Abstract

We introduce, analyze, and experimentally demonstrate a compact semiconductor-based scheme for a femtosecond-scale partial fourth-order coherence g(4) measurements. The scheme is based on a start-stop photon-counting Hanbury Brown and Twiss (HBT) interferometry of an upconversion-based Michelson-interferometer autocorrelation. The experimental realization employs second-harmonic generation in a semiconductor quantum-well structure, which may be further integrated with the HBT setup as a miniature photonic circuit, allowing compact characterization devices for photon-pair statistics in quantum photonics and quantum information processing.

© 2010 Optical Society of America

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  6. S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
    [CrossRef]
  7. E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
    [CrossRef]
  8. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
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    [CrossRef]
  16. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2009 (6)

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

A. Hayat, P. Ginzburg, and M. Orenstein, Phys. Rev. Lett. 103, 023601 (2009).
[CrossRef] [PubMed]

A. Hayat, A. Nevet, and M. Orenstein, Phys. Rev. Lett. 102, 183002 (2009).
[CrossRef] [PubMed]

2008 (2)

A. Hayat, E. Small, Y. Elor, and M. Orenstein, Appl. Phys. Lett. 92, 181110 (2008).
[CrossRef]

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

2006 (1)

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

2005 (1)

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

2004 (1)

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

2002 (1)

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

1999 (1)

1995 (1)

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

1992 (2)

D. J. Gauthier, Q. Wu, S. E. Morin, and T. W. Mossberg, Phys. Rev. Lett. 68, 464 (1992).
[CrossRef] [PubMed]

Y. Takagi, T. Kobayashi, K. Yoshihara, and S. Imamura, Opt. Lett. 17, 658 (1992).
[CrossRef] [PubMed]

1990 (1)

T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066 (1990).
[CrossRef] [PubMed]

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

Alibart, O.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Assanto, G.

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Assmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

Baldi, P.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Bayer, M.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Beattie, N. S.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Berger, V.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Beveratos, A.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Bocquillon, E.

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
[CrossRef]

Cooper, K.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Couteau, C.

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

De Rossi, A.

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Ducci, S.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Eisaman, M. D.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Elor, Y.

A. Hayat, E. Small, Y. Elor, and M. Orenstein, Appl. Phys. Lett. 92, 181110 (2008).
[CrossRef]

Fabre, C.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
[CrossRef]

Fainman, Y.

Fan, J.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Fasel, S.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Fiore, A.

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

Forchel, A.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Fukuzawa, T.

T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066 (1990).
[CrossRef] [PubMed]

Gauthier, D. J.

D. J. Gauthier, Q. Wu, S. E. Morin, and T. W. Mossberg, Phys. Rev. Lett. 68, 464 (1992).
[CrossRef] [PubMed]

Gies, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Ginzburg, P.

A. Hayat, P. Ginzburg, and M. Orenstein, Phys. Rev. Lett. 103, 023601 (2009).
[CrossRef] [PubMed]

Gisin, N.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Godard, A.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
[CrossRef]

Goldschmidt, E. A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

Hayat, A.

A. Hayat, A. Nevet, and M. Orenstein, Phys. Rev. Lett. 102, 183002 (2009).
[CrossRef] [PubMed]

A. Hayat, P. Ginzburg, and M. Orenstein, Phys. Rev. Lett. 103, 023601 (2009).
[CrossRef] [PubMed]

A. Hayat, E. Small, Y. Elor, and M. Orenstein, Appl. Phys. Lett. 92, 181110 (2008).
[CrossRef]

Höfling, S.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Hommel, D.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Hong, J. M.

T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066 (1990).
[CrossRef] [PubMed]

Hvam, J. M.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

Imamura, S.

Jahnke, F.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Kalden, J.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Kardynal, B. E.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Kistner, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Kobayashi, T.

Kruse, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Laflamme, R.

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

Lanco, L.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

Leo, G.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Likforman, J. -P.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

Lobo, C. J.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

Marcadet, X.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

Mazurenko, Y. T.

Mendez, E. E.

T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066 (1990).
[CrossRef] [PubMed]

Migdall, A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Morin, S. E.

D. J. Gauthier, Q. Wu, S. E. Morin, and T. W. Mossberg, Phys. Rev. Lett. 68, 464 (1992).
[CrossRef] [PubMed]

Mossberg, T. W.

D. J. Gauthier, Q. Wu, S. E. Morin, and T. W. Mossberg, Phys. Rev. Lett. 68, 464 (1992).
[CrossRef] [PubMed]

Nevet, A.

A. Hayat, A. Nevet, and M. Orenstein, Phys. Rev. Lett. 102, 183002 (2009).
[CrossRef] [PubMed]

Oba, K.

Orenstein, M.

A. Hayat, A. Nevet, and M. Orenstein, Phys. Rev. Lett. 102, 183002 (2009).
[CrossRef] [PubMed]

A. Hayat, P. Ginzburg, and M. Orenstein, Phys. Rev. Lett. 103, 023601 (2009).
[CrossRef] [PubMed]

A. Hayat, E. Small, Y. Elor, and M. Orenstein, Appl. Phys. Lett. 92, 181110 (2008).
[CrossRef]

Pepper, M.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Polyakov, S. V.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Ravaro, M.

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Razavi, M.

