Abstract—In this work we show the application of the angle-

resolved spectroscopic reflectivity and polarimetry to the

characterization of 2D photonic crystal structures in the form of

slabs. The method we introduce is based on the combination of i)

the numerical simulation of the interaction of the incident light

with the photonic crystal, ii) the experimental measurement and

iii) the comparison of both results. We show that the calculations

predict the coupling of the incident light to photonic modes

propagating inside the structure, related to the photonic bands of

the infinite photonic crystal. We demonstrate this coupling in the

experimental measurements for two kinds of samples consisting

of 2D photonic crystal slabs of micro- and nanostructured

photoresist on a silicon substrate.

I. INTRODUCTION

Optical characterization of photonic crystals is a

fundamental issue in the development of technologies for their

fabrication and future application. Much research has been

devoted to the development of characterization techniques that

measure the photonic band gap or identify the photonic bands

of such materials. A technique that permits the direct

measurement of the photonic bands is the angle-resolved

spectroscopy (ARS) [1]. This technique is based on the

identification of resonant features in the reflectivity spectra at

different angles of incidence and has been successfully applied

to the characterization of several 2D photonic structures [2].

An alternative method to ARS is angle-resolved polarimetry

(ARP). Polarimetry is a technique widely used in the optical

characterization of a great variety of samples [3], by making it

angle-resolved it can also be used for photonic band

measurement. In this work we show the characterization of

different photonic crystals composed of a layer of photoresist

onto a silicon substrate and patterned with an ordered array of

holes made by photolithographic techniques.

This work is organized as follows: in Sec. II we introduce

briefly the numerical method we applied to simulate the

measurements and we show some results of simulations. Then,

in Sec. III, we show the experimental results of the

measurements on different samples and we compare them with

the theoretical calculations. Finally, section IV draws our

conclusions.

II. SIMULATION METHOD FOR THE INTERACTION OF LIGHT

WITH THE PHOTONIC CRYSTAL SAMPLES

The samples we want to characterize consist of a micro- or

nanostructured slab of a photoresist onto a silicon substrate.

The micro- or nanostructuration consists of a periodic array of

holes in the slab, configuring thus a 2D photonic crystal slab.

In order to simulate the interaction of the incident light with

these samples in an angle-resolved spectroscopy experiment

we have used a numerical algorithm based on the scattering

matrix treatment proposed by Whittaker et al. [4]. This method

follows the same approach as the plane-wave expansion

(PWE) method for 2D photonic crystals but including

additional features to calculate the angular-dependent

reflectance or polarimetry spectra. The scattering matrix

method is based on the fact that the waves propagating inside

the structure can be expanded in a sum of plane waves.

However, in contrast with the standard PWE method for a 2D

structure, in this method the wave can propagate in a direction

not perpendicular to the scatterers. This is necessary to enable

the modelling of photonic modes inside the structure that can

couple with the incident light. For instance, for the magnetic

field component this expansion can be expressed as:

,(1)

where the G are the vectors of the reciprocal lattice, k the

Bloch wavevector (since the waves inside the structure must

fulfil the Bloch theorem), r the position in the x-y plane, z the

position along the scatterers and q the wavevector component

along the z direction. The introduction of the z component of

the wavevector (q) is necessary in order to allow the modelling

of waves that propagate inside the photonic crystal in an

oblique direction and that can couple to the photonic crystal

from the incident medium at a given angle.

