pd

Comparative analysis of absorbance calculationsfor integrated optical waveguide configurationsby use of the ray optics model and theelectromagnetic wave theory

Sergio B. Mendes and S. Scott Saavedra

Focusing on the use of planar waveguides as platforms for highly sensitive attenuated total reflectionspectroscopy of organic thin films, we extend the ray optics model to provide absorbance expressions forthe case of dichroic layers immobilized on the waveguide surface. Straightforward expressions arederived for the limiting case of weakly absorbing, anisotropically oriented molecules in the waveguide–cladding region. The second major focus is on the accuracy of the ray optics model. This model assumesthat the introduction of absorbing species, either in the bulk cladding or as an adlayer on the waveguidesurface, only causes a small perturbation to the original waveguide-mode profile. We investigate theaccuracy of this assumption and the conditions under which it is valid. A comparison to an exactcalculation by use of the electromagnetic wave theory is implemented, and the discrepancy of the rayoptics model is determined for various waveguide configurations. We find that in typical situations inwhich waveguide-absorbance measurements are used to study organic thin films ~klynl # 1021, hyl '1022! the discrepancy between the ray optics and the exact calculations is only a few percent ~2–3%!.© 2000 Optical Society of America

OCIS codes: 300.6490, 240.6490, 230.7390, 130.3120, 240.0310, 240.6690.

1. Introduction

Monolayers of organic molecules with a thickness of afew nanometers have been a subject of increasinginterest in recent years, as reported in the criticalreview of Swalen et al.1 Potential technological ap-

lications of these films include optical and electronicevices, sensors, and biomaterial surfaces.2–5 To ex-

ploit these possibilities fully, it is necessary to ad-dress several challenging problems. Among themare relations between the structural and the func-tional properties of molecules deposited as a very thinfilm on a solid support. The investigation of theserelations is a technically difficult challenge because ofthe high sensitivity required for monolayer detection,which should also be combined with nondestructive

The authors are with the University of Arizona, 1630 East Uni-versity Boulevard, Tucson, Arizona 85721. S. B. Mendes~sergiom@u.arizona.edu! is with the Optical Sciences Center. S.Scott Saavedra is with the Department of Chemistry.

Received 25 May 1999; revised manuscript received 18 October1999.

0003-6935y00y040612-10$15.00y0© 2000 Optical Society of America

612 APPLIED OPTICS y Vol. 39, No. 4 y 1 February 2000

sampling and in situ analysis. It has beendemonstrated6–8 that the planar optical waveguideoffers a highly sensitive platform for spectroscopi-cally probing organic films in the submonolayer re-gime. Applications include self-assembled films,polymers, Langmuir–Blodgett films, chemical sensorlayers, organic dye films, and protein monolayers.9–22

The inherent sensitivity of waveguide-absorbancemeasurements to the broadband spectral regime areextended with the recent development of a mul-tichannel, single-mode, planar waveguide-basedspectrometer.23

For mathematical analysis of waveguide struc-tures that comprise lossy media, as in the case of anabsorbing layer or cladding on a waveguide surface,we find in the literature several approaches. Thesecan be broadly classified as exact and approximate.Among the exact approaches for the calculation of theattenuation coefficient is early research on solutionsfor simple cases, as in that performed by Burke24 whoproduced a graphical solution for the symmetric,three-media, lossy waveguide in the TE polarization.The extension to asymmetric configurations, TMmodes, and automatic computer calculation was de-scribed by Kaminow et al.25 The application of the

peg

transfer-matrix method to multilayer waveguidesand the subsequent solution of a root-finding problemto determine the waveguide-bound modes have dom-inated recent research. Such procedures can befound in Chilwell and Hodgkinson26 and Offers-gaard.27 An interesting alternative for solving theeigenvalue problem has been given by Ghatak et al.28

by introduction of a prism coupler and by scanning ofthe coupling angle to find the resonant eigenvalues ofa multilayer waveguide structure. The integral ap-proach taken by Li29 guarantees calculation of allbound modes.

