We present an analytical model that describes the limiting spectral performance of arrayed-waveguide-grating (AWG) spectrometers that incorporate slow-light methods. We show that the loss-limited spectral resolution of a slow-light grating-based spectrometer scales as the loss–group-index ratio of the waveguide array. We further show that one can achieve a spectral resolution of a few GHz using currently available slow-light photonic crystal waveguides while greatly shrinking the on-chip footprint of the spectrometer.
© 2013 Optical Society of America
“Slow light” methods , which refer to a broad category of science and technology that aims at controlling the group index of optical materials, have recently been studied extensively [2–4] for applications such as optical buffers, interferometry, laser radar, nonlinear photonics, etc. Specifically, it has recently been shown that slow light methods can greatly enhance the performance of spectroscopic interferometers of various geometries [5–10]. Moreover, slow-light techniques can greatly reduce the size of a spectroscopic interferometer while maintaining high spectral resolution, and they therefore show promise for the development of miniaturized on-chip spectrometers. In photonic integrated circuits, arrayed waveguide gratings (AWGs)  are a common spectrometer geometry that is widely used in telecommunications and other areas. It has recently been shown that the spectral performance of an AWG can be greatly enhanced by replacing a normal waveguide array with the slow-light waveguide array [12, 13].
In ideal situations where the slow-light waveguides are lossless, the enhancement factor of the spectral resolution of a slow-light AWG is proportional to the group index of the waveguides [12, 13]. However, practical slow-light technologies, such as photonic crystal (PhC) line-defect waveguides [14–16], often possess associated loss, and it is unclear what the role of loss is in degrading the spectral performance of slow-light AWGs. In this paper, we derive analytically an expression for the spectral performance of a slow-light AWG in the presence of loss. We show that the fundamental limiting spectral resolution of a slow-light AWG scales as the loss–group-index ratio of the waveguides. We further show that by using currently available PhC slow-light technologies, one can achieve spectral resolutions comparable to or better than those achieved in conventional AWGs while significantly reducing the footprint of the device.
An AWG is typically comprised of three parts as shown in Fig. 1. The input signal field first passes through a free propagation region (FPR) where its beam size expands due to diffraction. The expanded field is then coupled into an array of waveguides, which are designed such that the length of neighboring waveguides differs by a fixed amount Δl = mλ0/neff, where λ0 is the designed central (vacuum) wavelength of the device, neff is the mode index of the waveguides, and m is the diffraction order of the waveguide grating. The outputs of these waveguides are spaced evenly with a spatial period of Λ along the input to a second FPR. The exiting fields from the waveguide array interfere and different wavelengths focus at different transverse locations of the other side of the second FPR, where they can be further separated by coupling into different output waveguides. A slow-light AWG has the same geometry as a conventional AWG, except that the neighboring waveguides differ by a fixed length Δl of slow-light waveguides whose mode group index ng ≡ neff + ωdneff/dω can be much larger than its mode index neff.
2. Performance of a slow-light AWG spectrometer
The diffraction equation of an AWG is given by ,Fig. 1, whose mode index is denoted by neff. The actual waveguide may contain sections of other types of waveguide, e.g., the conventional waveguides outside the purple region as plotted in Fig. 1. However, we assume here that the conventional waveguide sections are of the same length through out the waveguide array, and therefore the length and mode index of these conventional waveguides do not appear in the AWG diffraction equation of Eq. (1). By taking the derivative of Eq. (1) with respect to ω, one obtains its angular dispersion as follows ,
At the center frequency, the output fields from different waveguides of the array are in phase. Thus, the total field distribution at the exit plane of the waveguide array, i.e., the input to the second FPR, can be mathematically approximated by [see Fig. 2(a)]Fig. 2(a)].
