Abstract

The dispersive properties of materials, i.e., their frequency-dependent response to the interaction with light, in most situations determines whether an optical process can be observed. Although one can always search for a specific material with the sought-after properties, this material might be far from optimum or might not even exist. Therefore, it is of great interest to develop methods that could tune the dispersive properties of a medium independently of the working frequency band. Pulses with angular dispersion, or pulse-front tilt, precisely allow us to achieve this goal. In this tutorial, we show the basics of how angular dispersion can manage to tune the dispersion parameters that characterize the propagation of light in a medium, thus permitting the observation and application of various optical processes in nonlinear and quantum optics that could not be realized otherwise. To keep the focus on first principles, the list of topics addressed is not exhaustive. More specifically, we consider the role of angular dispersion for pulse stretching and compression, broadband second-harmonic generation, the generation of temporal solitons in nonlinear χ(2) media, the tunable generation of terahertz waves by means of optical rectification of femtosecond pulses, and the tuning of the frequency correlations and of the bandwidth of entangled paired photons.

© 2010 Optical Society of America

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2009

M. Hendrych, X. Shi, A. Valencia, J. P. Torres, “Broadening the bandwidth of entangled photons: a step towards the generation of extremely short biphotons,” Phys. Rev. A 79, 023817 (2009).
[CrossRef]

2008

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

J. Hebling, K. L. Yeh, M. C. Hoffmann, B. Bartal, K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25, B6–B19 (2008).
[CrossRef]

X. Shi, A. Valencia, M. Hendrych, J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. 33, 875–877 (2008).
[CrossRef] [PubMed]

J. Hebling, K.-L. Yeh, K. A. Nelson, M. C. Hoffmann, “High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 345–353 (2008).
[CrossRef]

Y. Zaouter, J. Boullet, E. Mottay, E. Cormier, “Transform-limited 100 μJ340 MW pulses from a nonlinear-fiber chirped-pulse amplifier using a mismatched grating stretcher-compressor,” Opt. Lett. 33, 1527–1529 (2008).
[CrossRef] [PubMed]

2007

M. C. Hoffmann, K. L. Yeh, J. Hebling, K. A. Nelson, “Efficient terahertz generation by optical rectification at 1035 nm,” Opt. Express 15, 11706–11713 (2007).
[CrossRef] [PubMed]

M. Hendrych, M. Mičuda, J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32, 2339–2341 (2007).
[CrossRef] [PubMed]

K. W. Chan, J. P. Torres, J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101(R) (2007).
[CrossRef]

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

2006

K. L. Vodopyanov, “Optical generation of narrow-band terahertz packets in periodically-inverted electro-optics crystals: conversion efficiency and optimal laser pulse format,” Opt. Express 14, 2263–2276 (2006).
[CrossRef] [PubMed]

J. Hebling, Z. Márton, “Theory of spectroscopic devices,” J. Opt. Soc. Am. B 23, 966–972 (2006).
[CrossRef]

2005

2004

S. Akturk, X. Gu, E. Zeek, R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004).
[CrossRef] [PubMed]

S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[CrossRef]

A. Valencia, G. Scarcelli, Y. H. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85, 2655–2657 (2004).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almasi, B. Bartal, J. Kuhn, “Tunable THz pulse generation by optical rectification of ultrashort pulses with tilted pulses,” Appl. Phys. B 78, 593–599 (2004).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, Science 306, 1330–1336 (2004).
[CrossRef] [PubMed]

X. Gu, S. Akturk, R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242, 599–604 (2004).
[CrossRef]

2002

T. Aichele, A. I. Lvovsky, S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[CrossRef]

F. Wise, P. Di Trapani, “The hunt for light bullets–spatiotemporal solitons,” Opt. Photon. News 13(2), 29–32 (February 2002).