Different approximate approaches have been pur-sued to provide direct expressions of the attenuationcoefficient in specific lossy waveguide structures. Acomplex index of refraction has been assumed in thewaveguide-characteristic equation for particularcases and limits.30,31 Another approach, which isfrequently employed in optical fiber configurations, isthe perturbation analysis.32–35 In this technique,the waveguide loss is calculated by the fraction of themodal power inside the absorbing region. Stewartand Culshaw36 used a ray optics approach and theFresnel reflection coefficient to describe the case ofbulk-cladding absorption in the waveguide configu-ration at the TE polarization.

Our focus here is twofold. First, absorbance cal-culations in waveguide structures with the ray opticsmodel are extended to thin dichroic layers of chro-mophores. This extension is important for studies ofmolecular films in which waveguide-based attenu-ated total reflection spectroscopies are used. Specif-ically, the orientation distribution of the molecules ina film can significantly affect the functional proper-ties of the ensemble, and orientation–function rela-tions in molecular films are being studied withwaveguide attenuated total reflection techniques inseveral laboratories. The derivations presentedhere yield straightforward expressions for the limit-ing case of weakly absorbing, anisotropically orientedensembles of molecules in the waveguide-claddingregion for both the TE and the TM polarizations. Inaddition, these derivations correct an error in pub-lished expressions37 of the dichroic ratio obtainedfrom waveguide-based attenuated total reflectionmeasurements.

Our second major focus is the accuracy of the rayoptics model. This model, as well as the other ap-proximate approaches discussed above, assumes thatthe introduction of absorbing species either in thebulk cladding or as an adlayer only causes a smallperturbation to the original waveguide-mode profile.The accuracy of this assumption has not been consid-ered in detail. Here we assess the accuracy of thisassumption and the conditions under which it is validby performing a comparison of absorbance calcula-tions using approximate ~ray optics! and exact ap-

roaches for typical waveguide configurations. Forxact calculations, we employ the method of Offers-aard27 because it provides a convenient extension of

the electromagnetic wave-vector theory to eigenvaluecalculation of the propagating guided modes in

waveguide structures containing dichroic or birefrin-gent layers.

2. Wave Model

In the wave model the waveguide structure is treatedas a multilayer stack with each layer characterizedby a thickness t and a dielectric constant tensor ε#. Adiagonal tensor, where all nondiagonal componentsare nulls, is used in the analysis. More general de-scriptions of the dielectric constant are straightfor-ward; however, the experimental situations to beaddressed here can be fully described by the diagonaltensor represented in Eq. ~1!.

ε# 5 ε0Snx2 0 0

0 ny2 0

0 0 nz2D , (1)

where ε0 is the vacuum dielectric constant.For an isotropic layer, the three diagonal tensor

components are equal. A dichroic or birefringentlayer is represented by a tensor with unequal diago-nal elements. In particular the dichroism is de-scribed by different values of the imaginary part ofthe complex refractive index na 2 ika, with a 5 x, y,and z.

The bound modes of the multilayer waveguidestructure are derived from Maxwell’s equations andboundary conditions. A transfer-matrix approach isused in the calculation of the tangential vector com-ponents of the electric and magnetic fields. By im-posing on the resulting fields a bounding-modecondition, a characteristic waveguide equation is ob-tained, as described by Offersgaard.27 Next the ef-fective indices of the guided modes are obtained fromthe characteristic waveguide equation by a Muller’snumerical procedure38 implemented in an automaticcomputer routine. The numerical error of this pro-cedure can be made extremely small with conven-tional personal computers and certainly muchsmaller than any experimental accuracy. When atleast one absorbing material is present in thewaveguide structure, the resulting effective index is acomplex number, N 5 NRe 2 iNIm, with its imaginarycomponent determining the attenuation of the prop-agating mode. In absorbance units, the attenuationis given by

A 54pL

l ln 10NIm, (2)

where L is the propagation length of the guided modeand l is the free-space wavelength.

