Abstract

Interferometers used for studies of spectral line profiles should produce instrumentally narrow and smooth interference fringes. We analyze in detail the fringe-formation process in a Fizeau wedge interferometer and develop a fringe-optimization method that can be used to find the optimum angle of incidence for producing the sharpest fringes in the detection plane. We show that the Fizeau fringes can be sharper than generally thought. The Fizeau interferometer can thus be used as a high-resolution spectrum analyzer that is suitable for both pulsed and cw light sources. The spectral resolution of such an analyzer can be made comparable with that of the more common Fabry–Perot interferometer. The Fizeau interferometer, however, does not require the same amount of alignment work and is thus more convenient to use with sources with variable bandwidths. Spectral measurement of pulsed lasers is an important application in which an analyzer of this kind is preferable.

© 1993 Optical Society of America

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References

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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. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 17.
  3. J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
    [CrossRef]
  4. K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
    [CrossRef]
  5. T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Sci. Instrum. 2, 837–840 (1969).
    [CrossRef]
  6. Y. H. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. 71, 1255–1263 (1981).
    [CrossRef]
  7. J. R. Rogers, “Fringe shifts in multiple-beam Fizeau interferometry,” J. Opt. Soc. Am. 72, 638–643 (1982).
    [CrossRef]
  8. L. A. Westling, M. G. Raymer, and J. J. Snyder, “Single-shot spectral measurements and mode correlations in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 1, 150–154 (1984).
    [CrossRef]
  9. P. H. Langenbeck, “Fizeau interferometer—fringe sharpening,” Appl. Opt. 9, 2053–2058 (1970).
    [CrossRef] [PubMed]
  10. W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), pp. 155–159.
  11. K. R. German, “High finesse étalons for laser diagnostics,” Lasers Optron., March1990, pp. 51–54.
  12. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), pp. 351–356.
  13. IMSL Math/Library, User’s Manual (IMSL, Inc., Houston, 1989), pp. 831–834.
  14. T. T. Kajava, H. M. Lauranto, and R. R. E. Salomaa, “Mode structure fluctuations in a pulsed dye laser,” Appl. Opt. 31, 6987–6992 (1992).
    [CrossRef] [PubMed]
  15. F. M. Dickey and T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
    [CrossRef]

1992 (1)

1990 (1)

K. R. German, “High finesse étalons for laser diagnostics,” Lasers Optron., March1990, pp. 51–54.

1984 (1)

1982 (1)

1981 (1)

1978 (1)

F. M. Dickey and T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

1970 (1)

1969 (1)

T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Sci. Instrum. 2, 837–840 (1969).
[CrossRef]

1953 (1)

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

1947 (1)

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

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).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), pp. 351–356.

Brossel, J.

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

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), pp. 155–159.

Dickey, F. M.

F. M. Dickey and T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

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).

German, K. R.

K. R. German, “High finesse étalons for laser diagnostics,” Lasers Optron., March1990, pp. 51–54.

Hall, T. A.

T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Sci. Instrum. 2, 837–840 (1969).
[CrossRef]

Harder, T. M.

F. M. Dickey and T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Kajava, T. T.

Kinosita, K.

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

Langenbeck, P. H.

Lauranto, H. M.

Meyer, Y. H.

Raymer, M. G.

Rogers, J. R.

Salomaa, R. R. E.

Snyder, J. J.

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 17.

Westling, L. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), pp. 351–356.

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).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

J. Phys. Soc. Jpn. (1)

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

J. Sci. Instrum. (1)

T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Sci. Instrum. 2, 837–840 (1969).
[CrossRef]

Lasers Optron. (1)

K. R. German, “High finesse étalons for laser diagnostics,” Lasers Optron., March1990, pp. 51–54.

Opt. Eng. (1)

F. M. Dickey and T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

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]

Other (4)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), pp. 351–356.

IMSL Math/Library, User’s Manual (IMSL, Inc., Houston, 1989), pp. 831–834.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 17.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), pp. 155–159.

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Figures (11)

Fig. 1
Fig. 1

Fizeau interferometer with wedge angle α and optical thickness L. With collimated illumination the interference pattern of a Fizeau interferometer results from the interference between plane wavefronts Wi that are cophasal at the wedge apex O.

Fig. 2
Fig. 2

Curves that represent the angles of incidence θ at which the phase difference between the directly transmitted and the nth beam is n = ±π/2. (L = 15 mm, α = 25 μrad, λ = 500 nm.) On the dotted line the phase difference between the beams vanishes. The most favorable conditions for constructive multiple-beam interference occur when the angle of incidence is chosen in such a way that the corresponding solid horizontal line stays between the limit curves as far as possible. With a detection distance of (a) x = 100 mm, θ ≈ 0.80 mrad and (b) with x = 500 mm, θ ≈ 1.5 mrad.

Fig. 3
Fig. 3

Limit curves of Fig. 2(a) transformed to be as horizontal as possible by the changing of the effective thickness of the wedge ΔLL/α = 0.30 mm). An angle of incidence of θ ≈ 1.4 mrad for the L = 15 mm wedge with x = 100 mm is suggested.

Fig. 4
Fig. 4

Limit curves for the L = 15 mm wedge with x = 100 mm and R = 0.95, optimized to produce maximum fringe height (θ = 1.224 mrad, ΔL/α = 0.249 mm).

