Abstract

We present an analytical treatment for the relatively new spectral disperser termed virtually imaged phased array (VIPA). Angular spectrum representation of the input Gaussian beam helps us obtain an exact analytic dispersion model and a dispersion law for a general VIPA by using the principle of multiple-beam interference. The consideration of the optical aberrations caused by refractions makes our model more accurate and practical than previous models. The validity of the proposed dispersion law has been validated theoretically by comparing with previous results. Some considerations of using a VIPA are also provided.

© 2015 Optical Society of America

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  1. M. Shirasaki, “Large angular dispersion by a virtually imaged phased array and its application to a wavelength demultiplexer,” Opt. Lett. 21(5), 366–368 (1996).
    [Crossref] [PubMed]
  2. M. Shirasaki, “Chromatic-dispersion compensator using virtually imaged phased array,” IEEE Photon. Technol. Lett. 9(12), 1598–1660 (1997).
    [Crossref]
  3. M. Shirasaki, “Virtually imaged phased array,” Fujitsu Sci. Tech. J. 35(1), 113–125 (1999).
  4. L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).
  5. S. Xiao and A. M. Weiner, “2-D wavelength demultiplexer with potential for ≥ 1000 channels in the C-band,” Opt. Express 12(13), 2895–2902 (2004).
    [Crossref] [PubMed]
  6. S. X. Wang, S. Xiao, and A. M. Weiner, “Broadband, high spectral resolution 2-D wavelength-parallel polarimeter for Dense WDM systems,” Opt. Express 13(23), 9374–9380 (2005).
    [Crossref] [PubMed]
  7. G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
    [Crossref]
  8. S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  12. K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
    [Crossref] [PubMed]
  13. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  17. P. Metz, J. Adam, M. Gerken, and B. Jalali, “Compact, transmissive two-dimensional spatial disperser design with application in simultaneous endoscopic imaging and laser microsurgery,” Appl. Opt. 53(3), 376–382 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  20. S. Xiao, A. M. Weiner, and C. Lin, “A dispersion law for virtually imaged phased-array spectral dispersers based on paraxial wave theory,” IEEE J. Quantum Electron. 40(4), 420–426 (2004).
    [Crossref]
  21. S. Xiao, “The spatial chirp effect and dispersion law of virtually imaged phased array (VIPA) wavelength demultiplexer,” M. S. Thesis, Purdue University, August, 2003.
  22. A. Mokhtari and A. A. Shishegar, “Rigorous vectorial Gaussian beam modeling of spectral dispersing performance of virtually imaged phased arrays,” J. Opt. Soc. Am. B 26(2), 272–278 (2009).
    [Crossref]
  23. A. Mokhtari and A. A. Shishegar, “Rigorous 3D vectorial Gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays,” Progress In Electromagnetics Research M 13, 1–16 (2010).
    [Crossref]
  24. D. J. Gauthier, “Comment on “Generalized grating equation for virtually imaged phased-array spectral dispersers”,” Appl. Opt. 51(34), 8184–8186 (2012).
    [Crossref] [PubMed]
  25. A. M. Weiner, “Reply to Comment on “Generalized grating equation for virtually imaged phased-array spectral dispersers”,” Appl. Opt. 51(34), 8187–8189 (2012).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  28. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7nd ed. (Cambridge University, 1999). pp. 313–386.
  29. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 54–72.
  30. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006). pp. 38–56.
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    [Crossref] [PubMed]
  33. S. Xiao, A. M. Weiner, and C. Lin, “Experimental and theoretical study of hyperfine WDM demultiplexer performance using the virtually imaged phased-array (VIPA),” J. Lightwave Technol. 23(3), 1456–1467 (2005).
    [Crossref]

2014 (2)

2013 (1)

2012 (3)

2011 (1)

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

2010 (3)

A. Mokhtari and A. A. Shishegar, “Rigorous 3D vectorial Gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays,” Progress In Electromagnetics Research M 13, 1–16 (2010).
[Crossref]

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010).
[Crossref]