3. Ray Optics Model

Unlike the electromagnetic wave approach, which in-cludes at the outset the absorbing media in the cal-culation, the ray optics model first assumes a losslesswaveguide structure. The effective refractive index,N, and the effective thickness, teff, of each guidedmode are calculated for the lossless structure. Next,absorbing species are introduced as a small pertur-

1 February 2000 y Vol. 39, No. 4 y APPLIED OPTICS 613

etTtcud

wtl

ewe

f

prsfiait

6

bation. Either a small extinction coefficient is intro-duced into the cladding material or a thin absorbinglayer is introduced between the waveguide and thecladding. A critical assumption is that the pertur-bation is small enough so that the previously calcu-lated quantities, N and teff, can be used in thequations to follow. The calculation of the attenua-ion coefficient, or absorbance, is straightforward.he introduction of an absorbing medium frustrates

he total internal reflection at the waveguide–ladding interface, as indicated in Fig. 1. The atten-ation of the propagating guided mode can then beescribed by the intensity reflection coefficient, R, at

the waveguide–cladding interface. If we define h asthe total number of reflections at the waveguide–cladding interface, then the absorbance can be writ-ten as

A 5 2log10~Rh! >

h~1 2 R!

ln 10, (3)

here we have used the weak perturbation assump-ion ~R > 1!. The total number of reflections calcu-ated from the ray optics model is given by

h 5~nw

2 2 N2!1y2

2NLteff

, (4)

where nw is the waveguide index of refraction. Thexpression of the waveguide effective thickness, teff,hich takes into account the Goos–Hanchen shift

ffects,31 is given by

teff,TE 5 t 1ly2p

~NTE2 2 nc

2!1y2 1ly2p

~NTE2 2 ns

2!1y2 , (5)

or the TE polarization, where nc and ns are the re-fractive index of cladding and substrate, respectively,and t is the waveguide thickness.39 For the TM po-larization the effective thickness is given by

teff,TM 5 t 1ly2p

qc~NTM2 2 nc

2!1y2 1ly2p

qs~NTM2 2 ns

2!1y2 , (6)

Fig. 1. Schematic representation of the waveguide configurationtreated with the ray optics model. Total internal reflection isfrustrated at the waveguide–cladding interface.

14 APPLIED OPTICS y Vol. 39, No. 4 y 1 February 2000

where

qc,s ; SNTM

nwD2

1 SNTM

nc,sD2

2 1. (7)

The intensity reflection coefficient, R, is the sole pa-rameter that remains to be determined for the absor-bance calculation in Eq. ~3!.

Before presenting the results for particularwaveguide configurations, a few comments and defi-nitions may be helpful. For calculations presentedhere the imaginary part of the index of refraction, k,also known as the extinction coefficient is used.This quantity is related to the molar absorptivity, ε,and the molar concentration, c, by the following ex-pression:

4pkl ln 10

5 εc. (8)

In close analogy with Beer’s law we define the equiv-alent pathlength, bwg, in a waveguide experiment bymeans of

bwg ;Aεc

. (9)

The equivalent pathlength describes the waveguideresponse, which, when combined with the materialoptical response, εc, provides the absorbance mea-sured in a waveguide platform. This quantity is re-lated to Harrick’s40,41 definition of evanescentpathlength, de, by

bwg 5 hde. (10)

Thus the key parameter to describe the absorbancemeasured in a waveguide is the product of the num-ber of reflections, h, and the evanescent pathlength,de. We emphasize that once the expression for bwg isdetermined, the absorbance can be easily calculatedfrom Eq. ~9!. In other words, bwg contains all theinformation regarding the waveguide mode that af-fects the absorbance, i.e., the intensity of the electricfield at the cladding interface, the evanescent depthof penetration, the effective index of refraction, andthe wavelength dependence ~modal and material dis-

ersion! of all these parameters. Therefore, in theemaining part of this section, we present the expres-ions of the equivalent pathlength for waveguide con-gurations of interest. First, the case of a bulkbsorbing cladding media is considered. Next, bothsotropic and dichroic absorbing layers deposited onhe waveguide surface are examined.