Fig. 5
Fig. 5

(a) Limit curves and the angle of incidence for an L = 100 mm wedge with x = 100 mm and R = 0.95 mm, optimized to produce maximum fringe height (θ = 0.639 mrad, ΔL/α = 0.559 mm). (b) Optimized limit curves and the angle of incidence for a L = 1 mm wedge with x = 100 mm and R = 0.95 (θ = 4.447 mrad, ΔL/α = 0.054 mm).

Fig. 6
Fig. 6

Calculated fringe profiles for an L = 15 mm wedge in the vertical detection plane at x = 100 mm (R = 0.95, α = 25 μrad, λ = 500 nm). (a) Optimized fringe (θ = 1.224 mrad) with a finesse of 42.1; (b) θ = 1.2 mrad, (c) θ = 1.3 mrad. (d) Narrowest fringe found with a finesse of 46 (θ = 1.5 mrad); (e) 0 = 0 mrad, (f) θ = 2.5 mrad.

Fig. 7
Fig. 7

(a) Optimized fringe profile for an L = 100 mm wedge (θ = 0.639 mrad) in the detection plane x = 100 mm. (b) Optimized fringe profile for an L = 1 mm wedge (θ = 4.447 mrad) in the detection plane x = 100 mm. The finesse of the fringes is 59.9 (R = 0.95, α = 25 μrad, λ = 500 nm). (c) Fringes from an L = 1 mm Fabry–Perot interferometer (α = 0) with a finesse of 62.1.

Fig. 8
Fig. 8

Optical design of the Fizeau spectrum analyzer, consisting of a spatial filter, collimating optics, and a Fizeau interferometer. The fringe pattern is detected with a linear photodiode array, externally controlled by a microcomputer through an array controller. The fringe pattern is seen on an oscilloscope screen and stored in the computer.

Fig. 9
Fig. 9

Measured Fizeau fringes for an L = 15 mm wedge with a detection distance of x = 100 mm (R = 0.95, α = 23 μrad, λ = 514.5 nm); (a) θ = 1.25 mrad, (b) θ = 0 mrad, (c) θ = 2.5 mrad.

Fig. 10
Fig. 10

(a) Experimentally optimized Fizeau fringes for an L = 100 mm wedge with a detection distance of x = 100 mm (θ = 0.6 mrad, R = 0.95, α = 21 μrad, λ = 514.5 nm). (b) Experimentally optimized Fizeau fringes for an L = 1 mm wedge with a detection distance of x = 100 mm (θ ≈ 4.5 mrad, R ≈ 0.95, α ≈ 17 μrad, λ = 514.5 nm).

Fig. 11
Fig. 11

Simulated spectral measurements with a Fizeau spectrum analyzer with FSR = 1500 MHz (L = 100 nm). (a) Measured spectrum when the true spectrum consists of three delta peaks separated by 200 MHz. (b) Measured spectrum when the true spectrum consists of three 100-MHz-wide Gaussian peaks separated by 200 MHz. The curves are obtained by convolving the true spectra with the instrumental fringe of Fig. 7(a).

Equations (20)

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v = FSR finesse = c 2 L F * ,
Δ y = λ / 2 α .
δ n = N 0 P N n P = y sin θ x cos θ + x cos ( θ 2 n α ) y sin ( θ 2 n α ) .
E [ L ( y ) ] = E 0 n = 0 N r 2 n exp ( i k δ n ) , I [ L ( y ) ] = I 0 | n = 0 N r 2 n exp ( i k δ n ) | 2 ,
θ θ θ 0 , x x cos θ 0 y sin θ 0 , y x sin θ 0 + y cos θ 0 ,
sin ( n α ) n α [ ( n α ) 3 / 6 ] + O ( n α ) 5 , cos ( n α ) 1 [ ( n α ) 2 / 2 ] + O ( n α ) 4 , 1 / tan ( α ) ( 1 / α ) ( α / 3 ) + O ( α ) 3 ,
δ n = 2 n α x ( sin θ n α cos θ ) + 2 n L [ cos θ + n α sin θ ( 2 n 2 + 1 ) 3 α 2 cos θ ] .
2 L cos θ = m λ .
| Δ δ n | = | 2 n α x ( sin θ n α cos θ ) + 2 n L [ n α sin θ ( 2 n 2 + 1 ) 3 α 2 cos θ ] | λ / 2 .
N log ( Δ I ) 2 log ( R ) .
L eff ( θ ) = L cos θ + x α sin θ L α 2 / 3 ,
Δ δ n = 2 n 2 α 2 x cos θ + 2 n L ( n α sin θ 2 n 2 3 α 2 cos θ ) .
θ ± = ( 2 n 3 + x L ) α ± λ 8 n 2 α L ,
θ θ θ 0 , x x cos θ 0 L [ ( 1 / α ) ( α / 3 ) ] sin θ 0 , L L cos θ 0 + x α sin θ 0 ,
L eff ( θ ) = L cos θ + x α sin θ L α 2 3 cos θ + Δ L .
Δ δ n = 2 ( n α ) 2 x cos θ + 2 n L ( n α sin θ 2 n 2 3 α 2 cos θ ) 2 n Δ L .
θ ± = ( 2 n 3 + x L ) α + Δ L α n L ± λ 8 n 2 α L .
I [ L ( y ) ] = I 0 | n = 0 N R n exp ( i k Δ δ n ) | 2
y m Fz = m λ / ( 2 α ) x sin θ Δ L / α ( 1 α 2 / 3 ) cos θ .
S m = Δ L / ( α cos θ ) + L α / 3 ,

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