Abstract

Analysis of transmission of a finite-diameter Gaussian beam by a Fizeau interferential wedge is presented. The fringe calculation is based on angular spectrum expansion of the complex amplitude of the incident wave field. The developed approach is applicable to any beam diameter and wedge thickness at any distance from the wedge and yields as a boundary case the fringes at plane-wave illumination. The spatial region of resonant transmission on the wedge surface is given by the width of the transmitted peak for plane-wave illumination. At higher coating reflectivity, the direction of the transmitted beam is deviated with respect to that of the incident beam. Evaluation of the spectral response based on the spectral width of the transmitted power curve is introduced as more realistic for a correct description of the application of a Fizeau wedge as an interferential selector in laser resonators.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics (Pergamon, 1980).
  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, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
    [CrossRef]
  4. B. V. Dorrio, A. Doval, C. Lopez, R. Soto, J. Blanco-Garcia, J. Fernandez, M. Perez-Amor, “Fizeau phase-measuring interferometry using the moiré effect,” Appl. Opt. 34, 3639–3643 (1995).
    [CrossRef]
  5. B. V. Dorrio, C. Lopez, J. Alen, J. Bugarin, A. Fernandez, A. Doval, J. Blanco-Garcia, M. Perez-Amor, 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. M. Morris, T. McIlrath, J. Snyder, “Fizeau wavemeter for pulsed laser wavelength measurement,” Appl. Opt. 23, 3862–3868 (1984).
    [CrossRef] [PubMed]
  7. C. Reiser, B. Lopert, “Laser wavemeter with solid Fizeau wedge interferometer,” Appl. Opt. 27, 3656–3660 (1988).
    [CrossRef] [PubMed]
  8. J. Snyder, T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World, February 1990, pp. 69–76.
  9. E. Alipieva, E. Stoykova, V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” in Laser Physics and Applications, P. Atanassov and S. Kartaleva, eds., Proc. SPIE4397, 129–133 (2001).
  10. T. Kajava, H. Lauranto, R. Salomaa, “Fizeau interferometer in spectral measurements,” J. Opt. Soc. Am. B 10, 1980–1989 (1993).
    [CrossRef]
  11. T. Kajava, H. Lauranto, A. Friberg, “Interference pattern of the Fizeau interferometer,” J. Opt. Soc. Am. A 11, 2045–2054 (1994).
    [CrossRef]
  12. Y. Meyer, M. Nenchev, “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
    [CrossRef] [PubMed]
  13. M. Nenchev, M. Martin, Y. Meyer, “Alternate wavelength DIAL dye laser using a reflecting interference wedge,” Appl. Opt. 24, 1957–1959 (1985).
    [CrossRef] [PubMed]
  14. M. Deneva, E. Stoykova, 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]
  15. M. Gorris-Neveux, M. Nenchev, R. Barbe, J.-C. Keller, “A two-wavelength, passively self-injection locked, cw Ti3+:Al2O3 laser,” IEEE J. Quantum Electron. 31, 1260–1263 (1995).
    [CrossRef]
  16. M. Deneva, D. Slavov, E. Stoykova, 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]
  17. M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405–408 (1991).
    [CrossRef]
  18. E. Stoykova, 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]
  19. E. Stoykova, 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]
  20. Y. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. 71, 1255–1261 (1981).
    [CrossRef]
  21. A. Pomeranski, U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrosk. 45 (No. 4), 773–779 (1978) (in Russian).
  22. M. Nenchev, 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]
  23. E. Stoykova, M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 28, 155–167 (1996).
    [CrossRef]
  24. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  25. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, 1997).
  26. E. Nichelatti, G. Salvetti, “Spatial and spectral response of a Fabry–Perot interferometer illuminated by a Gaussian beam,” Appl. Opt. 34, 4703–4712 (1995).
    [CrossRef] [PubMed]
  27. H. Lauranto, A. Friberg, T. Kajava, “Influence of the finite aperture and observation of higher longitudinal-order fringes in Fizeau interferometry,” Opt. Eng. 34, 2623–2630 (1995).
    [CrossRef]