A. Bulk Absorbance

A waveguide with an absorbing cladding medium isillustrated in Fig. 2. By using the Fresnel reflectioncoefficient at the waveguide–cladding interface andtaking the limiting case of a weakly absorbing me-dium, ~kcync! ,, 1, one calculates the intensity re-flection coefficient, R, for each polarization.Inserting this result into Eq. ~9! yields the following

t

A

aatw

a 42

ats

~

iptmtrsw

expressions for the equivalent pathlength in the caseof a bulk absorbing medium, bb,wg, in each polariza-ion: for TE modes,

bb,wg 5 F l

4p~NTE2 2 nc

2!1y2GH 2nc~nw2 2 NTE

2 !

teff,TENTE~nw2 2 nc

2!LJ ,

(11)

and for TM modes,

bb,wg 5 F l

4p~NTM2 2 nc

2!1y2G3 H 2nc nw

2~nw2 2 NTM

2 !~2NTM2 2 nc

2!

teff,TMNTM@nw4~NTM

2 2 nc2! 1 nc

4~nw2 2 NTM

2 !#LJ .

(12)

s seen in Eqs. ~11! and ~12!, the equivalent path-length in both TE and TM polarization is propor-tional to the depth of penetration, which appearsinside the square brackets.

B. Isotropic Adlayer Absorbance

A schematic representation of the absorbance due toan adlayer of thickness h and refractive index nl 2 ikldeposited on the waveguiding film is shown in Fig. 3,assuming kx 5 ky 5 kz 5 kl. The ray optics modelssumes a weakly absorbing medium ~klynl! ,, 1 andvery thin layer ~hyl! ,, 1. Under these assump-

ions, the intensity reflection coefficient of theaveguide–adlayer–cladding system is calculated,

Fig. 2. Schematic representation of the waveguide structure withan absorbing cladding medium.

Fig. 3. Schematic representation of the waveguide structure withan absorbing dichroic adlayer on the upper surface of the guidingfilm. Also indicated is the adopted coordinate system.

s by Macleod, to determine the equivalent path-length, bl,wg, in each polarization: for TE modes,

bl,wg 5 hH 2nl~nw2 2 NTE

2 !

teff,TENTE~nw2 2 nc

2!LJ , (13)

and for TM modes,

bl,wg 5 h

3 H2nl nw2~nw

2 2 NTM2 !@~1 1 ~ncynl!

4!NTM2 2 nc

2#

teff,TMNTM@nw4~NTM

2 2 nc2! 1 nc

4~nw2 2 NTM

2 !#LJ ,

(14)

Equations ~13! and ~14! show that the equivalentpathlength is proportional to the adlayer thickness,h. The proportionality constant is identical to thatin Eqs. ~11! and ~12!, assuming that nc 5 nl. Thedlayer absorbance is proportional to the adlayerhickness in the same proportion that the bulk ab-orbance is related to the depth of penetration.

C. Dichroic Adlayer Absorbance

Next we consider the absorbance in each polarizationdue to a dichroic adlayer deposited on the waveguidesurface with the optical constant as indicated in Fig. 3.The calculation of the extinction coefficient in eachCartesian direction, kx,y,z, from the molecular dipoleorientation is described in Appendix A. The intensityreflection coefficient of the waveguide–adlayer–cladding system follows the analysis reported by Ma-cleod,42 with the anisotropic coefficients taken fromHorowitz and Mendes.43 Assuming a thin andweakly absorbing adlayer, as before, the followingequivalent pathlength is obtained: for TE modes,

bl,wg 5 h2nl fy~nw

2 2 NTE2 !

teff,TENTE~nw2 2 nc

2!L, (15)

and for TM modes,

bl,wg 5 h

32nl nw

2~nw2 2 NTM

2 !@~NTM2 2 nc

2! fx 1 ~ncynl!4NTM

2 fz#

teff,TMNTM@nw4~NTM

2 2 nc2! 1 nc

4~nw2 2 NTM

2 !#L,

(16)

where the factor fi, defined in Appendix A, is the ratiobetween the extinction coefficient along a coordinateaxis and the extinction coefficient for a random mo-lecular orientation. In the isotropic limit we havefx 5 fy 5 fz 5 1, and Eqs. ~15! and ~16! reduce to Eqs.13! and ~14!, respectively.

