Abstract

Accurate measurement of large step heights using multiple-wavelength holographic interferometry is realized using laser diodes. Due to the high-resolution wavelength tunability of such lasers, a pair of holograms with a wavelength difference of less than 0.01nm is recorded and used to extract a phase difference having a large synthetic wavelength. Phase differences with synthetic wavelengths ranging from 2.5to73mm are extracted by using pairs of holograms with wavelength differences between 0.3 and 0.01nm. By combining the phase differences, measurements with a step height of 18mm and an rms error of 0.04mm could be achieved. The requirements for performing the phase unwrapping are discussed. Precise knowledge of the recording wavelengths is required to correctly perform this unwrapping.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
    [CrossRef]
  9. I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
    [CrossRef] [PubMed]
  13. M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917-1923 (1998).
    [CrossRef]
  14. A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37, 5416-5420 (1998).
    [CrossRef]
  15. G. Pedrini, P. Fröning, H. J. Tiziani, and M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460-3467 (1999).
    [CrossRef]
  16. Y.-Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539-4543 (1984).
    [CrossRef] [PubMed]
  17. Y.-Y. Cheng and J. C. Wyant, “Multiple-wavelength phase-shifting interferometry,” Appl. Opt. 24, 804-807 (1985).
    [CrossRef] [PubMed]
  18. P. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612-3616 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
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    [CrossRef]
  22. L. Paulsson, M. Sjödahl, J. Kato, and I. Yamaguchi, “Temporal phase unwrapping applied to wavelength-scanning interferometry,” Appl. Opt. 39, 3285-3288 (2000).
    [CrossRef]

2006 (3)

2004 (1)

2003 (1)

2001 (1)

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

2000 (2)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

L. Paulsson, M. Sjödahl, J. Kato, and I. Yamaguchi, “Temporal phase unwrapping applied to wavelength-scanning interferometry,” Appl. Opt. 39, 3285-3288 (2000).
[CrossRef]

1999 (2)

1998 (2)

1996 (1)

W. Nadeborn, P. Andrä, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260 (1996).
[CrossRef]

1991 (1)

1985 (2)

1984 (1)

1976 (1)

1969 (1)

L. O. Heflinger and R. F. Wuerker, “Holographic contouring via multifrequency lasers,” Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

1967 (1)

Andrä, P.

W. Nadeborn, P. Andrä, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260 (1996).
[CrossRef]

Asundi, A.

Cheng, Y.-Y.

Cuevas, F. J.

Dakoff, A.

de Groot, P.

Friesem, A. A.

Fröning, P.

Gass, J.

Goodman, J. W.

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

Gusev, M. E.

Haines, K.

Heflinger, L. O.

L. O. Heflinger and R. F. Wuerker, “Holographic contouring via multifrequency lasers,” Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

Hildebrand, B.

Ida, T.

Ishii, Y.

Y. Ishii, “Laser-diode interferometry,” in Progress in Optics, Vol. 46, E. Wolf, ed. (Elsevier, 2004), pp. 243-309.
[CrossRef]

Jüptner, W.

Kato, J.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

L. Paulsson, M. Sjödahl, J. Kato, and I. Yamaguchi, “Temporal phase unwrapping applied to wavelength-scanning interferometry,” Appl. Opt. 39, 3285-3288 (2000).
[CrossRef]

Kim, M. K.

Levy, U.

Malacara, D.

Marroquin, J. L.

Nadeborn, W.

W. Nadeborn, P. Andrä, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260 (1996).
[CrossRef]

Ohta, S.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

W. Nadeborn, P. Andrä, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260 (1996).
[CrossRef]

Parshall, D.

Paulsson, L.

Pedrini, G.

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

Servin, M.

Sjödahl, M.

Tiziani, H. J.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

Wensen, Z.

Wuerker, R. F.

L. O. Heflinger and R. F. Wuerker, “Holographic contouring via multifrequency lasers,” Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

Wyant, J. C.

Yamaguchi, I.

Yamashita, K.

Yaroslavsky, L. P.

Yokota, M.

Yonemura, M.

Yu, L.

Zhang, F.

Appl. Opt. (11)

A. A. Friesem and U. Levy, “Fringe formation in two-wavelength contour holography,” Appl. Opt. 15, 3009-3020 (1976).
[CrossRef] [PubMed]

Y.-Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539-4543 (1984).
[CrossRef] [PubMed]

Y.-Y. Cheng and J. C. Wyant, “Multiple-wavelength phase-shifting interferometry,” Appl. Opt. 24, 804-807 (1985).
[CrossRef] [PubMed]

P. de Groot, “Three-color laser-diode interferometer,” Appl. Opt. 30, 3612-3616 (1991).
[CrossRef] [PubMed]

A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37, 5416-5420 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917-1923 (1998).
[CrossRef]

G. Pedrini, P. Fröning, H. J. Tiziani, and M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460-3467 (1999).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

L. Paulsson, M. Sjödahl, J. Kato, and I. Yamaguchi, “Temporal phase unwrapping applied to wavelength-scanning interferometry,” Appl. Opt. 39, 3285-3288 (2000).
[CrossRef]

D. Parshall and M. K. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45, 451-459 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

L. O. Heflinger and R. F. Wuerker, “Holographic contouring via multifrequency lasers,” Appl. Phys. Lett. 15, 28 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

Opt. Lasers Eng. (2)

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

W. Nadeborn, P. Andrä, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260 (1996).
[CrossRef]

Opt. Lett. (4)

Other (2)

Y. Ishii, “Laser-diode interferometry,” in Progress in Optics, Vol. 46, E. Wolf, ed. (Elsevier, 2004), pp. 243-309.
[CrossRef]

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

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Figures (8)

Fig. 1
Fig. 1

Experimental geometry for the holographic interferometry.

