Abstract

We describe a plane-wave-expansion approach for calculation of the fringe pattern in transmission and reflection for a Gaussian monochromatic beam. Both positive and negative incidence, at which the incident light beam undergoes multiple reflections within the wedge in direction of increasing or decreasing wedge thickness respectively, are analyzed. It is shown that the two opposite incidences of the light beam are described by the same mathematical expressions; i.e., the transmitted/reflected fringe pattern at positive incidence is a continuation of the pattern at negative incidence at some distance from the wedge. Numerical simulations are made for a high-reflectivity-coating air-gap Fizeau interferential wedge with apex angle of 5100μrad and thickness of 5500μm as a useful optical element in laser resonator design. Experimental verification is also provided.

© 2009 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
  2. M. V. Mantravardi, “Newton, Fizeau and Haidinger interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley-Interscience, 1992), pp. 1-49.
  3. R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
    [CrossRef]
  4. B. Dorrio, A. Doval, C. Lopez, R. Soto, J. Blanco-Garcia, J. Fernandez, and M. Perez-Amor, “Fizeau phase-measuring interferometry using the moiré effect,” Appl. Opt. 34, 3639-3643 (1995).
    [CrossRef] [PubMed]
  5. B. Dorrio, C. Lopez, J. Alen, J. Bugarin, A. Fernandez, A. Doval, J. Blanco-Garcia, M. Perez-Amor, and J. Fernandez, “Multiplicative moiré two-beam phase-stepping and Fourier-transform methods for the evaluation of multiple-beam Fizeau patterns: a comparison,” Appl. Opt. 37, 1945-1952 (1998).
    [CrossRef]
  6. Y. Ishii, J. Chen, R. Onodera, and T. Nakamura, “Phase-shifting Fizeau interference microscope with a wavelength-shifted laser diode,” Appl. Opt. 42, 60-67 (2003).
    [CrossRef]
  7. M. Morris, T. Mollrath, and J. Snyder, “Fizeau wavemeter for pulsed laser wavelength measurement,” Appl. Opt. 23, 3862-3868 (1984).
    [CrossRef] [PubMed]
  8. C. Reiser and B. Lopert, “Laser wavemeter with solid Fizeau wedge interferometer,” Appl. Opt. 27, 3656-3660 (1988).
    [CrossRef] [PubMed]
  9. J. Snyder and T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World February 1990 pp. 69-76.
  10. E. Alipieva, E. Stoykova, and V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” Proc. SPIE 4397, 129-133 (2001).
    [CrossRef]
  11. T. Kajava, H. Lauranto, and R. Salomaa, “Fizeau interferometer in spectral measurements,” J. Opt. Soc. Am. B 10, 1980-1989 (1993).
    [CrossRef]
  12. T. Kajava, H. Lauranto, and A. Friberg, “Interference pattern of the Fizeau interferometer,” J. Opt. Soc. Am. A 11, 2045-2054 (1994).
    [CrossRef]
  13. Y. Meyer and M. Nenchev, “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119-121 (1981).
    [CrossRef] [PubMed]
  14. M. Nenchev and Y. Meyer, “Continuous-scanning system for single-mode wedge dye lasers,” Opt. Lett. 7, 199-200 (1982).
    [CrossRef] [PubMed]
  15. M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
    [CrossRef]
  16. M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
    [CrossRef]
  17. M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
    [CrossRef]
  18. M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
    [CrossRef]
  19. E. Stoykova and M. Nenchev, “Strong optical asymmetry of an interference wedge with unequal reflectivity mirrors and its use in unidirectional ring laser designs,” Opt. Lett. 19, 1925-1927 (1994).
    [CrossRef] [PubMed]
  20. E. Stoykova and M. Nenchev, “Fizeau wedge with unequal mirrors for spectral control and coupling in a linear laser oscillator-amplifier system,” Appl. Opt. 40, 5402-5412 (2001).
    [CrossRef]
  21. Y. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. 71, 1255-1261 (1981).
    [CrossRef]
  22. J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224-234 (1947).
    [CrossRef]
  23. A. Pomeranski and U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrok. 45, 773-779 (1978) (in Russian).
  24. E. Stoykova and M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 27, 155-167 (1996).
  25. M. Nenchev and E. Stoykova, “Interference wedge properties relevant to laser applications: transmission and reflection of the restricted light beams,” Opt. Quantum Electron. 25, 789-799 (1993).
    [CrossRef]
  26. E. Stoykova, “Transmission of a Gaussian beam by a Fizeau interferential wedge,” J. Opt. Soc. Am. A 22, 2756-2765 (2005).
    [CrossRef]
  27. J. Goodman, Introduction to Fourier Optics (Roberts, 2004).
  28. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge Univ. Press, 1997).
  29. E. Nichelatti and G. Salvetti, “Spatial and spectral response of a Fabry-Perot interferometer illuminated by a Gaussian beam,” Appl. Opt. 34, 4703-4712 (1995).
    [CrossRef] [PubMed]

