Abstract

In this paper, a cascade optical quantization scheme is proposed to realize all-optical analog-to-digital converter with efficiently enhanced quantization resolution and achievable high analog bandwidth of larger than 20 GHz. Employing the cascade structure of an unbalanced Mach-zehnder modulator and a specially designed optical directional coupler, we predict the enhancement of number-of-bits can be up to 1.59-bit. Simulation results show that a 25 GHz RF signal is efficiently digitalized with the signal-to-noise ratio of 33.58 dB and effective-number-of-bits of 5.28-bit.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrowband wavelength filter using LiNbO3 unbalanced Mach-Zehnder Interferometers,” J. Lightwave Technol. 14(11), 2530–2536 (1996).
    [Crossref]
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    [Crossref]
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  27. H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
    [Crossref]
  28. H. Nasu, T. Takagi, T. Shinagawa, M. Oike, T. Nomura, and A. Kasukawa, “A highly stable and reliable wavelength monitor integrated laser module design,” J. Lightwave Technol. 22(5), 1344–1351 (2004).
    [Crossref]

2014 (1)

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

2013 (5)

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Z. Kang, J. H. Yuan, Q. Wu, T. Wang, S. Li, X. Z. Sang, C. X. Yu, and G. Farrell, “Lumped time-delay compensation scheme for coding synchronization in the nonlinear spectral quantization-based all-optical analog-to-digital conversion,” IEEE Photon. J. 5(6), 7201109 (2013).
[Crossref]

Z. Kang, J. H. Yuan, S. Li, S. L. Xie, B. B. Yan, X. Z. Sang, and C. X. Yu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phys. B 22(11), 114211 (2013).
[Crossref]

Y. Wang, H. M. Zhang, Y. J. Dou, and M. Y. Yao, “Experimental evaluation of resolution enhancement of a phase-shifted all optical analog-to-digital converter using an electrical analog-to-digital converter array,” Chin. Opt. Lett. 11(8), 082301 (2013).
[Crossref]

K. Takahashi, H. Matsui, T. Nagashima, and T. Konishi, “Resolution upgrade toward 6-bit optical quantization using power-to-wavelength conversion for photonic analog-to-digital conversion,” Opt. Lett. 38(22), 4864–4867 (2013).
[Crossref] [PubMed]

2012 (5)

A. Khilo, S. J. Spector, M. E. Grein, A. H. Nejadmalayeri, C. W. Holzwarth, M. Y. Sander, M. S. Dahlem, M. Y. Peng, M. W. Geis, N. A. DiLello, J. U. Yoon, A. Motamedi, J. S. Orcutt, J. P. Wang, C. M. Sorace-Agaskar, M. A. Popović, J. Sun, G. R. Zhou, H. Byun, J. Chen, J. L. Hoyt, H. I. Smith, R. J. Ram, M. Perrott, T. M. Lyszczarz, E. P. Ippen, and F. X. Kärtner, “Photonic ADC: overcoming the bottleneck of electronic jitter,” Opt. Express 20(4), 4454–4469 (2012).
[Crossref] [PubMed]

W. Shile, W. Jian, Z. Lingjuan, Y. Chen, J. Chen, L. Dan, Z. Xilin, and Y. Zuoshan, “Multimode interference coupler based photonic analog-to-digital conversion scheme,” Opt. Lett. 37(17), 3699–3701 (2012).
[Crossref] [PubMed]

Y. Wang, H. M. Zhang, Q. W. Wu, and M. Y. Yao, “Improvement of photonic ADC based on phase-shifted optical quantization by using additional modulators,” IEEE Photon. Technol. Lett. 24(7), 566–568 (2012).
[Crossref]

Y. Miyoshi, S. Namiki, and K. I. Kitayama, “Performance evaluation of resolution-enhanced ADC using optical multiperiod transfer functions of NOLMs,” IEEE J. Sel. Top. Quantum Electron. 18(2), 779–784 (2012).
[Crossref]

T. Satoh, K. Takahashi, H. Matsui, K. Itoh, and T. Konishi, “10-GS/s 5-bit real-time optical quantization for photonic analog-to-digital conversion,” IEEE Photon. Technol. Lett. 24(10), 830–832 (2012).

