Abstract

A formula for the photoelectron-counting distribution for a two-photon detector is derived quantum mechanically assuming that the ionising transitions in the atoms of the detector take place through the simultaneous absorption of two photons. It is assumed that the incident light is quasimonochromatic. It is shown that the distribution of the photoelectrons is given by the average of a Poisson distribution, the parameter of the distribution being proportional to the time integral of the square of the instantaneous light intensity. Counting distributions for the thermal (gaussian) light and for some models of laser light are obtained for the limiting case when the counting-time interval <i>T</i> is short compared to the coherence time <i>T<sub>c</sub></i> of the light. An approximate formula for arbitrary time intervals for the counting distribution of thermal light is also proposed.

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  1. M. Goeppert-Mayer, Ann. Physik 9, 273 (1931).
  2. R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964); S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965); E. M. Logothetics and P. L. Hartman, 18, 581 (1967); and some of the references quoted in these papers.
  3. S. Kielich, Acta Phys. Polon. 30, 393 (1966); R. Wallace, Mol. Phys. 11, 457 (1966); and some of the references quoted therein.
  4. A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965); H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).
  5. L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).
  6. B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.
  7. G. S. Agarwal, Phys. Rev. 177, 400 (1969).
  8. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
  9. L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.
  10. J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), p. 211, and some of the references quoted in this review article.
  11. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quanwon Optics (W. A. Benjamin, Inc., New York, 1968).
  12. L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); 74, 233 (1959).
  13. The original derivation of Mandel was based on semiclassical arguments. The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches 1964, C. Dewitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65. See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
  14. E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964); G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).
  15. N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966); Phys. Rev. 151, 1076 (1966); See also R. Wallace, Phys. Rev. Letters 17, 397 (1966); Mol. Phys. 11, 457 (1966).
  16. The contribution of the A2 term is also easily taken into account. It may be shown that the final result is still given by Eq. (30) except that the quantum-efficiency parameter a changes slightly. The new value of α is [equation] Here 〈f|0〉 represents the scalar product between the ground-state wavefunction and the final-state (free-electron) wave-function.
  17. E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963). A detailed discussion of the diagonal representation is given in Ref. 11. See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).
  18. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.
  19. L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
  20. The representation of V in the form Eq. (24) assumes that the incident light is plane polarized. However, the results are easily generalized to the case of partially polarized light. If the incident light is of thermal origin, then as well known we may consider it as the superposition of two linearly polarized, statistically independent components with average intensities equal to ½(1+P)〈I〉 and ½(1−P)〈I〉, respectively, where P is the degree of polarization. It is also assumed that the condition of cross-spectral purity is satisfied. In this case, the final result (30) remains unchanged. (See also Sec. IIIA.) For the partially polarized light of nonthermal origin, the expression for the photoelectron counting distribution may be shown to be [equation] where [equation] and [equation]
  21. See Ref. 9, Appendix B. Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).
  22. See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.
  23. See, for example, Ref. 8, Eq. (4.41).
  24. A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169; R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, Jr., and V. Hochuli, Phys. Letters 25A, 272 (1967).
  25. G. Bedard, Phys. Letters 24A 613 (1967).
  26. H. Risken, Z. Physik 186, 85 (1965).
  27. G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
  28. S. O. Rice, Bell System Tech. J. 24, 46 (1945).

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. 177, 400 (1969).

Armstrong, J. A.

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169; R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, Jr., and V. Hochuli, Phys. Letters 25A, 272 (1967).

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), p. 211, and some of the references quoted in this review article.

Bebb, H. Barry

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965); H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).

Bedard, G.

G. Bedard, Phys. Letters 24A 613 (1967).

Bédard, G.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

Braunstein, R.

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964); S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965); E. M. Logothetics and P. L. Hartman, 18, 581 (1967); and some of the references quoted in these papers.

Chang, J. C.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).

Cohan, N. V.

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966); Phys. Rev. 151, 1076 (1966); See also R. Wallace, Phys. Rev. Letters 17, 397 (1966); Mol. Phys. 11, 457 (1966).

Glauber, R. J.

The original derivation of Mandel was based on semiclassical arguments. The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches 1964, C. Dewitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65. See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).

Goeppert-Mayer, M.

M. Goeppert-Mayer, Ann. Physik 9, 273 (1931).

Gold, A.

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965); H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).

Gradshteyn, I. S.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

Hameka, H. V.

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966); Phys. Rev. 151, 1076 (1966); See also R. Wallace, Phys. Rev. Letters 17, 397 (1966); Mol. Phys. 11, 457 (1966).

Keldysh, L. V.

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).

Kielich, S.

