Abstract

This article is based on a series of talks presented by personnel of The Perkin-Elmer Corporation at a symposium held in September 1967 and thereafter brought up to date for inclusion in the APPLIED OPTICS feature discussing Interferometry vs Spectroscopy.

© 1969 Optical Society of America

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Figures (77)

Fig. 2.1
Fig. 2.1

Fringe sharpness comparison.

Fig. 2.2
Fig. 2.2

Plane wave multiple beam interferometers.

Fig. 2.3
Fig. 2.3

Ray paths in Fabry-Perot and Fizeau interferometry.

Fig. 2.4
Fig. 2.4

Fringe profile comparison.

Fig. 2.5
Fig. 2.5

Position dependence of Fizeau fringes.

Fig. 2.6
Fig. 2.6

Fabry-Perot fringe profile.

Fig. 2.7
Fig. 2.7

Basic spherical wave interferometer.

Fig. 2.8
Fig. 2.8

Elimination of walk-off.

Fig. 2.9
Fig. 2.9

Interferogram of 30.5-cm f/4 lens.

Fig. 3.1
Fig. 3.1

Interferogram of two different aspheric surfaces. The interferogram describes the departure at a given state of correction of an in-process surface against the master surface.

Fig. 3.2
Fig. 3.2

Off-axis hologram interferometer. (1) Aspheric test surface, (2) collecting mirror, (3) hologram, (4) camera lens, (5) interferogram, (6) and (12) reference wavefront, (7) laser, (8) beam splitter, (9) small radius spherical mirror (ball), (10) object beam, (11) microscope objective. In the figure the reference wave is represented as spherical; however, a plano wavefront was also used.

Fig. 3.3
Fig. 3.3

On-axis hologram interferometer. The ground glass produces an extended source; the resulting interferograms can be interpreted in the same way as those with a point source. The ground glass produces an integrating effect. The spherical wavefront used as reference could also be a plano wavefront.

Fig. 3.4
Fig. 3.4

Interferograms in the off-axis configuration hologram interferometer (Fig. 3.2). (a) Equal phase field interferogram, between an aspheric surface and itself reconstructed from a hologram. The achievement of a null interferogram proves the ability to reposition the hologram and testing piece within interferometric tolerances. (b) Interferogram of aspheric surface against its reconstruction. Hologram translated with respect to its position during the recording to introduce the reference grid of fringes. Observe the curvature of the fringes. (c) Interferogram of the aspheric surface against itself after a few zones have been introduced by polishing. This interferogram was obtained with the interferometer adjusted as in (a). (d) Interferogram equivalent to (c), but in (b) adjustment. The asymmetric pattern is due to obliquity factors.

Fig. 3.5
Fig. 3.5

Interferograms in the on-axis configuration hologram interferometer (Fig. 3.3). Figures 3.5(a), (b), (c), and (d) are equivalent, respectively, to Figs. 3.4(a), (b), (c), and (d).

Fig. 3.6
Fig. 3.6

Three testing configurations.

Fig. 3.7
Fig. 3.7

Computed interferogram.

Fig. 3.8
Fig. 3.8

Hologram interferogram or moiré interferogram indicative of the profile of the parabolic surface of an x-ray telescope. Departures of fringes from the theoretical moiré represent errors in the parabolic surface. The errors on the interferogram can be related to errors in the surface by using the proper polar coordinate system in the interferogram.

Fig. 3.9
Fig. 3.9

Generation of a synthetic hologram as moiré between two interferograms. The mapping of one plane with respect to another is represented by lines parallel to their intersection, e.g., mapping of ∑3 with respect to ∑1 is represented by ∑3η1. Observe that the projection onto ∑2 of ∑3η1 is parallel to the moiré defined by ∑12 and ∑3η2 as it was supposed to be.

Fig. 3.10
Fig. 3.10

(a) Two-hologram interferometer (schematic). (b) Two-hologram interferometer.

Fig. 3.11
Fig. 3.11

Interferogram produced between two reconstructed wavefronts, one from a real hologram and the other from a synthetic hologram.

Fig. 3.12
Fig. 3.12

Map of errors from an interferogram between two wavefronts, real and synthetic. Each letter represents an increment of λ/10.

Fig. 3.13
Fig. 3.13

Interferogram produced by the moiré between two synthetic, on-axis holograms.

Fig. 4.1
Fig. 4.1

Electron microphotograph of scatter plate surface.

Fig. 4.2
Fig. 4.2

Optical schematic for scatter plate.

Fig. 4.3
Fig. 4.3

Wavefront distortion by a scatter plate.

Fig. 4.4
Fig. 4.4

Schematic of symmetrical scatter plate.

Fig. 4.5
Fig. 4.5

Influence of aberration.

