Abstract

The shapes of the reflected and transmitted fringes of an interferential wedge are studied theoretically and experimentally. It is shown that, for high reflectivities of wedge coatings, interesting interference patterns are obtained, such as bright narrow fringes in reflection. The fringe profiles are computed for different wedge geometries, coating reflectivities, and incidence angles in the cases of equal and unequal coating reflectivities. The localization of the fringes is studied in relation to the focusing effect that is due to the wedge. The interferential wedge properties of frequency selectivity and tunability are discussed briefly with regard to applications to spectroscopy and laser technology.

© 1981 Optical Society of America

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  1. H. Fizeau, “Recherches sur les modifications que subit la vitesse de la lumière dans le verre sous l’influence de la chaleur,” Ann. Chim. Phys. 66, 429–482 (1862).
  2. M. Hamy, “Sur les franges de reflexion des lames argentées,” J. Phys. 5, 789–809 (1906).
  3. J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
    [Crossref]
  4. J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
    [Crossref]
  5. K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 215–225 (1953).
    [Crossref]
  6. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  7. A. R. Cownie, “Formation of transmission like reflection fringes,” J. Opt. Soc. Am. 53, 425–428 (1963).
    [Crossref]
  8. D. Sen and P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–149 (1965).
    [Crossref]
  9. N. Barakat and F. G. Abouzakhm, “Phase modulation of the first beam in reflected multiple beam interference fringes,” Opt. Acta 12, 321–332 (1965).
    [Crossref]
  10. H. Dupoisot, “Mesure de l’épaisseur des couches minces par microscopie interferentielle à ondes multiples,” Thesis, University of Paris (1965); Ann. Phys. 3, 369–390 (1968).
  11. J. Pastor and P. H. Lee, “Transmission fringes in reflection multiple-beam interferometry,” J. Opt. Soc. Am. 58, 149–153 (1968).
    [Crossref]
  12. N. Aebischer, “Etude d’interférences en ondes multiples par diagramme complexe pour visualiser les franges en réflexion,” Thesis, University of Besançon (1969); Nouv. Rev. Opt. Appl. 1, 233–248 (1970).
  13. F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).
  14. J. J. Snyder, “Fizeau wavelength meter,” in Laser Spectroscopy III (Springer-Verlag, Berlin, 1977), pp. 419–420; “Compact static wavelength meter for both pulsed and cw lasers,” Sov. J. Quantum Electron. 8, 959–960 (1979); C. Cahen, J. P. Jegou, J. Pelon, P. Gildwarg, and J. Porteneuve, “Wavelength stabilization and control of the emission of pulsed dye lasers by means of a multi-beam Fizeau interferometer,” Rev. Phys. Appl. 16, 353–358 (1981).
    [Crossref]
  15. V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).
  16. Spectra-Physics, Laser Instruments Division, 1250 W. Middlefield Road, Mountain View, Calif. 94042.
  17. Y. H. Meyer and M. N. Nenchev, “Tuning of dye lasers with a reflecting Fizeau wedge,” Opt. Commun. 35, 115–119 (1980); “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
    [Crossref] [PubMed]
  18. D. Malacara, Optical Shop Testing (Wiley, New York, 1978) Sec. 6.3.

1980 (1)

Y. H. Meyer and M. N. Nenchev, “Tuning of dye lasers with a reflecting Fizeau wedge,” Opt. Commun. 35, 115–119 (1980); “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
[Crossref] [PubMed]

1972 (1)

V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).

1970 (1)

F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).

1968 (1)

1965 (2)

D. Sen and P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–149 (1965).
[Crossref]

N. Barakat and F. G. Abouzakhm, “Phase modulation of the first beam in reflected multiple beam interference fringes,” Opt. Acta 12, 321–332 (1965).
[Crossref]

1963 (1)

1953 (1)

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 215–225 (1953).
[Crossref]

1949 (1)

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[Crossref]

1947 (1)

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

1906 (1)

M. Hamy, “Sur les franges de reflexion des lames argentées,” J. Phys. 5, 789–809 (1906).

1862 (1)

H. Fizeau, “Recherches sur les modifications que subit la vitesse de la lumière dans le verre sous l’influence de la chaleur,” Ann. Chim. Phys. 66, 429–482 (1862).

Abouzakhm, F. G.

N. Barakat and F. G. Abouzakhm, “Phase modulation of the first beam in reflected multiple beam interference fringes,” Opt. Acta 12, 321–332 (1965).
[Crossref]

Aebischer, N.

N. Aebischer, “Etude d’interférences en ondes multiples par diagramme complexe pour visualiser les franges en réflexion,” Thesis, University of Besançon (1969); Nouv. Rev. Opt. Appl. 1, 233–248 (1970).

Barakat, N.

N. Barakat and F. G. Abouzakhm, “Phase modulation of the first beam in reflected multiple beam interference fringes,” Opt. Acta 12, 321–332 (1965).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Brossel, J.

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

Bunnagel, R.

F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).

Bushuk, B. A.

V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).

Cownie, A. R.

Dupoisot, H.

