Abstract

A vector representation of the formation of fringe profiles in a Fizeau (wedge) interferometer shows that, under certain conditions, off-axis illumination may lead to fringe sharpening. The incident angle is such that the beam is first reflected toward the apex of the wedge and, following a certain controllable number of reflections, is reflected away from the apex. A typical example is shown.

© 1970 Optical Society of America

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References

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  1. J. Pastor, H. Lee, J. Opt. Soc. Amer. 58, 149 (1968).
    [CrossRef]
  2. P. Langenbeck, J. Opt. Soc. Amer. 58, 1568 (1968).
  3. J. Brossel, Proc. Phys. Soc. 59, 224 (1947).
    [CrossRef]
  4. T. A. Hall, J. Sci. Instrum. (J. Phys. E) 2, 837 (1969).
    [CrossRef]
  5. G. Koppelmann, Habilitationsschrift (Tech. Univ.Berlin, 1966).
  6. E. Roberts, P. Langenbeck, Appl. Opt. 8, 11 (1969).
    [CrossRef]

1969 (2)

T. A. Hall, J. Sci. Instrum. (J. Phys. E) 2, 837 (1969).
[CrossRef]

E. Roberts, P. Langenbeck, Appl. Opt. 8, 11 (1969).
[CrossRef]

1968 (2)

J. Pastor, H. Lee, J. Opt. Soc. Amer. 58, 149 (1968).
[CrossRef]

P. Langenbeck, J. Opt. Soc. Amer. 58, 1568 (1968).

1947 (1)

J. Brossel, Proc. Phys. Soc. 59, 224 (1947).
[CrossRef]

Brossel, J.

J. Brossel, Proc. Phys. Soc. 59, 224 (1947).
[CrossRef]

Hall, T. A.

T. A. Hall, J. Sci. Instrum. (J. Phys. E) 2, 837 (1969).
[CrossRef]

Koppelmann, G.

G. Koppelmann, Habilitationsschrift (Tech. Univ.Berlin, 1966).

Langenbeck, P.

E. Roberts, P. Langenbeck, Appl. Opt. 8, 11 (1969).
[CrossRef]

P. Langenbeck, J. Opt. Soc. Amer. 58, 1568 (1968).

Lee, H.

J. Pastor, H. Lee, J. Opt. Soc. Amer. 58, 149 (1968).
[CrossRef]

Pastor, J.

J. Pastor, H. Lee, J. Opt. Soc. Amer. 58, 149 (1968).
[CrossRef]

Roberts, E.

E. Roberts, P. Langenbeck, Appl. Opt. 8, 11 (1969).
[CrossRef]

Appl. Opt. (1)

E. Roberts, P. Langenbeck, Appl. Opt. 8, 11 (1969).
[CrossRef]

J. Opt. Soc. Amer. (2)

J. Pastor, H. Lee, J. Opt. Soc. Amer. 58, 149 (1968).
[CrossRef]

P. Langenbeck, J. Opt. Soc. Amer. 58, 1568 (1968).

J. Sci. Instrum. (J. Phys. E) (1)

T. A. Hall, J. Sci. Instrum. (J. Phys. E) 2, 837 (1969).
[CrossRef]

Proc. Phys. Soc. (1)

J. Brossel, Proc. Phys. Soc. 59, 224 (1947).
[CrossRef]

Other (1)

G. Koppelmann, Habilitationsschrift (Tech. Univ.Berlin, 1966).

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Figures (9)

Fig. 1
Fig. 1

Fizeau multiple beam interferometer. Reflectivity of M is ρ, of T on air side r, and on glass side r. Phase change in reflection at M is ψ, at T on air side α, on glass side α, and in transmission β.

Fig. 2
Fig. 2

(a) Vector diagram representing first two vectors E1 and E2 from multiple beam series. Lengths are proportional to amplitude. As E1 rotates for ϕ, E2 rotates for 2ϕ. (b) Resultant of E1 and E2 in polar coordinates.

Fig. 3
Fig. 3

(a) Airy distribution: Vector diagram representing first six vectors from multiple beam series. As the first vector E1 has a phase ϕ, the nth vector has a phase nϕ. (b) Airy distribution: Resultant of first two, four, six, and fourteen vectors represented in polar coordinates. The curve obtained with infinitely many vectors takes on approximately the shape shown with the dashed line (Airy case).

Fig. 4
Fig. 4

(a) Fizeau distribution: Vector diagram representing first six vectors. Diagram shows the influence of phase defect. (b) Fizeau distribution: Resultant of first six vectors shown in Fig. 4(a) in polar coordinates. Diagram shows generation of side bands, decrease of, and shift of, maximum amplitude as a function of phase defect.

Fig. 5
Fig. 5

Fizeau interferometer with off-axis illumination. The n*th beam is retroreflected, if the incident beam includes an angle θ = n* ( = wedge angle).

Fig. 6
Fig. 6

Three vector chains (at ϕ = 0) for the Airy and the Fizeau case and for the off-axis Fizeau case. The latter illustrates the sharpening of Fizeau fringes through off-axis illumination.

Fig. 7
Fig. 7

Schematic increase of phase defect as a function of n for the normal Fizeau case (θ = 0) and the off-axis Fizeau case (θ = n*).

Fig. 8
Fig. 8

Large cavity Fizeau interferometer (M, T, CC1) where off-axis illumination produces sharper fringes than on-axis illumination. An auxiliary corner cube (CC2) forms a Twyman-Green interferometer with B as beam splitter and the glass-side of T as reference mirror. This combination shows the angle the transmission mirror T forms with the incident beam.

Fig. 9
Fig. 9

Fringe sharpening by off-axis illumination. Left side shows auxiliary corner cube CC2 indicating the angle θ which the beam splitter includes with the optical axis. Right side shows one edge of CC1 under test. (a) θ = 0, fringes show side bands and are broad. (b) θ optimized for sharper fringes.

Equations (13)

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E 0 , E 1 , E 2 E n .
E 1 = t 2 ρ exp { [ i ( 2 β + ψ ) + ϕ ] } , E 2 = t 2 ρ ( ρ r ) exp { i [ ( 2 β + ψ ) + ( ψ + α ) + 2 ϕ ] } ,
ϕ n = 2 π λ n t cos θ ( 1 n tan θ 2 n 2 + 1 3 2 ) .
ϕ n = ( 2 π / λ ) t n ( 2 π / λ ) t ( 2 3 n 3 2 ) = n ϕ τ n
ϕ n = n ϕ 1 3 ( λ π / p 2 ) n 3 t 0 ,
= λ / 2 p , λ = 6.3 × 10 5 cm , p = 0.1 cm , t 0 = 2 cm ,
ϕ n = n ϕ 0.013 n 3 .
n = 3 / ( 1 R ) .
τ n = 2 3 ( π / λ ) t 2 n 3 < π / 4 ( Brossel s 3 criterion ) .
n < ( 6 P 2 / t λ ) 1 3 .
( 1 R ) 3 ( t λ / 6 P 2 ) 1 3 .
0 n * n 3 d n = n * n n 3 d n , n * 0.8 n .
θ 0.8 ( 6 P 2 / t λ ) 1 3 ( λ / 2 P ) .

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