Abstract

Interferometry of wave fronts reflected off conical surfaces requires new interferometric schemes. The pencil beam interferometer is proposed as a metrology tool for conical as well as for spherical optical surfaces. The instrument employs two narrow pencil beams which scan the optical surface to be measured. An electronic fringe position readout system, which measures the location of the fringes that develop through interference of the pencil beams, provides high accuracy surface information. Metrology on a waxicon indicates a measurement precision of ±6 nm. Due to the differential nature of the instrument, alignment requirements are relatively modest.

© 1983 Optical Society of America

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References

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  1. G. C. Dente, Appl. Opt. 18, 2911 (1979).
    [CrossRef] [PubMed]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. F. Twyman, A. Green, British Pat.103,832 (1916).
  4. W. J. Bates, Proc. Phys. Soc. London 59, 940 (1947).
    [CrossRef]
  5. H. Fizeau, Ann. Chim. Phys. 3, 66, 1862 (1947).
  6. G. D. Dew, “The Measurement of Optical Flats,” J. Sci. Instrum. 43, 409 (1966).
    [CrossRef] [PubMed]
  7. J. C. Wyant, J. Opt. Soc. Am. 64, 1363 (1974).
  8. J. C. Wyant, “Interferometric Optical Metrology,” Laser Focus (May1982), p. 65.
  9. A. Offner, Appl. Opt. 2, 153 (1963).
    [CrossRef]
  10. P. M. Emmel et al., J. Opt. Soc. Am. 68, 1416 (1978)
  11. J. C. Wyant, P. K. O’Neill, Appl. Opt. 13, 2762 (1974).
    [CrossRef] [PubMed]
  12. A. E. Ennos, “High Accuracy Profile Measurement of Quasi-Conical Mirror Surfaces by Laser Auto-Collimation,” Precis. Eng. 4, No. 1, 5 (Jan.1982).
    [CrossRef]
  13. K. von Bieren, Appl. Opt. 12, 1642 (1973).
    [CrossRef] [PubMed]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  15. X. Jenkins, X. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).
  16. A. Gullstrand, “Tatsachen and Fiktionen in der Lehre von der optischen Abbildung,” Arch. Opt. 1, 1 (1907).
  17. H. Boegehold, Handbuch der Physik, Bol 18, 160Berlin (1927).
  18. J. C. Sturm, “Memoire sur L’optique,” Liouv. J. 3, 357 (1838).
  19. K. von Bieren, “Pencil Beam Interferometer for Aspherical Optical Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 343, 101 (1982).
  20. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1964).

1982

J. C. Wyant, “Interferometric Optical Metrology,” Laser Focus (May1982), p. 65.

A. E. Ennos, “High Accuracy Profile Measurement of Quasi-Conical Mirror Surfaces by Laser Auto-Collimation,” Precis. Eng. 4, No. 1, 5 (Jan.1982).
[CrossRef]

K. von Bieren, “Pencil Beam Interferometer for Aspherical Optical Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 343, 101 (1982).

1979

1978

P. M. Emmel et al., J. Opt. Soc. Am. 68, 1416 (1978)

1974

1973

1966

G. D. Dew, “The Measurement of Optical Flats,” J. Sci. Instrum. 43, 409 (1966).
[CrossRef] [PubMed]

1963

1947

W. J. Bates, Proc. Phys. Soc. London 59, 940 (1947).
[CrossRef]

H. Fizeau, Ann. Chim. Phys. 3, 66, 1862 (1947).

1907

A. Gullstrand, “Tatsachen and Fiktionen in der Lehre von der optischen Abbildung,” Arch. Opt. 1, 1 (1907).

1838

J. C. Sturm, “Memoire sur L’optique,” Liouv. J. 3, 357 (1838).

Bates, W. J.

W. J. Bates, Proc. Phys. Soc. London 59, 940 (1947).
[CrossRef]

Boegehold, H.

H. Boegehold, Handbuch der Physik, Bol 18, 160Berlin (1927).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Dente, G. C.

Dew, G. D.

G. D. Dew, “The Measurement of Optical Flats,” J. Sci. Instrum. 43, 409 (1966).
[CrossRef] [PubMed]

Emmel, P. M.

P. M. Emmel et al., J. Opt. Soc. Am. 68, 1416 (1978)

Ennos, A. E.

A. E. Ennos, “High Accuracy Profile Measurement of Quasi-Conical Mirror Surfaces by Laser Auto-Collimation,” Precis. Eng. 4, No. 1, 5 (Jan.1982).
[CrossRef]

Fizeau, H.

H. Fizeau, Ann. Chim. Phys. 3, 66, 1862 (1947).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Green, A.

F. Twyman, A. Green, British Pat.103,832 (1916).

Gullstrand, A.

A. Gullstrand, “Tatsachen and Fiktionen in der Lehre von der optischen Abbildung,” Arch. Opt. 1, 1 (1907).

Jenkins, X.

X. Jenkins, X. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

O’Neill, P. K.

Offner, A.

Sturm, J. C.

J. C. Sturm, “Memoire sur L’optique,” Liouv. J. 3, 357 (1838).

Twyman, F.

F. Twyman, A. Green, British Pat.103,832 (1916).

von Bieren, K.

K. von Bieren, “Pencil Beam Interferometer for Aspherical Optical Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 343, 101 (1982).

K. von Bieren, Appl. Opt. 12, 1642 (1973).
[CrossRef] [PubMed]

White, X.

X. Jenkins, X. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1964).

Wyant, J. C.

