1. Introduction

Optical spectroscopy, the precise determination of the wavelength composition of light, is arguably the most powerful analytical tool at the disposal of science. Optical spectrometers are required in the fields of chemical production, analysis of DNA and RNA macromolecules, high throughput screening of drugs for pharmaceutical purposes, medical diagnostics, and environmental testing [1]. Not surprisingly there has been great interest in microspectrometers [2] that can be integrated within lab-on-a-chip (LOC) and Micro-Total-Analysis-Systems (μTAS) [3]. One of the earliest works was by Goldman et al. [4], who reported spectral analysis based on waveguide grating couplers. Later, Yee et al. reported a hybrid spectrometer for chemical analysis [5], composed of silicon micromachined gratings and a CCD camera, in a system that was characterized by separate microfluidic and optical parts. For optimal performance, grating-based spectrometers require lenses or mirrors for collimation and focusing [2]. Micro-scale fabrication of these elements is often difficult, and this has prompted exploration of several alternative strategies. For example, Traut et al. [6] developed a miniaturized spectrometer by forming grating patterns on the top surface of an array of microlenses. In that design, each microlens in the array functioned as a dispersive and focusing element. In the design by Grabarnik et al. [7], a second grating was utilized as the focusing element. Adams et al. [8] proposed a non-grating approach based on monolithic integration of microfluidics and thin-film filters directly on top of image sensor arrays, but their experimental results were restricted to the filtering of excitation and fluorescence light. For many LOC applications, there remains a need for closer integration of the optical parts, including a diffraction-grating-based spectrometer, within the microfluidic platform.

Poly(dimethylsiloxane) (PDMS) is a low cost, biocompatible polymer increasingly used for the fabrication of microfluidic and LOC devices [9,10]. It is transparent in the UV–visible range (230 nm to 700 nm) and within selected bands of the near-IR, and hence is suitable for the fabrication of waveguides and other on-chip optical components. Given the popularity of optical detection for LOC sensing systems, there is a natural impetus to develop optical componentry (including spectral dispersion devices) in PDMS. Previous research includes work on PDMS-based waveguides [11] and lenses [12]. Researchers have also considered both hybrid and monolithic integration of spectral discrimination devices with PDMS microfluidics. For example, Domachuk et al. [13] embedded a microfluidic channel adjacent to a PDMS-based planar grating and used external lenses for focusing and collimation, while Yang et al. [14] reported a PDMS-based diffractive optical element that combined both spectral dispersion and focusing functions.

Waveguide-based spectrometers (and waveguide-based optics, generally speaking) are attractive for optofluidics and LOC systems, since they enable built-in alignment of optics and fluidics. While hybrid approaches, such as wafer bonding of PDMS microfluidics to silicon-based arrayed waveguide gratings (AWG) [15], have been described recently, there are very few reports of PDMS-based waveguide gratings. For example, Kee et al. [16] described a PDMS-based arrayed waveguide grating (AWG) with ~1 nm resolution but only 4 output channels and a correspondingly limited operating range (~640 to 645 nm). Here, we describe a PDMS-microsystem that monolithically integrates microfluidic channels, multimode waveguides, a collimating lens, and a slab-waveguide-based, focusing transmission grating. In addition to enabling a convenient chip layout, the use of a transmission grating (rather than a reflection grating) is partly motivated by the possibility for high spectral dispersion with large facet dimensions [17]. The grating described operates in second order with feature sizes greater than 3 μm, and incorporates a novel self-focusing design based on the use of cylindrical facets. Below, we provide a theoretical analysis and an experimental characterization of the spectrometer part of our chip. A wavelength resolution <10 nm over a free spectral range >200 nm in the visible band was predicted and verified by experimental results.

2. Design of the LOC spectrometer device

A typical spectrometer consists of a light source, a dispersive element (typically a diffraction grating), lenses or mirrors for light collimation and focusing, and a detector. In the present design, a parabolic lens is used to collimate the light while the necessary dispersion and focusing is accomplished by a curved transmission grating, as shown in Fig. 1.

 

Fig. 1 (a) Mask design layout of the LOC spectrometer device. In the fabricated chip, the filled black regions become hollow (air-filled) cavities. The teardrop shaped features at upper and lower left are microfluidic reservoirs. (b) Magnified view of the intersection point between the microfluidic channel and three waveguides. (c) Magnified view of the parabolic collimating lens together with the curved focusing transmission grating.

