Coded output photonic A / D converter based on photonic crystal slow-light structures

A photonic analog-to-digital converter (PADC) utilizing a slowlight photonic crystal Mach-Zehnder interferometer (MZI) is proposed, to enable the optically coded output of a PADC with reduced device size and power consumption. Assuming an index modulation for the MZI on the Taylor’s PADC structure, limiting factors in device size, speed, and effective number of bits are derived considering the signal transition time of the light and the slow light dispersion effects. Details of the device design and results of a time domain assessment of the device performance is described with discussions on the feasibility of sub-mm size, 20GS/s operation of the device having the ENOB (effective number of bits) > 5. ©2008 Optical Society of America OCIS codes: (000.0000) (130.0130) Integrated optics; (050.5298) Photonic crystals; (250.4110) Modulators; (230.4555) Coupled resonators; (200.4740) Optical processing. References and links 1. R. H. Walden, “Analog-to-Digital Converter Survey and Analysis,” IEEE J. Sel. Areas Commun. 17, 539 (1999). 2. M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-Performance Optical-FiberNonlinearity-based Optical Waveform Monitoring,” IEEE J. Lightwave Technol. 23, 2012-2022 (2005). 3. G. C. Valley, "Photonic analog-to-digital converters," Opt. Express 15, 1955 (2007). 4. H. F. Taylor, “An Optical Analog-to-Digital Converter-Design and Analysis,” IEEE J. Quantum Electron. 15, 210 (1979). 5. F. X. Kärtner, R. Amatya, M. Araghchini, J. Birge, H. Byun, J. Chen, M. Dahlem, N. A. DiLello, F. Gan, C. W. Holzwarth, J. L. Hoyt, E. P. Ippen, A. Khilo, J. Kim, M. Kim, A. Motamedi, J. S. Orcutt, M. Park, M. Perrott, M. A. Popović, R. J. Ram, H. I. Smith, G. R. Zhou, S. J. Spector, T. M. Lyszczarz, M. W. Geis, D. M. Lennon, J. U. Yoon, M. E. Grein, and R. T. Schulein, “Photonic analog-to-digital conversion with electronic-photonic integrated circuits,” Proc. SPIE 6898, 689806 (2008). 6. J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled Resonator Optical Waveguides: toward the slowing and storage of light,” Opt. Photon. News 16, 36-40 (2005). 7. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slowlight enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052-2059 (2002). 8. Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-Light Fourier Transform Interferometer,” Phys. Rev. Lett. 99, 240801 (2007) 9. A. R. Shroff and P. M. Fauchet, “Optical Jitter and Pulse Distortion in High Bit-rate, Slow-Light MachZehnder Interferometers,” OSA Slow and Fast Light topical meeting, Salt Lake City, UT, July (2007) 10. E. Parra and J. R. Lowell, “Toward Applications of Slow Light Technology,” Opt. Photon. News 18, 40 (2007). 11. R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guidedwave analog-to-digital converters,” Proc. IEEE 72, 802-819 (1984). 12. F. J. Leonberger, C. E. Woodward, D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits and Sys. 26, 1125-1131 (1979). 13. Y.-H. Ye, J. Ding, D. Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004) 14. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing with Photonic Crystal Waveguide Couplers,” IEEE J. Lightwave Technol. 19, 1970-1975 (2001). 15. U. Peschel, A. L. Reynolds, B. Arredondo, F. Lederer, P. J. Roberts, T. F. Krauss, and P. J. I. de Maagt, “Transmission and reflection analysis of functional coupled cavity components,” IEEE J. Quantum Electron. 38, 830-836 (2002). 16. A. Taflove, and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, (Boston, Artech House, 2000) 852. #97873 $15.00 USD Received 24 Jun 2008; revised 10 Aug 2008; accepted 10 Aug 2008; published 21 Aug 2008 (C) 2008 OSA 1 September 2008 / Vol. 16, No. 18 / OPTICS EXPRESS 13752 17. L. Yang, J. Motohisa, and T. Fukui, “Suggested procedure for the use of the effective-index method for high-index-contrast photonic crystal slabs,” Opt. Eng. 44, 078002 (2005). 18. F. Gan and F. X. Kartner, “High-Speed Silicon Electrooptic Modulator Design,” IEEE Photon. Technol. Lett. 17, 1007-1009 (2005). 19. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660-668 (2007).