2005 (1)

2003 (1)

2001 (2)

1998 (1)

1997 (1)

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

1996 (2)

M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
[CrossRef]

E. Stoykova and M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 27, 155-167 (1996).

1995 (3)

1994 (2)

1993 (2)

T. Kajava, H. Lauranto, and R. Salomaa, “Fizeau interferometer in spectral measurements,” J. Opt. Soc. Am. B 10, 1980-1989 (1993).
[CrossRef]

M. Nenchev and E. Stoykova, “Interference wedge properties relevant to laser applications: transmission and reflection of the restricted light beams,” Opt. Quantum Electron. 25, 789-799 (1993).
[CrossRef]

1992 (1)

R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
[CrossRef]

1991 (1)

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

1988 (1)

1984 (1)

1982 (1)

1981 (2)

1978 (1)

A. Pomeranski and U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrok. 45, 773-779 (1978) (in Russian).

1947 (1)

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Alen, J.

Alipieva, E.

E. Alipieva, E. Stoykova, and V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” Proc. SPIE 4397, 129-133 (2001).
[CrossRef]

Barbe, R.

M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
[CrossRef]

Blanco-Garcia, J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Brossel, J.

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Bugarin, J.

Chen, J.

Deleva, A.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Deneva, M.

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
[CrossRef]

Dorrio, B.

Doval, A.

Fernandez, A.

Fernandez, J.

Friberg, A.

Gizbrekht, A.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts, 2004).

Gorris-Neveux, M.

M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
[CrossRef]

Hansch, T.

J. Snyder and T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World February 1990 pp. 69-76.

Ishii, Y.

Józwicki, R.

R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
[CrossRef]

Kajava, T.

Keller, J.-C.

M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
[CrossRef]

Kujawinska, M.

R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
[CrossRef]

Lauranto, H.

Lopert, B.

Lopez, C.

Mantravardi, M. V.

M. V. Mantravardi, “Newton, Fizeau and Haidinger interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley-Interscience, 1992), pp. 1-49.

Meyer, Y.

Mollrath, T.

Morris, M.

Nakamura, T.

Nenchev, M.

E. Stoykova and M. Nenchev, “Fizeau wedge with unequal mirrors for spectral control and coupling in a linear laser oscillator-amplifier system,” Appl. Opt. 40, 5402-5412 (2001).
[CrossRef]

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
[CrossRef]

E. Stoykova and M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 27, 155-167 (1996).

M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
[CrossRef]

E. Stoykova and M. Nenchev, “Strong optical asymmetry of an interference wedge with unequal reflectivity mirrors and its use in unidirectional ring laser designs,” Opt. Lett. 19, 1925-1927 (1994).
[CrossRef] [PubMed]

M. Nenchev and E. Stoykova, “Interference wedge properties relevant to laser applications: transmission and reflection of the restricted light beams,” Opt. Quantum Electron. 25, 789-799 (1993).
[CrossRef]

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

M. Nenchev and Y. Meyer, “Continuous-scanning system for single-mode wedge dye lasers,” Opt. Lett. 7, 199-200 (1982).
[CrossRef] [PubMed]

Y. Meyer and M. Nenchev, “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119-121 (1981).
[CrossRef] [PubMed]

Nichelatti, E.