Fig. 2
Fig. 2

Hologram recording setup. OSA, optical spectrum analyzer; FC, fiber coupler; HM, half-mirrored beam splitter; OL1, 10 × objective lens; OL2, 5 × objective lens; L1, lens ( f = 60 mm ) ; L2, lens ( f = 70 mm ) ; L3, lens ( f = 170 mm ) ; L4, lens ( f = 60 mm ) .

Fig. 3
Fig. 3

Image reconstructed from a hologram.

Fig. 4
Fig. 4

Plot of the wavelength as a function of the injection current. A mode hop of 2 nm appears at an injection current of 76.0 mA .

Fig. 5
Fig. 5

Phase differences Ψ n . (a) Ψ 1 , λ 1 = 785.55 nm . (b) Ψ 2 , λ 2 = 785.56 nm . (c) Ψ 3 , λ 3 = 785.59 nm . (d) Ψ 4 , λ 4 = 785.61 nm . (e) Ψ 5 , λ 5 = 785.67 nm . (f) Ψ 6 , λ 6 = 785.80 nm .

Fig. 6
Fig. 6

Object height calculated from Δ φ 6 ( Λ 6 = 2.5 mm ) . (a) The entire distribution. (b) Plot of the object heights along the black line in panel (a) as a function of lateral position.

Fig. 7
Fig. 7

Areas A, B, and C for testing the measurement accuracy.

Fig. 8
Fig. 8

Plot of the standard deviation of the object heights in the test areas as a function of the synthetic wavelength.

Tables (5)

Tables Icon

Table 1 Synthetic Wavelengths ( λ 0 = 785.54 nm )

Tables Icon

Table 2 Average (Ave.) and Standard Deviation (S.D.) in Area A

Tables Icon

Table 3 Average (Ave.) and Standard Deviation (S.D.) after Median Filtering in Area A

Tables Icon

Table 4 Average (Ave.) and Standard Deviation (S.D.) after Median Filtering in Area B

Tables Icon

Table 5 Average (Ave.) and Standard Deviation (S.D.) after Median Filtering in Area C

Equations (32)

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u r ( x , y , z ) = 1 z exp ( i π x 2 + y 2 λ z ) ,
u o ( x , y , z ) = 1 z exp ( i π x 2 + y 2 λ z ) d x d y u o ( x , y , 0 ) exp { i 2 π ( x 2 + y 2 2 λ z x x + y y λ z ) } .
u o ( x , y ) = u o ( x , y , 0 ) exp ( i π x 2 + y 2 λ z ) ,
U o ( f x , f y ) = F ( u o ) = d x d y u o ( x , y ) exp [ i 2 π ( f x x + f y y ) ] .
u o ( x , y , z ) = 1 z exp ( i π x 2 + y 2 λ z ) U o ( x λ z , y λ z ) .
I ( x , y , z ) = 1 z 2 [ 1 + U o ( f x , f y ) 2 + U o ( f x , f y ) + U o * ( f x , f y ) ] .
u rec ( s Δ x , t Δ y ) = p = N x 2 N x 2 1 q = N y 2 N y 2 1 I ( p Δ x , q Δ y , z ) exp i 2 π ( p s N x + q t N y ) ,
Δ x = λ z Δ x N x , Δ y = λ z Δ y N y .
Δ φ n ( x , y ) = φ n φ 0 = 2 π Λ n ( L + x 2 + y 2 2 z ) 2 π Λ n L ,
L = ( 1 + cos θ ) h ( x , y ) x sin θ ,
Λ n = λ 0 λ n λ n λ 0 λ n ¯ 2 Δ λ n ,
λ ¯ n = λ 0 + λ n 2 , Δ λ n = λ n λ 0 .
Δ φ n = Ψ n + 2 π m n ,
Δ φ n = Ψ n + 2 π NINT ( α n Δ φ n 1 Ψ n 2 π ) ,
α n = Λ n 1 Λ n .
Ψ n = Ψ n + Ψ ε , n , Δ φ n = Ψ n + 2 π m n = Δ φ n Ψ ε , n ,
α n = α n + α ε , n .
Δ φ n = Ψ n + Ψ ε , n + 2 π NINT [ ( α n + α ε , n ) ( Δ φ n 1 + Ψ ε , n 1 ) Ψ n Ψ ε , n 2 π ] .
α n Δ φ n 1 Ψ n 2 π = Δ φ n Ψ n 2 π = m n
Δ φ n = Δ φ n + Ψ ε , n + 2 π NINT [ α ε , n ( Δ φ n 1 + Ψ ε , n 1 ) + α n Ψ ε , n 1 Ψ ε , n 2 π ] .
α ε , n Δ φ n 1 + α n Ψ ε , n 1 Ψ ε , n 2 π < 1 2
Ψ ε , n 2 π ε .
Δ φ n 1 = 2 π L α n Λ n .
ε < 1 α n + 1 ( 1 2 α ε , n α n L Λ n ) .
ε < 1 2 ( α n + 1 ) ,
α n < 1 2 ε 2 ε .
α ε , n < α n Λ n L ( 1 2 ε ( α n + 1 ) )
α ε , n α n L Λ n = ( Δ λ ε , n 1 Δ λ n 1 Δ λ ε , n Δ λ n ) L Δ λ n λ n ¯ 2 .
Δ λ ε , n Δ λ ε .
Δ λ ε < λ n ¯ 2 L ( 1 2 ( α n + 1 ) ε )
λ = { C × 0.0034 nm mA + 783.186 nm ( 45.0 mA C 76.0 mA ) C × 0.0036 nm mA + 785.268 nm ( 76.5 mA C 150 mA ) , }
h = Λ n Δ φ n 4 π .

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