Abstract

Interferometers that use different states of polarization for the reference and the test beams can modulate the relative phase shift by using polarization optics in the imaging system. Thus the interferometer can capture simultaneous images that have a fixed phase shift, which can be used for phase-shifting interferometry. As all measurements are made simultaneously, the interferometer is not sensitive to vibration. Fizeau interferometers of this type have an advantage compared with Twyman–Green-type systems because they are common-path interferometers. However, a polarization Fizeau interferometer is not strictly common path when both wavefronts are transmitted by an optic that suffers from birefringence. The two polarized beams see different phases owing to birefringence; as a result, an error can be introduced in the measurement. We study the effect of birefringence on measurement accuracy when different polarization techniques are used in Fizeau interferometers. We demonstrate that measurement error is reduced dramatically and can be eliminated if the reference and test beams are circularly polarized rather than linearly polarized.

© 2005 Optical Society of America

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References

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  1. M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), pp. 1–45.
  2. For example, see www.4dtechnology.com (FizCam series of interferometers) or www.engsynthesis.com (Intellium H1000 Fizeau interferometer).
  3. J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
    [CrossRef]
  4. See Information, p. 16 of the Schott Glass catalog at http://www.us.schott.com/optics_devices/english/download/catalog_optical_glass_informations_usa_2003.pdf .
  5. G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
    [CrossRef]
  6. E. Hecht, Optics (Addison-Wesley Longman, 1998), pp. 368–371.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, 1980), pp. 25–32.

1994 (1)

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Bao, N.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1980), pp. 25–32.

Brock, N.

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

Chung, P. S.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley Longman, 1998), pp. 368–371.

Jin, G.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Millerd, J.

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

Murty, M. V. R. K.

M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), pp. 1–45.

North-Morris, M.

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1980), pp. 25–32.

Wyant, J.

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

Opt. Eng. (1)

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase-shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Other (6)

E. Hecht, Optics (Addison-Wesley Longman, 1998), pp. 368–371.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1980), pp. 25–32.

M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), pp. 1–45.

For example, see www.4dtechnology.com (FizCam series of interferometers) or www.engsynthesis.com (Intellium H1000 Fizeau interferometer).

J. Millerd, N. Brock, M. North-Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” in Interferometry XII: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE5531, 304–314 (2004).
[CrossRef]

See Information, p. 16 of the Schott Glass catalog at http://www.us.schott.com/optics_devices/english/download/catalog_optical_glass_informations_usa_2003.pdf .

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Figures (9)

Fig. 1
Fig. 1

Simultaneous phase-shifting polarization Fizeau interferometer. Ref. Surf, Test Surf, reference and test surfaces; R, T, reference and test beams, respectively.

Fig. 2
Fig. 2

Phase-shifting interferometer with circularly polarized reference and test beams uses a linear polarizer as the phase shifter. Tran, transmission; OPD, optical path difference.

Fig. 3
Fig. 3

Illustration of the definitions of the global coordinate system and the wave plate’s local coordinate system. XGYG is the global coordinate system, in which the incident reference and test beams are in phase. XTYT is the wave plate local coordinate system with the fast axis along the XT direction. The test and reference beams are circularly polarized and in phase in the global coordinate system. They have equal intensity (exaggerated in the figure).

Fig. 4
Fig. 4

Illustration of the combined beam’s polarization. The dashed–dotted line illustrates the ideal linearly polarized beam when no birefringence ex0ists; the solid ellipsis illustrates the elliptically polarized beam when birefringence exists.

Fig. 5
Fig. 5

Illustration of definitions of the angles θ and ψ associated with elliptically polarized light. Also shown is a linear polarizer with a transmission axis that forms angle ω with the x axis.

Fig. 6
Fig. 6

Transmitted light intensity as a function of the linear polarizer’s rotation angle after the wave plate converts circular to linear polarization. Dashed curve, no error and linearly polarized light. Solid curve, birefringence has caused the light to be elliptically polarized. Note that the intensity has a phase shift and less contrast when the incident beam is elliptically polarized than when it is linearly polarized.

Fig. 7
Fig. 7

Maximum surface measurement error versus birefringence for both linear and circular polarization.

Fig. 8
Fig. 8

Simulated Fizeau measurements for a system with birefringence in the common part of the system. (a) Birefringence map. The fast-axis angle has a linear distribution along the y axis from 0° to 90° and the retardation has a linear distribution along the x axis from 0 to 60 nm. (b) Results of simulation for an ideal system with 25 nm rms surface irregularity. (c) Results for simulated phase-shift interferometry for 60 nm birefringence and the spatial distribution shown in (a). We calculated the measurement error by subtracting the ideal surface error from the simulated measurement. Note the reduction in fringe contrast as well as the phase error for both cases.

Fig. 9
Fig. 9

Plot of rms surface measurement error as a function of maximum retardation of the birefringence distribution shown in Fig. 8(a) for both linear and circular polarization. Circles, results of theoretical calculation by use of Eq. (16), which agree with the interferometric simulation results for circular polarization.

Equations (22)

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δ max = ϕ 2 π λ .
T G = ( 1 i ) , R G = ( 1 i ) .
B T = [ 1 0 0 exp ( i 2 ϕ ) ] .
T T = ( 1 i ) exp ( i α ) , R T = ( 1 i ) exp ( i α ) .
T T = B T T T exp ( i 2 γ ) = [ 1 i exp ( i 2 ϕ ) ] exp [ i ( 2 γ α ) ] ,
R T = B T R T = [ 1 i exp ( i 2 ϕ ) ] exp ( i α ) ,
R T + T T = { exp [ i ( 2 γ α ) ] + exp ( i α ) i exp [ i ( 2 ϕ + α ) ] i exp [ i ( 2 ϕ + 2 γ α ) ] } = exp ( i γ ) [ 2 cos ( γ α ) 2 sin ( γ α ) exp ( i 2 ϕ ) ] .
E = ( E x E y ) = [ A x A y exp ( i δ ) ] .
[ E x A x ] 2 + ( E y A y ) 2 2 ( E x A x ) ( E y A y ) cos ( δ ) = sin 2 ( δ ) .
tan ( θ ) = A y / A x .
tan ( 2 Ψ ) = tan ( 2 θ ) cos ( δ ) .
P = ( cos 2 ω cos ω sin ω cos ω sin ω sin 2 ω ) .
E trans = P [ A x A y exp ( i δ ) ] = [ A x cos ω + A y sin ω exp ( i δ ) ] ( cos ω sin ω ) .
I = | E trans | 2 = 1 / 2 ( A x 2 + A y 2 ) + 1 / 2 [ A x 4 + 2 A x 2 A y 2 × cos ( 2 δ ) + A y 4 ] 1 / 2 cos ( 2 ω 2 Ψ ) ,
2 Δ = 2 ( θ Ψ ) .
C = [ A x 4 + 2 A x 2 A y 2 cos ( 2 δ ) + A y 4 ] 1 / 2 A x 2 + A y 2 .
C = [ 1 sin 2 ( 2 θ ) sin 2 δ ] 1 / 2 .
tan ( 2 Δ ) = tan ( 2 θ 2 Ψ ) = 2 tan [ 2 ( γ α ) ] sin 2 ϕ 1 + tan 2 [ 2 ( γ α ) ] cos 2 ϕ ,
π 4 < ϕ < π 4 ;
tan ( 2 Δ max ) = sin 2 ϕ cos 2 ϕ .
2 Δ max ϕ 2 .
δ max = Δ max 2 π λ ϕ 2 4 π λ ,

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