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114 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 K. Peiponen and E. M. Vartiainen

Dispersion theory of the reflectivity of s-polarizedand p-polarized light

Kai-Erik Peiponen

Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland

Erik M. Vartiainen

Department of Electrical Engineering, Lappeenranta University of Technology, P.O. Box 20, FIN-53851Lappeenranta, Finland

Received April 4, 2005; revised August 3, 2005; accepted August 4, 2005

The dispersion theory for reflectivity of s- and p-polarized light is considered. The problem of phase retrievalfrom reflectance of p-polarized light, in the presence of complex zeros of the reflectivity in the upper half ofcomplex frequency plane, is addressed. In such a case the traditional Kramers–Kronig relations are not valid,but the maximum-entropy method will provide the required solution. © 2006 Optical Society of America

OCIS codes: 120.4530, 100.5070, 000.3860.

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. INTRODUCTIONeflection spectroscopy is one of the fundamental tools inptical materials research. This type of spectroscopy for aong time has constituted the means to investigate theomplex refractive index of opaque bulk media with theid of Fresnel formulas.1 Thus reflection spectroscopy haseen crucial in both basic research and industrial appli-ations for the assessment of the optical properties ofuch media as liquids.2 Reflection spectroscopy is a sim-ler alternative to a measurement technique such asllipsometry,3 which is a useful method for the assess-ent of the complex refractive index of media in solid

hase. When a reflectometer is used, the measured signals reflectance, which can be obtained for a fixed wave-ength and polarization of light by scanning the angle ofncidence; one can also record a spectrum by scanning thencident wavelength but fixing the angle of incidence andolarization of light. The latter case is usually more inter-sting, because it provides the possibility of obtaining theomplex refractive index of the medium as a function ofavelength. Typically, linearly polarized light, either s orpolarized, is employed in the measurement of the reflec-

ance. The possibility of wavelength scanning is really im-ortant, since then one may resolve spectral features ofhe medium owing to absorption, concentrations of differ-nt species, and dispersion. Unfortunately, when the re-ectance is recorded as a function of wavelength, the re-ectance being proportional to the intensity of lighteflected from an interface, the phase of the reflectedave is lost. Thus there is a problem in that we have onlyne known quantity (i.e., the reflectance) to derive twonknown quantities (i.e., the refractive index and the ex-inction coefficient) from the Fresnel equation. Fortu-ately, in the case of normal light incidence the role of

ight polarization is not a problem, and the phase re-rieval based on the utilization of the Kramers–KronigK–K) relations4 can be achieved in order to resolve the

0740-3224/06/010114-6/$15.00 © 2

omplex refractive index from reflectance. However,spnes5 has stated that “a K–K analysis should be at-

empted only if accurate reflectance data are availablever a very, very wide energy range.” Indeed, the limita-ion of the measured data to a relatively narrow spectralange has been “a bottle neck” in the K–K analysis, andorrect extrapolation of data beyond the measured spec-ral range is crucial.6 However, owing to a better under-tanding of principal value integrals7,8 an improved nu-erical efficiency in data inversion,9 and a realization of

he utility of multiply subtractive K–K (MSKK) relations,n both linear reflection spectroscopy10 and nonlinearpectroscopy,11–13 the K–K analysis has become more fea-ible. In particular, the utilization of MSKK relaxes theequirement of the knowledge of reflectance spectrumver a wide energy range.

The case of normal light incidence is the simplest onen respect to the phase retrieval by the K–K analysis. Thease of oblique light incidence is more problematic, espe-ially in the case of the reflection of p-polarized light. In-eed, it is possible that such a complex reflectivity pos-esses complex zeros in the upper half of the complexrequency plane. Thus the logarithm of the reflectivity,hich is crucial to the derivation of K–K relations for the

eflectivity, will be infinite in such singular points. Thismplies that the phase obtained by the K–K analysis wille erroneous,14 but one may try to find the correct phasesing a Blaschke product.15 Unfortunately, the Blaschkeroduct involves complex zeros of reflectivity. Data onuch complex valued singular points are not directly mea-urable, but they may be estimated with the aid of addi-ional optical data.14 Unfortunately, one does not alwaysnow a priori when the reflectance of p-polarized lightas complex zeros in the upper-half plane. Such an uncer-ain situation may be present, especially when we have tonvestigate, for instance, a new material, which we haveo previous information about its optical or microscopic

