Abstract

We present a study of coherent anti-Stokes Raman scattering (CARS) under focused beam excitation in a planar Fabry–Perot cavity using an image-dipole formalism. We give a comprehensive description of forward- and backward-CARS signal generation by introducing the expressions of the nonlinear induced polarization in the cavity as a function of their counterpart in free space. We show that the cavity gives rise to a backward-CARS signal and allows working with low numerical aperture collection objectives. We finally discuss the influence of the scatterer position in the cavity on the detected signal.

© 2009 Optical Society of America

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  1. B. Schrader, Infrared and Raman Spectroscopy (VCH, 1995).
    [CrossRef]
  2. J. P. Coffinet and F. de Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60-64 (1969).
    [CrossRef]
  3. J. J. Wynne, “Nonlinear optical spectroscopy of χ(3) in LiNbO3,” Phys. Rev. Lett. 29, 650-653 (1972).
    [CrossRef]
  4. P. R. Regnier and J. P.-E. Taran, “On the possibility of measuring gas concentrations by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240-242 (1973).
    [CrossRef]
  5. R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
    [CrossRef]
  6. M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman scattering microscope,” Opt. Lett. 7, 350-352 (1982).
    [CrossRef] [PubMed]
  7. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
    [CrossRef]
  8. J. Cooney and A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218-220 (1982).
    [CrossRef] [PubMed]
  9. S.-X. Qian, J. B. Snow, and R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-size droplets,” Opt. Lett. 10, 499-501 (1985).
    [CrossRef] [PubMed]
  10. H. Chew, D.-S. Wang, and M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56-66 (1984).
    [CrossRef]
  11. T.-W. Koo, S. Chan, and A. A. Berlin, “Single-molecule detection of biomolecules by surface-enhanced coherent anti-Stokes Raman scattering,” Opt. Lett. 30, 1024-1026 (2005).
    [CrossRef] [PubMed]
  12. C. Fabry, “Sur la localisation des franges d'interférences produites par les miroirs de Fresnel,” Acad. Sci., Paris, C. R. 110, 455-457 (1890).
  13. E. M. Purcell, “Spontaneous emission probabilities at radiofrequencies,” Phys. Rev. 69, 681 (1946).
    [CrossRef]
  14. P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
    [CrossRef]
  15. F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
    [CrossRef] [PubMed]
  16. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233-236 (1981).
    [CrossRef]
  17. R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
    [CrossRef] [PubMed]
  18. G. Gabrielse and H. Dehmelt, “Observation of inhibited spontaneous emission,” Phys. Rev. Lett. 55, 67-70 (1985).
    [CrossRef] [PubMed]
  19. D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
    [CrossRef] [PubMed]
  20. D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
    [CrossRef] [PubMed]
  21. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
    [CrossRef] [PubMed]
  22. F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
    [CrossRef] [PubMed]
  23. A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
    [CrossRef] [PubMed]
  24. Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
    [CrossRef] [PubMed]
  25. I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
    [CrossRef]
  26. H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
    [CrossRef]
  27. C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
    [CrossRef]
  28. H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
    [CrossRef]
  29. H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
    [CrossRef]
  30. H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356-2368 (1996).
    [CrossRef] [PubMed]
  31. H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
    [CrossRef]
  32. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910-916 (1989).
    [CrossRef]
  33. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus Acad. Sci. (Paris) 110, 1251-1253 (1890).
  34. L. G. Gouy, “Sur la propagation anormale des ondes,” Acad. Sci., Paris, C. R. 111, 33-35 (1890).
  35. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  36. H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792-1796 (1969).
    [CrossRef]
  37. P. W. Milonni and P. L. Knight, “Spontaneous emission between mirrors,” Opt. Commun. 9, 119-122 (1973).
    [CrossRef]
  38. J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
    [CrossRef]
  39. D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
    [CrossRef]
  40. M. Marrocco, “Coherent anti-Stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Phys. 17, 935-941 (2007).
    [CrossRef]
  41. D. Gachet, F. Billard, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a microcavity,” Opt. Lett. 34, 1789-1791 (2009).
    [CrossRef] [PubMed]
  42. H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
    [CrossRef]
  43. D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655-1666 (2008).
    [CrossRef]
  44. L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685-1694 (2000).
    [CrossRef]
  45. J.-X. Cheng and X. S. Xie, “Green's function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604-1610 (2002).
    [CrossRef]
  46. J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363-1375 (2002).
    [CrossRef]
  47. A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
    [CrossRef]
  48. M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447-4463 (1974).
    [CrossRef]
  49. A. Volkmer, “Vibrational imaging and microspectrometries based on coherent anti-Stokes Raman scattering microscopy,” J. Phys. D 38, R59-R81 (2005).
    [CrossRef]
  50. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  51. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
  52. The mirrors considered here made of a succession of high (H) and low (L) refractive index hafnium oxide HfO2(n=2.207) and silica SiO2(n=1.456) layers deposed on a silica substrate. The thicknesses (in nanometers) of the different layers are given by (H) 107; (L) 272; (H) 88; (L) 109; (H) 206; (L) 119; (H) 64; (L) 298; (H) 314; (L) 36; (H) 374; (L) 201; (H) 127; (L) 212; (H) 111; (L) 209; (H) 128; (L) 95.
  53. The orientation of the nonlinear induced dipoles is fixed by the excitation beams and the Raman depolarization ratio following Eq. .
  54. A. Kastler, “Atomes à l'intérieur d'un interféromètre Perot-Fabry,” Appl. Opt. 1, 17-24 (1962).
    [CrossRef]
  55. This is in good agreement with our medium reflectivity mirrors that alter marginally the emitter lifetime inside the cavity.
  56. G. Björk and Y. Yamamoto, “Spontaneous emission in dielectric planar microcavities,” in Spontaneous Emission and Laser. Oscillation in Microcavities, H.Yokoyama and K.Ujihara, eds. (Academic, 1995).
  57. D. Gachet, N. Sandeau, and H. Rigneault, “Far-field radiation pattern in coherent anti-Stokes Raman scattering (CARS) microscopy,” in Biomedical Vibrational Spectroscopy III: Advances in Research and Industry, A.Mahadevan-Jansen and W.H.Peetrich, eds., Proc. SPIE 6093, 609309 (2006).
    [CrossRef]
  58. Computing two-dimensional patterns such as the ones plotted in Fig. takes about 30 days with a PC working with a 2.5 GHz processor.
  59. Note that, as pointed out previously, because the ring structure does not rigorously follow a revolution symmetry (see Fig. ), the undergone jumps are smoother than in Fig. .
  60. M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spectrosc. (to be published).

