Abstract

Analytic expressions for spectral phase for optical systems are very important for the design of wide-bandwidth optical systems. We describe a general formalism for analytically calculating the spectral phase for arbitrary optical structure made up of nested pairs of plane-parallel interfaces that can be diffractive or refractive. Our primary application is the calculation of the spectral phase of a grism pair, which is then used to analyze the behavior of higher-order phase terms. The analytic expressions for the grism spectral phase provide insight into the tunability of the third-order phase of grisms as well as the fourth-order limits. Our exact and approximate expressions are compared with a raytracing model.

© 2008 Optical Society of America

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  1. E. Treacy, "Optical Pulse Compression With Diffraction Gratings," IEEE J. Quantum Electron. QE-5, 454-8 (1969).
    [CrossRef]
  2. O. E. Martinez, "3000 Times Grating Compressor with Positive Group Velocity Dispersion: Application to Fiber Compensation in 1.3-1.6 μm Region," IEEE J. Quantum Electron. QE-23, 59-64 (1987).
    [CrossRef]
  3. S. Kane and J. Squier, "Grating Compensation of 3rd-order material dispersion in the normal dispersion regime- sub-100-fs chirped pulse amplification using a fiber stretcher and grating-pair compressor," IEEE J. Quantum Electron. 31, 2052-2057 (1995).
    [CrossRef]
  4. W. A. Traub, "Constant-dispersion grism spectrometer for channeled spectra," J. Opt. Soc. Am. A 7, 1779-1791 (1990).
    [CrossRef]
  5. P. Tournois, "New diffraction grating pair with very linear dispersion for laser-pulse compression," Electron. Lett. 29, 1414-1415 (1993).
    [CrossRef]
  6. S. Kane and J. Squier, "Grism-pair stretcher compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification," J. Opt. Soc. Am. B 14, 661 (1997).
    [CrossRef]
  7. E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, "Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses," Opt. Lett. 31, 3363-3365 (2006).
    [PubMed]
  8. K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
    [CrossRef]
  9. D. M. Gaudiosi, E. Gagnon, A. L. Lytle, J. L. Fiore, E. A. Gibson, S. Kane, J. Squier, M. M. Murnane, H. C. Kapteyn, R. Jimenez, and S. Backus, "Multi-kilohertz repetition rate Ti : sapphire amplifier based on downchirped pulse amplification," Opt. Express 14, 9277-9283 (2006).
    [CrossRef] [PubMed]
  10. L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
    [CrossRef]
  11. J. J. Field, C. G. Durfee, J. A. Squier, and S. Kane, "Quartic-phase-limited grism-based ultrashort pulse shaper," Opt. Lett. 32, 3101-3103 (2007).
    [CrossRef] [PubMed]
  12. I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1-29 (2001). Part 1.
    [CrossRef]
  13. A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
    [CrossRef]
  14. R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Opt. Lett. 9, 150-152 (1984).
    [CrossRef] [PubMed]
  15. Wolfram, Mathematica, 6th ed. (Wolfram Research, Inc., Champaign, IL, 2007).

2007 (3)

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
[CrossRef]

J. J. Field, C. G. Durfee, J. A. Squier, and S. Kane, "Quartic-phase-limited grism-based ultrashort pulse shaper," Opt. Lett. 32, 3101-3103 (2007).
[CrossRef] [PubMed]

2006 (2)

1998 (1)

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

1997 (1)

1995 (1)

S. Kane and J. Squier, "Grating Compensation of 3rd-order material dispersion in the normal dispersion regime- sub-100-fs chirped pulse amplification using a fiber stretcher and grating-pair compressor," IEEE J. Quantum Electron. 31, 2052-2057 (1995).
[CrossRef]

1993 (1)

P. Tournois, "New diffraction grating pair with very linear dispersion for laser-pulse compression," Electron. Lett. 29, 1414-1415 (1993).
[CrossRef]

1990 (1)

1987 (1)

O. E. Martinez, "3000 Times Grating Compressor with Positive Group Velocity Dispersion: Application to Fiber Compensation in 1.3-1.6 μm Region," IEEE J. Quantum Electron. QE-23, 59-64 (1987).
[CrossRef]

1984 (1)

1969 (1)

E. Treacy, "Optical Pulse Compression With Diffraction Gratings," IEEE J. Quantum Electron. QE-5, 454-8 (1969).
[CrossRef]

Anzal, S.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Backus, S.

Braun, A.

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

Durfee, C.

Durfee, C. G.

Field, J. J.

Fiore, J. L.

Fork, R. L.

Gagnon, E.

Gaudiosi, D. M.

Gibson, E. A.

Gordon, J. P.

Huff, R.

Jimenez, R.

Kane, S.