Note that the equivalent pathlength, as describedn Eqs. ~11!–~16!, incorporates both explicit and im-licit wavelength dependencies. Therefore absorp-ion spectra performed in a waveguide configurationay exhibit differences from the corresponding spec-

ra obtained in a transmission configuration. A di-ect comparison of the spectra may incorrectlyuggest spectral shifts that are simply related to theavelength dependence of the waveguide mode. It

1 February 2000 y Vol. 39, No. 4 y APPLIED OPTICS 615

wwa

ipbrs

tama

~

tt

t1

6

is important to consider these dispersion effects inany spectral analysis before attempting to draw validconclusions. This observation is especially impor-tant in applications of waveguide spectroscopy inwhich observed spectral shifts are interpreted interms of the microenvironment of the chromophoresin the waveguide adlayer.

4. Comparison Analysis

In this Section, the electromagnetic-wave approachand the ray optics model are compared. The validityof the ray optics model and its weak perturbationassumptions are assessed for typical waveguide con-figurations.

A. Bulk Absorption

First we consider a Corning 7059 glass waveguide~nw 5 1.56, t 5 400 nm! deposited on a fused-silicasubstrate ~ns 5 1.46!. The cladding material is anaqueous solution in which absorbing species are dis-solved. The optical constants assumed for the solu-tion are nc 5 1.33 2 ikc. For each value of kc, theabsorbance is calculated with both the full-wavemodel of Section 2 and the ray optics model with Eqs.~11! and ~12!. The wavelength for the calculations

as 550 nm. Figure 4~a! shows the calculatedaveguide absorbance per unit length of mode prop-gation along the guide, Awgycm, plotted against the

extinction coefficient of the cladding solution, kc. Asseen in the plot, the ray optics and the wave modelsshow no significant difference at low to moderate val-ues of the extinction coefficient, demonstrating agree-ment between the two approaches. However, athigh values of kc, small deviations between the twoapproaches are observed as expected since the as-sumption for the derivation of the ray optics model~kcync ,, 1! is no longer valid. These deviations arellustrated in Fig. 4~b! where the relative error islotted; the relative error is defined as the differenceetween the absorbance values calculated with theay optics and the wave models, divided by the ab-orbance of the wave model. For kc 5 0.001, which

corresponds to Awgycm values of 3 ~TE! and 5 ~TM!,he discrepancy between the two calculations is onlypproximately 0.01%. Thus for experimentallyeasurable Awg values, the ray optics model is an

ccurate approximation.

B. Isotropic Adlayer Absorption

Next we examine a thin absorbing adlayer depositedon the upper surface of two waveguide configura-tions. The first configuration is the same waveguideas described above: a Corning 7059 glass film ~nw 51.56, t 5 400 nm! deposited on a fused-silica sub-strate ~ns 5 1.46!. The cladding media is waternc 5 1.33!, and the wavelength is 550 nm. The

adlayer is a 3-nm-thick film, which is a typical valuefor a protein monolayer.15 The real part of the re-fractive index of the adlayer is assumed the same asthe water superstrate ~nl 5 nc 5 1.33!. The extinc-ion coefficient, kl, for the adlayer is considered overhe range from 0 to 0.1. The relation between molar

16 APPLIED OPTICS y Vol. 39, No. 4 y 1 February 2000

absorptivity and extinction coefficient in the case of amolecular layer should be noted. For a moleculewith a cross-sectional area of 9 nm2, a single mono-layer corresponds to a surface coverage of 1.85 310211 molesycm2. For this case, an extinction coef-ficient of 0.1 corresponds to a molar absorptivity of161,000 M21 cm21. Thus the range of 0 to 0.1 for theextinction coefficient covers a broad range of molarabsorptivity and surface coverage.

In Fig. 5~a! the results calculated with the ray op-tics model are compared with those from the exactfull-wave approach. The discrepancy between thetwo approaches is expressed as the relative error, asdefined above, and is plotted in Fig. 5~b!. The rela-tive error is approximately 2% over a broad range ofthe extinction coefficient.

