2704 OPTICS LETTERS / Vol. 31, No. 18 / September 15, 2006

Z-probing of optical multilayers: theory

Claude Amra and Carole DeumieInstitut Fresnel Marseille, UMR CNRSTIC, Université Paul Cézanne, Ecole Généraliste Ingénieurs Marseille,

Université de Provence, Domaine Universitaire de St. Jérôme, 13397 Marseille Cedex 20, France

Received March 1, 2006; revised May 31, 2006; accepted June 2, 2006;posted June 30, 2006 (Doc. ID 68576); published August 25, 2006

A single optical technique is presented that allows direct and selective probing or imaging in the thicknessof multilayers. It is based on tunable interferences of polarized light. Adequate vertical resolution is pro-vided with this method. © 2006 Optical Society of America

OCIS codes: 120.3940, 120.5410, 120.6660, 310.1620, 310.6870.

Nondestructive probing of multilayers1,2 has alwaysbeen of crucial interest, in particular for applicationsin the fields of optics, microelectronics, and remotesensing. Indirect probing has extensively been ad-dressed by the thin-film community in the past fewdecades, which has led to the use of reverseengineering3 as a daily tool for most optical coatingcompanies. Such reverse engineering is most oftenbased on photometric or ellipsometric data4,5 versusthickness or wavelength. However, direct probing ofmultilayers remains a challenge, in particular, for se-lective imaging at specific interfaces or bulks withinmultilayers. Several successful techniques, such asconfocal microscopy and optical tomography,6 havebeen developed with this aim, but these techniquessuffer limitations due to inadequate z-resolution foroptical coatings. In this Letter we introduce a singletechnique that provides selective probing or imagingof the thickness of multilayers with subwavelengthvertical resolution. The idea is based on a recenttechnique7 developed for light scattering, which wereduce here to specular beams. A major application ismade possible by a specific procedure that consists ofthe annulment of a single term in a reflection ortransmission series of a multilayer.

First, consider a monochromatic polarized beam(Fig. 1) at incidence i and wavelength � on amultilayer. The incident electric field of the planewave is written as

E+ = ES+ + EP

+ = �AS+ + AP

+�exp�jk+ · ��, �1�

where �= �x ,y ,z� are the spatial coordinates and k+ isthe incident wave vector,

k+ = k0�sin�i�,0,cos�i��, �2�

with n0 as the refractive index of the superstrate andk=2�n0 /�. In Eq. (1) AS

+ and AP+ designate the proper

polarization states of the electric field, whose com-plex amplitudes are written as

AS+ = �AS

+�exp�j�S�, AP+ = �AP

+�exp�j�P�, �3�

with �S and �P as the polarization phase terms.Equations (1)–(3) are valid for elliptical ��S��P� orlinear ��S=�P� polarizations, depending on the rela-tive values of the phase terms. After the retardationplate, the polarization states are still perpendicular,

but their complex amplitudes are given as

0146-9592/06/182704-3/$15.00 ©

AS�+ = AS

+exp�j�S*�, AP�

+ = AP+exp�j�P

*�, �4�

where �* are the additional phase terms that can betuned with the retardation plate.

Consider now the field reflected by the multilayer(Fig. 1), whose amplitude reflection factors are com-plex numbers denoted by rS and rP. After the ana-lyzer is rotated at angle �, the field components areprojected and aligned, with an algebraic result givenby

Ar = rScos���AS�+ + rPsin���AP�

+. �5�

In the case of equal energy in each incident polariza-tion ��AS

+ � = �AP+ � �, Eq. (5) is reduced to

Ar = AS�+cos���f��r�, �6�

where f��r� is an interferential reflection state givenby

f��r� = f��rS,rP� = rS + �rP �7�

and � a complex number of modulus tg� and argu-ment ��+��*, with

�� = �P − �S, ��* = �P* − �S

* . �8�

At this step we notice that the specific positions orvalues of the analyzer and the retardation plate areallowed to reach any arbitrary complex �. In particu-lar, reflection can be eliminated as

Fig. 1. Incident polarized plane wave at incidence i on asample with normal z in the presence of a tunable retarda-

tion plate and a rotating analyzer.

2006 Optical Society of America

September 15, 2006 / Vol. 31, No. 18 / OPTICS LETTERS 2705

f��r� = 0 Û � = �c = − rS/rP, �9�

with �c given by

tg�c = �RS/RP�0.5, ��c* = � − �� − ��P − �S�, �10�

rS = RS0.5exp�j�S�, rP = RP

0.5exp�j�P�. �11�

In Eqs. (10) all parameters except �� depend on in-cidence angle i. Notice that these equations and prin-ciples are similar to those of ellipsometry, a well-known technique that provides an algebraic sum ofthe two reflected polarizations, with unknown pa-rameters (e.g., index, thickness) that can be ex-tracted by means of analyzer rotation or harmonicpolarization modulation. A considerable body of workwas developed in this field, with sophisticated ana-lytical and numerical reverse engineering, dependingon the complexity of samples (substrates or multilay-ers). In particular, Bosch and Monzonis4,5 pointed outthe equations to be solved in the case of multilayers,using a numerical procedure that allows one to ex-tract any combination of two parameters. However,the usual ellipsometry does not involve tunable staticretardation plates, so the annulment conditions arenot reached for all samples that are to be analyzed.We will see further why this annulment condition is akey for our procedure. In particular, the differencebetween our technique and the usual ellipsometrylies in the fact that our objective is to achieve direct(real-time) and selective probing or imaging (micros-copy) within multilayers, which constitutes a notice-able departure from reverse engineering. As an illus-tration, and depending on the application, reverseengineering can be necessary to extract a sample de-sign with high accuracy, which is necessary to predictthe annulment conditions that allow direct imagingwithin multilayers.