V. R. Supradeepa, E. Hamidi, D. E. Leaird, and A. M. Weiner, “New aspects of temporal dispersion in high-resolution Fourier pulse shaping: a quantitative description with virtually imaged phased array pulse shapers,” J. Opt. Soc. Am. B 27(9), 1833–1844 (2010).
[Crossref]

2009 (3)

2007 (1)

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[Crossref] [PubMed]

2006 (1)

G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
[Crossref]

2005 (2)

2004 (3)

S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
[Crossref]

S. Xiao, A. M. Weiner, and C. Lin, “A dispersion law for virtually imaged phased-array spectral dispersers based on paraxial wave theory,” IEEE J. Quantum Electron. 40(4), 420–426 (2004).
[Crossref]

S. Xiao and A. M. Weiner, “2-D wavelength demultiplexer with potential for ≥ 1000 channels in the C-band,” Opt. Express 12(13), 2895–2902 (2004).
[Crossref] [PubMed]

2003 (1)

1999 (1)

M. Shirasaki, “Virtually imaged phased array,” Fujitsu Sci. Tech. J. 35(1), 113–125 (1999).

1997 (1)

M. Shirasaki, “Chromatic-dispersion compensator using virtually imaged phased array,” IEEE Photon. Technol. Lett. 9(12), 1598–1660 (1997).
[Crossref]

1996 (1)

1991 (1)

1989 (1)

1962 (1)

Adam, J.

Bao, W.

Behnke, C.

Block, H.

Cao, S.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Capewell, D.

Chen, Z.

Cundiff, S. T.

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010).
[Crossref]

Diddams, S. A.

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[Crossref] [PubMed]

Ding, Z.

Eiselt, M.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Garrett, L.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Gauthier, D. J.

Gerken, M.

Gnauck, A.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Goda, K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[Crossref] [PubMed]

K. K. Tsia, K. Goda, D. Capewell, and B. Jalali, “Simultaneous mechanical-scan-free confocal microscopy and laser microsurgery,” Opt. Lett. 34(14), 2099–2101 (2009).
[Crossref] [PubMed]

Goosman, D. R.

Hamidi, E.

Hollberg, L.

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[Crossref] [PubMed]

Hong, W.

Jalali, B.

Keller, J. B.

Krantz, M.

Leaird, D. E.

Lee, G.-H.

G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
[Crossref]

Li, P.

Lin, C.

Mao, C.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Mbele, V.

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[Crossref] [PubMed]

McKinney, J. D.

S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
[Crossref]

McMillan, C. F.

Mei, S.

Metz, P.

Mokhtari, A.

A. Mokhtari and A. A. Shishegar, “Rigorous 3D vectorial Gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays,” Progress In Electromagnetics Research M 13, 1–16 (2010).
[Crossref]

A. Mokhtari and A. A. Shishegar, “Rigorous vectorial Gaussian beam modeling of spectral dispersing performance of virtually imaged phased arrays,” J. Opt. Soc. Am. B 26(2), 272–278 (2009).
[Crossref]

Parker, N. L.

Shen, W.

Shen, Y.

Shirasaki, M.

M. Shirasaki, “Virtually imaged phased array,” Fujitsu Sci. Tech. J. 35(1), 113–125 (1999).

M. Shirasaki, “Chromatic-dispersion compensator using virtually imaged phased array,” IEEE Photon. Technol. Lett. 9(12), 1598–1660 (1997).
[Crossref]

M. Shirasaki, “Large angular dispersion by a virtually imaged phased array and its application to a wavelength demultiplexer,” Opt. Lett. 21(5), 366–368 (1996).
[Crossref] [PubMed]

Shishegar, A. A.

A. Mokhtari and A. A. Shishegar, “Rigorous 3D vectorial Gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays,” Progress In Electromagnetics Research M 13, 1–16 (2010).
[Crossref]

A. Mokhtari and A. A. Shishegar, “Rigorous vectorial Gaussian beam modeling of spectral dispersing performance of virtually imaged phased arrays,” J. Opt. Soc. Am. B 26(2), 272–278 (2009).
[Crossref]

Supradeepa, V. R.