Introduction
The realization of a high speed (>10GS/s) electronic analog-to-digital (A/D) converter has become a serious challenge for circuit designers.With the fundamental physical limitations associated with aperture jitter, thermal noise, and comparator ambiguity [1], an increase in the universal measure of A/D converter performance (product of resolution and sampling rate) is near to the saturation regime.In this context, photonic means of digitizer, sampler [2], and A/D converters (PADC) have been a topic of intensive study as an attractive alternative for ultra-fast A/D converters.Especially for PADC, following serious research activities since early 1970s [3,4], various principles and forms of operation have been suggested -which can be classified 1) in terms of the degree of photonic implementation; photonic sampled, photonic quantized, photonic sampled & quantized, and photonic assisted [3], and 2) in terms of de-multiplexing means for analog signal; space, wavelength, time, and interferometric.
Of these, the interferometric PADC proposed by Taylor [4] has been a structure generating most intensive follow-up studies because of its scalability (as the required number of interferometers / sources / detectors increases linearly with the ADC bits/resolutions), and compatibility with electronic interfaces (using electrical input signal and providing quantized output).As an additional attribute, Taylor's PADC structure can be implemented with a low cost single wavelength laser source instead of a multi-wavelength source of higher functionality -which is an imperative for other competing approaches [5].Still, with the exponential dependence of the interferometer (MZI) length to the number of bits (proportional to 2 b ), and correspondingly with the increase in the power consumption for the phase modulation, most of the demonstrations of Taylor's PADC have focused on a low-bit and high sampling rate, not exceeding the effective number of bits (ENOB) of 4 (at 10GS/s) so far [3].
In this paper, we suggest a new photonic crystal A/D converter design based on Taylor's ADC concept.The uniqueness of our design is in the application of slow light advantages to the PADC with the introduction of photonic crystal Coupled Resonator Optical Waveguides (CROW) [6].By employing slow light, we obtain an efficient phase change with much shorter modulation length [7], to enable the reduction in the size / power consumption of the PADC (~ by factors of tens).To our knowledge, this is a new application of slow light, different from the traditionally proposed ones -such as interferometer [8,9], buffer, delay line, or synchronizer [6,10].For investigating the properties of the suggested photonic crystal slowlight PADC, we analyze the performance limiting factors in terms of light transit time in the modulation region, as well as the pulse broadening effect (differential time delay) from the slow-light dispersion.Results show the feasibility of achieving a PADC with 20 gigasamplings per second (GS/s) and an ENOB of more than 5, under a silicon platform.