H. Dupoisot, “Mesure de l’épaisseur des couches minces par microscopie interferentielle à ondes multiples,” Thesis, University of Paris (1965); Ann. Phys. 3, 369–390 (1968).

Fizeau, H.

H. Fizeau, “Recherches sur les modifications que subit la vitesse de la lumière dans le verre sous l’influence de la chaleur,” Ann. Chim. Phys. 66, 429–482 (1862).

Hamy, M.

M. Hamy, “Sur les franges de reflexion des lames argentées,” J. Phys. 5, 789–809 (1906).

Holden, J.

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[Crossref]

Kinosita, K.

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 215–225 (1953).
[Crossref]

Lee, P. H.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978) Sec. 6.3.

Meyer, Y. H.

Y. H. Meyer and M. N. Nenchev, “Tuning of dye lasers with a reflecting Fizeau wedge,” Opt. Commun. 35, 115–119 (1980); “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
[Crossref] [PubMed]

Moser, J.

F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).

Nenchev, M. N.

Y. H. Meyer and M. N. Nenchev, “Tuning of dye lasers with a reflecting Fizeau wedge,” Opt. Commun. 35, 115–119 (1980); “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
[Crossref] [PubMed]

Pastor, J.

Puntambekar, P. N.

D. Sen and P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–149 (1965).
[Crossref]

Rubinov, A. N.

V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).

Sen, D.

D. Sen and P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–149 (1965).
[Crossref]

Snyder, J. J.

J. J. Snyder, “Fizeau wavelength meter,” in Laser Spectroscopy III (Springer-Verlag, Berlin, 1977), pp. 419–420; “Compact static wavelength meter for both pulsed and cw lasers,” Sov. J. Quantum Electron. 8, 959–960 (1979); C. Cahen, J. P. Jegou, J. Pelon, P. Gildwarg, and J. Porteneuve, “Wavelength stabilization and control of the emission of pulsed dye lasers by means of a multi-beam Fizeau interferometer,” Rev. Phys. Appl. 16, 353–358 (1981).
[Crossref]

Spiegelhalter, F.

F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).

Tomin, V. I.

V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Ann. Chim. Phys. (1)

H. Fizeau, “Recherches sur les modifications que subit la vitesse de la lumière dans le verre sous l’influence de la chaleur,” Ann. Chim. Phys. 66, 429–482 (1862).

J. Opt. Soc. Am. (2)

J. Phys. (1)

M. Hamy, “Sur les franges de reflexion des lames argentées,” J. Phys. 5, 789–809 (1906).

J. Phys. Soc. Jpn. (1)

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 215–225 (1953).
[Crossref]

Opt. Acta (2)

D. Sen and P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–149 (1965).
[Crossref]

N. Barakat and F. G. Abouzakhm, “Phase modulation of the first beam in reflected multiple beam interference fringes,” Opt. Acta 12, 321–332 (1965).
[Crossref]

Opt. Commun. (1)

Y. H. Meyer and M. N. Nenchev, “Tuning of dye lasers with a reflecting Fizeau wedge,” Opt. Commun. 35, 115–119 (1980); “Single-mode dye laser with a double-action Fizeau interferometer,” Opt. Lett. 6, 119–121 (1981).
[Crossref] [PubMed]

Opt. Spectrosc. (1)

V. I. Tomin, B. A. Bushuk, and A. N. Rubinov, “Study of the gain curve and the triplet–triplet absorption in a rhodamine 6G solution laser,” Opt. Spectrosc. 32, 527–529 (1972); A. N. Rubinov and V. I. Nikolaev, “Stabilization and control of monopulse radiation spectrum of ruby laser,” Dokl. Akad. Nauk B. SSR 14, 20–24 (1970).

Optik (1)

F. Spiegelhalter, R. Bunnagel, and J. Moser, “Intensitätsverteilung von Fizeau–Streigen in reflektierten Licht bei Vielstrahl-interferenz,” Optik 31, 535–552 (1970).

Proc. Phys. Soc. London (2)

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London 62, 405–417 (1949).
[Crossref]

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

Other (6)

H. Dupoisot, “Mesure de l’épaisseur des couches minces par microscopie interferentielle à ondes multiples,” Thesis, University of Paris (1965); Ann. Phys. 3, 369–390 (1968).

J. J. Snyder, “Fizeau wavelength meter,” in Laser Spectroscopy III (Springer-Verlag, Berlin, 1977), pp. 419–420; “Compact static wavelength meter for both pulsed and cw lasers,” Sov. J. Quantum Electron. 8, 959–960 (1979); C. Cahen, J. P. Jegou, J. Pelon, P. Gildwarg, and J. Porteneuve, “Wavelength stabilization and control of the emission of pulsed dye lasers by means of a multi-beam Fizeau interferometer,” Rev. Phys. Appl. 16, 353–358 (1981).
[Crossref]

Spectra-Physics, Laser Instruments Division, 1250 W. Middlefield Road, Mountain View, Calif. 94042.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

N. Aebischer, “Etude d’interférences en ondes multiples par diagramme complexe pour visualiser les franges en réflexion,” Thesis, University of Besançon (1969); Nouv. Rev. Opt. Appl. 1, 233–248 (1970).