Nikolova, V.

E. Alipieva, E. Stoykova, and V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” Proc. SPIE 4397, 129-133 (2001).
[CrossRef]

Onodera, R.

Patrikov, T.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Perez-Amor, M.

Peshev, Z.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Pomeranski, A.

A. Pomeranski and U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrok. 45, 773-779 (1978) (in Russian).

Reiser, C.

Salbut, L.

R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
[CrossRef]

Salomaa, R.

Salvetti, G.

Slavov, D.

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

Smith, G. S.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge Univ. Press, 1997).

Snyder, J.

M. Morris, T. Mollrath, and J. Snyder, “Fizeau wavemeter for pulsed laser wavelength measurement,” Appl. Opt. 23, 3862-3868 (1984).
[CrossRef] [PubMed]

J. Snyder and T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World February 1990 pp. 69-76.

Soto, R.

Stoykova, E.

E. Stoykova, “Transmission of a Gaussian beam by a Fizeau interferential wedge,” J. Opt. Soc. Am. A 22, 2756-2765 (2005).
[CrossRef]

E. Stoykova and M. Nenchev, “Fizeau wedge with unequal mirrors for spectral control and coupling in a linear laser oscillator-amplifier system,” Appl. Opt. 40, 5402-5412 (2001).
[CrossRef]

E. Alipieva, E. Stoykova, and V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” Proc. SPIE 4397, 129-133 (2001).
[CrossRef]

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
[CrossRef]

E. Stoykova and M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 27, 155-167 (1996).

E. Stoykova and M. Nenchev, “Strong optical asymmetry of an interference wedge with unequal reflectivity mirrors and its use in unidirectional ring laser designs,” Opt. Lett. 19, 1925-1927 (1994).
[CrossRef] [PubMed]

M. Nenchev and E. Stoykova, “Interference wedge properties relevant to laser applications: transmission and reflection of the restricted light beams,” Opt. Quantum Electron. 25, 789-799 (1993).
[CrossRef]

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Tomashevski, U.

A. Pomeranski and U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrok. 45, 773-779 (1978) (in Russian).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Appl. Opt. (7)

IEEE J. Quantum Electron. (1)

M. Gorris-Neveux, M. Nenchev, R. Barbe, and J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260-1263 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

M. Deneva, D. Slavov, E. Stoykova, and M. Nenchev, “Improved passive self-injection locking method for spectral control of dye and Ti:Al2O3 lasers using two-step pulse pumping,” Opt. Commun. 139, 287-298 (1997).
[CrossRef]

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, and A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405-408 (1991).
[CrossRef]

Opt. Eng. (Bellingham) (1)

R. Józwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422-433 (1992).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

E. Stoykova and M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 27, 155-167 (1996).

M. Nenchev and E. Stoykova, “Interference wedge properties relevant to laser applications: transmission and reflection of the restricted light beams,” Opt. Quantum Electron. 25, 789-799 (1993).
[CrossRef]

Opt. Spektrok. (1)

A. Pomeranski and U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrok. 45, 773-779 (1978) (in Russian).

Proc. Phys. Soc. London (1)

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224-234 (1947).
[CrossRef]

Proc. SPIE (1)

E. Alipieva, E. Stoykova, and V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” Proc. SPIE 4397, 129-133 (2001).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Deneva, E. Stoykova, and M. Nenchev, “A novel technique for a narrow-line selection and wideband tuning of Ti3+:Al2O3 and dye lasers,” Rev. Sci. Instrum. 67, 1705-1714 (1996).
[CrossRef]

Other (5)

J. Snyder and T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World February 1990 pp. 69-76.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

M. V. Mantravardi, “Newton, Fizeau and Haidinger interferometers,” in Optical Shop Testing, D.Malacara, ed. (Wiley-Interscience, 1992), pp. 1-49.

J. Goodman, Introduction to Fourier Optics (Roberts, 2004).

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge Univ. Press, 1997).