By limiting the expansion of eq. (1) and applying Maxwell’s

equations a well-defined eigenvalue problem is obtained,

where the eigen values are the allowed q, for a given k and

wave frequency !. The solutions of this eigenvalue problem

Photonic band measurement by

angle-resolved spectroscopy and polarimetry

Zdenek Krála, Josep Ferré-Borrull

a*, Lluis F. Marsal

a, Member, IEEE, Josep Pallarès

a, Member, IEEE,

Enric Garcia-Caurelb, Martin Foldyna

b, and Santiago M. Olaizola

c, Member, IEEE

a Nanoelectronic and Photonic Systems, Universitat Rovira i Virgili, Tarragona, Spain

(phone: +34-977-558653; fax: +34-977-559605; e-mail*: josep.ferre@urv.cat). b Laboratoire de Physique des Interfaces et Couches Minces, Ecole Polytechnique, Palaiseau, France. c Department of Microelectronics, CEIT & TECNUN, University of Navarra, San Sebastian, Spain.

394978-1-4244-2839-7/09/$25.00 (∃)2009 ∗&&&

1ΣΠ∆ΦΦΕϑΟΗΤ ΠΓ ΥΙΦ 2009 4ΘΒΟϑΤΙ ∃ΠΟΓΦΣΦΟ∆Φ ΠΟ &ΜΦ∆ΥΣΠΟ %ΦΩϑ∆ΦΤ - ∋ΦΧ 11-13, 2009. 4ΒΟΥϑΒΗΠ ΕΦ ∃ΠΝΘΠΤΥΦΜΒ, 4ΘΒϑΟ.

are the photonic modes allowed to propagate inside the

sample. Once these modes are determined, it is possible to

calculate their coupling efficiency with the incident light using

a scattering matrix procedure and to obtain the reflection

coefficients for the two incident polarizations.

In this section we report on the calculations of reflectivity

and polarimetry spectra on two kinds of samples. The first

sample is a 2D photonic crystal slab composed of a photoresist

layer with a square lattice of square holes onto a silicon

substrate. The holes were produced by e-beam lithography and

subsequent etching. The lattice constant is a = 1"m and the

holes depth (photoresist layer thickness) is also 1"m. The

second sample is also a 2D photonic crystal slab composed of

a photoresist layer of 400nm thickness with a square lattice of

holes with lattice constant a = 400nm. In this case, the holes

were produced by the Laser Interference Lithography

technique[5].

Fig. 1 shows, for the first sample, the calculation of the

angle-resolved reflectivity spectra as a function of the angle of

incidence of the light and the normalized frequency. For the

calculation, it has been assumed that the light is incident along

the #M direction of the square lattice and is TE polarized. The

TE bands (second to sixth) for the 2D infinite photonic crystal

for the #M direction are overlapped to the graph for reference.

It can be noticed in the figure the resonant features that

indicate that at those specific incidence angles and wave

frequencies the incident light is coupling to photonic modes

within the photonic crystal. Three of such features can be

recognized, one beginning at a normalized frequency about

0.75 for $ =12º (the smallest angle in the graph), the second

one beginning at a normalized frequency about 0.80 for $=12º

and the third one finishing at a normalized frequency about

0.68 for $=70º (the biggest angle in the graph). The previous

works[6-8] using this method suggest that the two first

resonant features correspond to photonic modes with a parallel

wavevector equal to that of the second photonic band while the

third resonant feature corresponds to photonic modes related to

the fourth photonic band.

Fig. 2 shows an example of the calculation of angle-resolved

polarimetry spectra for the second sample. The polarimetry

spectra can be represented in two equivalent ways [9]: in the

form of the corresponding 2x2 complex Jones matrix or in the

form of the 4x4 real Mueller matrixes. In this work we have

chosen the Mueller matrix representation in order to enable the

comparison with the results obtained with the experimental

setup used to measure polarimetry spectra. We will restrict our

study to the matrix element M34, which is among the most

sensitive to the coupling of the incident light to photonic

modes in the photonic crystal slabs. Thus, Fig. 2 depicts the

M34 angle-resolved spectra for light incident along the #X (a)

and #M (b) directions of the square lattice. The lines

corresponding to the TE (solid lines) and TM (dashed lines)

photonic bands (starting with the second up to the seventh) of

the infinite photonic crystal structure are overlapped to the

graphs. In this case, it is interesting to see that for the #X

Fig. 1. Simulation of the angle-resolved reflectivity spectra for a 2D photonic

crystal slab consisting of a microstructured photoresist layer onto a silicon

substrate. The simulation corresponds to light incident along the #M

direction of the square lattice and with TE polarization. The photonic bands

for this direction and polarization are overlapped to the spectra.