2001

1998

1997

M. Deneva, D. Slavov, E. Stoykova, 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

M. Deneva, E. Stoykova, 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, M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 28, 155–167 (1996).
[CrossRef]

1995

H. Lauranto, A. Friberg, T. Kajava, “Influence of the finite aperture and observation of higher longitudinal-order fringes in Fizeau interferometry,” Opt. Eng. 34, 2623–2630 (1995).
[CrossRef]

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

B. V. Dorrio, A. Doval, C. Lopez, R. Soto, J. Blanco-Garcia, J. Fernandez, M. Perez-Amor, “Fizeau phase-measuring interferometry using the moiré effect,” Appl. Opt. 34, 3639–3643 (1995).
[CrossRef]

E. Nichelatti, G. Salvetti, “Spatial and spectral response of a Fabry–Perot interferometer illuminated by a Gaussian beam,” Appl. Opt. 34, 4703–4712 (1995).
[CrossRef] [PubMed]

1994

1993

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

M. Nenchev, 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

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

1991

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

1988

1985

1984

1981

1978

A. Pomeranski, U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrosk. 45 (No. 4), 773–779 (1978) (in Russian).

Alen, J.

Alipieva, E.

E. Alipieva, E. Stoykova, V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” in Laser Physics and Applications, P. Atanassov and S. Kartaleva, eds., Proc. SPIE4397, 129–133 (2001).

Barbe, R.

M. Gorris-Neveux, M. Nenchev, R. Barbe, 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, E. Wolf, Principles of Optics (Pergamon, 1980).

Bugarin, J.

Deleva, A.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, 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, 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, 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. V.

Doval, A.

Fernandez, A.

Fernandez, J.

Friberg, A.

H. Lauranto, A. Friberg, T. Kajava, “Influence of the finite aperture and observation of higher longitudinal-order fringes in Fizeau interferometry,” Opt. Eng. 34, 2623–2630 (1995).
[CrossRef]

T. Kajava, H. Lauranto, A. Friberg, “Interference pattern of the Fizeau interferometer,” J. Opt. Soc. Am. A 11, 2045–2054 (1994).
[CrossRef]

Gizbrekht, A.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, 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 (McGraw-Hill, 1968).

Gorris-Neveux, M.

M. Gorris-Neveux, M. Nenchev, R. Barbe, 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, T. Hansch, “Direct frequency approaches impact wavelength meters,” Laser Focus World, February 1990, pp. 69–76.

Józwicki, R.

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

Kajava, T.

Keller, J.-C.

M. Gorris-Neveux, M. Nenchev, R. Barbe, 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, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 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.

Martin, M.

McIlrath, T.

Meyer, Y.

Morris, M.

Nenchev, M.

E. Stoykova, 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, 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, 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, M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 28, 155–167 (1996).
[CrossRef]

M. Gorris-Neveux, M. Nenchev, R. Barbe, 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, 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, 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, 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, M. Martin, Y. Meyer, “Alternate wavelength DIAL dye laser using a reflecting interference wedge,” Appl. Opt. 24, 1957–1959 (1985).
[CrossRef] [PubMed]

Y. Meyer, 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, V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” in Laser Physics and Applications, P. Atanassov and S. Kartaleva, eds., Proc. SPIE4397, 129–133 (2001).

Patrikov, T.

M. Nenchev, A. Deleva, E. Stoykova, Z. Peshev, T. Patrikov, 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, 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, U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrosk. 45 (No. 4), 773–779 (1978) (in Russian).

Reiser, C.

Salbut, L.

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

Salomaa, R.

Salvetti, G.

Slavov, D.

M. Deneva, D. Slavov, E. Stoykova, 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 U. Press, 1997).

Snyder, J.

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

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

Soto, R.

Stoykova, E.

E. Stoykova, 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, 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, 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, M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 28, 155–167 (1996).
[CrossRef]

E. Stoykova, 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, 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, A. Gizbrekht, “Controlled time-delayed pulses operation of a two-wavelength combined dye-Ti:Al2O3 laser,” Opt. Commun. 86, 405–408 (1991).
[CrossRef]

E. Alipieva, E. Stoykova, V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” in Laser Physics and Applications, P. Atanassov and S. Kartaleva, eds., Proc. SPIE4397, 129–133 (2001).