Principle / theoretical analysis
Figure 1 illustrates a schematic diagram of Taylor's PADC [4].For each MZI, the necessary modulation region increases by a factor of 2 from the most significant bits.Optical outputs thus having a different period -as a function of the applied bias voltage strength (i.e, analog signal) -can be obtained from the output ports of different MZIs dedicated for each different bits, for the following conversion to a digitized gray coded output (with appropriate threshold / switch elements [11]).Worth to note, by adjusting the amount of modulation to the MZI, it is also possible to get conventional binary output [4].Meanwhile, assuming the simultaneous application of an analog signal to the entire electrode, the net phase change felt by the optical signal is not given by the instantaneous strength of the analog signal, but is rather replaced by its averaged value over the transit time T m (light signal propagation time in the modulated region L m of the m-bit MZI) [12].Different lengths of L m and T m for different bits of MZI thus lead to a non-uniform response to the input analog signal for its output.These non-uniform responses for different bits, thus impose a limitation on the maximum signal bandwidth guaranteed for error-free operation of the ADC.Writing T b as the transit time of light for the LSB MZI, where b the resolution of the PADC, we now can re-write the maximum signal bandwidth f s of the PADC [12] as a function of the index change ratio ρ = (Δn/n), for future reference purpose, (1), and (Δω / ω 0 ) = σ (Δn / n) = σρ -here, σ is the fraction of mode energy stored in the modulated region [7], n is the refractive index of the dielectric and f 0 is the frequency of the light source.Assuming a reasonable number of 1% refractive index modulation for λ 0 = 1550nm and σ = 0.6, the PADC bandwidth becomes limited by the light propagation time T b : to 11.5GHz at 6-bit resolution and to 1.44GHz at 8-bit resolution.
For the device size of PADC employing CROW, we now work on rewriting Eq. ( 1).Utilizing the dispersion relation of light in the CROW: ω(k) = ω 0 (1+κcos(k • Λ)) and υ g = dω/dk = κω 0 Λsin(k•Λ) [6] (κ the coupling coefficient; and ω 0 the resonant frequency of the CROW cavity).Equation ( 1) becomes (assuming the frequency of the signal light ~ ω 0 ) which is plotted in Fig. 2(a).Can be seen, for the size reduction of Taylor's PADC based on the CROW structure (determined by the modulation length required for the LSB MZI, the longest one) it is suggested to use the smallest κ available.However, understanding that a smaller coupling coefficient in the CROW results in a narrower bandwidth [6], we naturally expect a limitation in the smallest κ accessible for the reasonable operation of the PADC.Now, thus to analyze the penalty introduced from the restrictions in the signal bandwidth (alternatively, related to the device size, or κ), we start from the dispersion equation of infinite CROW structure (to note, this is reasonable and well-justified assumption considering the number of resonator cavities in our analysis, as can be seen in Fig. 2(a) and [13]).Focusing on the differential time delay of the signal -experienced with the CROW dispersion curvewhich ultimately affects the timing jitter of the output signal of the PADC, we calculate below the differential delay in the propagation time of light after passing the modulation region.With Δω being the frequency shift of the CROW structure experienced from the index change, m being the bit of the corresponding MZI (m=1~b for LSB~MSB), and L m and T m being the length / propagation time of light for the modulation region, the differential time delay (between modulation on / off states) of the light signal is given by, expressed in terms of κ , Now noting that kΛ≈ π /2 near the center of the CROW band and writing kΛ = π /2 + ΔkΛ , where ΔkΛ << 1, Eq. ( 4) then can be approximated, with Δk = Δω / υ g , as .
2 ) sin( Treating the effect of the signal transit time (Eq.( 2)) as the bit-dependent attenuation to the input sinusoid signal [12] and then applying the differential delays (Eq.( 5)) from the CROW dispersion considerations, we can now estimate the error in the ENOB (effective number of bits, the actual resolution one can derive from the ADC, contaminated with various noises) that originated from the introduction of CROW structure to Taylor's PADC platform.Applying a fast Fourier transform to the digitized output of the CROW PADC (with transit time and dispersion effects), and then calculating the signal-to-noise and distortion ratio (SINAD), we show the calculated ENOB = (SINAD-1.76)/6.02[1,3], in Fig. 2(b).Can be seen, the reduction of the ENOB with smaller values of κ is evident implying the performance-size tradeoff for Taylor's PADC, resulting from CROW bandwidth restrictions.For example, changing the κ from 0.02 to 0.005, it is possible to reduce the device size / power consumption by a factor of ~ 4, but only at the accompanied expense of an ENOB reduction of ~ 2 (for 10GHz, with 1 % index modulation).We also note that, there exists a maximum bound in the ENOB, dictated from the flight-time considerations only (without CROW, Eq. ( 2) -5 bits for 10GHz (or 20GS/s) and 2 bits for 200GHz (or 400GS/s).