Tkach, R.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Tsia, K. K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[Crossref] [PubMed]

K. K. Tsia, K. Goda, D. Capewell, and B. Jalali, “Simultaneous mechanical-scan-free confocal microscopy and laser microsurgery,” Opt. Lett. 34(14), 2099–2101 (2009).
[Crossref] [PubMed]

Vega, A.

Wang, C.

Wang, S. X.

Weiner, A. M.

A. M. Weiner, “Reply to Comment on “Generalized grating equation for virtually imaged phased-array spectral dispersers”,” Appl. Opt. 51(34), 8187–8189 (2012).
[Crossref]

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010).
[Crossref]

V. R. Supradeepa, E. Hamidi, D. E. Leaird, and A. M. Weiner, “New aspects of temporal dispersion in high-resolution Fourier pulse shaping: a quantitative description with virtually imaged phased array pulse shapers,” J. Opt. Soc. Am. B 27(9), 1833–1844 (2010).
[Crossref]

G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
[Crossref]

S. Xiao, A. M. Weiner, and C. Lin, “Experimental and theoretical study of hyperfine WDM demultiplexer performance using the virtually imaged phased-array (VIPA),” J. Lightwave Technol. 23(3), 1456–1467 (2005).
[Crossref]

S. X. Wang, S. Xiao, and A. M. Weiner, “Broadband, high spectral resolution 2-D wavelength-parallel polarimeter for Dense WDM systems,” Opt. Express 13(23), 9374–9380 (2005).
[Crossref] [PubMed]

S. Xiao and A. M. Weiner, “2-D wavelength demultiplexer with potential for ≥ 1000 channels in the C-band,” Opt. Express 12(13), 2895–2902 (2004).
[Crossref] [PubMed]

S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
[Crossref]

S. Xiao, A. M. Weiner, and C. Lin, “A dispersion law for virtually imaged phased-array spectral dispersers based on paraxial wave theory,” IEEE J. Quantum Electron. 40(4), 420–426 (2004).
[Crossref]

A. Vega, A. M. Weiner, and C. Lin, “Generalized grating equation for virtually-imaged phased-array spectral dispersers,” Appl. Opt. 42(20), 4152–4155 (2003).
[Crossref] [PubMed]

Xiao, S.

G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
[Crossref]

S. Xiao, A. M. Weiner, and C. Lin, “Experimental and theoretical study of hyperfine WDM demultiplexer performance using the virtually imaged phased-array (VIPA),” J. Lightwave Technol. 23(3), 1456–1467 (2005).
[Crossref]

S. X. Wang, S. Xiao, and A. M. Weiner, “Broadband, high spectral resolution 2-D wavelength-parallel polarimeter for Dense WDM systems,” Opt. Express 13(23), 9374–9380 (2005).
[Crossref] [PubMed]

S. Xiao and A. M. Weiner, “2-D wavelength demultiplexer with potential for ≥ 1000 channels in the C-band,” Opt. Express 12(13), 2895–2902 (2004).
[Crossref] [PubMed]

S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
[Crossref]

S. Xiao, A. M. Weiner, and C. Lin, “A dispersion law for virtually imaged phased-array spectral dispersers based on paraxial wave theory,” IEEE J. Quantum Electron. 40(4), 420–426 (2004).
[Crossref]

Yan, Y.

Yang, C.

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

Yang, L.

L. Yang, “Analytical treatment of virtual image phase array,” in Proceedings of Optical Fiber Communication Conference and Exhibit (OFC2002), pp. 321–322.
[Crossref]

Yu, H.