When the loss of the waveguides is negligible, the envelope function A(x) is equal to the envelope of the field at the entrance of the waveguide array, which is the mode profile of the input waveguide broadened through free propagation and which therefore can be approximated by a Gaussian function as
Different teeth of the comb function term in Eq. (5) represent different diffraction orders, which by definition do not overlap with each other. Meanwhile, the spatial width of the waveguide mode σ is also by design much smaller than NΛ. The value of w in Eq. (5) can be controlled by the size of the first FPR, and typically is comparable to or larger than NΛ such that light can be coupled into all waveguides in the array. Hence, in the ideal case in which the loss is negligible, the spot size of the output beam of an AWG is essentially limited by the width of the sinc function in Eq. (5), i.e., by the total number of waveguides in the array, through the following relation11] Eq. (7) that by using slow-light waveguides, the footprint of an AWG with a fixed spectral resolution scales as , where 1 ≪ i ≪ 2 depending on the choice between N and m.
When the waveguides are lossy, the transmission through waveguides of increasing lengths is no longer equal. The output envelope function is consequently modified asFig. 2(b), the extra negative exponential function can cause significant modification of the envelope function A(x) in terms of both the peak position and the effective width. Consequently, the output field profile at the focal plane of the second FPR is given by
One sees that the minimum spectral resolution for an AWG with lossy slow-light waveguides is always bounded by the velocity of light c times the loss–group-index ratio α/ng. This result is also consistent with previous studies of other types of interferometric spectrometers .
3. Numerical analysis
To better illustrate our analytical predictions, we plot in Fig. 3 the minimum spectral resolution of an AWG as a function of the group index and the loss coefficient, within reported values of the group index ranging from 10 to 200, and the loss coefficient ranging from 15 to 200 dB/cm.
In particular, we pick out three sets of values for ng and α as typical examples of currently available slow-light photonic crystal waveguides [15, 16]. Point A refers to a typical design of a dispersion engineered W1 flat-band photonic crystal waveguide , whose group index and loss coefficient are approximately 30 and 80 dB/cm, respectively. The minimum spectral resolution of an AWG using such a flat-band PhC waveguide is approximately 4 GHz. Note that the fundamental limiting spectral resolution of an AWG using such dispersion-engineered slow-light waveguides is already comparable to or less than that of conventional approaches, whose minimum spectral resolution is currently of the order of 5 GHz .
Note that the slow-light bandwidth Δλsl of such a flat-band slow-light waveguide over which the group index is maintained at 30 within 5% variation is approximately 10 nm . The working bandwidth of the slow-light AWG spectrometer is then determined by the smaller of Δλsl and its FSR. Note that since the footprint of an AWG scales at least linearly with 1/ng, a slow-light AWG using waveguides with ng ≈ 30 has a footprint at least 8 times smaller than that of an conventional AWG using conventional waveguides with ng ≈ 4. Using dispersion engineered designs, one can also achieve a larger group index, but it is typically associated with larger loss (see point B in Fig. 3) and narrower slow-light bandwidth [16, 17]. While the achievable spectral resolution is almost the same since the ratio of α/ng does not change significantly, the footprint can be further reduced because of the large value of ng.
Beside dispersion engineering, loss engineering on PhC waveguide designs has also been recently studied [15, 16, 19]. Using such approaches, one can achieve a group index of 30 with further reduced loss of 40 dB/cm. In such a case, the loss-limited spectral resolution (see point C in Fig. 3) can be further improved to be less than 2 GHz. As the design and fabrication process of PhC slow-light waveguides becomes further refined such that one can achieve group index and loss coefficient with values falling in the right-bottom portion of Fig. 3, one can potentially obtain spectral resolution of sub-GHz.
In summary, we have analytically derived an expression for the limiting spectral resolution of an AWG spectrometer in the presence of loss. We have shown that when the waveguide is lossy, the minimum achievable spectral resolution scales as the loss–group-index ratio of the waveguides used in the AWG. We have also shown that an AWG spectrometer using currently available loss-engineered PhC slow-light waveguides can already achieve spectral resolutions better than those of conventional approaches, with a much reduced footprint. As the design and fabrication of PhC waveguide is further improved with a further reduced loss–group-index ratio, one can potentially achieve spectral resolution of sub-GHz, sufficient for many spectroscopic applications such as chemical and biological substance recognition. Our analysis is applicable for other types of grating-based spectrometers, and it provides guidelines for developing on-chip slow-light spectroscopic devices to meet specific demands on the spectral performance. This project received support from the Defense Threat Reduction Agency-Joint Science and Technology Office for Chemical and Biological Defense (grant no. HDTRA1-10-1-0025).
References and links
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