…

Fig. 2. Simulation of angle-resolved polarimetry spectra (M34 component of

the Mueller matrix) for a 2D photonic crystal slab consisting of a

nanostructured photoresist layer onto a silicon substrate. a) light incident

along the #X direction of the square lattice and b) light incident along the

#M direction. The photonic bands for the corresponding lattice direction are

overlapped to the spectra (solid lines for TE polarization and dashed lines for

TM polarization).

395978-1-4244-2839-7/09/$25.00 (∃)2009 ∗&&&

1ΣΠ∆ΦΦΕϑΟΗΤ ΠΓ ΥΙΦ 2009 4ΘΒΟϑΤΙ ∃ΠΟΓΦΣΦΟ∆Φ ΠΟ &ΜΦ∆ΥΣΠΟ %ΦΩϑ∆ΦΤ - ∋ΦΧ 11-13, 2009. 4ΒΟΥϑΒΗΠ ΕΦ ∃ΠΝΘΠΤΥΦΜΒ, 4ΘΒϑΟ.

direction, a resonant feature starting at a normalized frequency

about 0.5 for an angle of incidence $=45º can be seen. In the

case of the #M direction one resonant feature can be

recognized starting at a normalized frequency about 0.8 for

$=45º. The spectra show also maxima and minima periodic

with the frequency. These maxima and minima correspond to

Fabry-Pérot interferences of the incident light inside the

photonic crystal slab.

III. EXPERIMENTAL RESULTS AND COMPARISON WITH

SIMULATIONS

The studied samples have lattice constants of 1 µm and 400

nm. The calculations from the previous section show that, for

these lattice constants, the first photonic bands lie in the near-

IR range for the sample with 1 µm lattice constant and in the

visible range for the sample with 400 nm lattice constant,

respectively. For this reason the samples were measured using

two different equipments: the sample with 1µm lattice constant

was measured in the near-IR region (10000-3000cm-1

) using a

commercial Fourier-transform IR (FTIR) spectrometer

(Bruker, model Vertex 70) equipped with a special reflectivity

attachmen. On the other hand, the sample with 450 nm lattice

constant was studied by angle-resolved polarimetry in the

visible range (350-830 nm) with a MM16 Mueller polarimeter

from Jobin-Yvon, equipped with an automatic goniometer for

the control of the angle of incidence and an automatic rotation

sample stage for the control of the direction of the incident

light with respect to the lattice.

The angle-resolved reflectivity spectra of the sample with

1µm lattice constant were measured over a range of incidence

angles ! between 12º to 70º in steps of 2º. A polished N-type

silicon wafer was used as absolute reflectance reference. The

measurements were performed for light incident along the #X

and #M square lattice directions, and with natural polarization.

Fig. 3 shows the result of the experimental measurement for

this sample and for the #M direction of the incident light

beam. In order to visualize correctly the photonic-band related

features, it is necessary a postprocessing of the raw

measurements obtained from the FT-IR spectrometer[6]. Thus,

the represented magnitude in the plot is the partial derivative

of the measured reflectivity with respect to the angle of

incidence. The photonic bands for this direction and for the TE

polarization have been overlapped to the graph. Several

photonic-band related resonant features can be clearly seen in

the spectra. One of them, ending at a wavenumber of about

7000 cm-1

for $=70º is clearly distinguished. This resonant

feature corresponds with the band observed in the calculation

and it is related to the fourth TE band.

