Abstract

We study the photon correlation properties of broadband parametric down-converted light. The measurement of the photon correlation is carried out thanks to a modified Hanbury Brown-Twiss interferometer based on two photon absorption in GaAs detector. Since this method is not affected by the phase matching conditions of the detecting apparatus (so called “final state post-selection”), the detection bandwidth can be extremely large. This is illustrated by studying, with the same apparatus, the degree of second order coherence of parametric light in both degenerate and non-degenerate cases. We show that our experiment is able to determine the coherent as well as the incoherent contributions to the degree of second order coherence of parametric light with a time resolution in the fs range scale.

© 2010 OSA

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  1. H. Z. Cummins, and E. R. Pike, Photon correlation spectroscopy and light beating spectroscopy (Plenum Press, New York, 1974).
  2. Y. Tanaka, N. Sako, T. Kurokawa, H. Tsuda, and M. Takeda, “Profilometry based on two-photon absorption in a silicon avalanche photodiode,” Opt. Lett. 28(6), 402–404 (2003).
    [CrossRef] [PubMed]
  3. D. Bouwmeester, A. Ekert, and A. E. Zeilinger, The Physics of Quantum Information (Springer-Verlag, New York, 2000).
  4. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
    [CrossRef]
  5. S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
    [CrossRef] [PubMed]
  6. M. Beck, “Comparing measurements of g(2)(0) performed with different coincidence detection techniques,” J. Opt. Soc. Am. B 24(12), 2972–2978 (2007).
    [CrossRef]
  7. D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968).
    [CrossRef]
  8. I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
    [CrossRef] [PubMed]
  9. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
    [CrossRef] [PubMed]
  10. F. Zäh, M. Halder, and T. Feurer, “Amplitude and phase modulation of time-energy entangled two-photon states,” Opt. Express 16(21), 16452–16458 (2008).
    [CrossRef] [PubMed]
  11. K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009).
    [CrossRef] [PubMed]
  12. B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007).
    [CrossRef]
  13. O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
    [CrossRef] [PubMed]
  14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
    [CrossRef]
  15. J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27(23), 2076–2078 (2002).
    [CrossRef]
  16. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).
  17. R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
    [CrossRef]
  18. K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
    [CrossRef]
  19. R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).
  20. B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
    [CrossRef] [PubMed]
  21. A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
    [CrossRef] [PubMed]

2009

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009).
[CrossRef] [PubMed]

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

2008

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

F. Zäh, M. Halder, and T. Feurer, “Amplitude and phase modulation of time-energy entangled two-photon states,” Opt. Express 16(21), 16452–16458 (2008).
[CrossRef] [PubMed]

2007

M. Beck, “Comparing measurements of g(2)(0) performed with different coincidence detection techniques,” J. Opt. Soc. Am. B 24(12), 2972–2978 (2007).
[CrossRef]

B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007).
[CrossRef]

2005

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

2003

2002

1990

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
[CrossRef] [PubMed]

1988

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
[CrossRef]

1986

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

1985

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[CrossRef] [PubMed]

1968

D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968).
[CrossRef]

1963

R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[CrossRef]

1956

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Abram, I.

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

Beck, M.

Ben-Aryeh, Y.

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
[CrossRef] [PubMed]

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

Dayan, B.

B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007).
[CrossRef]

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

Dolique, G.

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

Fabre, C.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

Feurer, T.

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[CrossRef] [PubMed]

Friesem, A. A.

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

Glauber, R.

R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[CrossRef]

Godard, A.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

Halder, M.

Hanbury-Brown, R.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Hong, C. K.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
[CrossRef] [PubMed]

Kurimura, S.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

Kurokawa, T.

Kuzucu, O.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

Mandel, L.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[CrossRef] [PubMed]

Mogi, K.

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
[CrossRef]

Murphy, T. E.

Naganuma, K.

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
[CrossRef]

O’Donnell, K. A.

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009).
[CrossRef] [PubMed]

Oudar, J. L.

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

Pe’er, A.

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

Raj, R. K.

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

Rosencher, E.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

Roth, J. M.

Sako, N.

Scarl, D. B.

D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968).
[CrossRef]

Serulnik, S.

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
[CrossRef] [PubMed]

Silberberg, Y.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

Takeda, M.

Tanaka, Y.

Tovstonog, S.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

Tsuda, H.

Twiss, R. Q.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

U’Ren, A. B.

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009).
[CrossRef] [PubMed]

Wong, F. N. C.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

Xu, C.

Yamada, H.

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
[CrossRef]

Zäh, F.

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988).
[CrossRef]

Nat. Phys.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[CrossRef]

Nature

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968).
[CrossRef]

Phys. Rev. A

B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007).
[CrossRef]

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005).
[CrossRef] [PubMed]

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008).
[CrossRef] [PubMed]

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009).
[CrossRef] [PubMed]

R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963).
[CrossRef]

I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986).
[CrossRef] [PubMed]

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[CrossRef] [PubMed]

Other

D. Bouwmeester, A. Ekert, and A. E. Zeilinger, The Physics of Quantum Information (Springer-Verlag, New York, 2000).

H. Z. Cummins, and E. R. Pike, Photon correlation spectroscopy and light beating spectroscopy (Plenum Press, New York, 1974).

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).

