1. Introduction

Optical spectroscopy for chemical and biological sensing is based on spectral structure induced by absorption, refraction, fluorescence or scattering [1]. Spectroscopic analysis of incoherent phenomena, such as fluorescence and Raman scattering, is inefficient because spatially diffuse sources couple poorly to spatial filtering spectrometers. Conventional spectrometers spatially filter to reduce ambiguity between spatial and spectral modes. To date, surface enhanced spectroscopy [2] has been the best approach to overcoming this mismatch. By localizing the fluorescing or scattering source, surface enhancement improves coupling between source and spectrometer. We propose an alternative approach to overcoming this mismatch using multimodal multiplex spectroscopy. Multiplex spectrometers measure weighted projections of multiple wavelength channels and have been common for the past half century [3]. Multimodal spectrometers are designed to measure the spectral density averaged over multiple spatial modes. Integrated multimodal spectroscopy is enabled by recent progress in photonic crystals and is to our knowledge first explicitly introduced in this report. A spectrometer capable of averaging spectral densities over many modes would improve the optical throughput of low coherence sources by several orders of magnitude (similar to the enhancements observed using surface enhancement) and could thereby enable volume Raman and fluorescence spectroscopy of diffuse sources, such as tissue and gases.

Multimodal multiplex spectrometers may be constructed using the spectral diversity of transmission through inhomogeneous photonic crystals. Several groups have used homogeneous photonic crystals to produce spectral filters and prisms. For example, Lin, et al., demonstrated a dispersive prism in two dimensional microwave photonic crystals [4]. Later, Kosaka et al., considered anomalous dispersion photonic crystals near resonance [5] and proposed use of the resulting superprism effect for wavelength demultiplexing applications [6]. The theory of the superprism effect is well developed [7, 8] and wavelength separation in a planar photonic crystal structure is demonstrated in good agreement with the theory [9]. In all these demonstrations, the incident optical beam has been spatially coherent and the photonic crystal has been assumed to be homogeneous.

Spatio-spectral structure in the transmittance of inhomogeneous disordered photonic crystals can convert spatially incoherent input signals with spatially uniform spectral density into spatially non-uniform spectral densities over a detection plane. The inhomogeneity induced in the transmitted spectrum can be sampled to measure multiplex spectral projections. These projections can be computationally inverted to estimate the mean spectrum over all the modes.

The proposed microspectromer consists of a photonic crystal mating directly to a detector array as shown in Fig. 1. The photonic crystal in this case is a 3D opal structure that is an inhomogeneous quasi-periodic array of microcavities that cause spectral variation in the near field. Advantages of the photonic crystal filter compared to an array of thin film filters are: insensitivity to angle of incidence, ability to characterize large etendue sources, and spectral diversity as a function of position on a small scale less than 10 microns.

 

Fig. 1. Proposed microspectromer based on spatio-spectral structure in the transmittance of inhomogeneous disordered photonic crystals for diffuse source characterization.

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2. Multimodal spectral diversity in photonic crystals

An optical field in an arbitrary state of coherence is described by a cross-spectral density, which can be reduced to a discrete set of spatial and spectral distributions by coherent mode decomposition [10]. Full specification of the field consists of a data cube describing the spatial distributions of the coherent modes as well as the spectral densities. In general, the coherent modes are determined by the nature of the source and by secondary scattering [11]. Spectrometers measure projections of the data cube. Representing the measurements as a vector m, we represent a projection as m=∑_n∫S_n(v)h_n(v)dv, where S_n(v) is the power spectral density of the n^th mode and h_n(v) is a vector describing the sensitivity of each measurement to the n^th spatial mode at the frequency v . h_n (v) is determined in the spectrometer by dispersive elements, such as prisms, gratings or interferometers. Conventional thin dispersive elements display spatio-spectral modal ambiguity. For example, a thin diffraction grating diffracts the field at wavelength λ ₁ incident at angle θ ₁ relative to the grating wavevector in the same direction as the field at wavelength λ ₂ incident at angle θ ₂ such that cosθ ₂=cosθ ₁+(λ ₁-λ ₂)/Λ, where Λ is the grating period. Spectroscopic measurements are not well-conditioned for estimating the spectra S_n(v) unless this ambiguity is broken. Conventional spectrometers take two approaches to removing the ambiguity. The most common approach is to spatially filter prior to and after dispersion, typically at the entrance and exit apertures (slits) of the spectrometer. This approach removes the ambiguity at a cost of dramatically reducing the light efficiency of the system (most modes are not transmitted to the detector.) The alternative approach is to image through the spectrometer, which is equivalent to spatially filtering many modes in parallel. This approach is effective but the optical and digital hardware required is bulky, expensive and, for non-mode specific spectroscopy, unnecessary.

