Abstract

We report new aspects of temporal dispersion in Fourier pulse shapers that contain spectral dispersers with a nonlinear frequency to space mapping. These effects are particularly important in high-resolution operation, since high-resolution dispersers typically exhibit pronounced nonlinear angular dispersion over relatively small bandwidths. In this paper we present a general discussion of the new effects followed by quantitative analysis and experimental verification for pulse shapers, which utilize a virtually imaged phased array (VIPA) as the spectral disperser. Compared to the well-known 4-F configuration, our results demonstrate a substantial modification to the placement of the optical components required to obtain zero temporal dispersion. Furthermore, spectral phase variations associated with nonzero dispersion coupled with contributions from multiple diffraction orders are shown to give rise to a dramatic new spectral interference effect, which can be used to monitor temporal dispersion purely in the spectral domain. We expect the effects we present in this paper to become prominent even for more conventional diffraction-grating-based pulse shapers for bandwidths sufficiently large that nonlinear spectral mapping becomes strong.

© 2010 Optical Society of America

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References

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  1. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
    [CrossRef]
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    [CrossRef]
  3. A. M. Weiner, Ultrafast Optics (Wiley, 2009).
    [CrossRef]
  4. H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
    [CrossRef]
  5. M. Shirasaki, “Large angular dispersion by a virtually imaged phased array and it’s application to a wavelength division multiplexer,” Opt. Lett. 21, 366–368 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  7. H. Ooi, K. Nakamura, Y. Akiyama, T. Takahara, T. Terahara, Y. Kawahata, H. Isono, and G. Ishikawa, “40-Gb∕s WDM transmission with virtually imaged phased array (VIPA) variable dispersion compensators,” J. Lightwave Technol. 20, 2196–2203 (2002).
    [CrossRef]
  8. M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
    [CrossRef]
  9. S. Xiao, J. D. McKinney, and A. M. Weiner, “Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper,” IEEE Photonics Technol. Lett. 16, 1936–1938 (2004).
    [CrossRef]
  10. E. Hamidi and A. M. Weiner, “Phase-only matched filtering of ultrawideband arbitrary microwave waveforms via optical pulse shaping,” J. Lightwave Technol. 20, 2355–2363 (2008).
    [CrossRef]
  11. 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] [PubMed]
  12. M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express 16, 2387–2397 (2008).
    [CrossRef] [PubMed]
  13. V. R. Supradeepa, C. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16, 11878–11887 (2008).
    [CrossRef] [PubMed]
  14. K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
    [CrossRef] [PubMed]
  15. G.-H. Lee and A. M. Weiner, “Programmable optical pulse burst manipulation using a virtually imaged phased array (VIPA) based Fourier transform pulse shaper,” J. Lightwave Technol. 23, 3916–3923 (2005).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
  17. X. Shijun, 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]
  18. Wolfram alpha,” http://www.wolframalpha.com.
  19. G.-H. Lee, “Demonstration of optical tunable dispersion compensation with a virtually-imaged phased-array based pulse shaper,” PhD dissertation, Purdue University (2006).
  20. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Optical Pulses (Kluwer, 2002).
    [CrossRef]

2009 (2)

A. M. Weiner, Ultrafast Optics (Wiley, 2009).
[CrossRef]

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

2008 (4)

E. Hamidi and A. M. Weiner, “Phase-only matched filtering of ultrawideband arbitrary microwave waveforms via optical pulse shaping,” J. Lightwave Technol. 20, 2355–2363 (2008).
[CrossRef]

M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express 16, 2387–2397 (2008).
[CrossRef] [PubMed]

V. R. Supradeepa, C. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16, 11878–11887 (2008).
[CrossRef] [PubMed]

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

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] [PubMed]

2006 (2)

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

G.-H. Lee, “Demonstration of optical tunable dispersion compensation with a virtually-imaged phased-array based pulse shaper,” PhD dissertation, Purdue University (2006).