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Figures (3)

Fig. 1
Fig. 1

(a) Two-photon absorption from valence band states to conduction band states in a direct gap semiconductor (e.g. GaAs). In a phototube, the electrons in the conduction band are emitted when reaching the extraction (or “vacuum”) level. Only photons arriving within time intervals shorter than the “virtual” state lifetime at midgap τH can induce TPA transitions. (b) The HBT apparatus is a Michelson interferometer with two arms: (Asph. L) is a 26 mm aspheric lens, (BS) beam splitter, (HPF) high pass filter, (M1) and (M2) mirrors and (PMT) is the GaAs photomultiplier tube. The source is based on a periodically poled lithium niobate (PPLN) crystal pumped at 780 nm by a mode-locked Ti:Sapphire laser delivering 10-ps pulses at a 80-MHz repetition rate. Estimated focal spot on the detector is 5 µm, far smaller than the detector size.

Fig. 2
Fig. 2

(a) DSOC spectrum, i.e., variation of the TPA photocounts as a function of the delay τ, of the TPA Michelson set-up of Fig. 1. The left inset shows the number of photoelectron counts as a function of the incident power. The quadratic behaviour clearly indicates a TPA process with an OPG in high gain regime. The right inset shows the spectrum of the degenerate OPG. (b) Zoom on small delay times τ which exhibits the DSOC g ( 2 ) ( τ ) features of the degenerate down-converted light. The red curve is TPALPF (τ) described in Eq. (2). (c) Theoretical modeling using the model described in the text.

Fig. 3
Fig. 3

(a) Zoom on the non-degenerate OPG part of the DSOC spectrum. The inset shows the spectrum of the non-degenerate OPG. (b) Zoom on the non-degenerated OPG part of the DSOC spectrum in the case where the idler wavelengths were attenuated by a dichroic mirror. The inset shows the spectrum after attenuation. (c) Theoretical modeling of the mutual DSOC spectrum of Fig. 3a. (d) Theoretical modeling of the mutual DSOC spectrum of Fig. 3b. The red curve is TPALPF (τ) described in Eq. (2).

Equations (18)

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g ( 2 ) ( τ ) = E ^ ( ) ( t + τ ) E ^ ( ) ( t ) E ^ ( + ) ( t ) E ^ ( + ) ( t + τ ) E ^ ( ) ( t ) E ^ ( + ) ( t ) 2
T P A L P F ( τ ) = 1 + 2 g ( 2 ) ( τ ) / g ( 2 ) ( 0 )     ;
a ^ ( z c , ω ) = [ μ ( z c , ω ) a ^ ( 0 , ω ) + i ν ( z c , ω ) a ^ ( 0 , ω p ω ) ] exp { i [ Δ k ( ω ) / 2 + k ( ω ) ] z c } .
Δ k ( ω ) = k ( ω p ) k ( ω ) k ( ω p ω ) 2 π Λ
μ ( z , ω ) = cosh [ γ ( ω ) z ] i Δ k ( ω ) 2 γ ( ω ) sinh [ γ ( ω ) z ]
ν ( z , ω ) = g ( ω ) γ ( ω ) sinh [ γ ( ω ) z ]
g ( ω ) = d eff c 2 ω [ ω p ω ] Z 0 I p n ( ω ) n ( ω p ω ) n ( ω p ) ,
γ ( ω ) = g ( ω ) 2 Δ k ( ω ) 2 / 4 .
a ^ ( z d , ω ) = 1 2 [ i M ( ω ) ( 1 + e i ω τ ) a ^ ( z c , ω ) ν ^ ( ω ) ( 1 e i ω τ ) ]
S T P C a ^ ( z d , t ) a ^ ( z d , t ) a ^ ( z d , t ) a ^ ( z d , t )
with   a ^ ( z , t ) = 1 2 π d ω a ^ ( z , ω ) e i ω t .
g ( 2 ) ( τ ) = g I ( 2 ) ( τ ) + g C ( 2 ) ( τ ) ,
g I ( 2 ) = 1 + 1 Φ 2 | 1 2 π 0 ω p d ω | ν ( z c , ω ) | 2 e i ω τ | 2 = 1 + | g ( 1 ) ( τ ) | 2
g ( 1 ) ( τ ) = Φ s Φ s + Φ i g s ( 1 ) ( τ ) + Φ i Φ s + Φ i g i ( 1 ) ( τ )
g I ( 2 ) ( τ ) = 1 + ( Φ s Φ ) 2 | g s ( 1 ) ( τ ) | 2 + ( Φ i Φ ) 2 | g i ( 1 ) ( τ ) | 2 + 2 Re [ Φ s Φ i Φ 2 | g s ( 1 ) ( τ ) | | g i ( 1 ) ( τ ) | e i ( ω s ω i ) τ ]
g C ( 2 ) ( τ ) = 1 Φ 2 | 1 2 π 0 ω p d ω μ ( z c , ω ) ν ( z c , ω ) e i [ φ ( ω ) + φ ( ω p ω ) ] e i ω τ | 2
μ ( z c , ω ) ν ( z c , ω )
g C ( 2 ) ( τ ) | 1 2 π 0 ω p d ω     [ 1 2 π d τ g ( 1 ) ( τ ) e i ω τ ] e i [ ϕ ( ω ) + ϕ ( ω p ω ) ] e i ω τ | 2

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