ^different angle ^{because the}

cell is wedge-shaped, ^{and thus can be} distinguished ^and ignored.

Prior to assembly, ^the prism ^{faces are treated to the} following procedure : (i) unidirectional polishing, ^with

3 >m diamond paste ^to microgroove the surfaces in a

direction perpendicular ^{to the} plane ^of figure 1 ; (ii) ultrasonic cleaning ⁱⁿ detergent ; (iii) ^ultrasonic

rinse in distilled water ; (iv) ^rinse (twice) in deionized

(18 Mfl) ^{water. The cell was then assembled and} expoxied ^{with a 350} >m spacer ^{at one} end, ^to provide

the required wedge angle. ^{Prior to} filling, the surface

reflectivity ^{was checked, to} verify ^{that it} corresponded

to the expected Fresnel reflection [12] ^{from a} simple

clean surface. This confirmed that the microgrooves

were insufficiently deep or numerous to significantly

affect the reflectivity.

After filling with 7CB the cell was studied carefully

between polarizers ^{each at 45° to the} microgroove (n) direction, ^and illuminated by monochromatic

light. Satisfactory ^uniform alignment ^{was indicated} by ^a

uniform set of parallel interference fringe parallel ^to

the ⁿ direction. Each bright fringe ^{occurs when the} path

difference between 0 and E rays is ^an integral ^number

of wavelengths.

The cell was mounted on a rotating ^{table and} arranged ^{as shown in} figure la. The whole apparatus,

including ^{laser and} detector, ^was placed ^{in a} tempera-

ture controlled box which could be stabilized to within 0.1 °C. Reflected intensities were measured by ^the photodiode ^for varying ^incident angles, (J 0 ^{as well as}

over the whole nematic range of 7CB (30-42.5 °C). ^{If V}

is the actual voltage response of the detector for ^a given angle ^and Vo the response ^to the direct laser beam,

then the intensity reflectivity off the F2/L.C. interface is given by :

where T,

Here, T ( i ) ^{is the} transmittance through ^{the two}

side-faces of the prism, having allowed for reflection losses. The amplitude reflectance, r ( i 1 ) , ^{at the F2/air}

faces is given simply by the Fresnel formulae [12] :

where

and subscripts ( j = 1, 2 ) ^{refer to} the first and second media (in ^{this case air and F2} glass).

The accuracy ^{of the} reflectivity measurements

(- 2 %) ^was mainly ^limited by ^the stability ^{of the}

laser except ^at the very lowest values of R (near ^to

Brewster reflection) where the lower limit of the accuracy in R ^{was ±} 10- 6.

Experimental results for R are shown in figure ²

which shows the trend of R with respect ^to temperature

for a given ^incident angle, 00

^{66.2°; and} figures ^{3 and}

4 which show the variation of R as a function of

00 ^{for a} given temperature.

Fig. ^2.

Experimental reflectivity ^{as a} function of temperat-

ure for constant incidence angle ( 80

66.2° ) . Dotted lines

are the theoretical Fresnel reflectivity ^between glass ^{and bulk}

L.C. (i.e. d

0). ^A single ^{set of} readings ^{in the} isotropic phase is also shown.

3. Theory.

An enlarged diagram ^{at the F2} glass/nematic ^interface

is shown in figure 1b, giving the nomenclature used in this paper. The light is incident from the glass, ^{which is} designated ^{medium zero,} having ^incident angle 8a ^and

refractive index no. The latter is known ^to high

accuracy, viz. no = 1.61656 ⁺ ( t - 20 °C ) ^{x 2.28 x}

10- 6 [13]. Medium 2 is the main bulk of the liquid crystal (7CB) ^{whose two indices,} n2o, n2E, ^{are also} known from the literature [14]. ^Medium 1 is the inter- facial region expected ^{to be oooooo 50-500 A thick,} having

refractive indices n10 ( z ) , n1E ( z ) which become

equal ^to the bulk values at z oooooo d, but vary ^{to some} unknown value No, NE at ^{the boundary} (z

0)

itself. The refractive indices are related to the local order parameter, S, by [15, 16] :

Fig. ^3.