Appl. Opt. (6)

Fujitsu Sci. Tech. J. (1)

M. Shirasaki, “Virtually imaged phased array,” Fujitsu Sci. Tech. J. 35(1), 113–125 (1999).

IEEE J. Quantum Electron. (1)

S. Xiao, A. M. Weiner, and C. Lin, “A dispersion law for virtually imaged phased-array spectral dispersers based on paraxial wave theory,” IEEE J. Quantum Electron. 40(4), 420–426 (2004).
[Crossref]

IEEE Photon. Technol. Lett. (3)

G.-H. Lee, S. Xiao, and A. M. Weiner, “Optical dispersion compensator with >> 4000-ps/nm tuning range using a virtually imaged phased array (VIPA) and spatial light modulator (SLM),” IEEE Photon. Technol. Lett. 18(17), 1819–1821 (2006).
[Crossref]

S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a VIPA direct space-to-time pulse shaper,” IEEE Photon. Technol. Lett. 16(8), 1936–1938 (2004).
[Crossref]

M. Shirasaki, “Chromatic-dispersion compensator using virtually imaged phased array,” IEEE Photon. Technol. Lett. 9(12), 1598–1660 (1997).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010).
[Crossref]

Nature (2)

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[Crossref] [PubMed]

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Progress In Electromagnetics Research M (1)

A. Mokhtari and A. A. Shishegar, “Rigorous 3D vectorial Gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays,” Progress In Electromagnetics Research M 13, 1–16 (2010).
[Crossref]

Other (6)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7nd ed. (Cambridge University, 1999). pp. 313–386.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 54–72.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006). pp. 38–56.

S. Xiao, “The spatial chirp effect and dispersion law of virtually imaged phased array (VIPA) wavelength demultiplexer,” M. S. Thesis, Purdue University, August, 2003.

L. Yang, “Analytical treatment of virtual image phase array,” in Proceedings of Optical Fiber Communication Conference and Exhibit (OFC2002), pp. 321–322.
[Crossref]

L. Garrett, A. Gnauck, M. Eiselt, R. Tkach, C. Yang, C. Mao, and S. Cao, “Demonstration of virtually-imaged phased-array device for tunable dispersion compensation in 16 times;10 Gb/s WDM transmission over 480 km standard fiber,” in Optical Fiber Communication Conference, 20004, 187–189 (2000).

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Figures (5)

Fig. 1
Fig. 1 Schematic geometry of the VIPA spectral disperser.
Fig. 2
Fig. 2 (a) Sketch map of optical paths in the incident plane used to calculate the optical path differences (OPD) for the neighboring transmitted rays. The OPD is ∆ = n′(OA + AB) − nOC = 2nhcosΘ ′. (b) In the auxiliary plane, the optical path from the output plane to the back focal plane can be represented by the chief ray (red line) passing through the center of the lens. The incident plane and the auxiliary plane will be in the same cross-section if and only if θy = 0.
Fig. 3
Fig. 3 Sectional view of a VIPA irradiated by the line-focused Gaussian beam for previous virtual sources model. (a) Considering the ideal reflection imaging of the beam waist, the geometrical spacing in the surrounding medium are ∆x0 = BC = 2htanφ′cosφ and ∆z0 = (S0A + AB)n′/nS0C = 2hnr cosφ′ . (b) For the diffracted rays with output angle of θ emitted from the adjacent sources, the OPD is describe as ∆′ = nS1C = n(S1DEF) = n(∆z0cosθ − ∆x0sinθ).
Fig. 4
Fig. 4 Comparison of diffracted rays in a solid VIPA for the proposed model and the previous model. (a) In previous model, the virtual source array (red points) is depicted by using the refraction and reflection of central ray (green line), and other diffracted rays (red dashed line) are assumed to be emitted from these virtual sources. In proposed model, all rays are depicted by their real refraction and reflection (blue line). (b) Referring to Fig. 3(b), the OPD of the previous model can be describe as ∆′ = nPH = n(∆z0cosθ −∆x0sinθ), and for the proposed model that is ∆ = nQK = n(∆zθ cosθ − ∆xθ sinθ) = 2nkhcos(φ + θ)′. With the help of another relation of S0K = EF + S0D = BC, we have ∆zθ sinθ + ∆xθ cosθ = 2htan(φ + θ)′cos(φ + θ). Then the spatial period ∆zθ and ∆xθ can be solved easily.
Fig. 5
Fig. 5 Absolute deviations in phase difference for different dispersion laws, normalized by 2π, (a) as a function of output angle θ with tilt angle φ = 4°, and (b) as a function of the tilt angle φ with output angle θ = 3°, while n′ = 1.5, n = 1.0, h = 1.5mm and λ = 1.55μm. The parameter values referenced from [33].