2009

2008

2007

M. Marrocco, “Coherent anti-Stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Phys. 17, 935-941 (2007).
[CrossRef]

2006

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
[CrossRef]

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

2005

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

T.-W. Koo, S. Chan, and A. A. Berlin, “Single-molecule detection of biomolecules by surface-enhanced coherent anti-Stokes Raman scattering,” Opt. Lett. 30, 1024-1026 (2005).
[CrossRef] [PubMed]

A. Volkmer, “Vibrational imaging and microspectrometries based on coherent anti-Stokes Raman scattering microscopy,” J. Phys. D 38, R59-R81 (2005).
[CrossRef]

2002

J.-X. Cheng and X. S. Xie, “Green's function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604-1610 (2002).
[CrossRef]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363-1375 (2002).
[CrossRef]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

2001

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

2000

1999

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
[CrossRef]

1998

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
[CrossRef]

H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
[CrossRef]

1996

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

1995

A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
[CrossRef] [PubMed]

1993

F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
[CrossRef] [PubMed]

1992

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

1991

J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
[CrossRef]

1989

1987

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

1985

R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
[CrossRef] [PubMed]

G. Gabrielse and H. Dehmelt, “Observation of inhibited spontaneous emission,” Phys. Rev. Lett. 55, 67-70 (1985).
[CrossRef] [PubMed]

S.-X. Qian, J. B. Snow, and R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-size droplets,” Opt. Lett. 10, 499-501 (1985).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

1984

1983

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

1982

1981

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

1976

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
[CrossRef]

1974

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447-4463 (1974).
[CrossRef]

R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
[CrossRef]

1973

P. R. Regnier and J. P.-E. Taran, “On the possibility of measuring gas concentrations by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240-242 (1973).
[CrossRef]

P. W. Milonni and P. L. Knight, “Spontaneous emission between mirrors,” Opt. Commun. 9, 119-122 (1973).
[CrossRef]

1972

J. J. Wynne, “Nonlinear optical spectroscopy of χ(3) in LiNbO3,” Phys. Rev. Lett. 29, 650-653 (1972).
[CrossRef]

1969

J. P. Coffinet and F. de Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60-64 (1969).
[CrossRef]

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792-1796 (1969).
[CrossRef]

1962

1959

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

1946

E. M. Purcell, “Spontaneous emission probabilities at radiofrequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

1890

C. Fabry, “Sur la localisation des franges d'interférences produites par les miroirs de Fresnel,” Acad. Sci., Paris, C. R. 110, 455-457 (1890).

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus Acad. Sci. (Paris) 110, 1251-1253 (1890).

L. G. Gouy, “Sur la propagation anormale des ondes,” Acad. Sci., Paris, C. R. 111, 33-35 (1890).

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Aktsipetrov, O. A.

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

Arakawa, Y.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

Becker, C.

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

Begley, R. F.

R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
[CrossRef]

Benisty, H.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
[CrossRef]

H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
[CrossRef]

Berlin, A. A.

Bethune, D. S.

Billard, F.

Björk, G.

G. Björk and Y. Yamamoto, “Spontaneous emission in dielectric planar microcavities,” in Spontaneous Emission and Laser. Oscillation in Microcavities, H.Yokoyama and K.Ujihara, eds. (Academic, 1995).

Bloembergen, N.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
[CrossRef]

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447-4463 (1974).
[CrossRef]

Byer, R. L.

R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
[CrossRef]

Cairo, F.

F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
[CrossRef] [PubMed]

Chan, S.

Chan, S. K.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Chang, R. K.

Cheng, J. -X.

Chew, H.

Childs, J. J.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

Coffinet, J. P.

J. P. Coffinet and F. de Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60-64 (1969).
[CrossRef]

Cooney, J.

de Martini, F.

F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
[CrossRef] [PubMed]

J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
[CrossRef]

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

J. P. Coffinet and F. de Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60-64 (1969).
[CrossRef]

De Neve, H.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
[CrossRef]

Dehmelt, H.