006 Optical Society of America

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K. Peiponen and E. M. Vartiainen Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 115

roperties at all. In this paper we apply the maximum en-ropy (ME) model,4,16 to the phase retrieval from the re-ectivity of p-polarized light in the case where complexeflectivity posseses zeros in the upper half of the complexrequency plane. We show by simulation that the phase ofhe complex reflectivity is obtained correctly by utilizationf the ME model, even in this pathological case. Further-ore, there is no need at all to know anything about the

resence or absence of complex zeros in such a phase-etrieval procedure. Thus an improvement of the tradi-ional K–K analysis requires the data correction by Bl-schke product, which is neither generally known norbtained from measurements. Moreover, the ME analysisan be performed using measured data only, i.e. there iso need for data extrapolation beyond the measuredange, a fact that has been shown in many studies andummarized in Ref. 4.

. KRAMERS–KRONIG RELATIONSf the reflectance measurement can be arranged so thatormal or almost-normal incidence of probe light is pos-ible, the problems related to the polarization of light cane avoided and traditional K–K analysis in reflectionpectroscopy may be employed as follows4:

����� = −2��

�P�

0

� ln�r����

�2 − ��2d�, �1�

ln�r����� − ln�r����� =2

�P�

0

�

������ 1

�2 − ��2

−1

�2 − ��2�d�, �2�

here � is the frequency, P denotes the Cauchy principalalue, r is the complex reflectivity, and � is the phase ofeflected electric field defined by

r��� = �r����exp i����. �3�

n the phase-retrieval procedure we have to exploit Eq.1), since the squared modulus of the reflectivity is re-orded during the measurement, i.e. the reflectance R�r�2. Note that in the case of K–K relation (2) the reflec-

ance has to a priori be known at one anchor point ��0. The key points in the derivation of the K–K relations

1) and (2), using complex contour integration, is theolomorphicity4 of the complex reflectivity in the upperalf of the complex frequency plane (we assume that the

ight field oscillates proportional to exp�−i�t�), strongnough fall off the reflectivity as a function of high fre-uency, and the symmetry properties of the reflectivity.Unfortunately, it is not always possible to exploit the

ormal angle of incidence. For instance, when recordingpectra from liquids (in practical metrology such as in in-ustrial environments) a prism as a probe window (whichan be rotated) is useful. Sometimes an oblique angle ofncidence is required. Such a requirement is present, e.g.,n the case of the surface-plasmon-resonance spectroscopyf liquids, which is based on the utilization of a prismoated by a thin metal film as a probe. Then a phase-

etrieval problem also appears.17 The case of obliquengle of incidence, and s- or p-polarized light, deals withhe Fresnel reflectances that can be written as follows foronmagnetic isotropic media:

Rs��� = cos � − �N2 − sin2 ��1/2

cos � + �N2 − sin2 ��1/22

, �4�

Rp��� = N2 cos � − �N2 − sin2 ��1/2

N2 cos � + �N2 − sin2 ��1/22

, �5�

here N=n+ ik is the complex refractive index (n is theeal refractive index, k is the extinction coefficient) and �s the angle of incidence. In the phase retrieval by the–K analysis, which is based on Eq. (1), one must note

hat the complex reflectivity may have complex zeros inpper-half plane; if so, the logarithm of the reflectivityill blow up. Such a possibility can be easily checked justy setting the reflectance of Eq. (4) [or Eq. (5)] equal toero [Eq. (5) will be treated in Section 3]. This means that

cos � − �N2 − sin2 ��1/2 = 0. �6�

bviously, Eq. (6) is fulfilled only when N=1; thus we doot have to worry about complex zeros in the upper-halflane, as in the case of Eq. (4). Hence the K–K analysisan be applied without hesitation for oblique incidence tos, since it fulfills the assumption imposed on the utilityf K–K relations. A detailed algebraic description of the–K analysis incorporated with Eq. (4) can be found inef. 4.