Now, instead of using the admittance method orothers to write the global reflection of the stack, weconsider the classical series of multiple reflections in-side the multilayer. With this aim, the multilayer ofFig. 2 has been arbitrarily divided into two submul-tilayers separated by a spacer layer. The top submul-tilayer is denoted (0), while the back one is denoted(1). All zones (0) and (1), as well as the spacer layer,are arbitrarily chosen. The reflection factor is givenas

r = r0 + t0t0�r1exp�j2�� + t0t0�r0�r12exp�j4��� + ¯

= nrn, �12�

r = r0 + t0t0�r1exp�j2���1/�1 − r0�r1exp�j2���� = r0 + Z,

�13�

where � is a dimensionless phase term that is char-acteristic of the spacer layer and is not dependent onpolarization,

� = �2�/���necos ii�sp, �14�

where �necos i�sp describes the apparent optical thick-ness of the spacer layer. In Eqs. (12) and (13) r0 and

t0 describe the reflection and transmission factors of

the top submultilayer (0), while r1 is given for the re-flection factor of the back submultilayer (1). In thecase of retrograde waves, similar factors are denotedby t0� and r0� (Fig. 2). The key point at this step is toapply the f transformation and try to eliminate anyarbitrary term in the reflection series. First, noticethat

f��iri� = if��ri�. �15�

Then, following Eqs. (13) and (15), a first result canbe written as

f�r� = f�r0� + f�Z�. �16�

Notice that the factor r0 is connected only with thetop region of the stack, while the Z factor involves thewhole stack (Fig. 2). Therefore, in the general case ofoblique incidence, and because the complex numbersr0 and Z will have different polarization behaviors,the annulment condition will be different for each ofthem. A specific value �z of � can be found to matchthe condition

f�z�Z� = 0 Þ f�z�r� = f�z�r0�. �17�

In this case the reflected signal originates from onlythe top part of the multilayer (Fig. 3), whatever theback zone design. Since the spacer layer was arbi-trary chosen, this technique allows us to probe anytop submultilayer inside the stack, with adequate op-tical resolution.

A more relevant result can be reached when weconsider the series given in Eq. (12):

f�r� = nf�rn�. �18�

As earlier in this paper and in the general case of ob-lique incidence, the annulment condition will be dif-ferent for each term of the series. Therefore the ftransformation would a priori allow us to eliminateany arbitrary single term or subseries of terms. Inparticular, if we keep only the second term of (12), we

Fig. 2. A spacer layer is arbitrary chosen within amultilayer and allows us to define two submultilayers (boldlines) located at the top (0) and at the back (1) of the figure.

Fig. 3. Situation similar to that of reflection from a uniquetop submultilayer (0).

obtain

2706 OPTICS LETTERS / Vol. 31, No. 18 / September 15, 2006

f��r − t0t0�r1exp�j2��� = 0 Þ f��r� = f�t0t0�r1exp�j2���.

�19�

Equation (19) shows that it is possible to collect lightthat is reflected from only the back zone of themultilayer (Fig. 4). Since the spacer layer was arbi-trary chosen, the technique allows us to probe anyback submultilayer inside the stack, with adequateoptical resolution.

In summary, we have described a single techniquethat allows direct and selective probing or imagingwithin the thickness of multilayers. It is based ontunable interferences of polarized light and involvessingle devices such as a rotating analyzer and a re-tardation plate. Although each surface of the stackcannot be probed alone, the technique offers the pos-sibility of probing any arbitrary top or back submul-tilayer within the stack. Before each particular prob-ing, specific values of the analyzer and retardationplates must be calculated to reach an annulment con-

Fig. 4. Situation where the back submultilayer is probed.

dition that depends on the stack design. Therefore, inthe case of high-precision optics, reverse engineeringcan be necessary to extract and check the sample de-sign, before the annulment conditions are calculatedfor selective imaging. In all cases and whatever theknowledge of the design, a two-dimensional scan (ro-tating analyzer and tunable retardation plate) allowsone to explore all annulment conditions. The nextstep will consist of the prediction and measurementof the efficiency and accuracy of the method for eachparticular probing, which will depend on the sub-stack designs.

C. Amra’s e-mail address is claude.amra@fresnel.fr.

References

1. F. J. Milligen, B. Bovard, M. R. Jacobson, J. Mueller, R.Potoff, R. L. Shoemaker, and H. A. Macleod, Appl. Opt.24, 1799 (1985).

2. B. Bovard, F. J. Milligen, M. J. Messerly, S. J. Saxe,and H. A. Macleod, Appl. Opt. 24, 1803 (1985).

3. A. V. Tikhonravov, M. K. Trubetskov, M. A. Kokarev, T.V. Amotchkina, A. Duparré, E. Quesnel, D. Ristau, andS. Gunster, Appl. Opt. 41, 2555 (2002).

4. S. Bosch and F. Monzonis, J. Opt. Soc. Am. A 12, 1375(1995).

5. S. Bosch and F. Monzonis, Semicond. Sci. Technol. 10,1634 (1995).

6. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L.Vabre, and C. Boccara, Appl. Opt. 43, 2874 (2004).

7. C. Amra, C. Deumié, and O. Gilbert, Opt. Express 13,10854 (2005).