G. Gabrielse and H. Dehmelt, “Observation of inhibited spontaneous emission,” Phys. Rev. Lett. 55, 67-70 (1985).
[CrossRef] [PubMed]

Dowling, J. P.

J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
[CrossRef]

Dumeige, Y.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Duncan, M. D.

Fabry, C.

C. Fabry, “Sur la localisation des franges d'interférences produites par les miroirs de Fresnel,” Acad. Sci., Paris, C. R. 110, 455-457 (1890).

Fainstein, A.

A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
[CrossRef] [PubMed]

Fedyanin, A. A.

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

Feld, M. S.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

Gabrielse, G.

G. Gabrielse and H. Dehmelt, “Observation of inhibited spontaneous emission,” Phys. Rev. Lett. 55, 67-70 (1985).
[CrossRef] [PubMed]

Gachet, D.

D. Gachet, F. Billard, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a microcavity,” Opt. Lett. 34, 1789-1791 (2009).
[CrossRef] [PubMed]

D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655-1666 (2008).
[CrossRef]

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
[CrossRef]

D. Gachet, N. Sandeau, and H. Rigneault, “Far-field radiation pattern in coherent anti-Stokes Raman scattering (CARS) microscopy,” in Biomedical Vibrational Spectroscopy III: Advances in Research and Industry, A.Mahadevan-Jansen and W.H.Peetrich, eds., Proc. SPIE 6093, 609309 (2006).
[CrossRef]

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus Acad. Sci. (Paris) 110, 1251-1253 (1890).

L. G. Gouy, “Sur la propagation anormale des ondes,” Acad. Sci., Paris, C. R. 111, 33-35 (1890).

Goy, P.

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

Gross, A.

Gross, M.

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

Haroche, S.

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

Harvey, A. B.

R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

Heinzen, D. J.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

Hilfer, E. S.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
[CrossRef] [PubMed]

Holtom, G. R.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
[CrossRef]

Hulet, R. G.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
[CrossRef] [PubMed]

Innocenti, G.

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

Ishikawa, A.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jacobovitz, G. R.

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

Jusserand, B.

A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
[CrossRef] [PubMed]

Kastler, A.

Kerker, M.

Kleppner, D.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
[CrossRef] [PubMed]

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

Knight, P. L.

P. W. Milonni and P. L. Knight, “Spontaneous emission between mirrors,” Opt. Commun. 9, 119-122 (1973).
[CrossRef]

Koo, T. -W.

Levenson, A.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Levenson, M. D.

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447-4463 (1974).
[CrossRef]

Lotem, H.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
[CrossRef]

Lu, W.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Lynch, R. T.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
[CrossRef]

Manuccia, T. J.

Marrocco, M.

M. Marrocco, “Coherent anti-Stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Phys. 17, 935-941 (2007).
[CrossRef]

M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spectrosc. (to be published).

Mataloni, P.

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

Mayer, M.

H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
[CrossRef]

Mériadec, C.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Mertz, J.

Meschede, D.

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

Milonni, P. W.

P. W. Milonni and P. L. Knight, “Spontaneous emission between mirrors,” Opt. Commun. 9, 119-122 (1973).
[CrossRef]

Monneret, S.

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

Monnier, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Morawitz, H.

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792-1796 (1969).
[CrossRef]

Moreaux, L.

Müller, G.

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

Murchikova, E. M.

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

Murra, D.

F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
[CrossRef] [PubMed]

Nichelatti, E.

M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spectrosc. (to be published).

Nishioka, M.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radiofrequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Qian, S. -X.

Raimond, J. M.

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

Regnier, P. R.

P. R. Regnier and J. P.-E. Taran, “On the possibility of measuring gas concentrations by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240-242 (1973).
[CrossRef]

Reintjes, J.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rigneault, H.

D. Gachet, F. Billard, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a microcavity,” Opt. Lett. 34, 1789-1791 (2009).
[CrossRef] [PubMed]

D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655-1666 (2008).
[CrossRef]

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
[CrossRef]

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

D. Gachet, N. Sandeau, and H. Rigneault, “Far-field radiation pattern in coherent anti-Stokes Raman scattering (CARS) microscopy,” in Biomedical Vibrational Spectroscopy III: Advances in Research and Industry, A.Mahadevan-Jansen and W.H.Peetrich, eds., Proc. SPIE 6093, 609309 (2006).
[CrossRef]

Sagnes, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Sandeau, N.

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
[CrossRef]

D. Gachet, N. Sandeau, and H. Rigneault, “Far-field radiation pattern in coherent anti-Stokes Raman scattering (CARS) microscopy,” in Biomedical Vibrational Spectroscopy III: Advances in Research and Industry, A.Mahadevan-Jansen and W.H.Peetrich, eds., Proc. SPIE 6093, 609309 (2006).
[CrossRef]

Sandre, O.

Schrader, B.

B. Schrader, Infrared and Raman Spectroscopy (VCH, 1995).
[CrossRef]

Scully, M. O.

J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
[CrossRef]

Snow, J. B.

Soboleva, I. V.

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

Sou, I. K.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Stanley, R.

H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
[CrossRef]

Taran, J. P.-E.

P. R. Regnier and J. P.-E. Taran, “On the possibility of measuring gas concentrations by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240-242 (1973).
[CrossRef]

Thierry-Mieg, V.