J. Hebling, G. Almasi, I. Z. Kozma, J. Kuhl, “Velocity matching by pulse front tilt for large area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002).
[CrossRef] [PubMed]

2001

V. Giovannetti, L. Maccone, S. Lloyd, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417–419 (2001).
[CrossRef] [PubMed]

S. Carrasco, J. P. Torres, L. Torner, F. Wise, “Walk-off acceptance for quadratic soliton generation,” Opt. Commun. 191, 363–370 (2001).
[CrossRef]

W. P. Grice, A. B. U’Ren, I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[CrossRef]

2000

J. P. Torres, S. Carrasco, L. Torner, E. W. Van Stryland, “Frequency doubling of femtosecond pulses in walk-off-compensated N-(4-nitrophenyl)-L-prolinol,” Opt. Lett. 25, 1735–1737 (2000).
[CrossRef]

X. Liu, K. Beckwitt, F. W. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
[CrossRef]

1999

X. Liu, L. J. Qian, F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[CrossRef]

1998

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, A. Piskarskas, “Observation of temporal solitons in second harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459–464 (1998).
[CrossRef]

B. Richman, S. Bisson, R. Trebino, E. Sidick, A. Jacobson, “Efficient broadband second-harmonic generation by dispersive achromatic nonlinear conversion using only prisms,” Opt. Lett. 23, 497–499 (1998).
[CrossRef]

1997

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071–1073 (1997).
[CrossRef] [PubMed]

T. E. Keller, M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534–1541 (1997).
[CrossRef]

1996

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

S. Szatmari, P. Simon, M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21, 1156–1158 (1996).
[CrossRef] [PubMed]

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
[CrossRef]

L. Torner, D. Mazilu, D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

R. Schieck, Y. Baek, G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

1995

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

1994

G. Szabó, Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994).
[CrossRef]

1992

M. Segev, B. Crosignani, A. Yariv, B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schieck, L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1992).
[CrossRef]

1990

T. R. Zhang, H. R. Choo, M. C. Downer, “Phase and group velocity matching for second harmonic generation of femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
[CrossRef] [PubMed]

G. Szabó, Zs. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

1989

O. E. Martínez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
[CrossRef]

1987

1986

O. E. Martínez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

O. E. Martínez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

O. E. Martínez, “Hybrid prism-grating ultrashort pulse compressor,” Opt. Commun. 83, 117–122 (1986).
[CrossRef]

1985

D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

1984

1982

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

1980

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

1975

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. L. Strizhevskii, “Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090–1098 (1975).
[CrossRef]

1969

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969).
[CrossRef]

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Figures (21)

Fig. 1
Fig. 1

Decomposition of a white-light beam into different colors due to the angular dispersion introduced by a prism, as originally depicted by I. Newton in Opticks[1].

Fig. 2
Fig. 2

Schematic of a grating surface and definition of angles: (+) refers to positives angles, and (−) to negative ones; m > 0 corresponds to positive diffraction orders, and m < 0 to negative diffraction orders.

Fig. 3
Fig. 3

Tilting of the front of the pulse. After the grating, the front of the pulse is no longer perpendicular to the direction of propagation.

Fig. 4
Fig. 4

The line of the loci of peak intensities in the x c t plane is tilted by an angle Φ.

Fig. 5
Fig. 5

The material whose dispersive properties are to be tailored is located between two gratings Gr1 and Gr2. Light enters the dispersive medium perpendicularly to its input face.

Fig. 6
Fig. 6

Schematic of a prism and definition of angles.

Fig. 7
Fig. 7

General scheme for pulse compression.

Fig. 8
Fig. 8

(a) General scheme for CPA and compression. (b) Device that can introduce positive or negative dispersion, depending on the length d. f is the focal length of the two lenses.

Fig. 9
Fig. 9

Detailed evolution of the SH pulse (a) with and (b) without GVM compensation. With GVM compensation with the help of pulse-front tilt, the SH quickly and efficiently builds up. Without GVM compensation, the nonlinear process is very inefficient, and the SH hardly appears. In addition, the lack of broadband phase-matching results in backconversion. Conditions: input FF peak intensity, 10 MW cm 2 ; FF input beam width, 3 mm ; FF input pulse duration, 100 fs ; wavelength of the fundamental wave, 1.6 μ m ; length of the NPP crystal [N-(4-nitrophenyl)-L-prolinol], 3 mm . Figure courtesy of J. P. Torres [12].