Tomashevski, U.

A. Pomeranski, U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrosk. 45 (No. 4), 773–779 (1978) (in Russian).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1980).

Appl. Opt.

IEEE J. Quantum Electron.

M. Gorris-Neveux, M. Nenchev, R. Barbe, 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.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

M. Deneva, D. Slavov, E. Stoykova, 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, 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.

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

H. Lauranto, A. Friberg, T. Kajava, “Influence of the finite aperture and observation of higher longitudinal-order fringes in Fizeau interferometry,” Opt. Eng. 34, 2623–2630 (1995).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

M. Nenchev, 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]

E. Stoykova, M. Nenchev, “Reflection and transmission of unequal mirrors interference wedge,” Opt. Quantum Electron. 28, 155–167 (1996).
[CrossRef]

Opt. Spektrosk.

A. Pomeranski, U. Tomashevski, “Spatial structure of multiple Fizeau interference,” Opt. Spektrosk. 45 (No. 4), 773–779 (1978) (in Russian).

Rev. Sci. Instrum.

M. Deneva, E. Stoykova, 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

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

E. Alipieva, E. Stoykova, V. Nikolova, “Wavemeter with Fizeau interferometer for CW lasers,” in Laser Physics and Applications, P. Atanassov and S. Kartaleva, eds., Proc. SPIE4397, 129–133 (2001).

M. Born, E. Wolf, Principles of Optics (Pergamon, 1980).

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 (McGraw-Hill, 1968).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Ray tracing for transmission of a Gaussian beam by an interferential Fizeau wedge.

Fig. 2
Fig. 2

Normalized distributions of maximum transmitted intensity for a 50 μ m wedge. (top: α W = 5 μ rad ; bottom: α W = 50 μ rad ). The thin solid lines depict R = 0.9 , the thick solid lines depict R = 0.995 , and the dashed lines depict the incident Gaussian beam f ( x ) = w 0 1 2 π exp 2 ( x x 0 ) 2 w 0 2 .

Fig. 3
Fig. 3

Patterns of transmitted intensity for a 50 μ m wedge with apex angle 50 μ rad at different wavelengths (left: Gaussian beam with w 0 = 200 μ m ; right: Gaussian beam with w 0 = 2500 μ m ). The numbers associated with each curve give the wavelength in nanometers. The dashed lines depict the incident Gaussian beam f ( x ) = w 0 1 2 π exp 2 ( x x 0 ) 2 w 0 2 .

Fig. 4
Fig. 4

Patterns of transmitted intensity for a 50 μ m wedge with apex angle 50 μ rad and R = 0.8 at increasing beam width: (a) w 0 = 3 × 10 4 μ m , (b) w 0 = 10 5 μ m , (c) w 0 = 3 × 10 5 μ m , (d) w 0 = 10 6 μ m , (e) w 0 = 10 6 μ m . The dashed lines depict the incident beam f ( x ) = w 0 1 2 π exp 2 ( x x 0 ) 2 w 0 2 . The patterns are obtained for a power-normalized beam, and the height of the peaks is diminishing with the rise of the beam diameter.

Fig. 5
Fig. 5

(a) 2-D spatial distribution of a normalized Gaussian beam f ( x ) = exp 2 ( x x 0 ) 2 w 0 2 with 2 w 0 = 100 μ m ; (b) 2-D spatial distributions of transmitted intensity behind the 50 μ m wedge for a normal incidence of the Gaussian beam in (a). Distributions are normalized to the peak intensity on the rear wedge surface; λ R = 0.6320975 μ m .

Fig. 6
Fig. 6

2-D spatial distributions of transmitted intensity behind the 50 μ m wedge with apex angle 50 μ rad for normal incidence of a 3 - cm -diameter beam. Distributions are normalized to the peak intensity on the rear wedge surface. The obtained distributions are identical to profiles at plane-wave illumination; λ R = 0.6334 μ m .

