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Optical Materials 29 (2006) 56–59

A photonic crystal sensor based on the superprism effect

Tushar Prasad a,*, Daniel M. Mittleman b, Vicki L. Colvin a

a Department of Chemistry, Rice University, Houston, TX 77005, USAb Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA

Available online 17 April 2006

Abstract

We describe an optical sensing technique, based on the superprism effect in three-dimensional photonic crystals formed by templatingcolloidal crystals. This method exploits the extreme sensitivity of the diffraction angle to the material properties, such as the refractiveindex contrast. We outline a theoretical method for analyzing this effect, and apply it to a particular sensing configuration. This sensorshould provide an increase by as much as three orders of magnitude in the sensitivity to small changes in index contrast, in comparisonwith other optical sensing methods.� 2006 Elsevier B.V. All rights reserved.

PACS: 42.70.Qs; 07.07.Df; 81.16.Dn

Keywords: Colloidal photonic crystal; Superprism; Optical sensor

1. Introduction

The development of three-dimensional photonic crystalswith stop bands in the visible and near-IR has attractedmuch attention recently, in part because of their potentialvalue in optical sensing applications [1]. The optical prop-erties of these periodic media are determined by the inter-ference of multiply diffracted waves, and are thereforevery sensitive to the material parameters such as the refrac-tive index and lattice spacing. This sensitivity can beexploited for the purposes of optical sensing, in a numberof different ways. Asher and co-workers have described acolloidal crystal hydrogel thin film, which can be used forsensing a variety of analytes by observing shifts in thewavelength of diffracted light [2–4]. A related techniqueinvolves measuring the wavelength diffracted from a vol-ume hologram written into a chemically sensitive polymer[5]. One can also use transmission spectroscopy to observesmall variations of the average refractive index within thepores of a macroporous polymer [6], an effect which has

0925-3467/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optmat.2006.03.008

* Corresponding author. Tel.: +1 713 348 3489; fax: +1 713 348 2578.E-mail address: tushar@rice.edu (T. Prasad).

been used for biosensing applications [7]. Very recently,an ultracompact biochemical sensor based on a two-dimen-sional photonic crystal microcavity was demonstrated forspectral shifts corresponding to minute changes in therefractive index [8]. Change in refractive index can alsobe used for fluid detection through intelligent design of aphotonic crystal waveguide coupler [9]. Chan et al. havedescribed an alternate approach, based on multilayers ofluminescent porous silicon with alternating porosity [10].For most of these examples, the minimum detectable shiftis on the order of a few nanometers, limited by the spectralwidth of the measured line. This corresponds to a change inrefractive index contrast or lattice spacing on the order of1%.

Here, we describe an optical sensor architecture basedon the angular deviation of light, rather than on its spectralproperties. This sensor exploits the superprism phenome-non, in which the propagation direction of a light ray insidea photonic crystal can be extremely sensitive to the materialparameters of the crystal. The superprism effect has beenstudied extensively in recent years by a number of research-ers [11–20], with a primary focus on applications such aswavelength division multiplexing and optical switching

0.0

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0.8

1.0

KWXΓLUXWave vector

Freq

uenc

y (

a/2

πc)

{100}

{010}

{001}

{111}

L

XXW

UX

K{110}{100}

{010}

{001}

{111}

L

XXW

UX

K{110}

Γ

ω

Fig. 1. The lowest eight bands in the band structure of the inverted FCCphotonic crystal lattice, computed using the plane wave method [28]. Thedielectric matrix has a refractive index of 1.59 (similar to polystyrene), andthe air spheres overlap slightly (r/a = 0.53) in order to model anexperimentally realizable sample. The two horizontal dashed linesrepresent the energies used to compute the dispersion surfaces in Fig. 2.Inset shows the first Brillouin zone of the FCC crystal with various high-symmetry points.

T. Prasad et al. / Optical Materials 29 (2006) 56–59 57

[16,20]. However, the application of this phenomenon tooptical sensing, which raises a unique set of considerations,has not been explored. For example, while many groupshave described the sensitivity of the effect to the incidentlaser wavelength [13,16–18], none have discussed the sensi-tivity to the dielectric contrast, which will be of paramountimportance in sensing applications.

