Abstract

We demonstrate a direct and sensitive technique for measuring two-photon absorption (TPA). An intensity-modulated femtosecond laser beam passes through a sample exhibiting TPA. A TPA signal at twice the modulation frequency is then generated and subsequently measured by a lock-in amplifier. The absolute TPA cross section of Rhodamine 6G at 800 nm is found to be 15.3±2.0×10-50 cm4 s/photon and agrees well with previously published results obtained with much higher intensity [J. Chem. Phys. 112, 9201 (2000)]. Our method may be especially useful in measuring nonlinear absorptions of nonfluorescent materials.

© 2002 Optical Society of America

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References

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  1. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  2. W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990).
    [CrossRef] [PubMed]
  3. P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
    [CrossRef]
  4. B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
    [CrossRef]
  5. S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
    [CrossRef] [PubMed]
  6. D. A. Parthenopoulos and P. M. Rentzepis, Science 245, 843 (1989).
    [CrossRef] [PubMed]
  7. C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
    [CrossRef]
  8. C. Xu and W. W. Webb, J. Opt. Soc. Am. B 13, 481 (1996).
    [CrossRef]
  9. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  10. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  11. The total TPA signal, PTPA, also contains components at dc and frequency f. If the magnitude of PTPA is much smaller than that of the incident beam, PI, the components of PTPA at dc and f are negligible compared with those of PI at the same frequencies. Hence, PT=PI-P2Fcos (2p2ft).
  12. W. Demtroder, Laser Spectroscopy (Springer, Berlin, 1996).
    [CrossRef]
  13. Equation 1 becomes P2F=(0.33Cdnp/tl f0)P21F when L> >Z0, indicating that the TPA signal is independent of the spot size.
  14. Both the experimental error (0.8 GM) and the uncertainty of the pulse temporal shape (15.3×12%=1.8 GM) contribute to the total error: 0.82+1.82=2.0 GM.
  15. P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).
  16. M. A. Albota, C. Xu, and W. W. Webb, Appl. Opt. 37, 7352 (1998).
    [CrossRef]
  17. When a laser beam is incident on the AOM driven by two RF frequencies, f1 and f2, the diffracted beam PI is amplitude modulated at f=f1-f2. A component P2F at 2f is inevitably generated by the third-order intermodulation process; i.e., PI=PAVG+P1F cos(2pft)+P2F cos(2p2ft). The ratio P2F/P1F is ~h/6, where h is the AOM diffraction efficiency D. Hecht, IEEE Trans. Sonics Ultrason. SU-24, 7 (1977). P2F/P1F is ~4×10-2 for h~25% in our setup.
    [CrossRef]
  18. S. V. Frolov and Z. V. Vardeny, Rev. Sci. Instrum. 69, 1257 (1998).
    [CrossRef]
  19. I. Kang, T. Krauss, and F. Wise, Opt. Lett. 22, 1077 (1997).
    [CrossRef] [PubMed]

2001

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

2000

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

1999

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

1998

S. V. Frolov and Z. V. Vardeny, Rev. Sci. Instrum. 69, 1257 (1998).
[CrossRef]

M. A. Albota, C. Xu, and W. W. Webb, Appl. Opt. 37, 7352 (1998).
[CrossRef]

1997

1996

1990

W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

1989

D. A. Parthenopoulos and P. M. Rentzepis, Science 245, 843 (1989).
[CrossRef] [PubMed]

1977

When a laser beam is incident on the AOM driven by two RF frequencies, f1 and f2, the diffracted beam PI is amplitude modulated at f=f1-f2. A component P2F at 2f is inevitably generated by the third-order intermodulation process; i.e., PI=PAVG+P1F cos(2pft)+P2F cos(2p2ft). The ratio P2F/P1F is ~h/6, where h is the AOM diffraction efficiency D. Hecht, IEEE Trans. Sonics Ultrason. SU-24, 7 (1977). P2F/P1F is ~4×10-2 for h~25% in our setup.
[CrossRef]

Albota, M. A.

Ananthavel, S. P.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Balaji, J.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Banerjee, S.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Barlow, S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Berland, K. M.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

Cumpston, B. H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Demtroder, W.

W. Demtroder, Laser Spectroscopy (Springer, Berlin, 1996).
[CrossRef]

DenBaars, S. P.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990).
[CrossRef] [PubMed]

Dong, C. Y.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

Dyer, D. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Ehrlich, J. E.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Erskine, L. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Frolov, S. V.

S. V. Frolov and Z. V. Vardeny, Rev. Sci. Instrum. 69, 1257 (1998).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Hecht, D.

When a laser beam is incident on the AOM driven by two RF frequencies, f1 and f2, the diffracted beam PI is amplitude modulated at f=f1-f2. A component P2F at 2f is inevitably generated by the third-order intermodulation process; i.e., PI=PAVG+P1F cos(2pft)+P2F cos(2p2ft). The ratio P2F/P1F is ~h/6, where h is the AOM diffraction efficiency D. Hecht, IEEE Trans. Sonics Ultrason. SU-24, 7 (1977). P2F/P1F is ~4×10-2 for h~25% in our setup.
[CrossRef]

Heikal, A. A.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Kang, I.