2011 (3)

2010 (2)

2009 (2)

2008 (1)

T. Nishitani, T. Konishi, and K. Itoh, “Resolution improvement of all-optical analog-to-digital conversion employing self-frequency shift and self-phase-modulation induced spectral compression,” IEEE J. Sel. Top. Quantum Electron. 14(3), 724–732 (2008).
[Crossref]

2007 (1)

2004 (1)

2003 (2)

Y. Han and B. Jalali, “Photonic Time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightwave Technol. 21(12), 3085–3103 (2003).
[Crossref]

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

1999 (1)

R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. 17(4), 539–550 (1999).
[Crossref]

1996 (1)

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrowband wavelength filter using LiNbO3 unbalanced Mach-Zehnder Interferometers,” J. Lightwave Technol. 14(11), 2530–2536 (1996).
[Crossref]

1976 (1)

D. S. Smith, H. D. Riccius, and R. P. Edwin, “Refractive indices of lithium niobate,” Opt. Commun. 17(3), 332–335 (1976).
[Crossref]

Avrutin, E. A.

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Bradley, E. M.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrowband wavelength filter using LiNbO3 unbalanced Mach-Zehnder Interferometers,” J. Lightwave Technol. 14(11), 2530–2536 (1996).
[Crossref]

Bryce, A. C.

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Byun, H.

Chen, J.

Chen, Y.

Chi, H.

Dagli, N.

Dahlem, M. S.

Dai, J.

Dai, Y. T.

Dan, L.

DiLello, N. A.

Dou, Y. J.

Dylewicz, M. H. R.

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Edwin, R. P.

D. S. Smith, H. D. Riccius, and R. P. Edwin, “Refractive indices of lithium niobate,” Opt. Commun. 17(3), 332–335 (1976).
[Crossref]

Eggleton, B. J.

Farrell, G.

Z. Kang, J. H. Yuan, Q. Wu, T. Wang, S. Li, X. Z. Sang, C. X. Yu, and G. Farrell, “Lumped time-delay compensation scheme for coding synchronization in the nonlinear spectral quantization-based all-optical analog-to-digital conversion,” IEEE Photon. J. 5(6), 7201109 (2013).
[Crossref]

Geis, M. W.

Golling, M.

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

Grein, M. E.

Han, Y.

Holzwarth, C. W.

Hou, L. P.

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Hoyt, J. L.

Ippen, E. P.

Itoh, K.

T. Satoh, K. Takahashi, H. Matsui, K. Itoh, and T. Konishi, “10-GS/s 5-bit real-time optical quantization for photonic analog-to-digital conversion,” IEEE Photon. Technol. Lett. 24(10), 830–832 (2012).

T. Konishi, K. Takahashi, H. Matsui, T. Satoh, and K. Itoh, “Five-bit parallel operation of optical quantization and coding for photonic analog-to-digital conversion,” Opt. Express 19(17), 16106–16114 (2011).
[Crossref] [PubMed]

T. Nishitani, T. Konishi, and K. Itoh, “Resolution improvement of all-optical analog-to-digital conversion employing self-frequency shift and self-phase-modulation induced spectral compression,” IEEE J. Sel. Top. Quantum Electron. 14(3), 724–732 (2008).
[Crossref]

Jalali, B.

Jian, W.

Jin, X.

Jin, X. F.

Kang, Z.