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The focal length of the collimating lens was set as 3 mm to match the lateral extent of the grating (2585μm). The radius of the lens is obtained from the well-known formula for spherical lenses:

As described previously [11], the optical waveguides are nominally configured to confine light by total internal reflection (TIR), between the high and low index PDMS layers in the vertical direction, and between the PDMS core and the adjacent air channels (side claddings, see Fig. 6 below) in the horizontal direction. Note, however, that the air channels themselves can act as ‘leaky’ waveguides, with propagation loss on the order of 1 dB/cm for the dimensions used here (~50 μm x ~50 μm) [18]. This was verified from analysis (not shown) of scattered light images (see Fig. 7 below, for example), and is very similar to losses reported for TIR-based channel waveguides in PDMS [11]. Such air-core waveguides can operate in a quasi-single-mode regime, since propagation loss scales quadratically with mode number [18]. Thus, a sufficient length of straight leaky waveguide preferentially suppresses high-order modes. For the optical characterization of the grating spectrometer (see below), laser light sources were coupled into the air-core leaky waveguides at the left edge of the chip (Fig. 1), whose inherent mode filtering resulted in an input field (at the start of the slab waveguide section prior to the lens) that better approximates the Gaussian input field assumed for numerical simulation purposes. This was motivated by a desire to simplify the comparison of experimental results with analytical and numerical predictions, and also allows us to assess the best-case resolution of the spectrometer. In practice, the existence of multiple waveguide modes will degrade the resolution of the instrument, as discussed in more detail below and elsewhere [19–21].

Note that the region between the lens and the diffraction grating is essentially a leaky, air-core slab waveguide, which contributes additional mode filtering prior to the grating. Since the spacing between lens and grating is on the order of 1 mm, the total propagation loss for low order modes in this region is expected to be quite small. It should be possible to reduce the loss of higher-order modes, for example by adding metal cladding layers in these regions. This would improve the light-gathering capability of the spectrometer, but reduce its resolving power. A complete exploration of these tradeoffs is left for future work.

2.1. The curved focusing phase transmission grating

We designed a grating where each facet is an arc or section of a circle, the radius of which is uniquely determined to obtain a common focal point among all facets (Fig. 2). By using curved grating facets, the focusing action is obtained without a lens. Furthermore, by eliminating the focusing lens, the number of interfaces that light must traverse is reduced. This is advantageous in cases where the spectral measurement of a faint signal such as a fluorescent source is needed.

 

Fig. 2 Geometry of the focusing grating/lens. The facets of the grating are sections of circles that act like lenses with a common focal point. X and Z are coordinates within the horizontal plane of the slab-waveguide system.

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The device shown in Fig. 2 is a slab waveguide phase transmission grating. The constructive interference of light from adjacent facets will occur when the difference in focal length for the two adjacent facets, ∆ f = f _j – f _{j + 1}, satisfies [17]:

 

Fig. 3 The detailed layout of the grating/lens device is shown. The focal point is chosen to lie at the same height as the first grating facet (i.e. at x = −1293 μm). The inset shows a magnified view of the central part of the curved grating, which can be approximated as a linear grating with mean facet period ~7.4 μm.

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3. Approximate analysis based on the grating equation

Analytical treatment of the curved, focusing phase grating is difficult [17]. However, as shown in the inset of Fig. 3, the central portion of the grating can be well-approximated as a straight grating with uniform facets. In practice (including the experiments described below), most of the incident light interacts with the central portion of the grating. Thus, a first order approximation based on standard grating theory can be employed for insight.

Basic details of a grating’s behavior can be readily obtained by using the grating equation from classical diffraction theory:

Directly from the geometrical layout of the central portion of the grating and assuming incidence along the optical axis (x = 0), we estimate θ₁ ~47.6 degrees and a mean grating period Λ ~7.4 μm. From the grating equation, this results in an estimate for the second-order (m = 2) diffraction angle (at the design wavelength) of θ₂ ~39.6 degrees. Note that this angle is relative to the effective grating normal at the central part of the grating; the predicted angle (of the second-order diffracted beam) relative to the optical axis is then simply θ₁ - θ₂ ~8 degrees, which is in very good agreement with the numerical simulations below.