A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
[CrossRef] [PubMed]

Thomas, J. E.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

Vidakovic, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Volkmer, A.

A. Volkmer, “Vibrational imaging and microspectrometries based on coherent anti-Stokes Raman scattering microscopy,” J. Phys. D 38, R59-R81 (2005).
[CrossRef]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363-1375 (2002).
[CrossRef]

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

von Freymann, G.

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

Walther, H.

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

Wang, D. -S.

Wegener, M.

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

Weisbuch, C.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
[CrossRef]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Wong, G. K. L.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Wong, K. S.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Wong, S.

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

Wynne, J. J.

J. J. Wynne, “Nonlinear optical spectroscopy of χ(3) in LiNbO3,” Phys. Rev. Lett. 29, 650-653 (1972).
[CrossRef]

Xie, P.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Xie, X. S.

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363-1375 (2002).
[CrossRef]

J.-X. Cheng and X. S. Xie, “Green's function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604-1610 (2002).
[CrossRef]

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
[CrossRef]

Yamamoto, Y.

G. Björk and Y. Yamamoto, “Spontaneous emission in dielectric planar microcavities,” in Spontaneous Emission and Laser. Oscillation in Microcavities, H.Yokoyama and K.Ujihara, eds. (Academic, 1995).

Yang, H.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Zhang, Z. -Q.

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
[CrossRef]

Acad. Sci., Paris, C. R.

C. Fabry, “Sur la localisation des franges d'interférences produites par les miroirs de Fresnel,” Acad. Sci., Paris, C. R. 110, 455-457 (1890).

L. G. Gouy, “Sur la propagation anormale des ondes,” Acad. Sci., Paris, C. R. 111, 33-35 (1890).

Appl. Opt.

Appl. Phys. Lett.

I. V. Soboleva, E. M. Murchikova, A. A. Fedyanin, and O. A. Aktsipetrov, “Second- and third-harmonic generation in birefringent photonic crystals and microcavities based on anisotropic porous silicon,” Appl. Phys. Lett. 87, 241110 (2005).
[CrossRef]

C. Becker, M. Wegener, S. Wong, and G. von Freymann, “Phase-matched nondegenerate four-wave mixing in one-dimensional photonic crystals,” Appl. Phys. Lett. 89, 131122 (2006).
[CrossRef]

P. R. Regnier and J. P.-E. Taran, “On the possibility of measuring gas concentrations by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240-242 (1973).
[CrossRef]

R. F. Begley, A. B. Harvey, and R. L. Byer, “Coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 25, 387-390 (1974).
[CrossRef]

Comptes Rendus Acad. Sci. (Paris)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus Acad. Sci. (Paris) 110, 1251-1253 (1890).

IEEE J. Quantum Electron.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part I: basic concepts and analytical trend,” IEEE J. Quantum Electron. 34, 1612-1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction--Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632-1643 (1998).
[CrossRef]

H. Yang, P. Xie, S. K. Chan, W. Lu, Z.-Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semiconductor photonic crystals,” IEEE J. Quantum Electron. 42, 447-452 (2006).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Publ. 1, 06013 (2006).
[CrossRef]

J. Opt. Soc. Am. A Opt. Image Sci. Vis.

H. Benisty, R. Stanley, and M. Mayer, “Method of source terms for dipole emission modification in modes of arbitrary planar structures,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 15, 1192-1201 (1998).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. D

A. Volkmer, “Vibrational imaging and microspectrometries based on coherent anti-Stokes Raman scattering microscopy,” J. Phys. D 38, R59-R81 (2005).
[CrossRef]

J. Raman Spectrosc.

M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spectrosc. (to be published).

Laser Phys.

M. Marrocco, “Coherent anti-Stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Phys. 17, 935-941 (2007).
[CrossRef]

Opt. Commun.

P. W. Milonni and P. L. Knight, “Spontaneous emission between mirrors,” Opt. Commun. 9, 119-122 (1973).
[CrossRef]

J. P. Dowling, M. O. Scully, and F. de Martini, “Radiation pattern of a classical dipole in a cavity,” Opt. Commun. 82, 415-419 (1991).
[CrossRef]

Opt. Lett.

Phys. Rev.

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792-1796 (1969).
[CrossRef]

E. M. Purcell, “Spontaneous emission probabilities at radiofrequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Phys. Rev. A

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748-1755 (1976).
[CrossRef]

Phys. Rev. B

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447-4463 (1974).
[CrossRef]

Phys. Rev. Lett.