. MAXIMUM ENTROPY METHODhe measurement of reflectance of p-polarized light issually as easy as that of s-polarized light. One may havetemptation to utilize K–K analysis on Rp without prior

nowledge of complex zeros of rp. The object usually is toesolve the complex refractive index from Eq. (5). The op-ical constants can be obtained by modifying the reflectiv-ty of p-polarized light given by

rp =cos � − ��−1 − �−2 sin2 ��1/2

cos � + �−1 − �−2 sin2 �=

cos � − �A + iB�

cos � + �A + iB�= C + iD,

�7�

here � is the relative permittivity of the medium. Thearameters C and D are supposed to be calculated withhe aid of a phase-retrieval procedure. Then the complexelative permittivity �=N2 can be solved as follows:

� =1

2�A + iB�−2 ±

1

2��A + iB�−4 − 4�A + iB�−2 sin2 ��1/2.

�8�

n the case of the complex zeros in the upper-half plane,he phase retrieval based on the K–K analysis will give anncorrect result. Only in the case in which no complex ze-os of rp exist in the upper-half plane can one expect to getcorrect result by the K–K analysis.It is interesting to note that if one actually measures

he reflectance of only s-polarized light, one can recon-

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116 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 K. Peiponen and E. M. Vartiainen

truct the reflectance of p-polarized light without measur-ng it. Indeed, Azzam18 has derived a remarkable result,amely,

rp =rs − cos 2�

1 − rs cos 2�. �9�

his property can be exploited, e.g., in practical datanalysis of testing the self-consistency between the mea-ured and calculated data of the reflectance of p-polarizedight.

Next we concentrate in depth on the problematic situ-tion in which the reflectivity rp has complex zeros. Theomplex zeros are obtained from Eq. (5) when we set

N2 cos � − �N2 − sin2 ��1/2 = 0. �10�

he solution is nontrivial and corresponds to the case ofrewster angle tan �=N. One solution of Eq. (10) can bebtained for a purely imaginary frequency variable. Theeason is that the relative permittivity �=N2 of a mediums a monotonically decreasing19 real function as a functionf the purely imaginary frequency.4 Then the complex ze-os of the reflectivity are located on the positive imagi-ary axis if the following condition is valid14:

�� � tan2 � � �static, �11�

here the upper limit is the static value of the permittiv-ty. Under the condition (11) the complex zeros yield sin-ularities to the logarithm of reflectance of the-polarized light. Naturally, one may try to avoid thisumbersome situation by choosing the angle of incidencen such a manner that the Eq. (11) is not valid. However,n the case of a sample with unknown optical properties,everal trials may need to be undertaken by changing thengle of incidence, scanning of the wavelength, and calcu-ating and testing the validity of the phase angle using,or instance, Eq. (9). Fortunately, ME analysis, which isescribed below, is free on such trials and iteration pro-esses because it tolerates the complex zeros. Before deal-ng with the ME model, we briefly investigate the tradi-ional way to handle the complex zeros of rp. If �j denotes

complex zero of reflectivity, j=1, . . . ,J, then the Bl-schke product is14,15

B��� = �j=1

J � − �j

� − �j* , �12�

here � *� denotes a complex conjugate. The true reflec-ivity is obtained by multiplying the complex reflectivity,btained by K–K (or by MSKK) analysis, by the Blaschkeroduct as follows:

rtrue��� = rK–K����j=1

J � − �j

� − �j* . �13�

rosse and Offermann have clearly demonstrated by us-ng real spectral data for crystals that the K–K analysisails but can be cured using the known Blaschke correc-ion term. Unfortunately, the complex zeros are not di-ectly obtained from measured reflectance, and thisauses the trouble. To overcome this difficulty, we use aimpler method than the optimization procedure of

rosse and Offermann and fit the reflectance data by theE model. It should be noted here that the concept of en-