Next we examine a different single-mode waveguideconfiguration with nw 5 1.80 and t 5 180 nm, keepingall other variables constant. A tightly confinedwaveguide ~relative to the 7059 waveguide! was cho-sen to determine if the degree of mode confinement hasany effect on the accuracy of the ray optics model.The calculated relative error is plotted in Fig. 6. Thediscrepancy of the ray optics model with respect to thefull-wave model is approximately only 3%.

Fig. 4. Calculations for a bulk absorbing cladding medium with avariable extinction coefficient, kc. ~a! Absorbance values calcu-lated with the ray optics ~dashed lines! and the wave ~solid lines!models. ~b! Relative error of the ray optics model compared withthe wave model calculation. The parameters used in the calcula-tions are wavelength l 5 550 nm, waveguide thickness t 5 400 nm,waveguide index of refraction nw 5 1.56, substrate index of refrac-ion ns 5 1.46, and real part of cladding index of refraction nc 5.33.

lAitdpawntc

t

a

da7

lm

ta

1

C. Dichroic Adlayer Absorption

Next the ray optics model and the full-wave approachare compared for a thin dichroic absorbing adlayerdeposited on the upper surface of a waveguide. Thesame glass waveguide structure described above isconsidered: a fused-silica substrate ~ns 5 1.46!, aCorning 7059 glass film ~nw 5 1.56, t 5 400 nm!, andwater ~nc 5 1.33! as the cladding material. The di-chroic adlayer is modeled as a 3-nm-thick film with acomplex and anisotropic refractive index of 1.33 2

Fig. 5. Calculations for an isotropic absorbing adlayer with avariable extinction coefficient, kl. ~a! Absorbance values calcu-ated by the ray optics ~dashed lines! and the wave ~solid lines!

odels. ~b! Relative error of the ray optics model compared withthe wave model calculation. The parameters used in the calcula-tions are wavelength l 5 550 nm, waveguide thickness t 5 400 nm,waveguide index of refraction nw 5 1.56, substrate index of refrac-ion ns 5 1.46, real part of the adlayer index of refraction nl 5 1.33,nd cladding index of refraction nc 5 1.33.

Fig. 6. Relative error of the ray model compared with the wavemodel for an absorbing adlayer. The waveguide configuration isthe same as described in Fig. 5, except for waveguide thickness t 580 nm and waveguide index of refraction nw 5 1.80.

ikx,y,z. The film is composed of molecules having ainear electronic absorption dipole, as described in theppendix A.1. A rotationally symmetric distribution

s assumed for the dipole orientation distribution inhe waveguide plane ~the x–y plane in Fig. 3!. Theichroism is due to a nonrandom distribution in theolar angle u, which is the angle between the electronicbsorption dipole and the axis perpendicular to theaveguide surface ~z axis!. Each Cartesian compo-ent of the extinction coefficient can then be written inerms of the extinction coefficient, k, for the isotropicase and the mean tilt orientation angle, u#, by use of

kx 5 ky 532 k sin2~u#!, (17)

kz 5 3k cos2~u#!. (18)

The isotropic condition ~kx 5 ky 5 kz 5 k! occurs athe magic angle, u# 5 54.74°.

The absorbance in the TE and the TM polarizationsre calculated as the tilt orientation angle u# is varied

from 0° to 90°. The calculations were performedwith both the full-wave and the ray optics methods,and the results are shown in Figs. 7~a! and 7~b! fork 5 0.01 and 0.1, respectively. The agreement be-tween the two methods is excellent except for regions~k 5 0.1 and TM polarization! where the absorbanceper centimeter reaches extremely high values ~.30cm21!. Thus in the regime where measurements areexperimentally feasible ~A cm21 , 10 cm21!, the dis-crepancy between the methods is minimal.