J. C. Wyant, “Interferometric Optical Metrology,” Laser Focus (May1982), p. 65.

J. C. Wyant, J. Opt. Soc. Am. 64, 1363 (1974).

J. C. Wyant, P. K. O’Neill, Appl. Opt. 13, 2762 (1974).
[CrossRef] [PubMed]

Ann. Chim. Phys.

H. Fizeau, Ann. Chim. Phys. 3, 66, 1862 (1947).

Appl. Opt.

Arch. Opt.

A. Gullstrand, “Tatsachen and Fiktionen in der Lehre von der optischen Abbildung,” Arch. Opt. 1, 1 (1907).

J. Opt. Soc. Am.

J. C. Wyant, J. Opt. Soc. Am. 64, 1363 (1974).

P. M. Emmel et al., J. Opt. Soc. Am. 68, 1416 (1978)

J. Sci. Instrum.

G. D. Dew, “The Measurement of Optical Flats,” J. Sci. Instrum. 43, 409 (1966).
[CrossRef] [PubMed]

Laser Focus

J. C. Wyant, “Interferometric Optical Metrology,” Laser Focus (May1982), p. 65.

Liouv. J.

J. C. Sturm, “Memoire sur L’optique,” Liouv. J. 3, 357 (1838).

Precis. Eng.

A. E. Ennos, “High Accuracy Profile Measurement of Quasi-Conical Mirror Surfaces by Laser Auto-Collimation,” Precis. Eng. 4, No. 1, 5 (Jan.1982).
[CrossRef]

Proc. Phys. Soc. London

W. J. Bates, Proc. Phys. Soc. London 59, 940 (1947).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

K. von Bieren, “Pencil Beam Interferometer for Aspherical Optical Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 343, 101 (1982).

Other

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

X. Jenkins, X. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

F. Twyman, A. Green, British Pat.103,832 (1916).

H. Boegehold, Handbuch der Physik, Bol 18, 160Berlin (1927).

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Figures (7)

Fig. 1
Fig. 1

Reflection off conical surface.

Fig. 2
Fig. 2

Huygens wavelets.

Fig. 3
Fig. 3

Reflection off conic surface.

Fig. 4
Fig. 4

Function [sinπ(β/λ)T]/[π(β/λ)T].

Fig. 5
Fig. 5

Funtion 1 4 π β λ T [ sin π β λ T π β λ T - cos π β λ T ].

Fig. 6
Fig. 6

Pencil beam interferometer.

Fig. 7
Fig. 7

Interferometer scan on conic surface. The conic surface is the outer cone of a diamond-turned waxicon.

Equations (18)

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u ( P 2 ) = A j λ Σ exp [ j 2 π λ ( r 20 + r 10 ) ] r 20 r 10 cos ( n , r 20 ) d s .
P 1 coordinates : x = 0 , y = s 1 sin θ 1 , z = s 1 cos θ 1 , P 2 coordinates : x = 0 , y = s 2 sin θ 2 , z = s 2 cos θ 2 .
r 20 + r 10 = s 1 + s 2 + 1 2 ( 1 s 1 + 1 s 2 - cos θ 1 + cos θ 2 ρ 1 ) x 2 + 1 2 ( cos 2 θ 1 s 1 + cos 2 θ 2 s 2 - cos θ 1 + cos θ 2 ρ 2 ) y 2 - ( sin θ 1 + sin θ 2 ) y .
x = y = 0 ,
θ 1 = - θ 2 .
u ( P 2 ) = A | ( 1 s 1 + 1 s 2 - 2 cos θ 1 ρ 1 ) ( 1 s 1 + 1 s 2 - 2 cos θ 1 ρ 2 ) | exp [ j 2 π λ ( s 1 + s 2 ) ] s 1 s 2 .
1 s 1 + 1 s 2 = 2 cos θ 1 ρ 1 ,
1 s 1 + 1 s 2 = 2 cos θ 1 ρ 2 .
Φ = 2 π λ ( s 1 + s 2 ) ,
u ( P 2 ) A ρ 1 ρ 2 2 s 1 s 2 exp [ j 2 π λ ( s 1 + s 2 ) ] ;
d U ¯ = ρ 1 ( y ) · ρ 2 ( y ) exp { j 2 π λ [ y β 1 + f ( y ) γ 1 ] } d s ,
U ( β ) = ρ T ( y ) · ρ S ( y ) × exp { j 2 π λ [ y ( β + β 1 ) + f ( y ) ( γ + γ 1 ) ] } d s .
u ( β ) = | cos θ 1 2 s 1 2 s 2 | ρ s ( y ) rect ( y t ) exp [ j 2 π λ y ( β + β 1 ) ] d y ,
ρ s = ρ 0 ( 1 + 1 2 y ρ 0 - 1 8 y 2 ρ 0 2 + ) .
U ( β ) = ρ 0 - + ( 1 + 1 2 y ρ 0 ) · rect ( y T ) · exp ( - j 2 π λ β y ) d s = ρ 0 T [ sin π β λ T π β λ T - j T ρ 0 1 4 π β λ T × ( sin π β λ T π β λ T - cos π β λ T ) ] .
f ¯ = 2 M λ F ,
Φ ( ξ ) = 4 π λ [ Δ ( ξ - M 2 ) - Δ ( ξ + M 2 ) ] ,
Δ ( ξ ) = j λ 8 π F ˜ - 1 [ F ˜ [ Φ ( ξ ) ] sin π f M ] ,

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