Experimental reflectivity ^{as a} function of angle ^{at a} temperature ^{of 32.3 °C.} (a) ^TE polarization (b) ^TM polarization. ^The large range of values in the latter ^case dictated a log plot ^which exaggerates ^the similarity ^between

fitted and Fresnel curves.

where åa Iii is the fractional polarizability anisotropy

of the L.C. molecules and nl n 2 ^{+ 2} n2 ) ^{/3 is the}

average refractive index. A similar equation also relates the bulk values n2o, n2E ^to n2 ^{and S.}

Since S = S ( z ) ^{is a function} [4-7] ^which probably decays ^{from a} high ^{value at the} boundary ^{to the bulk}

value over the skindepth, d, both indices n10, n1E ^must also be functions of _z, the distance from the boundary.

Therefore we have to deal with the problem ^{of E.M.}

wave propagation ^{in an} inhomogeneous ^or

stratified

medium, ^a subject ^{in which} fortunately the literature is

134

Fig. ^4.

^{As for} figure ^{3 but at a} temperature of 38.2 °C.

abundant [17-19], ^{and which has been reviewed} by

Jacobsen [17], and Abeles [18].

The optical properties of any thin film ^can be

represented by ^a

^{transfer matrix}

^M

Imij] - ^This

relates the fields at the entrance to the film ( z

0 ) ^to

those at the exit of the film ( z

d ) by [18] :

A similar formula applies ^{to the TM case with E and}

H interchanged in the column matrices. It can be shown that the amplitude reflectivity ^{is then} given by [17] :

with

We must also use the appropriate ^index in medium 2 for the two cases, i.e. n2 = n2E ( TE ) ^and n2 = n2o ( TM ) . The matrix elements are also different in the TE and TM cases as the wave equation ^{in the two}

main cases differs (see ^{below or Refs.} [17, 18]).

To ealculate the intensity reflectivity, ^R = I r12, ^{for a}

given ^index profile n1E ( z ) (TE case) ^or njo ( z ) (TM case) the matrix elements mij ^{need to be} computed.

Three approaches ^were used, ^{one outlined} below, ^the others described in Appendix ^1.

3.1. SERIES EXPANSION FOR SMALL WAVENUM- BERS.

The method described here is a brief summary from reference [17]. ^The differential equation ^{for both}

TE and TM EM waves of amplitude ^{U can be written}

as :

with

and

for a non-magnetic medium (A ---= I-LO)

^{and where}

nlE, o ( z ) sin 81 ( z )

no ^sin 00.

For small values of k = 2 v/A, ^i.e. A > d, ^the

solutions of U can be written as a series expansion ⁱⁿ

powers of k. The two particular solutions of interest here are F and f

i 0, ^{where :}

For particular ^solutions having ^the required boundary

conditions (F(O) = 1; ~(0) =0; F’ ( 0 ) = 0;

¢’ ( 0 ) = ky ( 0 ) /3 ( 0 ) ), substituting ⁱⁿ equation (7) yields:

with and

with

It can then be shown [17] that the matrix elements in

(6) ^are given by :

-

evaluated at z

d.

This method was the predominant method used in the analysis ^{since it was} relatively ^{fast in} computer time. However its accuracy needed ^to be checked. This

was done by comparing its results for the TE case of a

linear-E profile (n E = NE ( 1 + n’ z ) ^{which is one of}

the few profiles ^{whose wave} equation solution has been

rigorously ^derived (Appendix 1). ^This comparison

showed that the series expansion method, ^{when used to}

4th power in k (i.e. v=2) gave values of R which agreed

to within 0.5 % of the rigorous solution in Appendix ^la

for d = 1 000 A. For d = 1 200 Å _agreement fell to

-

1-2 % and then rapidly diverged ^{as d was increased.}

None of the profiles tried here was rigorously ^soluble

in the TM _case, so TM values of R were compared ^with

values obtained by ^direct stepwise integration ^{of the}

wave equation (Appendix 2). ^{This indicated a similar} degree of accuracy in the _{4th power} computations ^for

TM reflection also.

4. Results.

,

The general ^{trend of our} experimental ^{results is indi-}

cated by figure 2 which shows the reflectivity ^{for a} given ^incident angle ( 00 = 66.2* ) ^{as a} function of temperature, ^{between 30-41 °C. A} single measurement at 43.5° (where the 7CB is in the isotropic phase) ^{is also}

shown for comparison. ^The theoretical expectation ^for simple Fresnel reflection between F2 glass ^{and bulk} liquid crystal (i.e. d

0) is also shown.