Z. Kang, J. H. Yuan, Q. Wu, T. Wang, S. Li, X. Z. Sang, C. X. Yu, and G. Farrell, “Lumped time-delay compensation scheme for coding synchronization in the nonlinear spectral quantization-based all-optical analog-to-digital conversion,” IEEE Photon. J. 5(6), 7201109 (2013).
[Crossref]

Z. Kang, J. H. Yuan, S. Li, S. L. Xie, B. B. Yan, X. Z. Sang, and C. X. Yu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phys. B 22(11), 114211 (2013).
[Crossref]

Kärtner, F. X.

Kasukawa, A.

H. Nasu, T. Takagi, T. Shinagawa, M. Oike, T. Nomura, and A. Kasukawa, “A highly stable and reliable wavelength monitor integrated laser module design,” J. Lightwave Technol. 22(5), 1344–1351 (2004).
[Crossref]

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

Keller, U.

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

Khilo, A.

Kitayama, K. I.

Y. Miyoshi, S. Namiki, and K. I. Kitayama, “Performance evaluation of resolution-enhanced ADC using optical multiperiod transfer functions of NOLMs,” IEEE J. Sel. Top. Quantum Electron. 18(2), 779–784 (2012).
[Crossref]

Y. Miyoshi, S. Takagi, S. Namiki, and K. I. Kitayama, “Multiperiod PM-NOLM with dynamic counter-propagating effects compensation for 5-bit all-optical analog-to-digital conversion and its performance evaluations,” J. Lightwave Technol. 28(4), 415–422 (2010).
[Crossref]

Klenner, A.

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

Konishi, T.

K. Takahashi, H. Matsui, T. Nagashima, and T. Konishi, “Resolution upgrade toward 6-bit optical quantization using power-to-wavelength conversion for photonic analog-to-digital conversion,” Opt. Lett. 38(22), 4864–4867 (2013).
[Crossref] [PubMed]

T. Satoh, K. Takahashi, H. Matsui, K. Itoh, and T. Konishi, “10-GS/s 5-bit real-time optical quantization for photonic analog-to-digital conversion,” IEEE Photon. Technol. Lett. 24(10), 830–832 (2012).

T. Konishi, K. Takahashi, H. Matsui, T. Satoh, and K. Itoh, “Five-bit parallel operation of optical quantization and coding for photonic analog-to-digital conversion,” Opt. Express 19(17), 16106–16114 (2011).
[Crossref] [PubMed]

T. Nishitani, T. Konishi, and K. Itoh, “Resolution improvement of all-optical analog-to-digital conversion employing self-frequency shift and self-phase-modulation induced spectral compression,” IEEE J. Sel. Top. Quantum Electron. 14(3), 724–732 (2008).
[Crossref]

Li, S.

Z. Kang, J. H. Yuan, S. Li, S. L. Xie, B. B. Yan, X. Z. Sang, and C. X. Yu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phys. B 22(11), 114211 (2013).
[Crossref]

Z. Kang, J. H. Yuan, Q. Wu, T. Wang, S. Li, X. Z. Sang, C. X. Yu, and G. Farrell, “Lumped time-delay compensation scheme for coding synchronization in the nonlinear spectral quantization-based all-optical analog-to-digital conversion,” IEEE Photon. J. 5(6), 7201109 (2013).
[Crossref]

Li, Z.

Lin, J. T.

Lingjuan, Z.

Link, S. M.

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

Lyszczarz, T. M.

Mangold, M.

M. Mangold, S. M. Link, A. Klenner, C. A. Zaugg, M. Golling, B. W. Tilma, and U. Keller, “Amplitude noise and timing jitter characterization of a high-power mode-locked integrated external-cavity surface emitting laser,” IEEE Photon. J. 6(1), 1500309 (2014).

Marsh, J. H.

L. P. Hou, E. A. Avrutin, M. H. R. Dylewicz, A. C. Bryce, and J. H. Marsh, “160 GHz passively mode-locked AlGaInAs 1.55 μm strained quantum-well lasers with deeply etched intracavity mirrors,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1100409 (2013).
[Crossref]

Matsui, H.