2005 (1)

2004 (2)

X. Shijun, 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]

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

2002 (2)

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

1996 (1)

1990 (1)

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

1984 (1)

Akiyama, Y.

Balslev-Clausen, D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

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] [PubMed]

Fork, R. L.

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, 1145–1149 (2009).
[CrossRef] [PubMed]

Gordon, J. P.

Hamidi, E.

E. Hamidi and A. M. Weiner, “Phase-only matched filtering of ultrawideband arbitrary microwave waveforms via optical pulse shaping,” J. Lightwave Technol. 20, 2355–2363 (2008).
[CrossRef]

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] [PubMed]

Houxun, M.

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

Huang, C.

Ishikawa, G.

Isono, H.

Jalali, B.

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

Kato, K.

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Kawahata, Y.

Kirchner, M. S.

Leaird, D. E.

Lee, G.-H.

G.-H. Lee, “Demonstration of optical tunable dispersion compensation with a virtually-imaged phased-array based pulse shaper,” PhD dissertation, Purdue University (2006).

G.-H. Lee and A. M. Weiner, “Programmable optical pulse burst manipulation using a virtually imaged phased array (VIPA) based Fourier transform pulse shaper,” J. Lightwave Technol. 23, 3916–3923 (2005).
[CrossRef]

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 phase array (VIPA) and spatial light modulator,” IEEE Photonics Technol. Lett. 18, 1819–1821 (2006).
[CrossRef]

Lin, C.

X. Shijun, 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]

Martinez, O. E.

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] [PubMed]

McKinney, J. D.

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

Miller, P. J.

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

Mirkin, L.

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

Nakamura, K.

Nishi, I.

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Ooi, H.

Shijun, X.

X. Shijun, 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]

Shirasaki, M.

Supradeepa, V. R.

Suzuki, S.

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Takahara, T.

Takahashi, H.

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Terahara, T.

Thorpe, M. J.

Trebino, R.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Optical Pulses (Kluwer, 2002).
[CrossRef]

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, 1145–1149 (2009).
[CrossRef] [PubMed]

Weiner, A. M.

A. M. Weiner, Ultrafast Optics (Wiley, 2009).
[CrossRef]

E. Hamidi and A. M. Weiner, “Phase-only matched filtering of ultrawideband arbitrary microwave waveforms via optical pulse shaping,” J. Lightwave Technol. 20, 2355–2363 (2008).
[CrossRef]

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

V. R. Supradeepa, C. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16, 11878–11887 (2008).
[CrossRef] [PubMed]

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

G.-H. Lee and A. M. Weiner, “Programmable optical pulse burst manipulation using a virtually imaged phased array (VIPA) based Fourier transform pulse shaper,” J. Lightwave Technol. 23, 3916–3923 (2005).
[CrossRef]

X. Shijun, 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]

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

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Xiao, S.

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

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

Ye, J.

Electron. Lett. (1)

H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometre resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Shijun, 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]

IEEE Photon. Technol. Lett. (1)

M. Houxun, A. M. Weiner, L. Mirkin, and P. J. Miller, “All-order polarization-mode dispersion (PMD) compensation via virtually imaged phased array (VIPA)-based pulse shaper,” IEEE Photon. Technol. Lett. 20, 545–547 (2008).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

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

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

J. Lightwave Technol. (3)

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

Nature (2)

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] [PubMed]

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

Opt. Express (2)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[CrossRef]

Other (5)

Wolfram alpha,” http://www.wolframalpha.com.

G.-H. Lee, “Demonstration of optical tunable dispersion compensation with a virtually-imaged phased-array based pulse shaper,” PhD dissertation, Purdue University (2006).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Optical Pulses (Kluwer, 2002).
[CrossRef]

A. M. Weiner, Ultrafast Optics (Wiley, 2009).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

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Figures (12)

Fig. 1
Fig. 1

Schematic showing spectral dispersion by a VIPA due to interference between multiple virtual sources.

Fig. 2
Fig. 2

(A) input pulse, (B) input spectrum, (C) output pulse, (D) output spectrum. Note that spectra are plotted only over one free spectral range of the VIPA.