The capacity of photonic crystals to break the ambiguity between spatial and spectral modes was the core of their original conception [12, 13]. The idea has been to form “band gaps” such that no modes exist at specific frequencies and then to use dislocations or impurities to introduce localized states. Here we propose a new class of photonic crystal applications that do not require full band gaps, but do rely on inhomogeneous spectral properties due to crystal disorder. We use the local spatio-spectral distribution of fields in photonic crystals to build linear distributed devices for spectral estimation. As an example of a crystal appropriate for these applications, Fig. 2 is a true color image of a colloidal crystal formed of a quasi-periodic array of polymer spheres. The crystal is uniformly illuminated by a halogen light source subtending a solid angle of 0.1 steradians. The half angle is 10 degrees. One is accustomed to observing color separation with gratings, but only for narrow spatial bandwidth fields observed in the far field of the structure. As shown in Fig. 2, a photonic crystal can induce complex multidimensional spectral diversity in the near field of the device. This structure is illustrated in more detail in Fig. 3, which plots the spectral transmission at points distributed on a rectangular grid spaced by 100 microns. The photonic crystal structure is illuminated by the halogen source with an effective source spatial bandwidth of 25mm diameter. The distance of the source to the photonic crystal was 75mm. An Ocean Optics USB2000 fiber optic spectrometer was used to measure the transmission spectrum at a grid of points in a plane 0.5mm behind the opal structure.

 

Fig. 2. True color photographs of the photonic crystal opal structure illuminated by a white light source. (A) 20X magnification. (B) 4X magnification.

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The photonic crystal composites used in our measurements were prepared as described in [14]. Fabrication begins with a crystalline colloidal array composed of monodispersed crosslinked polystyrene spheres dispersed in water. The sphere diameter is 109±26 nm (mean and standard deviation) and the particle density is 10¹³–10¹⁴ cm^-3. The particles formed in a hydrogel using photoinitiated free radical polymerized methacrylate functionalized poly(ethylene glycol)(PEG) [15, 16]. Upon hydrogel encapsulation, the long range order of the particles is stable to ionic contamination and minor mechanical deformation. The opalescing hydrogel based film is removed from the glass slide assembly in which it is fabricated [16] and allowed to air dry for 2 days, then placed in a vacuum oven at 35 C. The resulting clear film is then swollen in a monomer solution of 2-methoxyethyl acrylate, 2-methoxyethyl methacrylate, or a mixture of the monomers for 2 days. Ethylene glycol dimethacrylate and DEAP are added to this solution and the formulation crosslinked by a 3-minute exposure to a UV lamp. All chemicals were purchased from Aldrich or Acros Organics.

 

Fig. 3. Transmission curves as a function of wavelength at points p1 through p7. The points are on a 100 micron spaced grid immediately behind the photonic crystal.

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Most photonic crystal analyses and applications focus on perfectly periodic structures with perhaps a few defects added to create localized states. Self-assembled colloidal crystals, in contrast, vary slightly in order and period. Such natural inhomogeneity enables multimodal spectroscopy. We searched for regions of the crystal with particularly strong spatio-spectral inhomogeneity. Figure 4 is a spectral diversity map of the crystal used in Fig. 2. The map shows the variance of the spectral transmission of each pixel relative to the mean spectral transmission. The map shows three regions with particularly high spectral variations. Figure 5 is a movie of the crystal images at different illumination wavelengths.

 

Fig. 4. Spectral diversity map of the opal structure. The standard deviation of transmission curves (Fig. 3) was used as a metric of spectral diversity for each point of the photonic crystal. Regions 1,2, and 3 in the plot exhibit strong spectral diversities, and those regions were chosen in our spectral estimation algorithm.

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Fig. 5. (800 KB movie) A series of images at different illumination wavelength. Click here to start the movie. One can see that there are three regions with strong pattern variations corresponding to those in Fig. 4.