Fig. 10
Fig. 10

Schematic diagram of a typical SHG configuration that uses pulses with pulse-front tilt. (a) The FF beam acquires pulse-front tilt and is focused into a PPLN crystal with a lens. A second grating is used to remove the angular dispersion introduced by the first grating. (b) Close-up view that illustrates the tilted quasi-phase-matching grating used. Figure courtesy of A. Schober [42].

Fig. 11
Fig. 11

SHG obtained with angular dispersion in PPLN. (a) Measured autocorrelation and (b) spectrum with angular dispersion. For the sake of comparison, (c) and (d) show the measured autocorrelation and spectrum when a crystal of identical length in an collinear configuration, with no pulse-front tilt, is used. Figure courtesy of A. Schober [42].

Fig. 12
Fig. 12

Simulated evolution of the peak intensity of the SH beam as a function of the crystal length. Solid curve, evolution according to Eqs. (61) with GVM compensation and no loss; dotted-dashed curve, evolution according to Eqs. (60) with GVM compensation and no loss; dashed curve, evolution according to Eqs. (61) with GVM compensation and loss; dotted curve, evolution according to Eqs. (61) with no GVM compensation. Inset, SH output pulse. Conditions: input FF peak intensity, 10 MW cm 2 ; FF input beam width, 3 mm ; FF input pulse duration, 100 fs . Figure courtesy of J. P. Torres [12].

Fig. 13
Fig. 13

Experimental setup to observe quadratic temporal solitons in BBO. (a) Schematic of the experiment. (b), (c) Highly elliptical spatial profiles of the pump beam. A cylindrical lens focuses the beam in the y direction. Figure courtesy of F. Wise [56]. © 2004 by the American Physical Society.

Fig. 14
Fig. 14

Experimental (a) temporal and (b) spatial widths of the solitons during propagation. The dashed curves represent the temporal and spatial widths of the wave if the propagation were dictated only by dispersion and diffraction that produce temporal and spatial broadening. The experimental black diamonds confirm that the temporal and spatial widths of the generated soliton remained constant. The insets show the temporal and spatial profiles at some selected distances. The peak intensity is 8 GW cm 2 , and Δ k = 60 π 25 mm . Figure courtesy of F. Wise [57].

Fig. 15
Fig. 15

Schematic of the configuration for noncollinear phase matching of optical and THz waves with pulse-front tilt.

Fig. 16
Fig. 16

Spectra of the THz pulses measured for different tilt angles ν. The maxima of the spectra are normalized. Figure courtesy of J. Hebling [62]. © 2004 by Springer.

Fig. 17
Fig. 17

Dependence of the energy and the frequency of the THz pulses on the tilt angle ν. The solid curves are guides to the eye. Figure courtesy of J. Hebling [62]. © 2004 by Springer.

Fig. 18
Fig. 18

Experimental setup used to demonstrate the control of frequency correlation and the bandwidth enhancement in SPDC by means of angular dispersion. G denotes gratings; PBS, polarization beam splitter; Mono, monochromators; D, single-photon counting modules; M, mirrors; &, coincidence electronics.

Fig. 19
Fig. 19

Shape S ( ω s , ω i ) of the frequency correlations measured experimentally (left) and predicted theoretically (right). (a), (b) no tilt, Φ = 0 ° ; (c), (d) anticorrelated photons, Φ = 38 ° ; (e), (f) uncorrelated photons, Φ = 20 ° ; (g), (h) correlated photons, Φ = 52 ° . Pump-beam bandwidth, Δ λ p = 2 nm ; nonlinear crystal length, L = 3.5 nm . Figure courtesy of M. Hendrych [77]. © 2009 by the American Physical Society.

Fig. 20
Fig. 20

Measured joint spectral density S ( ω s , ω i ) for a 2 mm type-II BBO crystal: (a) tilt angle Φ = 0 ° and (b) tilt angle Φ = Φ II max = 38 ° . The joint spectrum broadens sevenfold. Figure courtesy of M. Hendrych [77]. © 2009 by the American Physical Society.