J. J. Field, C. G. Durfee, J. A. Squier, and S. Kane, "Quartic-phase-limited grism-based ultrashort pulse shaper," Opt. Lett. 32, 3101-3103 (2007).
[CrossRef] [PubMed]

L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
[CrossRef]

D. M. Gaudiosi, E. Gagnon, A. L. Lytle, J. L. Fiore, E. A. Gibson, S. Kane, J. Squier, M. M. Murnane, H. C. Kapteyn, R. Jimenez, and S. Backus, "Multi-kilohertz repetition rate Ti : sapphire amplifier based on downchirped pulse amplification," Opt. Express 14, 9277-9283 (2006).
[CrossRef] [PubMed]

E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, "Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses," Opt. Lett. 31, 3363-3365 (2006).
[PubMed]

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

S. Kane and J. Squier, "Grism-pair stretcher compressor system for simultaneous second- and third-order dispersion compensation in chirped-pulse amplification," J. Opt. Soc. Am. B 14, 661 (1997).
[CrossRef]

S. Kane and J. Squier, "Grating Compensation of 3rd-order material dispersion in the normal dispersion regime- sub-100-fs chirped pulse amplification using a fiber stretcher and grating-pair compressor," IEEE J. Quantum Electron. 31, 2052-2057 (1995).
[CrossRef]

Kapteyn, H. C.

Kodate, K.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Komai, Y.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Kuznetsova, L.

L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
[CrossRef]

Lytle, A. L.

Martinez, O. E.

O. E. Martinez, "3000 Times Grating Compressor with Positive Group Velocity Dispersion: Application to Fiber Compensation in 1.3-1.6 μm Region," IEEE J. Quantum Electron. QE-23, 59-64 (1987).
[CrossRef]

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

Moritsuka, F.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Mourou, G. A.

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

Murnane, M. M.

Nakajima, K.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Norris, T.

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

Sosnowski, T.

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

Squier, J.

Squier, J. A.

Tournois, P.

P. Tournois, "New diffraction grating pair with very linear dispersion for laser-pulse compression," Electron. Lett. 29, 1414-1415 (1993).
[CrossRef]

Traub, W. A.

Treacy, E.

E. Treacy, "Optical Pulse Compression With Diffraction Gratings," IEEE J. Quantum Electron. QE-5, 454-8 (1969).
[CrossRef]

Van Rompay, P.

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

Watanabe, E.

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Wise, F.W.

L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
[CrossRef]

Appl. Phys. B (1)

L. Kuznetsova, F.W. Wise, S. Kane, and J. Squier, "Chirped-pulse amplification near the gain-narrowing limit of Yb-doped fiber using a reflection grism compressor," Appl. Phys. B 88, 515-518 (2007).
[CrossRef]

Electron. Lett. (1)

P. Tournois, "New diffraction grating pair with very linear dispersion for laser-pulse compression," Electron. Lett. 29, 1414-1415 (1993).
[CrossRef]

IEEE J. Quantum Electron. (3)

E. Treacy, "Optical Pulse Compression With Diffraction Gratings," IEEE J. Quantum Electron. QE-5, 454-8 (1969).
[CrossRef]

O. E. Martinez, "3000 Times Grating Compressor with Positive Group Velocity Dispersion: Application to Fiber Compensation in 1.3-1.6 μm Region," IEEE J. Quantum Electron. QE-23, 59-64 (1987).
[CrossRef]

S. Kane and J. Squier, "Grating Compensation of 3rd-order material dispersion in the normal dispersion regime- sub-100-fs chirped pulse amplification using a fiber stretcher and grating-pair compressor," IEEE J. Quantum Electron. 31, 2052-2057 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Braun, T. Sosnowski, S. Kane, P. Van Rompay, T. Norris, and G. A. Mourou, "Tunable third-order phase compensation by refraction from an intragrating-pair parallel plate," IEEE J. Sel. Top. Quantum Electron. 4, 426-429 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (3)

Opt. Rev. (1)

K. Nakajima, Y. Komai, E. Watanabe, F. Moritsuka, S. Anzal, and K. Kodate, "Fabrication of near-infrared volume phase holographic grism with high efficiency and high dispersion, and its application to a wavelength de-multiplexing device," Opt. Rev. 14, 201-207 (2007).
[CrossRef]

Other (2)

Wolfram, Mathematica, 6th ed. (Wolfram Research, Inc., Champaign, IL, 2007).

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1-29 (2001). Part 1.
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Schematics showing geometry for (a) a tilted window and (b) a tilted transmission grating pair.

Fig. 2.
Fig. 2.

Construction for calculating the phase of a prism pair. A tilted glass slab of thickness L 1 forms the outer boundaries of the prism pair. A slab of air, with thickness L 2, placed at an angle α to the glass slab defines the inner boundaries of the glass prisms. A ray propagates at angles θ 1-θ 4 through the first prism as shown. The line joining the prism tips has a length Lp and an angle θref with respect to the normal to the prism exit face. β is the angle between the tip-to-tip line and the wavelength-dependent ray direction at the first prism exit.

Fig. 3.
Fig. 3.

Schematic of a method to determine the value of θref . The first prism is pulled out of the beam to place an edge of the refracted beam. The second prism is also pulled out to pass a portion of the beam. A plot of the log of the transmitted spectrum shows a hard cutoff at the wavelength that passes from tip to tip.