Fig. 3
Fig. 3

(A) Schematic showing the different parameters varied (the lens-to-VIPA distance by a few centimeters, the transverse position of the VIPA by a few millimeters, and the mirror tilt angle by a few tenths of a degree). (B, D, F) and (C, E, G) are respectively the time domain cross-correlations and the corresponding spectra as the transverse position of the lens is varied. Similar behavior is observed with the variation of any of the three parameters. Spectra are plotted only over one free spectral range of the VIPA.

Fig. 4
Fig. 4

Figure showing the origin of temporal dispersion in grating and VIPA pulse shapers. (A) Temporal dispersion in a grating pulse shaper when not in 4-F configuration. (B) Nature of spectral dispersion by the VIPA due to interference between multiple virtual sources. (C) Location of the incident and reflected virtual sources in the 4-F configuration indicating the origin of additional dispersion. (D) Configuration showing the overlap between incident and reflected virtual sources—expected zero dispersion condition for VIPA pulse shapers.

Fig. 5
Fig. 5

Spectral dispersion of a VIPA (from [17]).

Fig. 6
Fig. 6

Schematic showing creation of higher-order spectral phases due to nonlinear sampling of spatial linear phase, where different spots represent equally spaced frequencies. (A) For linear spectral dispersers, the phase differences between equally spaced frequencies are constant. (B) For nonlinear spectral dispersers, the phase differences between equally spaced frequencies are now frequency dependent (leading to quadratic and higher-order spectral phase).

Fig. 7
Fig. 7

Spatial linear phase sampled nonlinearly in presence of multiple diffraction orders.

Fig. 8
Fig. 8

General configuration of a VIPA pulse shaper in the reflection geometry.

Fig. 9
Fig. 9

Schematic representation of spectral dispersion by the VIPA. Light is spectrally dispersed into multiple orders under a Gaussian intensity envelope.

Fig. 10
Fig. 10

Figure showing overlap conditions between input and reflected sources for varying L. (A) L = 0 , no overlap. (B) L < 0 , no overlap. (C) L > 0 , overlap.

Fig. 11
Fig. 11

Calculated and experimentally observed ripple periods for (A) variation in L. (B) Variation in θ m . (C) Variation in d.

Fig. 12
Fig. 12

(A)–(E) Observed spectrum of one VIPA free spectral range as the transverse position (L) of the lens is adjusted (while VIPA lens distance (d) and mirror angle ( θ m ) are maintained fixed). L is adjusted to move monotonically towards zero dispersion. (F) Cross-correlation signal showing time-domain output of the VIPA pulse shaper at zero dispersion.

Equations (35)