The results for the TE and TM cases are quite

different. TE reflectivity (of ^the E-ray) ^is considerably higher than the Fresnel value - in fact almost up ^to

100 % higher ^{near to} Tc. ^{The TM} reflectivity (of ^{the 0-} ray) however is very close ^to the Fresnel value, being

on average ^{about 7 %} lower than the Fresnel reflec-

tivity. Broadly ^similar experimental ^{results were ob-}

tained in reference [10] ^at normal incidence.

This asymmetry in the results means that a simple model, ^where only S ^varies from the bulk value in the interfacial region, ^is unlikely ^{if not} impossible, ^{to fit the} results ; ^for equations (3) ^and (4) ^{show that} increasing

S (near ^{to z}

0) ^{will cause} n1E ^to increase and nla ^to decrease. The glass refractive index, no ^{has a value}

midway between the 0 and E indices of the bulk

nematic, ^so increasing S alone would cause I no - nlr

for both 0 and E cases to increase. This, ^{for a} given ^d,

would cause an increase in reflectivity ^{for both TM and}

TE rays. The only way, ^{as we} shall see below, therefore, ^{to fit our} experimental ^{results is to} hypothesize ^that n1 (and ^{thus the} density) also varies from the bulk value and rises at the surface. Then the TM reflectivity - ^Fresnel reflectivity since for the 0-

Ray, ^{the two} effects almost cancel out (Eq. (3)), leading ^{to little} change ⁱⁿ nlo from the bulk value.

Typical ^{results on} which the actual fitting procedure

was performed ^{are shown in} figures ^{3 and 4 which show}

both TE and TM reflectivity ^{as a function of incident} angle, 00’ at constant temperatures ^{of 32.3 °C and} 38.2 °C respectively. ^Similar results in all were obtained for eight temperatures between 30-41 °C and also at

43.8 °C (isotropic).

To analyse ^{the results} exemplified ⁱⁿ figure ^{3 the} intensity reflectivity, ^R

r2, ^was computed using equation (6) and the 4th order series expansion ^method

outlined in section 2, using ^several postulated ^refractive

index profiles, n nip ( z ) ( P

^0, E ) for the inter- facial region (0 ^z d) , ^{and n}

n2P (bulk value)

for z > d. Profiles tried were :

Linear-n profile

Linear-E profile Hyperbolic profile Step profile

The latter profile is that of a simple uniform film of thickness d and is known rigorously [12].

Best fits to the experimental ^{data were obtained} by weighted least square fitting procedure by obtaining ^the

minimum of

where yi, Ri ^and (7-i ^{are the} experimental ^and theoretical

reflectivity respectively ^{for a} given ^{value of} 009 ^and ai i

the experimental ^error. Good almost identical fits were

obtained for the first three profiles ^{with the} ai ^cor-

reponding ^{to 2 %} error, whereas the step profile gave

significantly poorer fits. The best-fit curves, shown in

figure 3, ^thus yielded values of the surface indices

No ^and NE ^and skindepth, d, shown in table I for the linear-n profile. Results for the other two

good-fit » profiles ^were virtually identical within experimental

error. Also shown in table I, ^for comparison, ^are

interpolated bulk values from reference [14].

136

Table I.

Fitted results for ^linear-n profile.

The results show that for this particular ^nematic (7CB), the interfacial region ^{has a} skindepth ^which

remains roughly ^constant at d oooooo 700-1 000 A, ^within

experimental error over the whole nematic range. As

expected from the earlier discussion above, the surface

index, nlo, ^{was close} to, ^but greater ^{than the bulk value} whilst n1E ^is substantially greater.

We can now compare the order parameter, ^{S, and}

average refractive index ^at surface, N and in bulk, n2, through ^the equations below, obtained from

equations (3) ^and (4) :

and identical equations for the bulk v4lue S2 ^with NE, No ^{and N} replaced by the bulk indices n2E, n20 ^and n2.

The average refractive index is related ^to the density,

p, by [15] :

where k

3 m/4 Tra ^{is a constant with m} the molecu- lar mass.