Fig. 4.6
Fig. 4.6

Aperture location for enhanced contrast.

Fig. 4.7
Fig. 4.7

Fringe pattern of mirror.

Fig. 4.8
Fig. 4.8

Scatter plate decentered.

Fig. 4.9
Fig. 4.9

Scatter plate refocused.

Fig. 4.10
Fig. 4.10

Model of error surface.

Fig. 4.11
Fig. 4.11

Fringe pattern from autocollimated paraboloid.

Fig. 5.1
Fig. 5.1

Phase measuring interferometer.

Fig. 5.2
Fig. 5.2

Top: Vector diagram showing orthogonal components for circularly polarized wave. Bottom: Vector diagram for circularly polarized wave.

Fig. 5.3
Fig. 5.3

Vector description of an electric wave passing through a half-wave plate.

Fig. 5.4
Fig. 5.4

Frequency modulation of a circularly polarized beam by a rotating half-wave plate.

Fig. 5.5
Fig. 5.5

Optical frequency changer using fixed and rotating quarter-wave plates.

Fig. 5.6
Fig. 5.6

Phase-measuring interferometer, active optics experiment.

Fig. 5.7
Fig. 5.7

Interferometer components.

Fig. 5.8
Fig. 5.8

Image dissector signal.

Fig. 5.9
Fig. 5.9

Electronic waveforms.

Fig. 5.10
Fig. 5.10

Phase-measuring interferometer calibration.

Fig. 5.11
Fig. 5.11

Raster scan of fringe pattern with plane reflector in test beam.

Fig. 5.12
Fig. 5.12

Active optics experiment, fringe patterns after automatic alignment (numbers refer to sequential angular position of the frequency shifter quarter-wave plate).

Fig. 5.13
Fig. 5.13

Active optics experiment, pinhole images before and after automatic alignment. (a) Segments misaligned, (b) segments aligned.

Fig. 5.14
Fig. 5.14

Figure error profiles from the active optics experiment.

Fig. 6.1
Fig. 6.1

Twyman-Green multipass interferometer. Large wedge angle adjustment of the Fizeau part of the interferometer permits separating the orders of reflection.

Fig. 6.2
Fig. 6.2

Multipass interferograms of the same surface, having a square relief step of λ/8. First, second, third, fourth, and fifth order reflection interferograms are shown. Fringe orientation and spacing can be controlled by the reference mirror.

Fig. 6.3
Fig. 6.3

Multipass Twyman-Green interferometer with built-in polarized light luminance control reference light is reflected under Brewster’s angle at uncoated opaque glass, rotation of plane of polarization of illuminating light by means of λ/2 plate permits one to control luminance of reflected reference light.

Fig. 6.4
Fig. 6.4

Multipass interferometry applied to thin film thickness measurement. Luminance control at the higher orders was achieved by knife-edging (Foucault technique) of the reference beam. Fourth to seventh order of reflection are shown.

Fig. 6.5
Fig. 6.5

Fizeau multipass moiré interferometer with polarized light luminance control. Moiré is used to inspect the narrowly spaced fringes. (Fringe spacing is narrower because of large distance between light source images.)

Fig. 6.6
Fig. 6.6

(a) Multipass moiré Fizeau interferometer, laboratory setup with Perkin-Elmer Erectorset components and Perkin-Elmer 5200 helium–neon laser. (b) Closeup view of Fig. (a). Focal plane screen is seen in center of image. The multiple return light source images are seen at edge of screen. Zeroth order passes below screen, third order passes through slit in screen.

Fig. 6.7
Fig. 6.7

(a) Fizeau multiple beam interferegrams of test mirror. (b)–(f) Multipass moiré interferograms of same test mirror: (b) first order, λ/2 fringes; (c) second order, λ/4 fringes; (d) third order, λ/6 fringes; (e) fourth order, λ/8 fringes; (f) fifth order, λ/10 fringes.

Fig. 6.8
Fig. 6.8

Multipass moiré Fizeau interferograms in an axial alignment (moiré grid parallel to interference fringes) and a normal alignment (moiré grid at an angle to the interference fringes). The top two figures are customary Fizeau multiple beam interferograms. The others are MMF interferograms. The moiré grid plus diffusing screen were fixed in the image plane and photographed from behind.

Fig. 6.9
Fig. 6.9

Generation of second order moiré fringes by doubling the spatial frequency of the moiré grid lines, m1 = 2m2. rth order moiré fringes are interpreted as λ/2rn fringes, where n is the order of reflection (n = 1 in normal situations).

Fig. 6.10
Fig. 6.10

Fizeau multiple beam interferometer, modified to produce second order multiple beam fringes—G is an air-spaced grating that cuts out each second from the first most significant beams.