S. Kielich, Acta Phys. Polon. 30, 393 (1966); R. Wallace, Mol. Phys. 11, 457 (1966); and some of the references quoted therein.

Klauder, J. R.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quanwon Optics (W. A. Benjamin, Inc., New York, 1968).

Mandel, L.

L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); 74, 233 (1959).

See Ref. 9, Appendix B. Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).

Mehta, C. L.

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964); G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).

Mollow, B. R.

B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.

Ockman, N.

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964); S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965); E. M. Logothetics and P. L. Hartman, 18, 581 (1967); and some of the references quoted in these papers.

Rice, S. O.

S. O. Rice, Bell System Tech. J. 24, 46 (1945).

Risken, H.

H. Risken, Z. Physik 186, 85 (1965).

Ryzhik, I. M.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

Smith, A. W.

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), p. 211, and some of the references quoted in this review article.

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169; R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, Jr., and V. Hochuli, Phys. Letters 25A, 272 (1967).

Sudarshan, E. C. G.

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963). A detailed discussion of the diagonal representation is given in Ref. 11. See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quanwon Optics (W. A. Benjamin, Inc., New York, 1968).

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964); G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

Other (28)

M. Goeppert-Mayer, Ann. Physik 9, 273 (1931).

R. Braunstein and N. Ockman, Phys. Rev. 134, A499 (1964); S. Yatsiv, W. G. Wagner, G. S. Picus, and F. J. McClung, Phys. Rev. Letters 15, 614 (1965); E. M. Logothetics and P. L. Hartman, 18, 581 (1967); and some of the references quoted in these papers.

S. Kielich, Acta Phys. Polon. 30, 393 (1966); R. Wallace, Mol. Phys. 11, 457 (1966); and some of the references quoted therein.

A. Gold and H. Barry Bebb, Phys. Rev. Letters 14, 60 (1965); H. Barry Bebb and A. Gold, Phys. Rev. 143, 1 (1966).

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1966).

B. R. Mollow, Phys. Rev. 175, 1555 (1968) and some of the references quoted therein.

G. S. Agarwal, Phys. Rev. 177, 400 (1969).

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

L. Mandel in Progress in Optics II, E. Wolf, Ed. (North–Holland Publ. Co., Amsterdam, 1963), p. 181.

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), p. 211, and some of the references quoted in this review article.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quanwon Optics (W. A. Benjamin, Inc., New York, 1968).

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); 74, 233 (1959).

The original derivation of Mandel was based on semiclassical arguments. The corresponding quantum-mechanical derivation of the formula was given by R. J. Glauber, in Quantum Optics and Electronics, Les Houches 1964, C. Dewitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 65. See also P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964); G. Bédard, J. Opt. Soc. Am. 57, 1201 (1967).

N. V. Cohan and H. V. Hameka, Phys. Rev. Letters 16, 478 (1966); Phys. Rev. 151, 1076 (1966); See also R. Wallace, Phys. Rev. Letters 17, 397 (1966); Mol. Phys. 11, 457 (1966).

The contribution of the A2 term is also easily taken into account. It may be shown that the final result is still given by Eq. (30) except that the quantum-efficiency parameter a changes slightly. The new value of α is [equation] Here 〈f|0〉 represents the scalar product between the ground-state wavefunction and the final-state (free-electron) wave-function.

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963). A detailed discussion of the diagonal representation is given in Ref. 11. See also R. J. Glauber, Phys. Rev. 131, 2766 (1963).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1965), 3rd ed., Ch. X.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).

The representation of V in the form Eq. (24) assumes that the incident light is plane polarized. However, the results are easily generalized to the case of partially polarized light. If the incident light is of thermal origin, then as well known we may consider it as the superposition of two linearly polarized, statistically independent components with average intensities equal to ½(1+P)〈I〉 and ½(1−P)〈I〉, respectively, where P is the degree of polarization. It is also assumed that the condition of cross-spectral purity is satisfied. In this case, the final result (30) remains unchanged. (See also Sec. IIIA.) For the partially polarized light of nonthermal origin, the expression for the photoelectron counting distribution may be shown to be [equation] where [equation] and [equation]

See Ref. 9, Appendix B. Similar arguments are summarized in L. Mandel, Phys. Rev. 152, 438 (1966).

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1064.

See, for example, Ref. 8, Eq. (4.41).

A. W. Smith and J. A. Armstrong, Phys. Rev. Letters 16, 1169; R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, Jr., and V. Hochuli, Phys. Letters 25A, 272 (1967).

G. Bedard, Phys. Letters 24A 613 (1967).

H. Risken, Z. Physik 186, 85 (1965).

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).

S. O. Rice, Bell System Tech. J. 24, 46 (1945).

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