Finally the dichroic ratio, defined as the ratio of theTE and the TM absorbance values, is considered. Thedichroic ratio is a measure of the angular orientationdistribution of the molecules in the adlayer on thewaveguide surface, and it is independent of the molec-ular concentration, c, in the ray optics approach. Theichroic ratio was calculated with both the full-wavend the ray optics models as described above. In Fig.~c! the relative error is displayed for k 5 0.01. Ex-

cellent agreement between the two models is evident.In the same plot are included the results for the rela-tive error calculated from expressions currently beingused in the literature.37 These results show a relativeerror of 5%, which can be explained by the lack of acosine factor in the absorbance expressions reported byLee and Saavedra.37 In Fig. 7~d! we display the rel-ative error for the case of k 5 0.1. Again, the rayoptics show good agreement with the wave model ex-cept in the angular region where the predicted absor-bance values are extremely high.

5. Conclusions

We have derived straightforward expressions for cal-culating absorbance in typical planar waveguide ge-ometries by the ray optics model. The accuracy ofthe approximations used in the ray optics approachhas been assessed by comparison of results from ab-sorbance calculations with those obtained with anexact electromagnetic wave approach. From thisanalysis, we conclude that in typical situations inwhich waveguide absorbance measurements are used

1 February 2000 y Vol. 39, No. 4 y APPLIED OPTICS 617

ii w

d

6

to study organic thin films ~klynl ' 1021, hyl ' 1022!,the ray optics model provides accurate results.

Appendix A: Dichroic Extinction Coefficients

Following Hopf and Stegeman,44 the induced dipole,p, associated with a normal mode is written as

p 5 ql, (A1)

with

l 5 le, (A2)

where l is the effective charge separation per unitlength of q and e is a unit vector in the direction of thenduced dipole. The molecular polarizability tensors given by

a# 5ll

m~vo2 2 v2 2 ivG!

, (A3)

where v0 is the resonance frequency, G is the damp-ing constant, and m is the characteristic mass. Theextinction coefficient is linearly proportional to theimaginary part of the polarizability tensor throughthe relation

kij 5r

2ε0Im^a# ij&, (A4)

Fig. 7. Calculations for a dichroic adlayer with a variable dipolescribed in Eqs. ~17! and ~18!. Absorbance values calculated by

~a! k 5 0.01 and ~b! k 5 0.1. Relative error of the dichroic ratio foand ~d! k 5 0.1. The parameters used in the calculations are waveof refraction nw 5 1.56, substrate index of refraction ns 5 1.46, reof refraction nc 5 1.33.

18 APPLIED OPTICS y Vol. 39, No. 4 y 1 February 2000

where r represents the volume molecular density andthe brackets indicate an average over the molecularorientation distribution. Because of the symmetryconsiderations described below, it is preferable to rep-resent the ensemble average in the polar coordinatesystem,

^ f& 5** f ~u, f!N~u, f!sin~u!dudf

** N~u, f!sin~u!dudf

, (A5)

here N~u, f! is the molecular orientation distribution.

1. Linear-Polarized Dipole

In the case of a linear dipole the Cartesian compo-nents are represented by

Slx

ly

lz

D 5 Scos f 2sin f 0sin f cos f 0

0 0 1DS cos u 0 sin u

0 1 02sin u 0 cos u

DS00lD

5 Scos f sin usin f sin u

cos uDl. (A6)

ntation angle, u#. The extinction coefficients ~kx, ky, and kz! areray optics ~dashed curves! and the wave ~solid curves! models forray optics model and for expressions in Ref. @37# with ~c! k 5 0.01th l 5 550 nm, waveguide thickness t 5 400 nm, waveguide indexrt of the adlayer index of refraction nl 5 1.33, and cladding index

e oriethe

r theleng

al pa

a

w

tcp

t

a

ni

de

The orientation distribution of the dipoles in thex–y plane ~the waveguide plane as shown in Fig. 3! isssumed to be isotropic; therefore

N~u, f! 5 N~u!, (A7)

hich gives

^lx2& 5 ^ly

2& 512

l2* sin2~u!N~u!sin~u!du

* N~u!sin~u!du

, (A8)

^lz2& 5 l2

* cos2~u!N~u!sin~u!du

* N~u!sin~u!du

, (A9)

^lilj& 5 0, for i Þ j. (A10)

It is common practice to express the dichroism in

erms of an equivalent mean tilt angle. This angle isalculated by assuming a Dirac delta function for theolar dipole orientation distribution, N~u!