The use of the angular deviation of a laser for sensingoffers several important advantages over other sensingtechniques. It is generally much easier to detect small shiftsin the position of a laser beam than it is to detect smallchanges in wavelength. Position-sensitive detectors candetect shifts in the centroid location of a Gaussian beamas small as a few nanometers [21]; available commercialdevices achieve a few tens of nanometer resolution. In con-trast, small changes in the spectrum of diffracted or emittedradiation can be challenging to detect without multi-grat-ing spectrometers, which are inefficient and difficult toalign. Also, the photonic crystal sensor can in principlebe integrated directly on top of the position-sensitive detec-tor, for an extremely compact and efficient design. Asdescribed below, this method can provide orders of magni-tude greater sensitivity to shifts in the refractive index ofthe photonic crystal material.

2. Methods and apparatus

2.1. Fabrication techniques and numerical modeling

It is particularly advantageous to develop superprismsensors using colloidal photonic crystals, although theeffect described here applies to both two- and three-dimen-sional photonic crystals. Using a variety of templatingmethods, three-dimensional photonic crystals can beformed of a wide range of materials, including polymers,ceramics, semiconductors, and even metals [22–27]. As aresult, the photonic crystal substrate can be tailored to suitthe needs of a particular sensing application. Also, tem-plated colloidal crystals possess large and accessible inter-nal surface areas, and are therefore well suited forsensing applications.

The extreme sensitivity of propagation angle for fre-quencies near a photonic band edge has been recognizedfor a number of years [11]. Recently, several different theo-retical methods have been used to describe the anomalouslight propagation near a band edge, for both two-dimen-sional [12–14] and three-dimensional photonic crystals[16,20]. Because we are interested in the response of theband structure to very small changes in dielectric contrast,we have adapted the method of Kosaka et al. [13], whichrelies on the photonic band structure rather than onnumerical propagation methods described by Ochiai andSanchez-Dehesa [16].

Fig. 1 shows the band structure of the type of sample tobe modeled, computed using an available software package[28]. The photonic crystal is a macroporous film formed bytemplating a face-centered cubic (FCC) crystal of sub-

micron spheres [24,27]. When the spheres are removed,the resulting macroporous structure consists of an FCC lat-tice of interconnected air spheres, embedded in a dielectricmatrix. In our calculations, we assume a refractive index ofabout 1.59 for the dielectric, approximately equal to that ofpolystyrene [24]. The calculation accounts for the windowsbetween air spheres by allowing the spherical voids to over-lap slightly (r/a = 0.53). Though this structure does notprovide a high enough dielectric contrast for the formationof a full band gap, it does possess a substantial stop bandindicative of a partial gap along the {11 1} crystalline axis.This is an indication of strong band structure anisotropy,which is sufficient for the observation of the superprismeffect. We note that Fig. 1 displays the band structure inthe standard format, in which only the high symmetrydirections are represented. To perform the computationdescribed below, a complete band structure is required.In other words, we must calculate the band structure fromC to all possible points on the surface of the Brillouin zone,not merely to the high symmetry points [13,20]. We can ofcourse use the symmetry of the Brillouin zone to reduce thecomputational load.

From this full band structure, all possible values of wavevectors in the three-dimensional space for a particularenergy can be calculated. A plot of all these wave vectorsin k-space for a particular energy gives an equal-energy sur-face known as a dispersion surface. The shape of dispersionsurface depends on the chosen value of the energy, specifiedby the frequency of the incident light. For small values of fre-quency, far from the stop band, the dispersion surface isspherical with a radius given by c/nave, where nave is the aver-age (homogenized) refractive index [20]. At higher energyvalues, near the photonic band gap, the band structurebecomes anisotropic. As a result, the shape of dispersion sur-face deviates from spherical, although it retains the symme-try of the Brillouin zone. Two examples are shown in Fig. 2.

Fig. 2. Iso-energy (dispersion) surfaces computed from the full three-dimensional band structure (a small portion of which is shown in Fig. 1). On the left,the dispersion surface is computed for the third band, for an energy (in dimensionless units) of 0.8. On the right, the surface for the fourth band is shown,for an energy of 0.73. The anisotropy of these surfaces gives rise to the superprism effect.

Fig. 3. A schematic of the envisioned superprism sensor. An incident laserbeam lies in the plane defined by the {111} and f�1�12g axes; the f�110gaxis points out of the page. A slight change in the refractive index contrastleads to a large change in the internal propagation angle hp, whichproduces a displacement dx on a position-sensitive detector.