Fig. 21
Fig. 21

(a) Signal single counts for Φ = 0 ° ; Δ λ s = 5.2 nm . (b) Signal single counts for Φ = Φ II max = 38 ° ; Δ λ s = 37 nm . (c) Coincidences along the antidiagonal for Φ = 0 ° ; Δ Λ = 7.5 nm . (d) Coincidences along the antidiagonal for Φ = Φ II max = 38 ° ; Δ Λ = 52 nm . Solid curves represent the theoretical prediction; squares are the experimental data.

Tables (1)

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Table 1 Conditions to Obtain Spatial, Temporal, and Spatiotemporal Solitons

Equations (85)

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sin ϵ ¯ + sin θ ¯ = m λ d ,
ϵ = α θ + γ Δ λ ,
α = cos θ 0 cos ϵ 0
γ = m d cos ϵ 0
E ( x , y , z , t ) = 1 2 A ( x , y , z , t ) exp ( i k 0 z i ω 0 t ) + h.c ,
A ( x , y , z , t ) d q x d q y d Ω a ( q x , q y , Ω , z ) exp ( i Ω t + i q x x + i q y y ) .
q x = p x α tan Φ α c Ω ,
tan Φ = c k 0 ( ϵ ω ) ω 0 .
a ( q x , q y , Ω , z 1 = 0 ) a ( q x α tan Φ α c Ω , q y , Ω , z 2 = 0 ) .
a ( q x , q y , Ω , z 2 = 0 ) a ( q x , q y , Ω , z 2 = 0 ) exp ( i k 0 Ω z 2 ) ,
A ( x 2 , y 2 , z 2 , t ) d q x d q y d Ω a ( q x α tan Φ α c Ω , q y , Ω , z 2 = 0 ) exp ( i k 0 Ω z 2 ) exp ( i Ω t + i q x x 2 + i q y y 2 ) .
A ( x 1 , y 1 , z 1 = 0 , t ) A ( α x 2 , y 2 , z 2 , t [ k 0 z 2 + tan Φ c x 2 ] ) .
t ( k 0 z 2 + tan Φ c x 2 ) = 0 .
tan ν tan Φ c k 0 .
tan Φ = n 0 λ 0 γ .
n 1 ϵ 1 = n 2 ϵ 2 ,
a ( q x , q y , Ω , z ) = a ( q x , q y , Ω , z = 0 ) exp [ i ( k ( ω ) k 0 ) z ] exp ( i | q | 2 2 k 0 z ) exp ( i q x z tan ρ ) ,
k ( ω ) = k 0 + k 0 Ω + 1 2 k 0 Ω 2 .
a ( q x , q y , Ω , z = L ) a ( q x α tan Φ α c Ω , q y , Ω , z = 0 ) exp ( i | q | 2 2 k 0 L ) exp ( i q x L tan ρ ) exp ( i k 0 Ω L + i 2 k 0 Ω 2 L ) .
a ( q x , q y , Ω ) a ( α q x + Ω tan Φ c , q y , Ω ) .
A ( x , y , z = L , t ) d q x d q y d Ω a ( q x , q y , Ω , z = 0 ) exp ( i q x x ) exp ( i q y y ) exp { i Ω ( t k 0 L + α q x L tan Φ k 0 c tan Φ tan ρ c L ) } exp { i Ω 2 [ k 0 ( tan Φ c ) 2 1 k 0 ] L 2 } exp ( i α q x tan ρ L ) exp ( i α 2 q x 2 2 k 0 L ) exp ( i q y 2 2 k 0 L ) .