For spectral analysis, another key parameter is the angular dispersion of the grating. From the grating equation with the angle of incidence θ₁ fixed, it follows that:

4. Numerical simulation of the microspectrometer

The integrated planar microspectrometer, fabricated in PDMS as described below, is a multimode (MM) waveguide device. In a MM waveguide device, the polarization dependence can be neglected for simplified analyses [19]. Moreover, since the smallest features of the present device (i.e. the grating facets) are large in comparison to the operational wavelengths, a scalar electromagnetic simulation considering only the TE modes is expected to be sufficiently accurate. Consider a Cartesian coordinate system where x and z are the horizontal coordinates in the plane of the slab and y is the vertical (out-of-plane) coordinate. Strictly speaking, the field for each wavelength at a point (x, y, z) within the slab is a linear combination of vertical modes supported by the slab waveguide [19]. As discussed above and in Section 6 below, however, we used the air channels as leaky waveguides for launching the laser beams used to experimentally characterize the grating. This ensured that the field launched into the slab waveguide section prior to the collimating lens was quasi-single-moded in the vertical direction. Based on this assumption, only the fundamental slab waveguide mode was considered in the scalar simulation of the grating. Furthermore, a ‘hard boundary’ assumption [19] was used to obtain the relevant parameters of the fundamental mode, which is justified by the large thickness of the PDMS core. As mentioned above, excitation of higher-order modes is expected to degrade the resolution of the spectrometer. However, it should be noted that the use of air-core waveguides for delivery of excitation and emission light, and thus as a mode filter prior to the grating, is a realistic option in practice, and is a subject we hope to explore in greater detail in future work. To approximate the fundamental mode of the square hollow input waveguide, a 40μm diameter Gaussian field was assumed as the input field at the start of the slab waveguide section.

Grating-based microspectrometers typically operate in the Fresnel diffraction regime, and thus can be treated using the Fresnel diffraction integral [22]:

4.1. Rayleigh-Sommerfeld I diffraction formula

The Rayleigh-Sommerfeld diffraction model was first proposed by A. Sommerfeld [23]. The so-called Rayleigh-Sommerfeld diffraction integrals I & II can be written in terms of Hankel functions, as outlined by M. Totzek [24]. A step-by-step derivation of the Rayleigh-Sommerfeld I diffraction integral in terms of the first-order Hankel function of the first kind can be found in our recent publication [25]. The Rayleigh-Sommerfeld I diffraction integral in terms of the Hankel function is given by:

This integral expresses the fact that the field at every output point is due to the sum of fields at all input points. The simulation routine (encoded in Matlab) uses a Gaussian electric field as input and calculates the resulting electromagnetic field intensity at the output plane, which is located at a distance | ρ − ρ^′ | from the input. Since there are several interfaces (lens, curved grating and output plane), the output obtained at the first interface is treated as the input for the next interface and so on.

4.2. Simulation of multiple wavelengths in the 2nd diffraction order

Figure 4 shows the intensity profiles at the output plane of the spectrometer, for a 40 μm diameter input Gaussian beam and five different wavelengths ranging from 532 to 758 nm, centered around the design wavelength of 645 nm. For clarity, the zero, first and second diffraction orders are labeled.

 

Fig. 4 The intensity profile at the output plane is shown, for a 40 μm input Gaussian beam and wavelengths ranging from 532 to 758 nm. The zero, first and second diffracted orders are labeled accordingly.

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Consistent with the discussion in Section 2, the second-order diffracted beam at the design wavelength (λ = 0.645 μm, which is shown in red) appears at x = −1293 μm, directly in line with the first grating facet (see Fig. 3). At the lower left corner of Fig. 4, the zero order beams for all wavelengths are shown overlapping at a single position. The explanation is that the position of the zero order beam (regardless of wavelength) is determined only by Snell’s law of refraction (n₁.sinθ₁ = n₂.sinθ₂) applied to the air-PDMS interface along the line of the grating. Here n₁ = 1, n₂ = 1.43 (i.e. assumed wavelength-independent), θ₁ is the angle of incidence, and θ₂ is the angle of refraction. According to the simulation, the position of the zero order beams is at −2734μm. Since the horizontal distance (on the optical axis) from the grating centre to the output plane is 9224 μm, we can substitute θ₂ = [θ₁–tan⁻¹(2734 / 9224)] into Snell’s law to obtain an estimate for the effective incident angle, θ₁ = 47.6°, which is in agreement with the geometrical estimate from Section 3.This provides further evidence that good qualitative predictions are obtained by approximating the curved grating as a straight grating, in turn approximated by the central ~150 grating facets in the vicinity of the optical axis.