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3. Spectral estimation

The spatial spectral diversity of the colloidal crystal was used to build a multimodal spectrometer. The goal of the spectrometer is to measure the mean spectral density $\overline{S}\left(v\right)=\frac{1}{N}\sum _{n=1}^N{S}_n\left(v\right)$ . We are particularly interested in multimodal sources with spectra encoded by identical physical processes, for such sources there is no systematic variation in the power spectrum from one mode to the next. Assuming that variations of the modal power spectra from the mean modal spectrum are random with zero mean, we may assume that 〈S_n(v)〉=S̄(v). The measurements can be characterized by a multimodal spectral response h(v)=∑_n h_n(v) and 〈m〉=∫S̄(v)h(v)dv. Bandlimits on the spatio-spectral variation in the transmission of the photonic crystal allow source-measurement transformation to be expressed in discrete form as m=H̿s, where m is an m-tuple measurement vector. H̿ is an m-by-n spectral filter response matrix, each row of which represents the spectral filter response of a specific pixel. Each column of H̿ represents the characteristic vector corresponding to a particular spectral channel. s is an n-tuple source vector, representing the mean spectral density at each wavelength.

10,000 contiguous pixel measurements covering regions 1–3 of Fig. 4 were used to estimate source spectra with 5 nm resolution over 500–650 nm wavelength range. Because the spectral filter response H̿ is not known in advance, a set of calibration sources is used to characterize H̿. A ½ meter focal length Acton Research grating monochromator with a halogen lamp input illumination source was used for calibration. The output beam from the monochromator illuminated a diffuser such that the effective source for the photonic crystal was a uniform diffuse spot with a diameter of 12mm. The photonic crystal was placed 30mm from the diffuser so that each point on the photonic crystal subtends a full angle of 22 degrees. The image size of the photonic crystal was 5mm square. Images were captured using a Roper Scientific CoolSnap monochrome camera with a 1.2X relay lens. A series of narrow band spectra (each of 8 nm width spaced in 2 nm steps over the 500–650 nm spectral range) were generated from the monochromator and their corresponding filter responses were recorded on the CCD camera. These training spectra formed a banded spectral intensity matrix. The transfer function matrix H̿ is estimated using non-negative least squares optimization.

The calibrated photonic crystal was used to estimate the spectra of unknown sources over the wavelength range from 500 to 650 nanometers (nm) at resolutions varying from 2 to 20 nm. Since the number of spectral channels estimated (between 8 and 75) is much less than the number of pixel measurements (10,000) pixel, the measurements over-determine the spectrum. Over-determined problems do not have globally consistent solutions due to the presence of noise, but one can find a solution in the least squares sense. In finding the solution, we add the additional constraint that the spectral density is non-negative, which makes direct linear least squares inversion impossible. Instead we used the Matlab Optimization Toolbox to solve the general nonlinear optimization problem: $\underset{s}{min}{\parallel \stackrel{═}{\mathit{Hs}}-\mathbf{m}\parallel }_2$ , such that s≥0, where ║▯║₂ denotes the Euclidean norm.

Reconstruction at resolutions varying from 2 to 20 nm were attempted, 2 nm reconstruction failed to achieve high fidelity. Example spectral reconstruction results are shown in Fig. 6 with 5 nm resolution. Figure 6(a) shows the spectrum of a 15 inch LCD computer monitor set to a uniform screen color and apertured to 12 mm on the same optical path as the calibration signal. Figure 6(b) shows the spectrum of a mercury neon discharge lamp illuminating the same diffuser as was used for calibration.

4. Conclusion

We are not prepared to comment on the ultimate spectral resolution of photonic crystal microspectrometers, but the resolution and modal acceptance demonstrated in the first device promising for the development of spectrometers consisting simply of a photonic crystal mounted directly on a detector array. Demonstrated advantages of the first photonic crystal microspectrometer include high optical throughput and a capacity to characterize diffuse sources.

 

Fig. 6. Spectra reconstruction. (A) Liquid crystal display (LCD) spectrum reconstruction. (B) Neon lamp spectrum reconstruction. In both (A) and (B), the red lines are true spectra taken by Ocean Optics USB2000 optic fiber spectrometer; the cyan bar plots are numerically reconstructed spectra using the photonic crystal spectrometer.

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