Using equation (13) ^{and one} known value of the

density of the nematic (p

^{0.995 at 36 °C} [21]) ^{we can}

now obtain estimates of the density ^at the surface (with

n

N ) and in bulk (with n

n2). The results of these calculations are shown in table II, together ^{with those}

for the order parameter ^from equation (12) using ^a

value of a /Aa = 1.34. The latter was obtained by comparing absolute values of S obtained by ^Raman scattering [22] with the bulk values [14] through equation (12). The accuracy of this estimate is probably

not of importance here, ^{as we are} mainly interested in

comparative parameters ^at surface and in bulk. In summary, ^our results indicate an interfacial region ^of

thickness - 850 A with a surface value of S from 10 %

(at ^30.5 °C) ^{to 20 %} (at ^40.9 °C) greater than that in the

bulk, ^{and surface} density ^{of order 11/2 %} greater ^than the bulk density.

Table II.

Derived interfacial parameters.

5. Discussion.

It should be emphasised that the refractive index

profiles considered are probably’not ^the only ^ones

which would fit the data. The method used, although discriminating against ^the simple step profile, ^{is not}

very shape-dependent for similar profiles. ^{With these}

reservations our results indicate a skindepth (- ⁸⁵⁰ A)

substantially higher than that indicated by ^theories

based on the Landau-de Gennes expansion of the free- energy [3-6]. ^The theory developed by ^Allender

et al. [6] suggests skindepths ^{of order E =}

Eo [ T*l ( T - T* ) ]’/’ where eo - ^{5 A for 7CB} [9].

This yields ^{a value of E - 53 A for our} single ^{set of}

measurements in the isotropic region which compares

favourably ^{with best fits} giving d - ^{30-100 A. Lack of}

accurate refractive index data in the isotropic phase precluded ^{more accurate} estimates of d, ^{so further}

measurements were not pursued ^{in the} isotropic phase.

Two drawbacks arise in the above-mentioned theoretical approach : (i) ^{the theories are confined to}

the neighbourhood ^{of the} clearing temperature, TC (ii) ^no provision is made for a possible change ⁱⁿ density ^{near to} the wall. Both these objections ^however

are avoided by ^the theory developed by ^{Telo da}

Gama [7] ^{for the} homeotropic ^{case which not} only envisages ^but predicts ^{a rise in} density ^{near to the wall.}

However his results for the skindepth ^are comparable

with E obtained by ^the simpler theories, ^{and it seems} unlikely ^{that the} parallel alignment ^case (yet ^{to be} published) ^would produce ^{results over an order of} magnitude greater.

A possible explanation of both the large skindepths

and rise in density ^{in the} interfacial region ^indicated by

these measurements could be the formation of a thin solid phase ^or quasi-smectic layer ^at the wall surface.

Observation of smectic layers ^for octylcyanobiphenyl (8CB) ^{above its} smectic/nematic transition have been observed by Rosenblatt [23] by observing ^{anomalies in}

the Frdedericksz transition critical field. A tendency ^for

both 8CB and 5CB to form smectic layers ^{at the} nematic/vapour surface has been inferred from surface tension measurements [24]. Indications of smectic ten- dencies by ^{5CB and 7CB} (which ^{do not exhibit a}

smectic phase) ^{even in the} bulk of the fluid have been

shown by ^the X-ray diffraction results of Leadbetter et al. [25], who infer local smectic order extending

over 150 molecules in the ^case of 7CB. Such large

local ordering ^{however was not} observed for MBBA and EBBA. Thus 7CB (and ^{to a lesser extent} 5CB), being ^{similar to} the smectic 8CB, ^{is liable to have its}

smectic tendencies enhanced in the neighbourhood ^of

an alilgning boundary, ^{and show an} interfacial region having quite ^a large skindepth. ^{It is} quite likely

however that a nematic such as MBBA with little smectic tendencies will show much smaller skindepths.

Finally, ^{work on the} interfacial region ^{at a} nematic/vapour free surface is of interest at this point.

Als-Neilson et al. [26], using X-ray diffraction, ^show significant ^smectic ordering ^{near to} the nematic free surface for 80CB above the smectic/nematic transition.

Other work by Pershan et al. [27] ^and Ocko et al. [28]

use the deviation of the X-ray reflectivity ^{at the}

interface from the Fresnel value to show similar conclu- sions for 12CB and 80CB.