Fig. 6.11
Fig. 6.11

Fizeau multiple beam interferogram (second order, λ/4 fringes) of surface with various layers that are overcoated to measure their thickness.

Fig. 6.12
Fig. 6.12

Fizeau interferogram (first order, λ/2 fringes) at same surface as in Fig. 6.11. Exactly the same interferometer alignment—only the focal plane grating has been removed.

Fig. 6.13
Fig. 6.13

Second order Fizeau interferogram (magnified from section indicated in Fig. 6.11).

Fig. 6.14
Fig. 6.14

Densitograms of multiple beam fringes. (a) λ/2 fringes shown in Fig. 6.12; (b) λ/4 fringes shown in Fig. 6.13.

Fig. 6.15
Fig. 6.15

Schematic of gradual buildup of walk-off in the multipass interferometer, as the number of reflection increases. Errors are integrated in depth (desired) and in width (undesired).

Fig. 6.16
Fig. 6.16

Multipass interferograms (second, third, and fourth order) compared with a regular Twyman-Green (first order) and a multiple beam Fizeau interferogram.

Fig. 6.17
Fig. 6.17

Grazing incidence multipass interferometer used for Fig. testing large surfaces with small optical components.

Fig. 6.18
Fig. 6.18

Grazing incidence multipass interferograms of a 75-cm diam mirror with about 1λ convexity

Fig. 6.19
Fig. 6.19

Wollaston-type differential interferometer employed to look at a multipass beam.

Fig. 6.20
Fig. 6.20

Differential (or shearing) interferograms of the same surface, having a square relief step of λ/8. First through fifth order of reflection interferograms are shown. S is direction of shear. The interferograms correspond directly to the multipass Twyman-Green interferograms shown in Fig. 6.2.

Fig. 6.21
Fig. 6.21

Fifth order of reflection shearing interferogram. W = direction of walk-off; S = direction of shear (same test mirror as used in Fig. 6.20).

Fig. 7.1
Fig. 7.1

Schematic of tunable laser Fizeau interferometer.

Fig. 7.2
Fig. 7.2

Field of view, tunable laser Fizeau interferometer.

Fig. 7.3
Fig. 7.3

Assembly details of tunable laser Fizeau interferometer.

Fig. 7.4
Fig. 7.4

Interferometer in environmental chamber.

Fig. 7.5
Fig. 7.5

Procedure for locking on fringes.

Fig. 7.6
Fig. 7.6

Composite of fringes.

Fig. 7.7
Fig. 7.7

Operation of position lock-on servo.

Fig. 7.8
Fig. 7.8

Electrooptical schematic of frequency servo.

Fig. 7.9
Fig. 7.9

Scanning etalon display.

Equations (52)