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of cavity-enhanced single-atom spontaneous emission,” Phys. Rev. Lett. 50, 1903-1906 (1983).
[CrossRef]

F. de Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, “Anomalous spontaneous emission time in a microscopic optical cavity,” Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

R. G. Hulet, E. S. Hilfer, and D. Kleppner, “Inhibitedspontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137-2140 (1985).
[CrossRef] [PubMed]

G. Gabrielse and H. Dehmelt, “Observation of inhibited spontaneous emission,” Phys. Rev. Lett. 55, 67-70 (1985).
[CrossRef] [PubMed]

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320-1323 (1987).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

F. Cairo, F. de Martini, and D. Murra, “QED-vacuum confinement of inelastic quantum scattering at optical frequencies: a new perspective in Raman spectroscopy,” Phys. Rev. Lett. 70, 1413-1416 (1993).
[CrossRef] [PubMed]

A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Raman scattering enhancement by optical confinement in a semiconductor planar microcavity,” Phys. Rev. Lett. 75, 3764-3767 (1995).
[CrossRef] [PubMed]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

J. P. Coffinet and F. de Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60-64 (1969).
[CrossRef]

J. J. Wynne, “Nonlinear optical spectroscopy of χ(3) in LiNbO3,” Phys. Rev. Lett. 29, 650-653 (1972).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142-4145 (1999).
[CrossRef]

Proc. R. Soc. London, Ser. A

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Other

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

The mirrors considered here made of a succession of high (H) and low (L) refractive index hafnium oxide HfO2(n=2.207) and silica SiO2(n=1.456) layers deposed on a silica substrate. The thicknesses (in nanometers) of the different layers are given by (H) 107; (L) 272; (H) 88; (L) 109; (H) 206; (L) 119; (H) 64; (L) 298; (H) 314; (L) 36; (H) 374; (L) 201; (H) 127; (L) 212; (H) 111; (L) 209; (H) 128; (L) 95.

The orientation of the nonlinear induced dipoles is fixed by the excitation beams and the Raman depolarization ratio following Eq. .

This is in good agreement with our medium reflectivity mirrors that alter marginally the emitter lifetime inside the cavity.

G. Björk and Y. Yamamoto, “Spontaneous emission in dielectric planar microcavities,” in Spontaneous Emission and Laser. Oscillation in Microcavities, H.Yokoyama and K.Ujihara, eds. (Academic, 1995).

D. Gachet, N. Sandeau, and H. Rigneault, “Far-field radiation pattern in coherent anti-Stokes Raman scattering (CARS) microscopy,” in Biomedical Vibrational Spectroscopy III: Advances in Research and Industry, A.Mahadevan-Jansen and W.H.Peetrich, eds., Proc. SPIE 6093, 609309 (2006).
[CrossRef]

Computing two-dimensional patterns such as the ones plotted in Fig. takes about 30 days with a PC working with a 2.5 GHz processor.

Note that, as pointed out previously, because the ring structure does not rigorously follow a revolution symmetry (see Fig. ), the undergone jumps are smoother than in Fig. .

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

B. Schrader, Infrared and Raman Spectroscopy (VCH, 1995).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

(a) Energy diagrams of the resonant and nonresonant components of the CARS process. (b) Coherent building of the CARS signal generated by an object in free space. Left: the pump and Stokes fields are focused on an object and a nonlinear polarization is locally induced. Right: resultant Fwd- and Epi-CARS signals emitted by the object. (c) Studied configuration. A CARS scatterer is set between two parallel mirrors M 1 and M 2 that constitute a Fabry–Perot cavity for the anti-Stokes wavelength. The mirrors are designed to be transparent for the pump and Stokes excitation beams but reflecting for the generated anti-Stokes signal. The pump and Stokes beams are focused through a 0.6 NA objective. (d) Intensity P 0 ( 3 ) 2 of the nonlinear induced polarization in the vicinity of the focus of the objective.

Fig. 2
Fig. 2

Spectral features of the mirrors that constitute the Fabry–Perot cavity. Power reflectivity as a function of the wavelength and the incidence θ ( n   sin ( θ ) = k x / k 0 ) on the mirrors for the (a) TE and (b) TM polarization modes, with fused silica (refractive index n = 1.45 ) and N,N-dimethylformamide (refractive index n = 1.43 ) as the respective substrate and superstrate for the mirrors. (c) Power reflectivity as a function of the incidence k x / k 0 on the mirrors at wavelength λ = 660   nm [solid lines in (a) and (b)] for TE (blue) and TM (red) modes.

Fig. 3
Fig. 3

Image-dipole method. (a) The initial dipole p 0 is set in the cavity constituted by the M 1 and M 2 mirrors (located, respectively, located at z = e / 2 and z = e / 2 ). Its images p j are found by operating successive axial symmetries by the M 1 and M 2 mirrors. (b) Successive images of the initial excitation volume (centered at z = z foc ) are found by following the same procedure. The points M j are the images of the point M 0 located in the excitation volume. For the sake of clarity, only the first images are represented although an infinity of images has to be taken into account.

Fig. 4
Fig. 4

Amplitude P ( 3 ) of the equivalent nonlinear induced polarization for specific configurations of the Fabry–Perot cavity. (a) Free space, (b) antiresonant cavity, (c) cos-type resonant cavity, and (d) sin-type resonant cavity. The amplitudes are normalized with respect to the nonlinear induced polarization amplitude in the free space. The pump and Stokes exciting lasers are focused through a 0.6 NA microscope objective in a n = 1.43 refractive index medium and the anti-Stokes wavelength is λ as = 660   nm .

Fig. 5
Fig. 5

Fwd-CARS signal emitted in normal incidence by an object with transverse dimensions ( 2 × 2 μ m ) as a function of its length. The object is set in a L opt = 20 (solid blue line), 20.5 (dotted red line) or 20.25 λ as (dashed black line) Fabry-Perot cavity, where L opt is the optical thickness of the cavity (with λ as = 660   nm ). The Fwd-CARS signal is normalized with respect to the free space signal (emitted by an object with the same dimensions). The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 6
Fig. 6

Fwd far-field radiation patterns in the Fourier space ( k x , k y ) of a 2 × 2 × 8 μ m object (a) in free space and (b) in a L opt = 20 λ as ( λ as = 660   nm ) Fabry–Perot cavity. The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 . The incident polarization of the lasers is given by the transverse arrow and k 0 is given by k 0 = 2 π / λ as . All graphs are normalized with respect to their own maxima.