6. Conclusion.

Preliminary investigations have been made into the wall/nematic interfacial region ^{of 7CB} by measuring ^the optical reflectivity of the interface. Substantial differ-

ences from the Fresnel reflectivity ^{of a} simple ^walllbulk

nematic boundary ^{were observed for the} E-rays,

whereas for the 0-rays ^the reflectivity ^was slightly

lower than the Fresnel value. The experimental ^results

could only ^{be fitted} by assuming ^an increase in density

as well as order parameter ^{in the} interfacial region. ^The

interfacial region ^was thus deduced to have a thickness,

d - 850 A with order parameter ^{10-20 %} higher ^{at the}

wall than in bulk, ^{and a} density rise of order 1112 %.

These results are difficult to explain ^{on a} simple

Landau-de Gennes nematic model, but could be ex-

plained by the formation of a quasi-smectic ^A layer ^at

the surface of the aligning ^wall.

Acknowledgments.

Thanks are due to Mr. David Lang ^whose help ⁱⁿ developing the detector and making preliminary ^meas-

urements are greatly appreciated, ^{and to Mr. Bob}

Dawson for fine technical work in making much of the apparatus. ^{I am also} grateful ^to Drs. M. Bradshaw

(R.S.R.E.), ^{D. Dunmuir} (Dept. ^of Chemistry, ^Shef-

field University) and T. Faber (Cavendish Laboratory, University ^of Cambridge) ^for helpful correspondence

and refractive index data on 7CB. I acknowledge ^also

the support, ^both logistic ^and financial, of my -col-

leagues ^{in the} Dept. ^of Physics, University ^of Strathclyde.

Appendix ^1.

TE REFLECTION FOR A LINEAR-E PROFILE, nl = N2 ( 1- n’ z ) . ^{- This is one of the few profiles which}

is soluble analytically, and is treated in the book by

Wait [19].

The differential equation for the TE electric field in medium (1) 1Elf + k2(n, (Z)2 _ S2 ) E = 0] ^be-

comes :

where k

vacuum wavenumber

2 03C0/03BB ; C

(N 2 _ S2) 1/2 ^{where S}

^{no sin 00} and the E suffixes have been dropped since this section is TE throughout.

Making the substitution :

we obtain Stoke’s differential equation :

The two independent ^{solutions of this} equation ^are

the Airy functions Ai ( t ) ^{and Bi} ( t ) [20].

The article by ^Wait only ^{treats an} infinitely ^thick layer (yielding [ R I = 1), ^{but we can} easily ^{extend the}

result above to a thin layer of thickness d. The general

formulae for the matrix elements has been obtained by Abeles [18], ^which, ignoring ^constant factors and as-

suming ^a non-magnetic ^{medium, are} given by :

Here Ul, U2 ^{are two} independent solutions of the wave

equation for the index profile ⁱⁿ question. Thus for the linear-E profile ^{we must make the} substitutions :

where t is given by (A.2).

The primes ⁱⁿ equation (A.4) ^denote differentiation with respect ^{to z ; thus :}

where the dot superscript ^indicates differentiation with respect ^{to t.}

Appendix ^2.

TM REFLECTION - GENERAL WAVE EQUATION INTE- GRATION. - For TM reflection it is easier ^to consider the H field since there is just ^one component perpen- dicular to the plane of incidence. There is only ^a single

transmitted wave in medium (2) whose time indepen-

dent form is :

138

where k2 = n2 k ; ^a = no ^sin 80 = n2 ^sin 02, and the y- axis is along the interface boundary ^{in the} plane ^of

incidence.

We therefore seek solutions in medium (1) ^{of the}

form :

Both C ( z ) ^and S ( z ) obey ^{the time} independent

wave equation which for the case where n1 = n1 (z) only ^is [12, 18] :

The procedure ^{is then as follows :}

1. Starting ^{with the correct} boundary conditions at the (1)/(2) interface viz :

(assuming ^that n1 ( d) = n2) ^{the functions} C ( z ) , S ( z ) ^{are evaluated} by step-by-step integration ^of (A.8) ^{until z}

0 is reached.

2. This yields ^the computed ^values C ( 0 ) , S ( 0 ) , C’(0),8’(0).

3. Correct boundary conditions at the zero/(l) ^face

are [12] :

The field Ho ^{in medium} (0) is made up of ^an incident and reflected wave having amplitudes ^{I and R} _respect- ively :

and

Combining equations (A.7), (A.9) ^and (A.10) ^and setting ^z

^{0 at the} boundary :

Combining equations (A.12) ^and (A.13) :

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