ropy comes purely from the information theory,20 and inhe present case the question is by no means about realntropy, which is a thermodynamic concept. Actually, theE estimate can be obtained without the concept of en-

ropy at all.We first deal with a mathematical model that may help

o understand the series expansion, which is the startingoint of the ME analysis. It is well known that rationalunctions provide relatively good estimates of spectra. Alassical example is the Lorentzian model, which has for aong time constituted a standard line-shape model in bothinear and nonlinear spectra analysis. Press et al.21 statedhat a rational function of a complex variable is a goodhoice regarding the description of power spectra. Thiseans that we can assume that a rational function of the

ype

�rp�z��2 = � k=−K

Kakzk

l=−L

Lblz

l �2

�14�

s useful in the spectra analysis of the reflectance. Theew complex variable z is restricted to the boundary of anit circle.The interesting mathematical feature with the quotient

n Eq. (14) is that the polynomials in numerator and de-ominator, without common factors, are holomorphicunctions in the unit circle. Thus the complex reflectivityust be treated as a meromorphic function.4 In otherords, the zeros of the denominator are the complex polesf the reflectivity. The simplest approximation of Eq. (14)an be achieved by setting the nominator equal to 1. Thuse get

�rp�z��2 �1

� l=−L

Lblz

l�2=

�zL�2

b−L�1 + ¯ +bL

b−Lz2L�2 .

�15�

n the last expression on the right-hand side of Eq. (15),he nominator is also equal to 1, since z lies on the bound-ry of a unit circle. We can rename the dummy index andbtain the following expression

�rp�z��2 ��d0�2

�1 + m=1

Mdmzm�2

, �16�

here �d0�2=1/ �b−L�2. Equation (16), which we obtainedhrough a simple treatment of a complex valued mero-orphic function, is a result and the starting point of theE analysis. Equation (16) is usually derived using aore-complicated formalism based on the utilization of

he entropy concept, calculus of variations, and Lan-range multipliers22,23 or on complex analysis.4 The MEethod was originally proposed for time-signal analysis

y Burg.24 However, Vartiainen et al.25,26 initiated the ap-lication of ME model in the case of phase-retrieval prob-ems related to power spectra, such as wavelength-ependent reflectance spectra. The power spectrum can

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K. Peiponen and E. M. Vartiainen Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 117

e fitted with high accuracy using the ME model. The MEnalysis has also found applications in phase-retrievalroblems in nonlinear optical spectroscopy. We mention aew examples in the field of nonlinear optics such as co-erent anti-Stokes Raman scattering spectroscopy,27

hird-harmonic generation spectroscopy,28 and sum-requency vibrational spectroscopy.23

In practical data analysis the spectrum at measuredange ��1 ,�2� is first compressed into a narrow interval0, 1] by defining a new variable as follows:

� =� − �1

�2 − �1. �17�

he coefficients dk are obtained from a set of Yule–Walkerquations4

k=1

M

dkC�n − k� = ��d0�2, n = 0

0, n = 1, . . . ,M� , �18�

here the autocorrelations C�m� are obtained from the in-egral

C�m� =�0

1

�rp����2 exp�2�im��d�. �19�

Now we can set the range ��1 ,�2� onto the unit circley defining z=exp�−i2��� and write a ME model for theomplex reflectivity as a function of the normalized fre-uency as follows:

rp��� ��d0�exp�i�����

� m=0

Mdm exp�− i2�m���

, �20�

here ����=arg�rp�. We can now proceed toward the finalE model of rp by defining a ME phase as ���arg�M d exp�−i2�m��� and by writing

ig. 1. Real and imaginary parts of the Lorentzian relative per-ittivity, �=��+ i��, that was used in the modeling of the complex

eflectivity. The parameters are ��=1.0, �p=3.0 eV, �0=2.0 eV,nd =0.1 eV.

m=0 m

rp��� ��d0�exp�i����� − �����

� m=0

Mdm exp�− i2�m���exp�− i����

=�d0�exp�i�����

m=0

Mdm

* exp�i2�m��, �21�

here �=�− is a so-called error phase. It should beoted that is obtainable solely from the ME fitting of theeflectance, i.e., from Eqs. (18) and (19). Thus, by usinghe ME model of Eq. (21), we can decompose � into twoerms, as �=�+, where � is the only unknown quantity.ypically yields the same spectral features as �,hereas � is a smooth background term. Therefore � cane presented as a polynomial interpolation