In keeping with the discussion in Section 3, we can thus use the diffraction equation [Eq. (3)] to obtain an analytical estimate of the position of any wavelength in any given order. For example, consider the second-order (m = + 2) diffraction of λ = 0.532 μm light. Using n₁ = 1, n₂ = 1.43, θ₁ = 47.6°, m = 2, λ = 0.532 μm and Λ = 7.4 μm for the mean value of the grating period near the optical axis, we obtain θ₂ = 38.1° as the angle of diffraction. As above, the angle with respect to the optical axis follows as θ₂ - θ₁ = −9.54°. The position on the output plane is then obtained as tan (−9.54°)$\times$ 9224 μm = −1551 μm. As shown in Fig. 4, the simulated results are in nearly exact agreement with this analytical prediction.

4.3. Simulation of multiple orders at a single wavelength

Figure 5 shows the predicted intensity profile at the output plane, for the same Gaussian input beam mentioned above, and for λ = 0.532 μm. Note that the horizontal axis encompasses 10 diffracted orders and that the intensity is plotted on a logarithmic scale. The position of each order at the output plane (X_m) is indicated on the plot. As expected from the design criteria above, the 2nd diffracted order (i.e. the design order) is located at X₂ = −1551 μm and has the highest intensity. Given the distance from the grating centre to the output plane mentioned above (Z = 9224 μm), the angle (θ_m) between a given order and the optical axis is obtained from tan(θ_m) = X_m/Z. The relative angles between successive orders were extracted from the simulation in this way, and are compared to the experimental results in Section 6 below.

 

Fig. 5 The intensity profile at the output plane is plotted, for an input Gaussian beam and λ = 0.532μm. The horizontal axis was scaled to encompass 10 diffracted orders as indicated by the labels, including the m = + 2 design order centered at – 1551μm.

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5. Method of fabrication

The main details of our fabrication process, which enables the integration of optical and fluidic components in a monolithic LOC platform, have been described elsewhere [11]. Briefly, the planar system [Fig. 6(a)] comprises three layers of PDMS. The upper and lower layers are Sylgard-184^®, which has a refractive index of 1.41, while the central layer is a higher index (n ~1.43) formulation of PDMS. The higher index PDMS is patterned using a soft-lithography molding technique, to simultaneously define both fluidic channels and optical waveguide channels (i.e. waveguide cores) within the same layer. Because of the index contrast with the upper and lower cladding layers, light can be confined within the central “optofluidic” layer, also referred to as the core layer. Furthermore, channel waveguides are enabled by the in-plane refractive index offset between the PDMS core and the adjacent air channels (i.e. ‘side claddings’, see Fig. 6 and Fig. 1 above). As briefly discussed in Section 2, the side claddings themselves were found to function as excellent air-core, leaky waveguides while providing a desirable suppression of high-order modes. For the experimental characterization of the grating described below, these air-core waveguides were used for the in-coupling of laser light.

 

Fig. 6 (a) A schematic illustration of the integration strategy is shown. The diagram represents the cross-sectional view of the 3-layer PDMS system with waveguides and microfluidic channels patterned in the higher-index, central PDMS layer. (b) SEM image of the grating facets on the silicon master. (c) SEM image of the grating facets transferred to PDMS using a soft-lithography process.

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A single silicon master (see Fig. 1) was used to pattern the entire optofluidic chip, including microchannels, waveguides, the lens, and the curved focusing transmission grating. From a fabrication tolerance perspective, the curved grating represents the most critical aspect of the fabrication process. In particular, it is important that the grating facets on the silicon master are smooth and vertical. As shown in Fig. 6(b), this was accomplished using the well-known “Bosch^®” deep reactive ion etching (DRIE) process. Furthermore, proper operation of the grating relies on an accurate transfer of these facets into the PDMS waveguide core layer. As shown in Fig. 6(c), excellent replication of the grating features was achieved using an optimized soft-lithography process [11].