Fig. 7
Fig. 7

Fwd and Epi far-field radiation patterns along the k x direction as a function of the length of an object with transverse dimensions ( 2 × 2 μ m ) . (a) Fwd and (b) Epi CARS in free space, (c) Fwd and (d) Epi CARS in a L opt = 20 λ as ( λ as = 660   nm ) Fabry–Perot cavity. All the radiation patterns are normalized with respect to the Fwd-CARS signal emitted by a 8 μ m long object in the free space in normal incidence ( k x = 0 ) . The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 8
Fig. 8

Fwd- and Epi-CARS signals collected as the function of the collection NA and the length of an object with transverse dimensions ( 2 × 2 μ m ) . The object emits either in the free space (a),(b) or in a (c) L opt = 20 λ as ( λ as = 660   nm ) Fabry–Perot cavity. All the signals are normalized with respect to the Fwd-CARS signal emitted by a 8 μ m long object in free space and collected by a 0.6 NA objective. The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 9
Fig. 9

Fwd- and Epi-CARS signals collected as the function of the length of an object with transverse dimensions ( 2 × 2 μ m ) set in a L opt = 20 λ as ( λ as = 660   nm ) Fabry–Perot cavity for several values of the collection objective NA. Those curves correspond to the cuts operated in Fig. 8. The Fwd- and Epi-CARS signals generated in the cavity (solid black lines) are compared to the Fwd- and Epi-CARS signals generated in the free space (respectively, dashed blue and dotted red lines). All the signals are normalized with respect to the Fwd-CARS signal emitted by a 8 μ m long object in free space and collected by a 0.6 NA objective. The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 10
Fig. 10

Epi- to Fwd-CARS signal ratio as the function of the length of an object with transverse dimensions ( 2 × 2 μ m ) when the signals are collected with 0.6 NA collection objectives. The object is set either in a L opt = 20 λ as ( λ as = 660   nm ) Fabry–Perot cavity (dashed red line) or in free space (solid blue line). The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 11
Fig. 11

Modification of the Fwd- and Epi-CARS far-field radiation patterns along the k x direction as a function of the cavity detuning Δ L opt for different values of the Fabry–Perot cavity optical thickness L opt . (a) L opt = 100 λ as , (b) 40 λ as and (c) 20 λ as ( λ as = 660   nm ) . All the radiation patterns are normalized with respect to the signal emitted in free space in the Fwd direction in normal incidence. The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 12
Fig. 12

Fwd-CARS signal enhancement by the Fabry–Perot cavity (with respect to the free space signal) as a function of the cavity detuning Δ L opt for different collection objective NAs and values of the Fabry-Perot cavity optical thickness L opt . (a) NA = 0.6 , (b) 0.45, (c) 0.3, and (d) 0.1. L opt = 100 λ as (black triangles), 40 λ as (blue squares), and 20 λ as (red dots) ( λ as = 660   nm ) . The signals are normalized with respect to the Fwd-CARS signals generated in free space and collected with an objective with the same NA. The object and the exciting lasers (focused through a 0.6 NA microscope objective) are centered into the cavity. The object's refractive index is n = 1.43 .

Fig. 13
Fig. 13

Effect of a longitudinal shift of the object in the cavity. The longitudinal extension of the object L is (a) very small compared to the anti-Stokes wavelength ( L λ as / 2 n ) , (b) comparable to λ as / 2 n , or (c) larger than λ as / 2 n . Blue curve: intensity of the equivalent nonlinear induced polarization ( λ as / 2 n periodical).

Fig. 14
Fig. 14

(a)–(c) Modification of the Fwd- and Epi-CARS far-field radiation patterns along the k x direction as a function of the object's longitudinal shift z obj (relative to the center of the cavity) in a L opt = 20 λ as Fabry–Perot cavity ( λ as = 660   nm ) . The object has ( 2 × 2 μ m ) transverse dimensions and its length L is (a) 25 nm, (b) 325 nm, or (c) 1 μ m . (d) Modulation of the collected Fwd-CARS signal as a function of the object length L when the object is shifted in the cavity. The modulation is defined as the ratio of the minimum to the maximum of the collected signal when the object is shifted over a λ as / 2 n range in the cavity. Fwd-CARS signal collection NA: 0.1 (red dots), 0.45 (blue squares), or 0.6 (black triangles). Dashed lines: multiples of λ as / 2 n . The object's refractive index is n = 1.43 and the exciting lasers are focused through a 0.6 NA microscope objective and centered on the object.