N~u! 5 d~u 2 u#!, (A11)

herefore

^lx2& 5 ^ly

2& 512 l2 sin2~u#!, (A12)

^lz2& 5 l2 cos2~u#!. (A13)

However, for a random distribution of polar tiltngles, N~u! 5 1 is inserted into Eqs. ~A8! and ~A9! to

get

^lx2& 5 ^ly

2& 5 ^lz2& 5 ~1y3!l2. (A14)

These results allow the extinction coefficient foreach Cartesian component to be expressed in terms ofthe extinction coefficient, k, of the randomly orientedensemble of dipoles by

kx 5 ky 5 ~3y2!k sin2~u#!, (A15)

kz 5 3k cos2~u#!. (A16)

At 54.74°, the so-called magic angle, the compo-ents of the extinction coefficient correspond to an

sotropic medium, which means that kx 5 ky 5 kz.Values of the mean tilt angle that are far from themagic angle indicate strong dichroism.

Slx

ly

lz

D 5 Sl1~cos f cos a 2 sin f sin a cosl1~sin f cos a 1 cos f sin a cos

l1~sin a sin

2. Circular-Polarized Dipole

For two orthogonal dipoles, l1 and l2, their spatialorientation is described by a set of independent an-gular rotations ~Euler angles!:

Slx

ly

lz

D 5 Scos f 2sin f 0sin f cos f 0

0 0 1DS1 0 0

0 cos u 2sin u0 sin u cos u

D3 Scos a 2sin a 0

sin a cos a 00 0 1

DSl1

l2

0D . (A17)

In the right-hand side of Eq. ~A17!, the first matrixapplied to the dipole vectors represents a rotation inthe plane of the dipoles. For a porphyrin molecule,this represents a rotation of the dipoles in the molec-ular plane. The second matrix implements a rota-tion on the polar angle u. The third matrix appliedto the vectors is a rotation on the azimuthal angle w.The Cartesian components are then given by

For an ensemble of circularly polarized molecules,the distribution along a is uniform. We also assumea uniform distribution ~uniaxial! in the x–y plane.Under these assumptions, the molecular orientationdistribution becomes

N~u, f, a! 5 N~u!, (A19)

which gives

^lx2& 5 ^ly

2& 514

l2* N~u!@1 1 cos2~u!#sin~u!du

* N~u!sin~u!du

, (A20)

^lz2& 5

12

l2* N~u!sin2~u!sin~u!du

* N~u!sin~u!du

, (A21)

^lx ly& 5 ^lx lz& 5 ^ly lz& 5 0, (A22)

where the definition l2 [ l12 1 l2

2 is used.Applying a simple consistency test for a random

istribution, where all dipole orientations should bequally probable, we obtain

^lx2& 5 ^ly

2& 5 ^lz2& 5 ~1y3!l2. (A23)

l2~2cos f sin a 2 sin f cos a cos u!l2~2sin f sin a 1 cos f cos a cos u!l2~cos a sin u!

D . (A18)

u! 1

u! 1

u! 1

1 February 2000 y Vol. 39, No. 4 y APPLIED OPTICS 619

t

p

tion spectrometry and IOW-Raman spectroscopy,” Thin Solid

6

Again, by expression of the dichroism in terms ofan equivalent mean angle

N~u! 5 d~u 2 u#!, (A24)

he following expressions are obtained:

^lx2& 5 ^ly

2& 514 l2@1 1 cos2~u#!#, (A25)

^lz2& 5

12 l2 sin2~u#!. (A26)

The extinction coefficients in each Cartesian com-onent is then given by

kx 5 ky 5 ~3y4!k@1 1 cos2~u#!#, (A27)

kz 5 ~3y2!k sin2~u#!. (A28)

As a final note, the factor fi is defined as the ratiobetween the extinction coefficient in each Cartesiandirection and the extinction coefficient for an ensem-ble of randomly oriented dipoles:

fi ;ki

k. (A29)

The definition is applied to both the linear- and thecircular-polarized dipoles, and these factors are usedin the calculations of dichroic absorption in Section 3.

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