For the sample with 400nm lattice constant angle-resolved

polarimetry was applied in the visible range (350-830 nm). A

schematic setup of the Mueller polarimeter can be found

elsewhere [10]. The angle of incidence ! was varied from 45º

to 70º in steps of 2° and the measurements were also

performed along the #X and #M principal lattice orientations.

Fig. 3. Experimental measurement of the angle-resolved reflectivity spectra

for a microstructured 2D photonic crystal slab. The measurement was carried

out with light incident along the #M direction of the square photonic lattice

and with natural polarization. The bands for the TE polarization for this lattice direction are overlapped to the spectra.

Fig. 4. Experimental measurement of angle-resolved polarimetry spectra

(M34 component of the Mueller matrix) for a nanostrucutred 2D photonic

crystal slab. a) light incident along the #X direction of the square lattice and

b) light incident along the #M direction. The photonic bands for the

corresponding lattice direction are overlapped to the spectra (solid lines for

TE polarization and dashed lines for TM polarization).

396978-1-4244-2839-7/09/$25.00 (∃)2009 ∗&&&

1ΣΠ∆ΦΦΕϑΟΗΤ ΠΓ ΥΙΦ 2009 4ΘΒΟϑΤΙ ∃ΠΟΓΦΣΦΟ∆Φ ΠΟ &ΜΦ∆ΥΣΠΟ %ΦΩϑ∆ΦΤ - ∋ΦΧ 11-13, 2009. 4ΒΟΥϑΒΗΠ ΕΦ ∃ΠΝΘΠΤΥΦΜΒ, 4ΘΒϑΟ.

Fig. 4 shows the experimental results of the M34 component

for this sample for light incident along the #X (a) and #M (b)

directions of the square lattice. The photonic bands for the TE

(solid lines) and TM (dashed lines) polarizations are

overlapped to the measurements. It is important to note that the

measurement setup performs the simultaneous measurement of

the 16 Mueller matrix components. In contrast with the FT-IR

reflectivity measurements, no post-processing is applied to the

polarimetry measurements.

The spectra for the two directions of the square lattice show

clear oscillations due to Fabry-Pérot interferences in the

photonic crystal slab. Overlapped to such oscillations several

resonant features can be identified. More concisely, for the #X

direction a resonant feature beginning at a normalized

frequency about 0.54 for the incidence angle $=45º can be

recognized. This feature is predicted by the calculations shown

in Fig. 2a) and corresponds to the second photonic band.

Nevertheless, it is difficult to affirm whether this resonant

feature corresponds to a TE or TM mode, since the TE and TM

bands for this material are very similar due to the small index

contrast between the photoresist and the air.

In the case of the measurement along the #M direction, the

simulations predict a weak coupling to photonic modes related

to the fourth photonic band in the region of normalized

frequencies between 0.7 and 0.8. This weak coupling hardly

visible in the measurement for this lattice direction, probably

because of the strength of the oscillations due to Fabry-Pérot

interferences.

IV. CONCLUSIONS

In this work we have shown the application of angle

resolved reflectivity and polarimetry to the study of photonic

crystal slabs. The approach we propose is the combination of i)

the numerical simulation of the interaction of the incident light

with the photonic crystal slabs, ii) the measurement of the

spectra with different instruments and iii) the comparison of

both results. We have shown this by studying two kinds of

samples composed of micro- and nanostructured layers of

photoresist on a silicon substrate. The simulations show how

the incident light couples to photonic modes propagating

inside the photonic crystal slabs and how this coupling is

translated into resonant features in the angle-resolved spectra.

Concerning the polarimetry measurements, the calculations

show that the different components of the Mueller matrix have

different sensitivity to the coupling of the incident light to the

photonic modes.

The measurement of angle-resolved reflectivity spectra,

with the adequate postprocessing, shows such coupling to the

photonic modes with a good agreement with the calculations.

On the other hand, the measurement of angle-resolved

polarimetry spectra shows also some of the resonant features

predicted by the numerical simulation, although in this case,

the Fabry-Pérot oscillations make difficult the recognition of

such features.