Equations (25)

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E( x,y;0 )= E 0 exp( x 2 w 0x 2 y 2 w 0y 2 ),
E( x,y;z )= + + A( f x , f y )exp[ j2π( f x x+ f y y+ 1 ( λ f x ) 2 ( λ f y ) 2 z ) ]d f x d f y ,
A( f x , f y )= + + E( x,y;0 )exp[ j2π( f x x+ f y y ) ]dxdy .
A( f x , f y )= A 0 ( θ x , θ y )= E 0 exp[ k 2 4 ( w 0x 2 sin 2 θ x + w 0y 2 sin 2 θ y ) ],
P( θ x , θ y ;z )= A 0 ( θ x , θ y )exp[ jk( xsin θ x +ysin θ y +z 1 sin 2 θ x sin 2 θ y ) ],
cosΘ= e n e k =cosφ 1 sin 2 θ x sin 2 θ y sinφsin θ x .
δ=2kh { n 2 [ cosφ 1 sin 2 θ x sin 2 θ y sinφsin θ x ] 2 +( n 2 n 2 ) } 1 2 .
δ l =k( d z cosγ+ f cosγ d x cosη sinγ ),
I t = A 0 2 T 2 1+ ( R 1 R 2 ) M 2 ( r 1 r 2 ) M cos(Mδ) 1+ R 1 R 2 2 r 1 r 2 cosδ .
I t = E 0 2 (1 r 2 2 ) (1 r 1 M r 2 M ) 2 ( 1 r 1 r 2 ) 2 exp[ 2 f c 2 W sin 2 θ x 2 π 2 W 2 λ 2 sin 2 θ y ] 1+ F M sin 2 ( Mδ /2 ) 1+ F 1 sin 2 ( δ/2 ) ,
2kh { n 2 [ cosφ 1 sin 2 θ x sin 2 θ y sinφsin θ x ] 2 +( n 2 n 2 ) } 1 2 =2mπ.
2 n khcos(φ+θ ) =2kh n 2 n 2 sin 2 ( φ+θ ) =2mπ,
( Δλ ) θ =λ λ 0 = 2h m [ n 2 n 2 sin 2 ( φ+θ ) n 2 n 2 sin 2 φ ],
dθ dλ = 2[ n 2 n 2 sin 2 ( φ+θ ) ] n 2 λsin2( φ+θ ) ,
(Δλ) FSR = λ m = λ 2 2h n 2 n 2 sin 2 ( φ+θ ) .
R= λ Δλ = 2mπ ε M = 2kh n 2 n 2 sin 2 ( φ+θ ) ε M .
2kh[ n cos θ / cos φ ntan φ sin( φ+θ ) ]=2mπ.
kh[ 2 n cos φ 2ntan φ cosφθ n 2 cos 2 φ n cos φ θ 2 ]=2mπ.
2kh( n cos φ cosθntan φ cosφsinθ )=2mπ.
kh[ 2 n cos φ 2ntan φ cosφθ n cos φ θ 2 ]=2mπ,
2kh[ n / cos φ ntan φ sin( φ+θ ) ]=2mπ,
kh[ 2 n cos φ 2ntan φ cosφθ cos φ θ 2 / n ]=2mπ.
Δξ= 2h n r 2 sin 2 ( φ+θ ) [ sinφcos( φ+θ )( n r 2 1 )sinθ ]2htan φ cosφ,
Δη= 2h n r 2 sin 2 ( φ+θ ) [ cosφcos( φ+θ )+( n r 2 1 )cosθ ]2 n r hcos φ .
Δδ=kh[ cos φ n n 2 cos 2 φ n cos φ ] θ 2 +o( θ 2 ).

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