Miles, E. W.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrowband wavelength filter using LiNbO3 unbalanced Mach-Zehnder Interferometers,” J. Lightwave Technol. 14(11), 2530–2536 (1996).
[Crossref]

Miyoshi, Y.

Y. Miyoshi, S. Namiki, and K. I. Kitayama, “Performance evaluation of resolution-enhanced ADC using optical multiperiod transfer functions of NOLMs,” IEEE J. Sel. Top. Quantum Electron. 18(2), 779–784 (2012).
[Crossref]

Y. Miyoshi, S. Takagi, S. Namiki, and K. I. Kitayama, “Multiperiod PM-NOLM with dynamic counter-propagating effects compensation for 5-bit all-optical analog-to-digital conversion and its performance evaluations,” J. Lightwave Technol. 28(4), 415–422 (2010).
[Crossref]

Motamedi, A.

Nagashima, T.

Namiki, S.

Y. Miyoshi, S. Namiki, and K. I. Kitayama, “Performance evaluation of resolution-enhanced ADC using optical multiperiod transfer functions of NOLMs,” IEEE J. Sel. Top. Quantum Electron. 18(2), 779–784 (2012).
[Crossref]

Y. Miyoshi, S. Takagi, S. Namiki, and K. I. Kitayama, “Multiperiod PM-NOLM with dynamic counter-propagating effects compensation for 5-bit all-optical analog-to-digital conversion and its performance evaluations,” J. Lightwave Technol. 28(4), 415–422 (2010).
[Crossref]

Nasu, H.

H. Nasu, T. Takagi, T. Shinagawa, M. Oike, T. Nomura, and A. Kasukawa, “A highly stable and reliable wavelength monitor integrated laser module design,” J. Lightwave Technol. 22(5), 1344–1351 (2004).
[Crossref]

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

Nejadmalayeri, A. H.

Nishitani, T.

T. Nishitani, T. Konishi, and K. Itoh, “Resolution improvement of all-optical analog-to-digital conversion employing self-frequency shift and self-phase-modulation induced spectral compression,” IEEE J. Sel. Top. Quantum Electron. 14(3), 724–732 (2008).
[Crossref]

Niu, J.

Nomura, T.

H. Nasu, T. Takagi, T. Shinagawa, M. Oike, T. Nomura, and A. Kasukawa, “A highly stable and reliable wavelength monitor integrated laser module design,” J. Lightwave Technol. 22(5), 1344–1351 (2004).
[Crossref]

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

Oike, M.

H. Nasu, T. Takagi, T. Shinagawa, M. Oike, T. Nomura, and A. Kasukawa, “A highly stable and reliable wavelength monitor integrated laser module design,” J. Lightwave Technol. 22(5), 1344–1351 (2004).
[Crossref]

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

Orcutt, J. S.

Peng, M. Y.

Perrott, M.

Popovic, M. A.

Ram, R. J.

Riccius, H. D.

D. S. Smith, H. D. Riccius, and R. P. Edwin, “Refractive indices of lithium niobate,” Opt. Commun. 17(3), 332–335 (1976).
[Crossref]

Sander, M. Y.

Sang, X. Z.

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Chin. Phys. B (1)

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IEEE Photon. J. (2)

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[Crossref]

IEEE Photon. Technol. Lett. (3)

H. Nasu, T. Takagi, M. Oike, T. Nomura, and A. Kasukawa, “Ultrahigh wavelength stability through thermal compensation in wavelength-monitor integrated laser modules,” IEEE Photon. Technol. Lett. 15(3), 380–382 (2003).
[Crossref]

Y. Wang, H. M. Zhang, Q. W. Wu, and M. Y. Yao, “Improvement of photonic ADC based on phase-shifted optical quantization by using additional modulators,” IEEE Photon. Technol. Lett. 24(7), 566–568 (2012).
[Crossref]

T. Satoh, K. Takahashi, H. Matsui, K. Itoh, and T. Konishi, “10-GS/s 5-bit real-time optical quantization for photonic analog-to-digital conversion,” IEEE Photon. Technol. Lett. 24(10), 830–832 (2012).