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ϕ 0 - ϕ n = n π .
ϕ 0 - ϕ n = m π .
( ϕ 0 - ϕ n ) - ( ϕ 0 - ϕ n ) = N π ,
( ϕ 0 - ϕ 0 ) - ( ϕ n - ϕ n ) = N π ,
ϕ 0 - ϕ 0 = m π ,             ϕ n - ϕ n = n π ,             m - n = N .
ϕ 0 - ϕ 0 = N π .
E 2 = K 1 ( 0 ) E 0 exp ( - i ω t ) × exp ( 2 π i R / λ ) exp [ i φ ( 0 ) ] ,
E 3 = K 1 ( θ ) E 0 exp ( 2 π i R θ / λ ) exp [ i φ 1 ( θ ) ] .
E 6 = K 2 ( 0 ) K 1 ( θ ) E 0 exp ( 4 π i R θ / λ ) × exp [ i φ 1 ( θ ) ] exp [ i ψ ( θ ) ] .
ψ ( θ ) = ( 2 π t / λ ) ( n - 1 ) csc θ ,
E 9 = E 2 K 2 ( θ ) exp ( 2 π i R / λ ) exp [ i φ 2 ( θ ) ] exp [ i ψ ( θ ) ] = E 0 K 1 ( 0 ) K 2 ( θ ) exp ( 4 π i R / λ ) exp [ i φ 2 ( θ ) ] × exp [ i ψ ( θ ) ] .
I θ = E θ 2 = E 6 2 + E 9 2 + 2 E 6 E 9 = E 0 2 { K 2 2 ( 0 ) K 1 2 ( θ ) + K 1 2 ( 0 ) K 2 2 ( θ ) + 2 K 2 ( 0 ) K 1 ( θ ) K 1 ( 0 ) K 2 ( θ ) × cos [ ( 4 π / λ ) ( R θ - R ) + φ 1 ( θ ) - φ 2 ( θ ) ] } .
I θ = K 2 ( 0 ) K 2 ( θ ) [ 1 + cos ( 4 π / λ ) ( R θ - R ) ] .
R e = R { 1 + [ s 2 r 2 / R 2 ( R 2 + s 2 ) ] } 1 / 2 R [ 1 + ( s 2 r 2 / R 4 ) ] 1 / 2 ,             R 2 s 2 ,
L 1 = 2 ( R 2 + s 2 ) 1 / 2 - 2 { [ R 2 + ( s 2 r 2 / R 2 ) ] 1 / 2 - R } ,
L 1 - L 2 = - ( s 2 r 2 / R 3 ) [ 1 - 5 4 ( s 2 r 2 / R 4 ) + ] .
Δ ¯ = 4 π s 0 2 0 π / 2 0 s 0 Δ s d s d ξ ,
Δ ¯ = - ( r 2 s 0 2 / 2 R 3 ) [ 1 - ( 5 r 2 s 0 2 / 6 R 2 ) + ] .
= - ( J r 2 / R 2 ) [ 1 + 1 2 ( r 2 / R 2 ) + ] .
d = 2 R λ [ d n / ( d r ) ] ,
φ 2 d / π s 0 ,             d s 0 .
M = exp - [ π n ( Δ λ / λ ) ] 2 ,
S rel = ( Δ P / P ) T = 1 λ ,
S rel , n = n ( Δ P / P ) T = n λ ,
λ / 2 fringes : 0 , ϕ , 2 ϕ , 3 ϕ , 4 ϕ ,             without focal plane stop , λ / 4 fringes : 0 , 2 ϕ , 3 ϕ , 4 ϕ , λ / 6 fringes : 0 , 3 ϕ , 4 ϕ , }             without focal plane stop
( Δ P / P ) 1 = 2 h / λ ,             ( Δ P / P ) 2 = 4 h / λ .
W x , n = [ n 2 + n ( n - 2 ) ] S β ,
h / ( x ) = 1 2 λ / P x , n ;
h min = ( λ / 2 ) ( Δ p / p ) = λ / 200 ,
λ / 2 P x , n W x , n < λ / 2 ( Δ P / P ) = λ / 200 ,
100 W x , n < P ¯ x , n ,
100 ( n - 1 ) S λ < P ¯ x , n 2 .
( 100 ) ( 4 ) ( n - 1 ) ρ < ( f / ξ n ) 2 ,
P n , max < λ g / 2 n r λ = g / 2 n r .
( Δ P / P ) = ( 4 h cos α ) / λ .
( Δ P / P ) n = ( 4 h cos α n ) / λ .
I = I max [ 1 / ( 1 + F 2 sin 2 δ ) ] ,             F 2 = 4 R / ( 1 - R ) 2 ,
I max = [ T / ( 1 - R ) ] 2 ,
δ = ( m + x ) π ,             m is an integer x 1.
I = 1 [ 1 + ( F 2 / 2 ) ] - ( F 2 / 2 ) cos 2 π x 1 [ 1 + ( F 2 / 2 ) ] - ( F 2 / 2 ) { 1 - [ ( 2 π x ) 2 / 2 ] } = 1 1 + ( π F x ) 2 .
S d I d x = - π F 2 sin 2 π x { [ 1 + ( F 2 / 2 ) ] - ( F 2 / 2 ) cos 2 π x } 2 - 2 x ( π F x ) 2 [ 1 + ( π F x ) 2 ] 2 .
Δ S / S = ( 2 π cot ( 2 π x ) - 2 π F 2 sin 2 π x [ 1 + ( F 2 / 2 ) ] - ( F 2 / 2 ) cos 2 π x ) × Δ x = 0.
cos 2 ( 2 π x ) + cos ( 2 π x ) - 2 = - ( 2 / F 2 ) cos ( 2 π x ) ,
[ cos ( 2 π x ) + 2 ] [ cos ( 2 π x ) - 1 ] = - ( 2 / F 2 ) cos ( 2 π x ) .
cos ( 2 π x ) = 1 - 2 F 2 cos ( 2 π x ) cos ( 2 π x ) + 2 = 1 - 2 F 2 1 1 + [ 2 / cos ( 2 π x ) ] .
cos ( 2 π x ) 1 - [ ( 2 π x ) 2 / 2 ] = 1 - [ 2 / ( 3 F 2 ) ] .
( π F x ) 2 = 1 3 .
I = 3 4 .
S 3 / 4 = - [ 1 3 / ( 1 + 1 3 ) 2 ] 2 ( 3 ) 1 / 2 π F = - [ 3 ( 3 ) 1 / 2 / 8 ] π F .
( π F x 1 / 2 ) 2 = 1 ,
S 1 / 2 = - 1 4 2 π F = - 1 2 π F .
F = 1 / 2 x 1 / 2 = 1 2 π F .

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