Fig. 4.
Fig. 4.

a) A Tournois grism pair (no refraction at prism entrance) b) A reflection grism pair (dark). The lower grism is unfolded, showing the equivalent transmission grism (α 1=α 2) c) Modular construction to calculate the phase of the grism pair shown in panel (b). A pair of transmission gratings is at an angle α 1 to a tilted glass slab. A slab of air, with thickness Lair , is placed at an angle α 2 to the glass slab, to define the inner boundaries of the glass prisms. d) A reflection grism (and the equivalent transmission grism) where the beams pass through different prism faces.

Fig. 5.
Fig. 5.

Comparison of the calculation of the ratio of third- to second-order spectral phase for a grism pair (1480 grooves/mm, BK7 glass apex angles α 1=α 2=45°). Solid lines: calculation based on analytic derivatives of Eq 10, for several reference angles θref =50°,70°,75°,80° (from right to left in decreasing grayscale). Larger θref corresponds to greater prism insertion. Dashed line: calculation based on approximation (Eq 15). Dots: calculation from raytrace for several combinations of θref and θ 1. Inset: raytrace for θref =70° and θ 1=15°, for the spectral range 750nm (blue) to 850nm (red).

Fig. 6.
Fig. 6.

Calculation of higher-order phase of a grism pair (1480 lines/mm, BK7, apex angles 45o) vs. reference angle θref . Solid lines: ϕ 3/ϕ 2 (fs) for incident angles θ 1=10° (blue), θ 1=15° (red), and θ 1=20° (green). Lines terminate on the left when θref is too small to accommodate the full 750–850nm bandwidth. Horizontal gray line indicates the value of ϕ 3/ϕ 2 required to compensate BK7 glass. Dashed lines: ϕ 4/ϕ 2 (fs 2, scaled down by 2). Vertical lines guide the eye to indicate the value of residual ϕ 4/ϕ 2 when θref is set to compensate BK7 glass.

Equations (21)

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ϕ ( ω ) = k ( ω ) · r = ω n ( ω ) c k ̂ ( ω ) · r .
ϕ 1 ( ω ) ϕ ω = j L j ( v g ) j
k ω = ( v g ) 1 = n + ω n c .
ϕ w ( ω ) = k 0 L w ( n 2 cos θ 2 n 1 cos θ 1 )
n 2 sin θ 2 n 1 sin θ 1 = m λ d
ϕ gr ( ω ) = k 0 L gr n 2 2 ( m λ d n 1 sin θ 1 ) 2 k 0 L gr 1 ( λ d sin θ 1 ) 2
ϕ ( ω ) = ϕ gr ( ω ) + k 0 L w ( n g cos θ 4 ( ω ) cos θ 3 ( ω ) )
ϕ p ( ω ) = k 0 L 1 ( n 2 cos θ 2 n 1 cos θ 1 ) k 0 L 2 ( n 2 cos θ 3 n 1 cos θ 4 )
ϕ p ( ω ) = k 0 L p n 1 [ cos β cos ( θ 1 + θ ref α ) ] + ϕ 0 ,
ϕ grism ( ω ) = k 0 L p n 1 cos β ,
ϕ = k 0 L g cos θ d + k 0 n L p sin θ ref sin σ k 0 L p cos ( α θ ref ) ,
ϕ = k 0 L g cos θ d k 0 L g cos α + k 0 ( n 1 ) L p sin θ ref sin α .
ϕ ( ω ) ϕ 0 + ϕ 1 Δ ω + 1 2 ! ϕ 2 ( Δ ω ) 2 + 1 3 ! ϕ 3 ( Δ ω ) 3 +
ϕ 1 = ϕ m θ sin θ + ϕ m cos θ ϕ 2 = ( ϕ m θ + 2 ϕ m θ ) sin θ + ( ϕ m θ 2 + ϕ m ) cos θ ϕ 3 = ( ϕ m ( θ 3 θ ) 3 ϕ m θ 3 ϕ m θ ) sin θ + ( 3 ϕ m θ θ 3 ϕ m θ 2 + ϕ m ) cos θ
ϕ 2 ( L p c ) ω ( θ ) 2
ϕ 3 3 ( L p c ) θ ( θ + ωθ )
ϕ p ( ω ) = k 0 L p n 2 ( cos θ 2 cos ( α θ ref ) cos θ 3 cos θ ref )
k 0 L p n 1 ( cos θ 1 cos ( α θ ref ) cos θ 4 cos θ ref )
ϕ p ( ω ) = k 0 L p n 1 ( cos β cos θ 1 cos ( α θ ref ) )
+ k 0 L p n 2 ( cos θ 2 cos ( α θ ref ) cos ( θ 3 θ ref ) )
ϕ p ( ω ) = k 0 L p n 1 ( cos β cos θ 1 cos ( α θ ref ) ) k 0 L p n 2 sin θ 2 sin ( α θ ref )

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