J. Lightwave Technol. (4)

Opt. Commun. (1)

D. S. Smith, H. D. Riccius, and R. P. Edwin, “Refractive indices of lithium niobate,” Opt. Commun. 17(3), 332–335 (1976).
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Opt. Express (4)

Opt. Lett. (4)

Other (2)

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IEEE Instrumentation and Measurement Society, “IEEE standard for Digitizing Waveform Recorders,” IEEE Std 1057–1994(R2001) (2001).

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of the proposed COQ-ADC. Inset (a) illustrates the power transfer function and coding results of the first-stage quantization in the case of two channels (N = 2), and inset (b) illustrates the combined power transfer function and coding results of the two quantization stages with N = 2 and M = 2. PSOQ: phase-shifted optical quantization; Mux: multiplexer; Demux: demultiplexer; PD: Photodiodes; COMP: comparator.
Fig. 2
Fig. 2 Performance of resolution enhancement with different M.
Fig. 3
Fig. 3 (a) The structure of UMZM, and (b) extra phase shift as a function of input wavelength (ΔL = 40 μm, wavelength range: 1.54~1.56 μm).
Fig. 4
Fig. 4 (a) The structure of the designed directional coupler, and (b) the power transfer functions and coding results of the second-stage quantization.
Fig. 5
Fig. 5 (a) The dynamic of the optical field along the propagation direction (λ = 1.55 μm), and (b) the power fraction of three output ports and the ground port as a function of the propagation length.
Fig. 6
Fig. 6 The calculated power fraction of the symmetric and asymmetric structures as a function of input wavelength at (a) port1, (b) port2, and (c) port3, respectively. (d) the power fluctuations of the symmetric and asymmetric structures as a function of input wavelength at different output ports.
Fig. 7
Fig. 7 (a) Input sinusoidal wave signal, and (b) the output temporal profiles of the UMZM.
Fig. 8
Fig. 8 The detected waveforms at (a) port1, (b) port2, and (c) port3 of the directional coupler.
Fig. 9
Fig. 9 (a) The digitized result and sinusoidal fitted curve, and (b) the errors between the digitized values and the fitted curve.
Fig. 10
Fig. 10 (a) τmax as a function of the input frequency with different NOB, and (b) αmax as a function of the NOB.
Fig. 11
Fig. 11 The ENOB of the proposed COQ-ADC as a function of the phase jitter with different number of channels.
Fig. 12
Fig. 12 (a) The amounts of walk-off and β1(λ) as a function of wavelength, and (b) the maximum amount of the inter-channel walk-off and supportable maximum sampling rate as a function of the length of UMZM.

Tables (1)

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Table 1 Selected Wavelengths and Corresponding Phase-shifts

Equations (12)

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H( t )= [ 1+cos( πV( t ) V π + 2π n e ΔL λ ) ] 2
SNR RMS_signal RMS_errors = A 2 1 M n=1 M ( y n y n ' ) 2
ENOB= SNR1.76 6.02
δA=Asin[ 2πf( t+τ ) ]Asin( 2πft )
δA=2πfAτcos( 2πft )
τ max = ( 2 NOB+1 πf ) 1
Δ 1 =arcsin( α 1+α ) P FS 2π
α< sin( π/ 2 NOB ) 1sin( π/ 2 NOB )
ENOB= SNR1.76 6.02 = 10 log 10 { ( P FS 2 2 ) 2 1 P FS [ i=1 N 2×3×( 0 Δ/2+ Δ 2_i p 2 dp+ Δ/2+ Δ 2_i Δ ( pΔ ) 2 dp ) ] }1.76 6.02
23δθ P FS 2π < Δ 2
δT 1 2 R s = T 2
δT=[ β 1 ( λ i ) β 1 ( λ j ) ]( L 1 + L 2 ) i,j1,2...N;ij

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