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Figures (16)

Fig. 1
Fig. 1

Schematic representation of positive and negative incidence in reflection and transmission for a Fizeau wedge.

Fig. 2
Fig. 2

Ray tracing in transmision for negative and positive incidence.

Fig. 3
Fig. 3

8 -bit gray-level contour map of the resonant transmitted intensity. From top to bottom: First row, e = 20 μ m , w 0 = 100 μ m ; second row, e = 100 μ m , w 0 = 100 μ m ; third row, e = 200 μ m , w 0 = 100 μ m ; fourth row, e = 200 μ m , w 0 = 200 μ m . Calculation starts at z i neg , which corresponds to 0 in the plots.

Fig. 4
Fig. 4

Resonant transmitted intensity distribution: e = 200 μ m , w 0 = 200 μ m , R = 0.8 .

Fig. 5
Fig. 5

Power Fourier spectrum of the resonant transmitted intensity in Fig. 4: e = 200 μ m , w 0 = 200 μ m , R = 0.8 .

Fig. 6
Fig. 6

8 -bit gray-scale maps of the wrapped phase of the complex amplitude of the transmitted beam: e = 20 μ m , w 0 = 100 μ m .

Fig. 7
Fig. 7

Fringe profiles at z i neg = ( x 0 q w 0 ) tan ( θ α ) (left curve) and z i pos = ( x 0 + q w 0 ) tan ( θ + α ) (right curve) for e = 100 μ m , w 0 = 100 μ m .

Fig. 8
Fig. 8

Fringe profiles in transmission close to the focal point and at 5 m from the wedge at negative incidence for two incident beams with diameters (a) 1 cm and (b) 2 cm .

Fig. 9
Fig. 9

Ray tracing in reflection for negative and positive incidence.

Fig. 10
Fig. 10

Ray tracing in reflection for negative and positive incidence: representation equivalent to Fig. 9.

Fig. 11
Fig. 11

8 -bit gray-level contour map of the resonant reflected intensity. From top to bottom: first row, e = 20 μ m , w 0 = 100 μ m ; second row, e = 100 μ m , w 0 = 100 μ m ; third row, e = 200 μ m , w 0 = 100 μ m ; fourth row, e = 200 μ m , w 0 = 200 μ m . Calculation starts at z i neg , which corresponds to 0 in the plots.

Fig. 12
Fig. 12

Resonant intensity distribution in reflection: e = 200 μ m , w 0 = 100 μ m , and R = 0.8 .

Fig. 13
Fig. 13

Power Fourier spectrum of the resonant distribution of the reflected intensity in Fig. 12: e = 200 μ m , w 0 = 200 μ m , and R = 0.8 .

Fig. 14
Fig. 14

Intensity distribution in reflection far from the resonance for e = 200 μ m , w 0 = 100 μ m and (a) R = 0.8 and (b) R = 0.96 .

Fig. 15
Fig. 15

Fringe profiles in reflection at z i neg = ( x 0 q w 0 ) tan ( θ α ) (thin curve) and z i pos = ( x 0 + q w 0 ) tan ( θ + α ) (thick curve) for e = 100 μ m .

Fig. 16
Fig. 16

Photographed light spots of the reflected beam for a 200 μ m air-gap wedge with reflectivity of the coatings 0.9 and a wedge angle α W = 50 μ rad at illumination with a He–Ne laser. (a), (b), (c) Negative incidence on the wedge surface; (a) in front of the focal point, (b) at the focal point, and (c) behind the focal point. (d) Positive incidence on the wedge surface.