Kao, F.-J.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Kawata, S.

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

Keller, S.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Krauss, T.

Kuebler, S. M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Kumar, G. R.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Lee, I.-Y. S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Liang, J.-C.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Mack, M. P.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Maiti, S.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Marker, S. R.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Masters, B. R.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

McCord-Maughon, D.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Mishra, U.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Parthenopoulos, D. A.

D. A. Parthenopoulos and P. M. Rentzepis, Science 245, 843 (1989).
[CrossRef] [PubMed]

Perry, J. W.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Philip, R.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Qin, J.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Rentzepis, P. M.

D. A. Parthenopoulos and P. M. Rentzepis, Science 245, 843 (1989).
[CrossRef] [PubMed]

Rockel, H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Rumi, M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Sengupta, P.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

So, P. T. C.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990).
[CrossRef] [PubMed]

Sun, C.-K.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Sun, H-B.

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

Takada, K.

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

Tanaka, T.

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Vardeny, Z. V.

S. V. Frolov and Z. V. Vardeny, Rev. Sci. Instrum. 69, 1257 (1998).
[CrossRef]

Wang, J.-C.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

Webb, W. W.

Wei, T.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wise, F.

Wu, X-L

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

Xu, C.

Annu. Rev. Biomed. Eng.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, Annu. Rev. Biomed. Eng. 2, 399 (2000).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

C.-K. Sun, J.-C. Liang, J.-C. Wang, F.-J. Kao, S. Keller, M. P. Mack, U. Mishra, and S. P. DenBaars, Appl. Phys. Lett. 76, 439 (2000).
[CrossRef]

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

IEEE Trans. Sonics Ultrason.

When a laser beam is incident on the AOM driven by two RF frequencies, f1 and f2, the diffracted beam PI is amplitude modulated at f=f1-f2. A component P2F at 2f is inevitably generated by the third-order intermodulation process; i.e., PI=PAVG+P1F cos(2pft)+P2F cos(2p2ft). The ratio P2F/P1F is ~h/6, where h is the AOM diffraction efficiency D. Hecht, IEEE Trans. Sonics Ultrason. SU-24, 7 (1977). P2F/P1F is ~4×10-2 for h~25% in our setup.
[CrossRef]

J. Chem. Phys.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, J. Chem. Phys. 112, 9201 (2000).

J. Opt. Soc. Am. B

Nature

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X-L Wu, S. R. Marker, and J. W. Perry, Nature 398, 51 (1999).
[CrossRef]

S. Kawata, H-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001).
[CrossRef] [PubMed]

Opt. Lett.

Rev. Sci. Instrum.

S. V. Frolov and Z. V. Vardeny, Rev. Sci. Instrum. 69, 1257 (1998).
[CrossRef]

Science

W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990).
[CrossRef] [PubMed]

D. A. Parthenopoulos and P. M. Rentzepis, Science 245, 843 (1989).
[CrossRef] [PubMed]

Other

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

The total TPA signal, PTPA, also contains components at dc and frequency f. If the magnitude of PTPA is much smaller than that of the incident beam, PI, the components of PTPA at dc and f are negligible compared with those of PI at the same frequencies. Hence, PT=PI-P2Fcos (2p2ft).

W. Demtroder, Laser Spectroscopy (Springer, Berlin, 1996).
[CrossRef]

Equation 1 becomes P2F=(0.33Cdnp/tl f0)P21F when L> >Z0, indicating that the TPA signal is independent of the spot size.

Both the experimental error (0.8 GM) and the uncertainty of the pulse temporal shape (15.3×12%=1.8 GM) contribute to the total error: 0.82+1.82=2.0 GM.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (4)

Fig. 1
Fig. 1

Illustration of the loss modulation scheme: (a) time dependence and the power spectra of the pulse train, (b) intensity-modulated input beam, (c) TPA signal, and (d) transmission. A frequency of f=10 MHz is chosen for simplicity. The TPA signal is exaggerated.

Fig. 2
Fig. 2

Experimental setup: BS1, BS2, beam splitters; AOM1, AOM2, acousto-optic modulators; M’s, mirrors; DL, delay line; S, sample; L, sample path length; PD, photodiode. The center of the sample lies at the focal point of the beam.

Fig. 3
Fig. 3

(a) Logarithmic plot of the TPA signal, P2F, versus P1F for R6G in methanol. The slope of 1.95 indicates the quadratic dependence of P2F on P1F. (b) Ratio of P2F/P1F versus P1F. The plot shows the linear dependence predicted by Eq. (2). δ of 15.3±2.0 GM is obtained by fitting of the measured slope to Eq. (2).

Fig. 4
Fig. 4

Shot-noise limit of this scheme. The logarithmic plots of P2F/P1F versus P1F due to shot noise (broken lines, for different bandwidths) and TPA (solid lines, for different β) are shown.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

P2F=0.66Cδnτλf0arctanLn2Z0P1F2,
P2FP1F=0.66Cδnτλf0arctanLn2Z0P1F.

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