Results / discussion
Without loss of two-dimensional square dielectric rod photonic crystal (PC) has been used as the basic platform, for the study of a slow-light PADC under Taylor's principle.Dielectric rod of radius r = 0.20a, with a lattice constant of a = 550nm, and an effective refractive index n eff (Si) of 3.23 were adopted.For the construction of the CROW, the dispersion relation (Fig. 3(a)) was calculated using the PWEM (Plane Wave Expansion Method).The radius of the defect in the CROW cavity was set at r = 0.08a with a defect period of 3a, to provide a coupling coefficient κ of 0.0142 -considering restrictions both in the ENOB (related to differential time delay) and device size.Worth to mention, care has been taken in the wideband design of the 3dB directional splitter (Fig. 3(b)) [14] and the CROW-PC waveguide adapter (Fig. 3(c)).For the CROW-PC waveguide adapter, coupled mode theory (CMT) has been applied [15] to give a coupling efficiency > 95%.With the large problem size extending to more than 100μm (=L LSB for a 3bit PADC), 2D FDTD analysis [16] was carried out employing an effective index method [17].Push-pull index modulation (+/− 0.496%) was used to drive the MZI assuming a DC bias to both arms of the modulator, with the signal frequency set at the CROW center frequency f = 0.3543(c/a).Figure 4 shows the operation of the 3-bit PADC (having different modulation lengths for each MZI), for a linearly increasing (within t = 6.88ps~27.5ps)analog input signal.To note, outputs of each MZI can be digitized by setting decision thresholds, to produce coded output.For the current example, the threshold was set at 43% of the normalized LSB peak power.Confirming the basic operation, now further to analyze the PADC operations at an ultrahigh sampling rate, high-frequency (100 ~ 350GHz) sinusoidal analog signal has been tested to get the ENOB values at corresponding frequencies -following the procedures detailed earlier in this section.Figure 5(a) shows 100GHz analog input signal overlaid on the digitized output.Also shown in Fig. 5(b) is the obtained ENOB of the 3-bit PADC structure (simulation bit limited by the computational capacity) plotted together with the theoretical ENOB values derived in Section 2. With the selection of a reasonable value for κ (0.0142), the differential time delay penalty from the CROW dispersion was minimized, leaving the signal transit time as the dominant source of error in the ENOB of the CROW PADC operation.Worth to mention, to counter the signal transit time and thus to get a higher ENOB (>3) for the PADC at these operation speed (>100GHz), a traveling-wave MZI with longer modulation length can be considered, possibly using Kerr nonlinearity, and co-propagating control optical analog signals.
Finally critical to mention, for the current study assuming electrical modulation, a realistic modulation speed and the material aspects of dielectrics need to be counted.For modulating the index of silicon with an electrical signal, a plasma dispersion effect (with a response time up to 20GHz) can be used [18,19].Assuming 1% of the index change ratio ρ in air-hole type Silicon PC structures, ENOB = 5 for the PADC operation was estimated at a 20GS/s sampling speed (from Fig. 2(a), with 10GHz analog signal, with κ = 0.02 and at resolution b = 6, σ = 0.6, and Λ = 2 ㎛), giving the length of longest MZI smaller than 700μm.

Conclusion
We proposed and analyzed a new and unique photonic A/D converter design using slow-light photonic crystal structures.Theoretical analysis on the performance limitation has been developed / and carried out considering both the signal transit time and CROW dispersion effect, to point out the trade-off in device size and penalties in the ENOB, associated with the differential time delay (from the CROW dispersion).Numerical assessment of the device performance shows the feasibility of a slow-light PADC with a reduced device size (up to a factor of 10 (~υ g / c, when compared to those MZI employing a conventional waveguide) and higher ENOB / speed of operations (ENOB = 5, 20GS/s for a silicon platform using a plasma dispersion effect).

Fig. 1 .
Figure1illustrates a schematic diagram of Taylor's PADC[4].For each MZI, the necessary modulation region increases by a factor of 2 from the most significant bits.Optical outputs thus having a different period -as a function of the applied bias voltage strength (i.e, analog signal) -can be obtained from the output ports of different MZIs dedicated for each different bits, for the following conversion to a digitized gray coded output (with appropriate threshold / switch elements[11]).Worth to note, by adjusting the amount of modulation to the MZI, it is also possible to get conventional binary output[4].For later use, we write the relationship between the resolution (b) of the Taylor's PADC, and the required phase change (L LSB Δk, L LSB being the length of the modulation region for the least significant bit: LSB, Δk being the differential of wave vector from the waveguide index change) in the LSB-MZI;

Fig. 4 .
Fig. 4. Demonstration of a 3-bit CROW PADC under a linear refractive index modulation (Media 1).The green and yellow regions show the modulation region (push-pull with different sign) of each MZI.For the figures on the right, the red dashed lines show the decision threshold.

Fig. 5 .
Fig. 5. (a).Plot of the digitized output (black, magenta) from a 3-bit CROW PADC compared with its analog sinusoid input signal (yellow: 100GHz, cyan: 350GHz).(b) ENOB of the 3-bit PADC at different frequencies of analog input signals.The ENOB from the FDTD simulation (black) is compared to the ENOB values calculated from theory (in Section 2)