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

Reitzenstein, S.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Ritchie, D. A.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Rosencher, E.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Nat. Phys. 5, 267 (2009).
[CrossRef]

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

Schneider, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Seurin, Y.

M. Ravaro, Y. Seurin, S. Ducci, G. Leo, V. Berger, A. De Rossi, and G. Assanto, J. Appl. Phys. 98, 063103 (2005).
[CrossRef]

Shields, A. J.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Small, E.

A. Hayat, E. Small, Y. Elor, and M. Orenstein, Appl. Phys. Lett. 92, 181110 (2008).
[CrossRef]

Stevenson, R. M.

Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295, 102 (2002).
[CrossRef]

Sun, P.

Takagi, Y.

Tanzilli, S.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, New J. Phys. 6, 163 (2004).
[CrossRef]

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

van der Poel, M.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

van Houwelingen, J. A. W.

L. Lanco, S. Ducci, J.-P. Likforman, X. Marcadet, J. A. W. van Houwelingen, H. Zbinden, G. Leo, and V. Berger, Phys. Rev. Lett. 97, 173901 (2006).
[CrossRef] [PubMed]

Veit, F.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, Science 325, 297 (2009).
[CrossRef] [PubMed]

Vinter, B.

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

Weihs, G.

E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Phys. Rev. A 79, 035801 (2009).
[CrossRef]

Weill, D.

A. Fiore, E. Rosencher, B. Vinter, D. Weill, and V. Berger, Phys. Rev. B 51, 13192 (1995).
[CrossRef]

Wiersig, J.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
[CrossRef] [PubMed]

Wu, Q.

D. J. Gauthier, Q. Wu, S. E. Morin, and T. W. Mossberg, Phys. Rev. Lett. 68, 464 (1992).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Ultrafast partial g ( 4 ) measurement setup. The laser pulses enter the waveguide following a Michelson interferometer with a femtosecond resolution. SHG output is collected, filtered, and measured through an HBT setup containing two Si APD detectors connected to a time-interval (start-stop) counter.

Fig. 2
Fig. 2

g ( 2 ) measurement of the input laser pulses. The blue squares represent the measurements by overall counts in one of the Si APD detectors versus time delay ( τ 1 ) in the Michelson interferometer. The black solid curve represents a theoretical fit according to FWHM of 440 fs of sech pulses. The inset is the measured spectrum of the SHG (blue triangles) compared to the measured femtosecond laser spectrum plotted versus half the wavelength (red solid curve).

Fig. 3
Fig. 3

g ̃ ( 4 ) ( τ 1 , τ 2 ) measurement versus the time delay in the Michelson interferometer ( τ 1 , femtosecond resolution) and the time delay between Si APD detections ( τ 2 , nanosecond resolution). Calculated g ̃ ( 4 ) for τ 2 = 0 is given according to Eq. (9) (red solid curve).

Equations (9)

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g ( 4 ) ( t 1 , t 2 , t 3 , t 4 ) = E ̂ ( ) ( t 1 ) E ̂ ( ) ( t 2 ) E ̂ ( ) ( t 3 ) E ̂ ( ) ( t 4 ) E ̂ ( + ) ( t 4 ) E ̂ ( + ) ( t 3 ) E ̂ ( + ) ( t 2 ) E ̂ ( + ) ( t 1 ) E ̂ ( ) ( t 1 ) E ̂ ( + ) ( t 1 ) E ̂ ( ) ( t 2 ) E ̂ ( + ) ( t 2 ) E ̂ ( ) ( t 3 ) E ̂ ( + ) ( t 3 ) E ̂ ( ) ( t 4 ) E ̂ ( + ) ( t 4 ) ,
g ( 2 ) ( 0 ) = 2 | a a a a | 2 / 2 | a a | 2 2 = 1 / 2 ,
g ( 4 ) ( 0 ) = 2 | a a a a a a a a | 2 / 2 | a a | 2 4 = 0.
E total ( t ) = E 1 ( t ) e i ω t + E 1 ( t τ 1 ) e i ω t e i ω τ 1 ,
E 2 ( t ) e i 2 ω t = η SH ( E 1 2 ( t ) + E 1 2 ( t τ 1 ) e i 2 ω τ 1 + 2 E 1 ( t ) E 1 ( t τ 1 ) e i ω τ 1 ) e i 2 ω t ,
I 2 ( τ 1 ) = η SH ( I 1 2 ( t ) + I 1 2 ( t τ 1 ) + 4 I 1 ( t ) I 1 ( t τ 1 ) ) ,
g ( 2 ) ( τ 1 ) = I 1 ( t ) I 1 ( t τ 1 ) I 1 ( t ) I 1 ( t τ 1 ) = I 2 ( τ 1 ) / η SH 4 I 1 ( t ) 2 I 1 2 ( t ) 2 I 1 ( t ) 2 ,
G ̃ ( 4 ) ( τ 1 , τ 2 ) = I 2 ( t , τ 1 ) I 2 ( t τ 2 , τ 1 ) ,
g ̃ ( 4 ) ( τ 1 , 0 ) = 2 η SH 2 { I 1 4 ( t ) I 1 ( t ) 4 + 2 [ 4 g ( 4 ) ( 0 , 0 , 0 , τ 1 ) + 4 g ( 4 ) ( τ 1 , τ 1 , τ 1 , 0 ) + 9 g ( 4 ) ( 0 , 0 , τ 1 , τ 1 ) ] } ,

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