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Δ z = 2 t cos θ i ,
Δ ξ = 2 t sin θ i .
L + F 2 tan 2 θ m d tan θ i = 0.
E n ( ξ ) = ( R ) n E 0 exp ( ( ξ Δ ξ n + L ) 2 w 0 2 ) ,
Δ ξ n = n * 2 t sin θ i ,
E out ( x , k ) n = 0 exp ( i k ( Δ z n d ) ) exp ( i k ( Δ z n d ) 2 F 2 x 2 ) × E n ( ξ ) exp ( i k x ξ F ) d ξ ,
Δ z n = n * 2 t cos θ i .
E out ( x , k ) exp ( i k d ) exp ( i k L F x ) exp ( i k d 2 F 2 x 2 ) n = 0 exp ( i k Δ z n ) exp ( i k Δ z n 2 F 2 x 2 ) ( R ) n E 0 exp ( ( ξ Δ ξ n ) 2 w 0 2 ) exp ( i k x ξ F ) d ξ .
E out ( x , k ) exp ( i k d ) exp ( i k L F x ) exp ( i k d 2 F 2 x 2 ) exp ( x 2 w 2 ) ( 1 1 R exp ( i k ( 2 t cos θ i 2 t sin θ i x F t cos θ i x 2 F 2 ) ) ) ,
exp ( i k L x F ) exp ( i k d x 2 2 F 2 ) .
k ( 2 t cos θ i 2 t sin θ i x F t cos θ i x 2 F 2 ) = 2 m π
2 t cos θ i 2 t sin θ i x F t cos θ i x 2 F 2 = m λ .
exp ( i k tan 2 θ m x ) .
exp ( i k ( 2 L + F tan 2 θ m ) x F ) exp ( i k d x 2 F 2 ) .
x F = sin θ i + sin θ i 1 m ( λ λ m ) cos θ i t sin 2 θ i cos θ i ,
m ( λ λ m ) cos θ i t sin 2 θ i < m Δ λ cos θ i 2 t sin 2 θ i λ cos θ i 2 t sin 2 θ i ,
m ( λ λ m ) cos θ i t sin 2 θ i < λ cos θ i 2 t sin 2 θ i 0.2.
x F 1 2 t sin θ i m ( λ λ m ) cos θ i 8 t 2 sin 3 θ i m 2 ( λ λ m ) 2 cos 2 θ i 16 t 3 sin 5 θ i m 3 ( λ λ m ) 3 5 cos 3 θ i 128 t 4 sin 7 θ i m 4 ( λ λ m ) 4 .
exp ( i k ( L + F 2 tan 2 θ m ) t sin θ i m ( λ λ m ) )
exp ( i k ( L + F 2 tan 2 θ m d tan θ i ) cos θ i 4 t 2 sin 3 θ i m 2 ( λ λ m ) 2 )
exp ( i k ( L + F 2 tan 2 θ m d tan θ i ) cos 2 θ i 8 t 3 sin 5 θ i m 3 ( λ λ m ) 3 )
exp ( i k ( L + F 2 tan 2 θ m d tan θ i ) 5 cos 3 θ i 64 t 4 sin 7 θ i m 4 ( λ λ m ) 4 )
x F = tan θ i ( 1 α 1 )
exp ( i 2 k tan θ i ( L + F 2 tan 2 θ m d tan θ i ) ( 1 α 1 ) ) exp ( i k d tan 2 θ i α ) .
L + F 2 tan 2 θ m d tan θ i = 0.
ψ m + 1 linear ( λ ) ψ m linear ( λ ) = 2 π λ ( L + F 2 tan 2 θ m ) 2 t sin θ i [ ( m + 1 ) λ 2 t cos θ i m λ + 2 t cos θ i ] ,
ψ m + 1 linear ( λ ) ψ m linear ( λ ) = 2 π ( L + F 2 tan 2 θ m ) 2 t sin θ i .
ψ m + 1 quadratic ( λ ) ψ m quadratic ( λ ) = 2 π λ ( L + F 2 tan 2 θ m d tan θ i ) cos θ i 4 t 2 sin 3 θ i [ ( ( m + 1 ) λ 2 t cos θ i ) 2 ( m λ 2 t cos θ i ) 2 ] ,
ψ m + 1 quadratic ( λ ) ψ m quadratic ( λ ) = 2 π ( L + F 2 tan 2 θ m d tan θ i ) cos θ i 4 t 2 sin 3 θ i [ ( 2 m + 1 ) λ ( 4 t cos θ i ) ] ,
[ ψ m + 1 ( λ + P ) ψ m ( λ + P ) ] [ ψ m + 1 ( λ ) ψ m ( λ ) ] = 2 π .
2 π ( L + F 2 tan 2 θ m d tan θ i ) cos θ i 4 t 2 sin 3 θ i [ ( 2 m + 1 ) P ] = 2 π .
P λ m t sin 3 θ i ( L + F 2 tan 2 θ m d tan θ i ) cos 2 θ i .
P λ m t sin 3 θ i cos 2 θ i 1 Δ L .
P λ m t sin 3 θ i cos 2 θ i 1 F 2 Δ ( tan 2 θ m ) λ m t sin 3 θ i cos 2 θ i 1 F Δ θ m ,
P λ m t sin 2 θ i cos θ i 1 Δ d .

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