Abstract

We present a simplified all-reflective Fourier transform spectrometer with a split-mirror configuration for use over a broad spectral range with spatially coherent sources. The device is particularly well suited for measurement of broadband laser-like light, with resolution limited by beam size and collimation. Spectra are taken in the near-UV and the mid-IR, a total span of 4.6 octaves, including an octave spanning spectrum. Potential sources of error are investigated both theoretically and experimentally.

© 2007 Optical Society of America

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  6. M. J. Padgett, A. R. Harvey, A. J. Duncan, and W. Sibbett, "Single-pulse, Fourier-transform Spectrometer having no moving parts," Appl. Opt. 33, 6035-6040 (1994).
    [CrossRef] [PubMed]
  7. M. J. Padgett and A. R. Harvey, "A static Fourier-Transform Spectrometer based on Wollaston Prisms," Rev. Sci. Instrum. 66, 2807-2811 (1995).
    [CrossRef]
  8. G. Zhan, K. Oka, T. Ishigaki, and N. Baba, "Birefringent imaging spectrometer," Appl. Opt. 41, 734-738 (2002).
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  9. Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
    [CrossRef]
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  11. A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  25. F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
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    [CrossRef]

2006 (2)

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

2005 (3)

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
[CrossRef]

T. Harimoto, Y. Takeuchi, and M. Fujita, "Curve-fitting method for pure spectrum measurement of femtosecond laser pulses by a pinhole pair," Opt. Express 13, 5689-5693 (2005).
[CrossRef] [PubMed]

2004 (1)

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004).
[CrossRef]

2002 (4)

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, "Birefringent imaging spectrometer," Appl. Opt. 41, 734-738 (2002).
[CrossRef] [PubMed]

T. Kobayashi and A. Baltuska, "Sub-5 fs pulse generation from a noncollinear optical parametric amplifier," Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

2000 (1)

1995 (1)

M. J. Padgett and A. R. Harvey, "A static Fourier-Transform Spectrometer based on Wollaston Prisms," Rev. Sci. Instrum. 66, 2807-2811 (1995).
[CrossRef]

1994 (1)

1992 (1)

1991 (1)

M. L. Junttila, J. Kauppinen, and E. Ikonen, "Performance limits of Stationary Fourier Spectrometers," J. Opt. Soc. Am. A. 8, 1457-1462 (1991).
[CrossRef]

1987 (1)

A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

1985 (1)

1984 (2)

1982 (1)

J. W. Brault, "Fourier-Transform Spectrometry in relation to other passive Spectrometers," Philos. Trans. R. Soc. London, Ser. A. 307, 503-511 (1982).
[CrossRef]

1966 (1)

1892 (1)

A. A. Michelson, "On the application of interference methods to Spectroscopic Measurements - II," Philos. Mag. 34, 280-299 (1892).

Agostini, P.

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004).
[CrossRef]

Attwood, D.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Attwood, D. T.

Baba, N.

Backus, S.

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Baltuska, A.

T. Kobayashi and A. Baltuska, "Sub-5 fs pulse generation from a noncollinear optical parametric amplifier," Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

Barnes, T. H.

Bartels, R. A.

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Benabid, F.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Berrill, M.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Birks, T. A.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Brault, J. W.

J. W. Brault, "Fourier-Transform Spectrometry in relation to other passive Spectrometers," Philos. Trans. R. Soc. London, Ser. A. 307, 503-511 (1982).
[CrossRef]

Christov, I. P.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Cohen, O.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Couny, F.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Cox, G.

A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

DiMauro, L. F.

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004).
[CrossRef]

Du, F.

Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
[CrossRef]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Duncan, A. J.

Fujita, M.

Gaudiosi, D. M.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Green, H.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Grisham, M.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Hamm, P.

Harimoto, T.

Harris, C. J.

A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

Harvey, A. R.

M. J. Padgett and A. R. Harvey, "A static Fourier-Transform Spectrometer based on Wollaston Prisms," Rev. Sci. Instrum. 66, 2807-2811 (1995).
[CrossRef]

M. J. Padgett, A. R. Harvey, A. J. Duncan, and W. Sibbett, "Single-pulse, Fourier-transform Spectrometer having no moving parts," Appl. Opt. 33, 6035-6040 (1994).
[CrossRef] [PubMed]

Ikonen, E.

M. L. Junttila, J. Kauppinen, and E. Ikonen, "Performance limits of Stationary Fourier Spectrometers," J. Opt. Soc. Am. A. 8, 1457-1462 (1991).
[CrossRef]

Ishigaki, T.

Jacobsen, C.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Junttila, M. L.

M. L. Junttila, "Stationary Fourier-Transform Spectrometer," Appl. Opt. 31, 4106-4112 (1992).
[CrossRef] [PubMed]

M. L. Junttila, J. Kauppinen, and E. Ikonen, "Performance limits of Stationary Fourier Spectrometers," J. Opt. Soc. Am. A. 8, 1457-1462 (1991).
[CrossRef]

Kaindl, R. A.