A ( y , z = L , t ) = d q y d Ω a ( q y , Ω , z = 0 ) exp ( i q y y ) exp ( i q y 2 2 k 0 L ) exp { i Ω [ t ( k 0 + tan Φ tan ρ c ) L ] } exp { i Ω 2 [ k 0 ( tan Φ c ) 2 1 k 0 ] L 2 } .
k 0 , eff = k 0 + tan Φ tan ρ c ,
k 0 , eff = k 0 1 k 0 ( tan Φ c ) 2 .
k 0,eff = k 0 + tan Φ tan ρ c ,
k 0,eff = k 0 1 k 0 ( tan Φ c ) 2 .
sin θ ¯ = n ( λ ) sin δ ¯ 1 ,
n ( λ ) sin δ ¯ 2 = sin ϵ ¯ .
ϵ = α θ + γ Δ λ ,
α = cos θ 0 cos ϵ 0 cos δ 20 cos δ 10 ,
γ = sin C cos ϵ 0 cos δ 10 ( n λ ) λ 0 ,
ϵ = θ + 2 ( n λ ) λ 0 Δ λ .
tan Φ = 2 λ 0 ( n λ ) λ 0 .
A ( x , Ω ) = A 0 exp ( α 2 x 2 w 0 2 y 2 w 0 2 ) exp ( Ω 2 T 0 2 4 + i tan Φ c Ω x ) ,
A ( x , Ω ) = A 0 exp [ ( x ξ Ω ) 2 w 0 2 ] exp ( Ω 2 T 0 2 4 + i δ Ω 2 2 + i μ Ω x ) .
I ( x , t ) exp [ 2 ( t δ v ¯ x μ x ) 2 τ ¯ 2 ] exp ( 2 x 2 w ¯ 0 2 ) ,
v ¯ = ξ ξ 2 + T 0 2 w 0 2 4 ,
τ ¯ = ( T 0 2 + 4 ξ 2 w 0 2 + 4 δ 2 T 0 2 + 4 ξ 2 w 0 2 ) 1 2 ,
w ¯ 0 = [ 1 w 0 2 v ¯ 2 ( T 0 2 4 + ξ 2 w 0 2 ) ] 1 2 .
A ( x , Ω ) exp ( Ω 2 T 0 2 4 + i z Ω c ) exp { ( x α z μ Ω k 0 ) 2 w 0 2 + 2 i α 2 z k 0 } exp { i μ 2 z 2 k 0 Ω 2 } ,
A ( t , z = 0 ) = A 0 exp ( t 2 T 1 2 ) a ( Ω , z = 0 ) exp ( Ω 2 T 1 2 4 ) ,
A ( t , z ) = A ( t , z = 0 ) exp ( i γ f P ( t ) z ) ,
A ( t , z ) = A 0 exp [ t 2 T 1 2 ( 1 + 2 i γ f P 0 z ) ] ,
a ( Ω , z ) exp { Ω 2 T 1 2 4 [ 1 + ( 2 γ f P 0 z ) 2 ] + i Ω 2 γ P 0 z T 1 2 2 [ 1 + ( 2 γ f P 0 z ) 2 ] } .
B 2 = B 1 [ 1 + ( 2 γ f P 0 z ) 2 ] 1 2 .
α SPM = γ f P 0 T 1 2 z [ 1 + ( 2 γ f P 0 z ) 2 ] .
A ( t , L ) = A 0 T 1 T 1 2 2 i k eff L exp ( t 2 T 1 2 2 i k eff L ) ,
P 2 = P 1 1 + ( 2 k eff L T 1 2 ) 2 ,
T 2 = T 1 1 + ( 2 k eff L T 1 2 ) 2 .
H ( q ) = exp { i f d k 0 | q | 2 } .
β = f d k 0 ( tan Φ c ) 2 .
k 2 = 2 k 1 .
k 2 = k 1 ,
L gvm = T 0 | k 2 k 1 | ,
I 2 ( L ) I 1 ( 0 ) I 1 ( 0 ) L 2 sinc 2 ( Δ k L 2 ) = I 1 ( 0 ) L 2 sin 2 ( Δ k L 2 ) ( Δ k L 2 ) 2 ,
n 1 ( ϵ ( λ 1 ) , λ 1 ) = n 2 ( ϵ ( λ 2 ) , λ 2 ) ,
( ϵ λ ) λ 10 = ( n 1 λ ) λ 10 1 2 ( n 2 λ ) λ 20 ( n 2 θ ) θ 0 ( n 1 θ ) θ 0 ,
k 1 , eff = k 1 + tan Φ tan ρ 1 c ,
k 2 , eff = k 2 + tan Φ tan ρ 2 c .