Tables (1)

Tables Icon

Table 1 Values for z i neg and z i pos

Equations (28)

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

k p = ( k p x , k p z ) , k p x = k sin ( 2 p α W ) , k p z = k cos ( 2 p α W ) , p = 0 , 1 , 2 , ,
g ( x ) = g ( x , z 0 ) = G 0 ( α ) exp ( i 2 π α x ) d α ,
G 0 ( α ) = ( 2 π ) 1 4 w 0 exp [ ( π w 0 α ) 2 ] exp ( i 2 π x 0 ) .
g ( x , z ) = G 0 ( α ) exp { i 2 π [ α x + γ ( z z 0 ) ] } d α .
g ( x , z ) = exp [ i k ( z z 0 ) ] G 0 ( α ) exp [ i π λ ( z z 0 ) α 2 ] exp ( i 2 π α x ) d α .
k p = ( k p x , k p z ) , k p x = k sin ( η + 2 p α W ) , k p z = k cos ( η + 2 p α W ) , p = 0 , 1 , 2 , ,
A T P W ( x , z , η ) = T p = 0 R p exp [ i p ( ϕ + ϕ ) ] exp { i k [ sin ( η + 2 p α W ) x + cos ( η + 2 p α W ) z ] } ,
τ ( x , z , η ) = A T P W ( x , z , η ) a η ( x , z ) = T p = 1 R p 1 exp [ i 2 ( p 1 ) π ] exp [ i 2 π λ ( x ψ p + z ψ p ) ] ,
ψ p = cos ( θ p + η ) cos ( η ) ,
ψ p = sin ( θ p + η ) sin ( η ) ,
φ p ( x , z , η ) = 2 π λ ( P N p P N 1 ) = 2 π λ O P [ sin ( χ + θ p η ) sin ( χ η ) ] = 2 π λ ( x ψ p + z ψ p ) .
A T G ( x , z ) = T exp [ i 2 π λ ( z z 0 ) ] p = 1 R p 1 exp [ i 2 ( p 1 ) π ] G ( α ) exp [ i π λ ( z z 0 ) α 2 ] × exp ( i 2 π α x ) exp [ i φ p ( x , z , η ) ] d α .
ψ p = [ 1 1 2 ( λ α ) 2 ] ν p λ α ν p ,
ψ p = [ 1 1 2 ( λ α ) 2 ] ν p + 2 λ α ν p ,
A T G ( x , z ) = a T exp [ i 2 π λ ( z z 0 ) ] p = 1 R p 1 exp ( b α 2 ) × exp [ i ( p p α 2 + q p α + r p ) ] d α ,
a = ( 2 π ) 1 4 w 0 , b = π 2 w 0 2 , p p = π ( z z 0 + x ν p + z ν p ) λ ,
q p = 2 π ( x + x ν p z ν p + x 0 ) , r p = 2 π λ ( x ν p + z ν p ) .
exp ( b α 2 ) Θ 1 , 2 ( p p α 2 + q p α + r p ) d α = Ω p × Θ 1 , 2 ( ϵ p ) ,
Ω p = π b 2 + p p 2 4 exp [ b q p 2 4 ( b 2 + p p 2 ) ] ,
ϵ p = 1 2 arctan p p b p p ( q p 2 4 p p r p ) 4 b 2 r p 4 ( b 2 + p p 2 ) .
A T G ( x , z ) = a T exp [ i 2 π λ ( z z 0 ) ] p = 1 R p 1 Ω p ( cos ϵ p + i sin ϵ p ) .
I T G ( x , z ) = a 2 T 2 ( S 1 2 + S 2 2 ) ,
A F = e cos ( θ 2 α W ) sin 2 α W · tan θ + tan ( θ 2 α W ) 1 + tan θ tan α W
F C = e cos ( θ + 2 α W ) sin 2 α W tan θ + tan ( θ + 2 α W ) 1 tan θ tan α W .
A F + F C = A C = A O + O C = 2 A O = 2 e sin θ tan α W .
P T = I T G ( x , z ) d x x 0 q w 0 x 0 + q w 0 I T G ( x , z ) d x ,
ρ ( x , z , η ) = A R P W ( x , z , η ) a η ( x , z ) = r + T r p = 1 R p exp [ i ( 2 p 1 ) π ] exp [ i 2 π λ ( x ψ p + z ψ p ) ] ,
I R G ( x , z ) = a 2 { R Ω 1 2 2 T ( S 1 cos ϵ 1 + S 2 sin ϵ 1 ) Ω 1 + T 2 R ( S 1 2 + S 2 2 ) } ,

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