Equations (54)

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χ ( 3 ) = χ R ( 3 ) + χ NR ( 3 ) .
P ( 3 ) ( r , ω as ) = χ ( 3 ) ( r ; ω as , ω p , ω p , ω s ) E p ( r , ω p ) : E p ( r , ω p ) : E s ( r , ω s ) .
P ( 3 ) ( r , ρ R ) = 6 [ χ x x y y ( 3 ) R ( r ) S ( r , ρ R ) + χ x x y y ( 3 ) NR ( r ) S ( r , 1 / 3 ) ] ,
S ( r , ρ R ) = [ E p ( r ) E s ( r ) ] E p ( r ) + ρ R 1 ρ R [ E p x 2 ( r ) + E p y 2 ( r ) + E p z 2 ( r ) ] E s ( r ) ,
M j = ( x 0 , y 0 , ( 1 ) j z 0 + j e )
p j = K D j R | j | / 2   exp [ i ( φ 0 j φ refl ) ] p 0 ,
D j = ( 1 0 0 0 1 0 0 0 ( 1 ) j ) .
p 0 2 + 2 j = 1 + R j p 0 2 ,
1 + R 1 R p 0 2 .
E as 0 ( k ) = E as 0 TE ( k ) + E as 0 TM ( k ) ,
E as j α ( k ) = { K R α | j | / 2   exp [ i φ j α ( k ) ] exp [ i ( j e + z 0 ) k z ̂ ] D j E as 0 α ( k ) if   j   is   even K R α | j | / 2   exp [ i φ j α ( k ) ] exp [ i ( j e z 0 ) k z ̂ ] D j E as 0 α ( k ) if   j   is   odd , }
φ j α ( k ) = φ 0 j φ refl α ( k ) .
E as , tot ( k ) = E as , tot TE ( k ) + E as , tot TM ( k ) ,
E as , tot α ( k ) = K   exp ( i φ 0 ) [ 1 R α ( k ) ] 1 + R α 2 ( k ) 2 R α ( k ) cos { 2 [ e k z ̂ φ refl α ( k ) ] } { [ 1 + R α ( k ) ] exp ( i z 0 k z ̂ ) E as 0 α ( k ) + 2 R α 1 / 2 ( k ) cos [ e k z ̂ φ refl α ( k ) ] exp ( i z 0 k z ̂ ) D 1 E as 0 α ( k ) } .
E 0 ( Fwd ) = A P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) ,
E 0 ( Epi ) = A P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) ,
ϕ G ( z 0 ) = 2 φ p ( z 0 ) φ s ( z 0 ) .
φ 0 = k as z 0 + ϕ G ( z 0 ) .
E tot ( Fwd ) = A P 0 ( 3 ) ( x , y , z 0 ) [ 1 R 1 + R ] 1 / 2 1 R 1 + R 2 2 R   cos [ 2 ( φ e φ refl ) ] [ ( 1 + R ) exp ( i k as z 0 ) + 2 R 1 / 2 cos ( φ e φ refl ) exp ( i k as z 0 ) ] ,
E tot ( Epi ) = A P 0 ( 3 ) ( x , y , z 0 ) [ 1 R 1 + R ] 1 / 2 1 R 1 + R 2 2 R   cos [ 2 ( φ e φ refl ) ] [ ( 1 + R ) exp ( i k as z 0 ) + 2 R 1 / 2   cos ( φ e φ refl ) exp ( i k as z 0 ) ] ,
E tot ( Fwd ) = E tot ( Epi ) = 2 [ 1 + R 1 R ] 1 / 2 cos ( k as z 0 ) A P 0 ( 3 ) ( x , y , z 0 ) .
P equ , Fwd ( 3 ) ( x , y , z 0 ) = 2 [ 1 + R 1 R ] 1 / 2 cos ( k as z 0 ) P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) ,
P equ , Epi ( 3 ) ( x , y , z 0 ) = 2 [ 1 + R 1 R ] 1 / 2 cos ( k as z 0 ) P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) .
φ equ , Fwd = { 2 k as z 0 + ϕ G ( z 0 ) if   cos ( k as z 0 ) > 0 2 k as z 0 + ϕ G ( z 0 ) + π if   cos ( k as z 0 ) < 0 } ,
φ equ , Epi = { ϕ G ( z 0 ) if   cos ( k as z 0 ) > 0 ϕ G ( z 0 ) + π if   cos ( k as z 0 ) < 0 } .
E tot ( Fwd ) = E tot ( Epi ) = 2 i [ 1 + R 1 R ] 1 / 2 sin ( k as z 0 ) A P 0 ( 3 ) ( x , y , z 0 ) ,
P equ , Fwd ( 3 ) ( x , y , z 0 ) = 2 i [ 1 + R 1 R ] 1 / 2 sin ( k as z 0 ) P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) ,
P equ , Epi ( 3 ) ( x , y , z 0 ) = 2 i [ 1 + R 1 R ] 1 / 2 sin ( k as z 0 ) P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) .
φ equ , Fwd = { 2 k as z 0 + ϕ G ( z 0 ) π 2 if   sin ( k as z 0 ) > 0 2 k as z 0 + ϕ G ( z 0 ) + π 2 if   sin ( k as z 0 ) < 0 } ,
φ equ , Epi = { ϕ G ( z 0 ) π 2 if   sin ( k as z 0 ) > 0 ϕ G ( z 0 ) + π 2 if   sin ( k as z 0 ) < 0 } .