Kapteyn, H. C.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Kauppinen, J.

M. L. Junttila, J. Kauppinen, and E. Ikonen, "Performance limits of Stationary Fourier Spectrometers," J. Opt. Soc. Am. A. 8, 1457-1462 (1991).
[CrossRef]

Kawata, S.

Knight, J. C.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Kobayashi, T.

T. Kobayashi and A. Baltuska, "Sub-5 fs pulse generation from a noncollinear optical parametric amplifier," Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

Kogelnik, H.

Learner, R. C. M.

A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

Li, T.

Liu, Y.

Liu, Y.W.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Lu, Y. Q.

Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
[CrossRef]

Michelson, A. A.

A. A. Michelson, "On the application of interference methods to Spectroscopic Measurements - II," Philos. Mag. 34, 280-299 (1892).

Minami, S.

Murnane, M. M.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

Oka, K.

Okamoto, T.

Padgett, M. J.

M. J. Padgett and A. R. Harvey, "A static Fourier-Transform Spectrometer based on Wollaston Prisms," Rev. Sci. Instrum. 66, 2807-2811 (1995).
[CrossRef]

M. J. Padgett, A. R. Harvey, A. J. Duncan, and W. Sibbett, "Single-pulse, Fourier-transform Spectrometer having no moving parts," Appl. Opt. 33, 6035-6040 (1994).
[CrossRef] [PubMed]

Paul, A.

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood, "Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement," Opt. Lett. 27, 707-709 (2002).
[CrossRef]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y.W. Liu, D. Attwood, and C. Jacobsen, "Generation of spatially coherent light at extreme ultraviolet wavelengths," Science 297, 376-378 (2002).
[PubMed]

Popmintchev, T.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Reagan, B.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Reimann, K.

Rocca, J. J.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
[CrossRef] [PubMed]

Russell, P. S.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. Russell, "Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres," Nature 434, 488-491 (2005).
[CrossRef] [PubMed]

Shannon, C.

C. Shannon, "Communication in the presence of noise," Proc. IEEE 72, 1192-1201 (1984).
[CrossRef]

Sibbett, W.

Takeuchi, Y.

Thorne, A. P.

A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

Walker, B. C.

D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
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Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
[CrossRef]

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[CrossRef]

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[CrossRef]

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M. L. Junttila, J. Kauppinen, and E. Ikonen, "Performance limits of Stationary Fourier Spectrometers," J. Opt. Soc. Am. A. 8, 1457-1462 (1991).
[CrossRef]

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A. P. Thorne, C. J. Harris, I. Wynnejones, R. C. M. Learner, and G. Cox, "A Fourier-Transform Spectrometer for the Vacuum ultraviolet - design and performance," J. Phys. E:J. Sci. Instrum. 20, 54-60 (1987).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Q. Lu, F. Du, Y. H. Wu, and S. T. Wu, "Liquid-crystal-based Fourier optical spectrum analyzer without moving parts," Jpn. J. Appl. Phys. 44, 291-293 (2005).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Opt. Express (1)

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D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, "High-order harmonic generation from ions in a capillary discharge," Phys. Rev. Lett. 96 203001 (2006).
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[CrossRef]

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[CrossRef]

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J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the crossed-beam spectrometer. S, slit used to sample the interferogram; D, detector appropriate to the wavelength; dotted box, scanning translation stage.

Fig. 2.
Fig. 2.

Spectra taken with the crossed beam spectrometer (solid), compared to spectra taken with grating spectrometers (dashed). (a) Spectrum of CW Ti:Saph Oscillator (b) Spectrum of Mode-Locked Ti:Saph Oscillator (c) Octave-spanning spectrum of Ti:Saph oscillator & frequency doubled oscillator light (d) Spectrum of MIR CO2 laser with (solid) and without (dash-dot) envelope correction. Bandwidths of (a) and (d) are limited by spectral resolution.

Fig. 3.
Fig. 3.

Interferogram and beam halves captured to facilitate envelope removal. (a) Captured interferogram (b) Captured left and right beam halves, and (c) Interferogram with envelope and offset removed.

Fig. 4.
Fig. 4.

Comparison of measured, calculated, and simulated errors as described in Section 4 for uncollimated beams (a) and collimated beams where a steering error is introduced (b).

Equations (4)

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E ˜ 1,2 ( x , ω ) = A 1,2 A ˜ ( ω ω 0 ) U 1,2 ( x ) exp [ ± i 2 π f x x ]
E 1,2 ( x , t ) = A 1,2 A ( t τ ( x ) ) U 1,2 ( x ) exp [ i ω 0 ( t τ ( x ) ) ] .
I ( x ) = I 1 ( x ) + I 2 ( x ) + 2 τ ¯ p I 0 1 I 0 2 U 1 ( x ) U 2 ( x ) Re { γ ( 2 τ ( x ) ) exp [ i 2 ω 0 τ ( x ) ] }
I ( v ) = D ( v ) + R ( v ) [ S ( v v 0 ) + S ( v + v 0 ) ]

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