( ϵ λ ) λ 10 = ( n 1 λ ) λ 10 1 2 ( n 2 λ ) λ 20 n 1 ( tan ρ 2 tan ρ 1 ) ,
ρ ( θ ) = θ tan 1 { n o 2 n e 2 ( θ ) tan θ } ,
n 1 , 2 θ = n 1 , 2 tan ρ 1 , 2 .
i A 1 z + i k 1 A 1 t k 1 2 2 A 1 t 2 i tan ρ 1 A 1 x + 1 2 k 1 [ 2 A 1 x 2 + 2 A 1 y 2 ] + i Γ 1 2 A 1 + K 1 A 1 * A 2 exp ( i Δ k z ) = 0 ,
i A 2 z + i k 2 A 2 t k 2 2 2 A 2 t 2 i tan ρ 2 A 2 x + 1 2 k 2 [ 2 A 2 x 2 + 2 A 2 y 2 ] + i Γ 2 2 A 2 + K 2 A 1 2 exp ( i Δ k z ) = 0 ,
i A 1 z + i k 1 , eff A 1 t k 1 , eff 2 2 A 1 t 2 + i Γ 1 2 A 1 + K 1 A 1 * A 2 exp ( i Δ k z ) = 0 ,
i A 2 z + i k 2 , eff A 2 t k 2 , eff 2 2 A 2 t 2 + i Γ 2 2 A 2 + K 2 A 1 2 exp ( i Δ k z ) = 0 .
i a 1 ξ + 1 2 2 a 1 τ 2 + 1 2 L dis L dif [ α 2 2 a 1 s 2 + 2 a 1 η 2 ] + i 2 L dis L abs a 1 + 2 L dis L nl a 1 * a 2 exp ( i 2 π L dis L coh ξ ) = 0 ,
i a 2 ξ + 1 2 L dis L dis 2 a 2 τ 2 + 1 4 L dis L dif [ α 2 2 a 2 s 2 + 2 A 2 η 2 ] i 2 L dis L gvm a 2 τ i 2 L dis L w A 2 s + i 2 L dis L abs a 2 + 2 L dis L nl a 1 2 exp ( i 2 π L dis L coh ξ ) = 0.
A ( z , t ) = d ω a ( ω ) exp { i [ k opt ( ω 0 + ω ) k opt 0 ] z i ω t } ,
P THz NL ( Ω ) = ϵ 0 χ ( 2 ) d ω a ( ω + Ω ) a * ( ω ) exp { i [ k opt ( ω + Ω ) k opt ( ω ) ] z } .
k THz ( Ω ) = k opt ( ω + Ω ) k opt ( ω )
k THz ( Ω ) = Ω n THz ph c ,
n THz ph = n opt gr ,
n opt , eff gr = n opt gr + tan Φ tan ρ .
k THz ( Ω ) = k opt ( ω + Ω ) k opt ( ω ) .
n THz ph cos γ ¯ = n opt gr ,
n THz ph sin γ ¯ = n opt ph ω 0 ( ϵ ω ) ω 0 .
tan γ ¯ = n opt ph n opt gr ω 0 ( ϵ ω ) ω 0 .
tan ν = tan Φ n opt gr .
| Ψ = d Ω s d Ω i d q s d q i Ψ ( Ω s , Ω i , q s , q i ) | ω s 0 + Ω s , q s s | ω i 0 + Ω i , q i i ,
Ψ ( Ω s , Ω i , q s , q i ) = E p ( Ω s + Ω i , q s + q i ) sinc ( Δ k L 2 ) exp ( i s k L 2 )
Δ k = ( k p , eff k s , eff ) Ω s + ( k p , eff k i , eff ) Ω i 1 2 k s , eff Ω s 2 1 2 k i , eff Ω i 2 1 2 k p , eff ( Ω s + Ω i ) 2 ,
Δ k ( k i , eff k s , eff ) Ω s 1 2 ( k s , eff + k i , eff ) Ω s 2 .
Φ II max = tan 1 { c ( k i k s ) tan ρ s tan ρ i } ,
Φ I max = tan 1 c 2 k s k s 0 ,

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