6. Experimental results

In order to verify the functionality of the grating/lens combination, we first launched a green laser into one of the air-core leaky waveguides aligned with the optical axis. This ensured that the light field was nearly single-moded at the start of the slab waveguide section prior to the lens, and was approximately aligned with the centre of the grating. Diffraction effects were studied by imaging the light scattered from the waveguide core layer onto a color camera mounted on top of a standard zoom microscope. This circumvented the problem of preparing a high-quality facet at the output plane of the chip, although end-facet light collection was also verified (not shown) and would likely be the most desirable configuration in practice. As expected and shown in Fig. 7(a), the input light was dispersed into a number of diffraction orders. As described in Section 2, the grating is designed to diffract light preferentially into the second diffraction order. Consistent with this ‘blaze’ condition, the second-order beam was observed to be significantly brighter than those of adjacent orders. As an initial assessment of the grating, we measured the angles between consecutive diffracted orders, and compared the results to the predictions of the numerical simulations from Section 4. The comparison is summarized in Table 1, and reveals excellent agreement between experimental and numerical results.

 

Fig. 7 Scattered light images captured by a color camera are shown. The images correspond to diffraction of a green laser, λ = 532 nm (a), a red laser, λ = 632 nm (b), and an amber laser, λ = 594 nm (c). The light path, including the input waveguide, spherical lens interface, and diffraction grating interface, are most clearly visible in part (c).

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Table 1. - Relative angles between adjacent orders

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We performed similar experiments using other laser sources, including a red laser with wavelength λ = 632 nm [Fig. 7(b)] and an amber laser with wavelength λ = 594 nm [Fig. 7.(c)]. Data from these experiments was used to quantify the dispersion and resolution of the grating. A key figure of merit for a spectrometer is the resolving power (RP), commonly expressed as RP = λ/ dλ, where dλ is the resolution, in turn defined by the full width half maximum (FWHM) or −3db power bandwidth for the fringe of interest. Most microspectrometers reported in the literature have a RP in the range of ~10 -100 [2].

In order to experimentally assess the optical resolution of the grating, we observed the second-order diffraction fringes produced by two laser sources. Figure 8 plots the average pixel intensity versus distance along the output plane for the second order diffracted modes associated with 594 and 532 nm laser light. These lasers were coupled into the same air–core leaky waveguide, and scattered light images were captured near the output end of the device. Column-wise averaging of the pixel intensity was used to reduce noise. Device features of known size were used to scale the images and to enable a mapping between pixel number and spatial coordinates. Based on this mapping, the resolution was estimated from dλ ~dx/(Δx/Δλ), where Δλ is the known wavelength spacing between the two lasers. Resolution as good as ~6nm was extracted from such measurements (i.e. RP ~100) ; a typical example is shown in Fig. 8. For a grating, RP ~mN, where N is the number of grating facets contributing to the interference fringes. From the experimental estimate of the second-order RP above, this suggests ~50 facets are effectively illuminated by the input laser beam, which is reasonably well supported by close inspection of the images in Fig. 7. However, it is also likely that fabrication defects contributed to the lowering of the measured RP. Optimization for these details is left for future work.

 

Fig. 8 Average pixel intensity plotted versus vertical distance x along the output plane for the 2nd order diffracted modes of 594 nm (left peak) and 532 nm laser light (right peak).

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7. Concluding remarks

To our knowledge, this is the first report of a slab-waveguide-based transmission grating fabricated directly in PDMS. The parameters of the microspectrometer (RP~100 and a free spectral range >200 nm in the visible region) compare favorably to most other microspectrometers reported in the literature. Moreover, these performance specifications should be adequate to address many applications of current interest in LOC analysis systems, including visible-band fluorescence and absorptance spectroscopy. Furthermore, since the device is fabricated directly in PDMS, it offers good potential for monolithic integration within LOC and optofluidic microsystems.

We expect that the RP of the grating could be improved by optimization of the light collection and collimation prior to the grating, to increase the number of illuminated facets. Furthermore, it should be possible to fabricate significantly smaller grating facets in PDMS using an optimized soft-lithography process, thereby increasing the spectral dispersion and RP of the spectrometer. In principle, the focusing of the grating could be improved by the use of aspheric facets, although this would necessitate careful consideration of the fabrication tolerances associated with the soft-lithography process. These refinements are left for future work.