ACKNOWLEDGMENTS

This work was supported by Spanish Ministry of Education

and Science (MEC) under grants number TEC2006-06531 and

HOPE CSD2007-00007 (Consolider-Ingenio 2010). Josep

Ferré-Borrull acknowledges the Ramón y Cajal fellowship

from the MEC. Z. Kral acknowledges the Grant 2007-BE2-

00163.

REFERENCES

[1] V. N. Astratov, M. S. Skolnick, S. Brand, T. F. Krauss, O. Z. Karimov,

R. M. Stevenson, D. M. Whittaker, I. Culshaw, and R. M. De la Rue,

“Experimental technique to determine the band structure of two-

dimensional photonic lattices,” IEE Proc.-Optoelectronics vol. 145, pp.

398-402, December 1998.

[2] M. Galli, M. Agio, L.C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli,

M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Liu, P.

Bellutti, “Spectroscopy of photonic bands in macroporous silicon

photonic crystals,” Phys. Rev. B, vol. 65, p. 113111, March 2002.

[3] D. E. Aspnes, “Expanding horizons: new developments in ellipsometry

and polarimetry,” Thin Solid Films, vol. 455-56, pp. 3-13, May 2004.

[4] D.M. Wittaker and I.S. Culshaw,“Scattering-matrix treatment of

patterned multilayer photonic structures,“ Phys. Rev. B, vol. 60, pp.

2610-2618, July 1999.

[5] M. Ellman, A. Rodríguez, N. Pérez, M. Echeverria, Y.K. Verevkin, C.S.

Peng, T. Berthou, Z. Wang, S.M. Olaizola and I. Ayerdi, “High-power

laser interference lithography process on photoresist: Effect of laser

fluence and polarisation,” Applied Surface Science,

doi:10.1016/j.apsusc.2008.07.201.

[6] Z. Kral, J. Ferre-Borrull, T. Trifonov, L. F. Marsal, A. Rodriguez, J.

Pallarès, R. Alcubilla, “Mid-IR characterization of photonic bands in 2D

photonic crystals on silicon,” Thin Solid Films, vol. 516, pp. 8059-8063,

September 2008.

[7] Z. Kral, J. Ferre-Borrull, J. Pallarès, T. Trifonov, A. Rodriguez, R.

Alcubilla, L. F. Marsal, “Characterization of 2D macroporous silicon

photonic crystals: Improving the photonic band identification in angular-

dependent reflection spectroscopy in the mid-IR, Materials Science and

Engineering: B, vol. 147, pp 179-182, February 2008.

[8] Z. Kral, L. Vojkuvka, E. Garcia-Caurel, J. Ferre-Borrull, L. F. Marsal, J.

Pallares, “Calculation of Angular-Dependent Reflectance and

Polarimetry Spectra of Nanoporous Anodic Alumina-Based Photonic

Crystal Slabs,” Photonics and Nanostructures - Fundamentals and

Applications, to be published.

[9] R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light.

Amsterdam: North-Holland, 1987.

[10] E. Garcia-Caurel, A. De Martino and B. Drevillon, “Spectroscopic

Mueller polarimeter based on liquid crystal devices,” Thin Solid Films

vol. 455-456, pp. 120-123, May 2004.

397978-1-4244-2839-7/09/$25.00 (∃)2009 ∗&&&

1ΣΠ∆ΦΦΕϑΟΗΤ ΠΓ ΥΙΦ 2009 4ΘΒΟϑΤΙ ∃ΠΟΓΦΣΦΟ∆Φ ΠΟ &ΜΦ∆ΥΣΠΟ %ΦΩϑ∆ΦΤ - ∋ΦΧ 11-13, 2009. 4ΒΟΥϑΒΗΠ ΕΦ ∃ΠΝΘΠΤΥΦΜΒ, 4ΘΒϑΟ.