58 T. Prasad et al. / Optical Materials 29 (2006) 56–59

These dispersion surfaces can be used to determine thepropagation direction of light inside the photonic crystal,for a specified incident light ray [13,20]. This is accom-plished by noting that the transverse component of thewave vector is conserved across the boundary of the crys-tal. This condition, along with the direction of the incidentray, defines a point on the dispersion surface, which is theinternal wave vector. The propagation direction is obtainedby computing the normal to the dispersion surface at theend point of the propagation wave vector, since the groupvelocity is given by VG = $kx(k). If the dispersion surfaceis not spherical, then the gradient can be a sensitive func-tion of the input parameters.

2.2. Sensor configuration

Once the internal propagation direction is determined, itis possible to compute the shift in the position of the trans-mitted ray if the geometry of the sample is specified. Anexample geometry, compatible with common fabricationmethods, is shown in Fig. 3. If the two sample surfacesare parallel, then the transmitted ray propagates parallelto the incident ray but shifted by an amount dx that is pro-portional to the thickness d of the film. If this shift is largerthan �10 nm, then it can be detected by a position-sensitivedetector. This signal can then be used as an indicator ofchanges in the refractive index contrast or the lattice spac-ing of the photonic crystal.

As an illustration, we compute the beam displacement asa function of the refractive index of the polymer for severaldifferent wavelengths. These results are shown in Fig. 4. Inthese calculations, a film thickness of 10 lm is assumed.The incident angles hin = 39� and /in = 0� are specified rel-ative to the {111} surface normal and the f�1�12g axes,respectively. Because of the symmetry of the photonic lat-tice, beams with /in = 0� remain in the plane of incidenceeven inside the film [20]. These results demonstrate thatsubstantial shifts can be expected for refractive indexchanges smaller than 1%, which might arise from the pres-

ence of an analyte adsorbed into the pores of the photoniclattice. Based on the aforementioned sensitivity of position-sensitive detectors, we extrapolate a minimum detectableindex shift of a few parts in 105. This represents roughlya factor of 1000 improvement over previous optical sensingschemes [4,5,10]. It should be pointed out that these calcu-lations require computation of a new dispersion surface,not only for each energy, but also for each value of refrac-tive index.

We note that this effect is sensitive to the wavelength ofthe incident light. For example, for the situation illustratedin Fig. 4, if the spheres have a diameter of 450 nm then thespecified wavelengths are spaced by �0.33 nm each. Thislevel of spectral purity can readily be achieved using a vari-ety of commercial laser sources. Further, if the index of thepolymer is slightly shifted from its nominal value, one can

1.580 1.585 1.590 1.595 1.600

-10

-5

0

5

10

Ω = 0.80.8005

0.8010

Beam

cen

troid

pos

ition

(μm

)

Polymer refractive index

Fig. 4. Calculated beam position as a function of the refractive index ofthe macroporous medium, near its nominal value of 1.59 (correspondingto the index of polystyrene). The beam positions are shown relative to anarbitrary zero position, for three different incident laser wavelengths(shown in dimensionless units). This calculation is performed for incidentangles of hin = 39� and /in = 0�, and assumes a film thickness of 10 lm. Ateach wavelength, a beam shift of more than 10 lm is predicted for arefractive index change of less than 1%. For X = 0.8005, the calculationhas been computed for a finer step size of Dn = 0.0002, to demonstrate thelinearity of the beam displacement within the transition region.

T. Prasad et al. / Optical Materials 29 (2006) 56–59 59

compensate by choosing a slightly different laser wave-length, and still maintain the high sensitivity.

3. Conclusion

In conclusion, we have described a new sensing schemewhich relies on the sensitivity of the propagation angle oflight in a photonic crystal, rather than on its spectral prop-erties. We discuss this new application of the superprismeffect in the context of three-dimensional templated pho-tonic crystals. These can be fabricated from an impressivearray of materials, providing a wide versatility with respectto the nature of the sensing application. Since the positionof a laser beam can be measured with extreme precision,this technique should provide a substantial increase in thesensitivity of optical sensing methods.

Acknowledgement

This work has been supported by the National ScienceFoundation (Grant #0103174) through the Center for Bio-

logical and Environmental Nanotechnology at RiceUniversity.

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