Abstract

I correct an error made by Vega et al. [Appl. Opt. 42, 4152 (2003)], who derived the spectral dispersion properties of a virtually imaged phased-array etalon using a ray-based, multibounce interference analysis. I demonstrate that the corrected dispersion law is in agreement with the results obtained by paraxial wave theory [Xiao et al., IEEE J. Quantum Electron. 40, 420 (2004)].

© 2012 Optical Society of America

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  1. C. Dufour, “Use of the Fabry–Perot interferometer for the study of feeble satellites,” Rev. Opt. 24, 11–18 (1945).
    [CrossRef]
  2. C. F. McMillan, N. L. Parker, and D. R. Goosman, “Efficiency enhancements for Fabry–Perots used in velocimetry,” Appl. Opt. 28, 826–827 (1989).
    [CrossRef]
  3. C. McMillan and L. Steinmetz, “Striped Fabry–Perots: improved efficiency for velocimetry,” Proc. SPIE 1346, 113–120 (1991).
    [CrossRef]
  4. D. R. Goosman, “Formulas for Fabry–Perot velocimeter performance using both stripe and multifrequency techniques,” Appl. Opt. 30, 3907–3923 (1991).
    [CrossRef]
  5. M. Shirasaki, “Large angular dispersion by a virtually imaged phased array and its application to a wavelength demultiplexer,” Opt. Lett. 21, 366–368 (1996).
    [CrossRef]
  6. M. Shirasaki, “Virtually imaged phased array (VIPA) having air between reflective surfaces,” U.S. patent 5,969,866(19October1999).
  7. A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
    [CrossRef]
  8. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445, 627–630 (2007).
    [CrossRef]
  9. P. Bowlan and R. Trebino, “Complete single-shot measurement of arbitrary nanosecond laser pulses in time,” Opt. Express 19, 1367–1377 (2011).
    [CrossRef]
  10. J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express 20, 3110–3117 (2012).
    [CrossRef]
  11. A. Vega, A. M. Weiner, and C. Lin, “Generalized grating equation for virtually-imaged phased-array spectral dispersers,” Appl. Opt. 42, 4152–4155 (2003).
    [CrossRef]
  12. 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, 420–426 (2004).
    [CrossRef]

2012 (1)

2011 (2)

2007 (1)

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

2004 (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, 420–426 (2004).
[CrossRef]

2003 (1)

1996 (1)

1991 (2)

C. McMillan and L. Steinmetz, “Striped Fabry–Perots: improved efficiency for velocimetry,” Proc. SPIE 1346, 113–120 (1991).
[CrossRef]

D. R. Goosman, “Formulas for Fabry–Perot velocimeter performance using both stripe and multifrequency techniques,” Appl. Opt. 30, 3907–3923 (1991).
[CrossRef]

1989 (1)

1945 (1)

C. Dufour, “Use of the Fabry–Perot interferometer for the study of feeble satellites,” Rev. Opt. 24, 11–18 (1945).
[CrossRef]

Bowlan, P.

Cundiff, S. T.

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, 627–630 (2007).
[CrossRef]

Dufour, C.

C. Dufour, “Use of the Fabry–Perot interferometer for the study of feeble satellites,” Rev. Opt. 24, 11–18 (1945).
[CrossRef]

Goosman, D. R.

Hollberg, L.

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

Lin, C.

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, 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, 4152–4155 (2003).
[CrossRef]

Mbele, V.

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

McMillan, C.

C. McMillan and L. Steinmetz, “Striped Fabry–Perots: improved efficiency for velocimetry,” Proc. SPIE 1346, 113–120 (1991).
[CrossRef]

McMillan, C. F.

Parker, N. L.

Shirasaki, M.

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

M. Shirasaki, “Virtually imaged phased array (VIPA) having air between reflective surfaces,” U.S. patent 5,969,866(19October1999).

Steinmetz, L.

C. McMillan and L. Steinmetz, “Striped Fabry–Perots: improved efficiency for velocimetry,” Proc. SPIE 1346, 113–120 (1991).
[CrossRef]

Trebino, R.

Vega, A.

Weiner, A. M.

J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express 20, 3110–3117 (2012).
[CrossRef]

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[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, 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, 4152–4155 (2003).
[CrossRef]

Willits, J. T.

Xiao, S.

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, 420–426 (2004).
[CrossRef]

Appl. Opt. (3)

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, 420–426 (2004).
[CrossRef]

Nature (1)

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

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (1)

C. McMillan and L. Steinmetz, “Striped Fabry–Perots: improved efficiency for velocimetry,” Proc. SPIE 1346, 113–120 (1991).
[CrossRef]

Rev. Opt. (1)

C. Dufour, “Use of the Fabry–Perot interferometer for the study of feeble satellites,” Rev. Opt. 24, 11–18 (1945).
[CrossRef]

Other (1)

M. Shirasaki, “Virtually imaged phased array (VIPA) having air between reflective surfaces,” U.S. patent 5,969,866(19October1999).

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Figures (2)

Fig. 1.
Fig. 1.

VIPA spectral disperser. A collimated multiwavelength input beam is focused through the transparent window of the etalon input face and onto the etalon output face with a cylindrical lens. Interference within the etalon causes each wavelength to propagate at different directions with high efficiency. The amplitude reflection and transmission coefficients for the input (output) faces are denoted by r1, t1 (r2, t2). Typically, r11. The interior of the etalon and the surrounding medium are assumed to be air for simplicity.

Fig. 2.
Fig. 2.

Ray paths used to determine the VIPA resonances. (a) Geometry used by Vega et al. in the ray-based analysis of the VIPA spectral dispersion. (b) Corrected geometry indicating the direction of the central ray of the focused beam (angle θi) and for the primary input/output ray in the beam (angle θi+θλ). The primary ray is not refracted as it is transmitted through the VIPA output face.

Equations (5)

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2kL[1cos(θi)tan(θi)sin(θλ)]=2mπ,
2kL[cos(θi)sin(θi)θλ+12tan(θi)sin(θi)θλ2]=2mπ.
2kL[cos(θi)sin(θi)θλ12cos(θi)θλ2]=2mπ.
2kL[cos(θi)]=2mπ.
2kL[cos(θi)sin(θi)θλ12cos(θi)θλ2]=2mπ,

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