���� = k=0

L

Bk�k, �22�

here the coefficients are obtained from a Vandermondeystem4:

�1 �0 ¯ �0

L

1 �1 ¯ �1L

] ] � ]

1 �L ¯ �LL��

B0

B1

]

BL

� = ����0�

���1�

]

���L�� . �23�

hus the error phase is obtained if the complex reflectiv-ty is known at some anchor points ���k� ,k=0,1, . . . ,L.ang and Huang23 gave three relatively general criteriaor how to avoid experimental phase measurements in or-er to choose the anchor points in nonlinear optics. Theirriteria are based on the symmetry properties and behav-or of the spectra at the wings.

To increase the linearity of the error phase, anotherompression of the spectrum into a narrower spectralange is performed using the following data-fittingrocedure28:

ig. 2. Real and imaginary parts of complex reflectivity, rp=rp�irp�, used in the ME analysis. The angle of incidence is taken toe �=50°.

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118 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 K. Peiponen and E. M. Vartiainen

�rp����2 → �rp��;K��2 = ��rp��1��2, 0 � � � wK��1�

�rp����2, wK��1� � � � wK��2�

�rp��2��2, wK��2� � � � 1� ,

�24�

here

wK��� =1

2K + 1� � − �1

�2 − �1+ K� , �25�

nd

� =wK��� − �K��1�

wK��2� − wK��1�, �26�

here K is a positive integer.Now one may ask what the difference is between the

–K analysis and the ME analysis. The main difference,n the present case is that in the K–K analysis we dealith the logarithm of the reflectivity, whereas with theE analysis we deal with the reflectivity itself. Complex

eros are not singular points of the reflectivity itself, thushe ME analysis works!

Next we demonstrate the capability of ME procedure toesolve the correct phase of rp in the case of complex ze-os. For this purpose we make use of single-resonanceorentzian permittivity defined by

���� = �� +�p

2

�02 − �2 − i �

, �27�

here �0 is the resonance frequency, �p is the plasma fre-uency, and is the line width. We can choose the angle ofncidence in Eq. (11) in such a manner that the complexeros appear on the positive imaginary axis. It is evidentrom Eq. (27) that ��i0�=�static and e�i��=��. In Fig. 1 theeal and imaginary parts of the relative permittivity usedn our simulation are shown. The following real andmaginary parts of the reflectivity of the p-polarized light

ig. 3. The true phase function of the complex reflectivity �thick curve) and the true error phase � (thin curve). The esti-ate for � (open circles) is computed with the aid of two anchor

oints, whose locations are indicated by the arrows. The therebybtained estimate for � is shown by square dots. The ME modelarameters are K=1 and M=450.

alculated using Eqs. (5) and (27) are shown in Fig. 2. Inig. 3 the phase ����=arg�rp���� (thick curve) is shown to-ether with the corresponding ME error-phase functionnd ���� (thin curve). An estimated error phase (openircles) was obtained using two anchor points at frequen-ies indicated by the arrows in Fig. 3. The true errorhase is close to a linear function of �, whereas the ob-ained estimate ���� for the true phase shows an excellentt.

. CONCLUSIONSn this paper we investigated the dispersion theory of therequency-dependent reflectivity of s- and p-polarizedight. We have shown that maximum entropy model cane applied in the problematic case when the reflectivity of-polarized light has complex zeros in upper half of com-lex frequency plane. This is the case when the K–K andSKK relations are not valid, owing to the invalidity of

he assumption of holomorphicity, which is crucial in theerivation of such dispersion relations. In the applicationf ME model one is free from the worry of the complex ze-os, since ME procedure resolves the correct phase re-ardless of the existence of the complex zeros.

K.-E. Peiponen is the corresponding author; his e-mailddress is peiponen@cc.joensuu.fi.

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