E tot ( Fwd ) = [ 1 R 1 + R ] 3 / 2 A P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) ,
E tot ( Epi ) = [ 1 R 1 + R ] 3 / 2 A P 0 ( 3 ) ( x , y , z 0 ) exp ( i k as z 0 ) .
P equ ( 3 ) ( x , y , z 0 ) = [ 1 R 1 + R ] 3 / 2 P 0 ( 3 ) ( x , y , z 0 ) ,
Δ φ ( Fwd ) = ϕ G ( z ) ,
Δ φ ( Epi ) = 2 k as z + ϕ G ( z ) .
Δ φ ( Fwd ) = { k as z + ϕ G ( z ) if   cos ( k as z ) > 0 k as z + ϕ G ( z ) + π if   cos ( k as z ) < 0 } ,
Δ φ ( Epi ) = { k as z + ϕ G ( z ) if   cos ( k as z ) > 0 k as z + ϕ G ( z ) + π if   cos ( k as z ) < 0 } .
Δ φ ( Fwd ) = { k as z + ϕ G ( z ) π 2 if   sin ( k as z ) > 0 k as z + ϕ G ( z ) + π 2 if   sin ( k as z ) < 0 } ,
Δ φ ( Epi ) = { k as z + ϕ G ( z ) π 2 if   sin ( k as z ) > 0 k as z + ϕ G ( z ) + π 2 if   sin ( k as z ) < 0 } .
Δ φ ( Fwd ) = Δ φ ( Epi ) = ϕ G ( z )   if   cos ( k as z ) ± 1   or   sin ( k as z ) ± 1.
e k z ̂ φ refl α ( k ) = m π ,
P ( Ω ) = Ω d P ( Ω ) d Ω d Ω ,
P ( θ max ) = 2 π 0 θ max d P d Ω ( θ ) sin ( θ ) d θ .
P ( k x max ) = 2 π k as 0 k x max d P d Ω ( k x ) k x ( k as 2 k x 2 ) 1 / 2 d k x .
E as , tot α ( k ) = E as , tot , 2 p α ( k ) + E as , tot , 2 p 1 α ( k ) ,
E as , tot , 2 p α ( k ) = p = 0 + E as , 2 p α ( k ) + p = 1 + E as , 2 p α ( k ) ,
E as , tot , 2 p 1 α ( k ) = p = 1 + [ E as , 2 p 1 α ( k ) + E as , 1 2 p α ( k ) ] .
E as , tot , 2 p α ( k ) = K   exp ( i φ 0 ) { p = 0 + R α p ( k ) exp { 2 i p [ e k z ̂ φ refl α ( k ) ] } + p = 1 + R α p ( k ) exp { 2 i p [ e k z ̂ φ refl α ( k ) ] } } exp ( i z 0 k z ̂ ) E as 0 α ( k ) .
E as , tot , 2 p α ( k ) = K   exp ( i φ 0 ) { 1 1 R α ( k ) exp { 2 i [ e k z ̂ φ refl α ( k ) ] } + R α ( k ) exp { 2 i [ e k z ̂ φ refl α ( k ) ] } 1 R α ( k ) exp { 2 i [ e k z ̂ φ refl α ( k ) ] } } exp ( i z 0 k z ̂ ) E as 0 α ( k ) ,
E as , tot , 2 p α ( k ) = K   exp ( i φ 0 ) 1 R α 2 ( k ) 1 + R α 2 ( k ) 2 R α ( k ) cos { 2 [ e k z ̂ φ refl α ( k ) ] } exp ( i z 0 k z ̂ ) E as 0 α ( k ) .
E as , tot , 2 p 1 α ( k ) = K   exp ( i φ 0 ) R α 1 / 2 { p = 1 + R α p ( k ) exp { i ( 2 p 1 ) [ e k z ̂ φ refl α ( k ) ] } + p = 1 + R α p ( k ) exp { i ( 2 p 1 ) [ e k z ̂ φ refl α ( k ) ] } } exp ( i z 0 k z ̂ ) D 1 E as 0 α ( k ) ,
E as , tot , 2 p 1 α ( k ) = K   exp ( i φ 0 ) R α 1 / 2 { R α ( k ) exp { i [ e k z ̂ φ refl α ( k ) ] } 1 R α ( k ) exp { 2 i [ e k z ̂ φ refl α ( k ) ] } + R α ( k ) exp { i [ e k z ̂ φ refl α ( k ) ] } 1 R α ( k ) exp { 2 i [ e k z ̂ φ refl α ( k ) ] } } exp ( i z 0 k z ̂ ) D 1 E as 0 α ( k ) ,
E as , tot , 2 p 1 α ( k ) = K   exp ( i φ 0 ) 2 R α 1 / 2 [ 1 R α ( k ) ] cos [ e k z ̂ φ refl α ( k ) ] 1 + R α 2 ( k ) 2 R α ( k ) cos { 2 [ e k z ̂ φ refl α ( k ) ] } exp ( i z 0 k z ̂ ) D 1 E as 0 α ( k ) .
E as , tot α ( k ) = K   exp ( i φ 0 ) [ 1 R α ( k ) ] 1 + R α 2 ( k ) 2 R α ( k ) cos { 2 [ e k z ̂ φ refl α ( k ) ] } { [ 1 + R α ( k ) ] exp ( i z 0 k z ̂ ) E as 0 α ( k ) + 2 R α 1 / 2 ( k ) cos [ e k z ̂ φ refl α ( k ) ] exp ( i z 0 k z ̂ ) D 1 E as 0 α ( k ) } .

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