D. Malacara, Optical Shop Testing (Wiley, New York, 1978) Sec. 6.3.

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Figures (8)

Fig. 1
Fig. 1

Multiple reflected and transmitted rays in a wedge for incidence θ0.

Fig. 2
Fig. 2

Dephasing of the waves interfering at point P after multiple reflection in a wedge.

Fig. 3
Fig. 3

Computed ray tracing for reflection and transmission from a wedge. a, Negative incidence; b, positive incidence; c, evaluation of Fn; d, evidence of quasi-focusing at Fn ~ 3 cm of 200 rays in a 100-μm wedge with α = 4 × 10−4 rad.

Fig. 4
Fig. 4

Fringes of the interferential wedge observed within a reflected He–Ne laser beam in a plane at several distances D from the wedge (R = 0.99, R′ = 1, e = 60 μm, α = 10−4 rad, θ0 = 0.2 rad, F ~ 0/α = 12 cm). a, D > F; b, DF; c, DF; d, D < F.

Fig. 5
Fig. 5

Computed shape of the fringes in the wedge plane. Reflected (top curve) and transmitted (lower curve) intensity for an incident intensity of unity as a function of the thickness of an interferential wedge. The number of steps and the size of each step are indicated in the bottom left corner together with the summation number n. In all cases R = R′, and the incidence is normal. The horizontal bar corresponds to a spectral interval of 0.1 Å (λ ≡ L = 0.602 nm). a–f, Effect of varying the reflectivity R in a 200-μm wedge with αA = 3 × 10−5 rad; g–i, fringes calculated with 20-, 79-, and 636-μm-thick wedges and R = 0.99, α = 3 × 10−5; j–m, intensity distribution along a fixed 1.2-mm length on the Y axis for different wedge angles.

Fig. 6
Fig. 6

Computed shape of the fringes in the wedge plane with R = R′ = 0.99, e = 20 μm, α = 3 × 10−5 rad. a–k, effect of varying the incidence θ0I; m, Fabry–Perot fringe with n = 800 (Aα = 0).

Fig. 7
Fig. 7

Computed shape of the reflected fringes in the wedge plane with unequal coating reflectivities. a–c, Evidence of a single bright fringe, effect of varying R1; d–f, optimization of the wedge angle A to obtain maximum peak intensity with the incidence θ0I = −0.02 rad; e–g, effect of varying αA with e = 200 μm and incidence θ0I = −0.008 rad.

Fig. 8
Fig. 8

Selection of wavelength λ1 with a slit S placed at point M1 in a plane Δ at distance D from the wedge.

Tables (1)

Tables Icon

Table 1 Incidence θM Used to Produce the Single Fringe in Reflection in the Wedge Plane for Different Thicknesses

Equations (19)

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θ n = θ 0 ± 2 ( n - 1 ) α ,
θ n = θ 0 ± ( 2 n - 1 ) α ,
δ n = P N n - P N 1 = P O [ sin ( θ n ± γ ) ± sin ( θ 1 ± γ ) ] ,
δ n = ± X p ( cos θ n - cos θ 1 ) + Y p ( sin θ n - sin θ 1 )
δ n = e ( sin θ n - sin θ 1 ) / tan α ± X p [ tan θ 1 ( sin θ n - sin θ 1 ) - cos θ n + cos θ 1 ] ,
φ n = 2 π e ( sin θ n - sin θ 0 ) / ( λ tan α ) .
φ n = 2 π e ( - sin θ n + sin θ 0 ) / ( λ tan α ) ;
φ n = 2 π e ( sin θ n + sin θ 0 ) / ( λ tan α ) .
A R = A 0 { r + t t r 2 r n = 2 ( r r ) n exp i [ φ n + ( n - 2 ) ϕ + ( n - 1 ) ϕ ] } .
I R = I 0 R [ 1 - 2 τ ρ R S 1 + τ 2 ρ 2 R 2 ( S 1 2 + S 2 2 ) ] ,
S 1 = n = 2 ρ n cos φ n ,             S 2 = n = 2 ρ n sin φ n .
A T = A 0 t t { 1 + ( r r ) - 1 n = 2 ( r r ) n × exp i [ φ n + ( n - 1 ) ( ϕ + ϕ ) ] } .
I T = A T 2 = I 0 T T [ 1 + 2 ρ - 1 S 1 + ρ - 2 ( S 1 + S 2 ) ] ,
H n = p = 1 n h p 2 e n [ θ 0 - ( n - 1 ) α ] .
F n = H n 2 ( n - 1 ) α = e n ( n - 1 ) - 1 ( θ 0 / α - n + 1 ) e ( θ 0 / α - n ) .
k λ 2 e cos θ 0 ,
I R = I 0 R [ 1 - 2 1 - R R 2 S 1 + ( 1 - R 2 ) R 4 ( S 1 2 + S 2 2 ) ]
I T = I 0 ( 1 - R ) [ 1 + 2 R S 1 + 1 R 2 ( S 1 2 + S 2 2 ) ] ,
S 1 = n = 2 R n cos φ n ,             S 2 = n = 2 R n sin φ n .