Fig. 7
Fig. 7

Percentage of transmitted power as a function of the wavelength at different diameters of the incident Gaussian beam for a 50 μ m wedge at R = 0.9 and α W = 50 μ rad . The dashed line corresponds to plane-wave illumination.

Fig. 8
Fig. 8

Transmission peak width as a function of reflectivity. Squares, e = 5 μ m ; triangles, e = 500 μ m ; dashed lines depict the dependence δ x PW = c ( log R ) α W with c = 1 μ m .

Fig. 9
Fig. 9

Excess of the transmission peak width over its width at e = 5 μ m with the increase of the wedge thickness. Squares, α W = 10 μ rad ; triangles, α W = 50 μ rad .

Tables (2)

Tables Icon

Table 1 Percentage of the Resonant Transmitted Power as a Function of Reflectivity of the FW Coatings ( e = 19.89 μ m , α W = 50 μ rad , λ = 0.6328 μ m )

Tables Icon

Table 2 Spectral Width of the Wedge Transmission in Picometers

Equations (29)

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 = 1 , 2 , .
A PW ( x , z ) = T p = 0 R p exp [ i p ( ϕ + ϕ ) ] exp { i k [ sin ( 2 p α W ) x + cos ( 2 p α W ) z ] } ,
A PW ( x , z ) = T p = 1 N R p 1 exp [ i ( p 1 ) ( ϕ + ϕ ) ] exp [ i k δ p ( x , z ) ] ,
δ p ( x , z ) = x ( sin ξ p sin ξ 1 ) + z ( cos ξ p cos ξ 1 ) ,
δ p ( x , z ) = x sin ( θ 2 p α W ) x sin θ + z cos ( θ 2 p α W ) z cos θ .
g ( x ) = g ( x , z 0 ) = G 0 ( α ) exp ( i 2 π α x ) 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 .
τ ( x , z , η ) = T p = 1 R p 1 exp [ i 2 ( p 1 ) π ] exp [ i φ p ( x , z , η ) ] .
φ p ( x , z , η ) = 2 π λ ( x ψ p + z ψ p ) ,
ψ p = cos ( ξ p + η ) cos ( ξ 1 + η ) ,
ψ p = sin ( ξ p + η ) sin ( ξ 1 + η ) .
A T ( 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 + λ α ν p ,
exp { i 2 π λ [ 1 2 ( λ α ) 2 ( z z 0 ) + ( λ α ) x + x ψ p + z ψ p ] } = exp { i 2 π λ [ 1 2 ( λ α ) 2 ( z z 0 ) + ( λ α ) x + x ν p 1 2 ( λ α ) 2 x ν p + x ν p λ α + z ν p 1 2 ( λ α ) 2 z ν p z ν p λ α ] } = exp { i 2 π λ [ 1 2 ( z z 0 + x ν p + z ν p ) ( λ α ) 2 + ( x + x ν p z ν p ) ( λ α ) + x ν p + z ν p ] } .
G 0 ( α ) = 2 π 4 w 0 exp [ ( π w 0 α ) 2 ] exp ( i 2 π x 0 ) .
A T ( x , z ) = a T exp [ i 2 π λ ( z z 0 ) ] p = 1 R p 1 exp ( b α 2 ) exp [ i 2 π λ ( p p α 2 + q p α + r p ) ] d α ,
a = 2 π 4 w 0 , b = π 2 w 0 2 , p p = 1 2 ( z z 0 + x ν p + z ν p ) λ 2 ,
q p = ( x + x ν p z ν p + x 0 ) λ , r p = x ν p + z ν p .
exp ( b α 2 ) cos sin [ 2 π λ ( p p α 2 + q p α + r p ) ] d α = Ω p × cos ε p sin 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 ( x , z , λ ) = 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 ( x , z , λ ) = a 2 T 2 ( S 1 2 + S 2 2 ) ,
S 1 = p = 1 R p 1 Ω p cos ε p , S 2 = p = 1 R p 1 Ω p sin ε p .
ρ ( Ω ) = I T ( x , Ω ) d x .
ρ ( Ω ) x 0 q w 0 x 0 + q w 0 I T ( x , Ω ) d x .
δ λ PW = d λ d x δ x PW .

Metrics