Abstract

Fringe formation in the two-grating interferometer is analyzed in the presence of a small parallelism error between the diffraction gratings assumed in the direction of grating shear. Our analysis shows that with partially coherent illumination, fringe contrast in the interference plane is reduced in the presence of nonzero grating tilt with the effect proportional to the grating tilt angle and the grating spatial frequencies. Our analysis also shows that for a given angle between the gratings there is an angle between the final grating and the interference plane that optimizes fringe contrast across the field.

© 2008 Optical Society of America

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References

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  1. F. J. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227-230 (1959).
    [CrossRef]
  2. E. N. Leith and B. J. Chang, “Space-invariant holography with quasicoherent light,” Appl. Opt. 12, 1957-1963 (1973).
    [CrossRef] [PubMed]
  3. E. N. Leith and B. J. Chang, “Image formation with an achromatic interferometer,” Opt. Commun. 23, 217-219 (1977).
    [CrossRef]
  4. B. J. Chang, R. C. Alferness, and E. N. Leith, “Space-invariant achromatic grating interferometers: theory,” Appl. Opt. 14, 1592-1600 (1975).
    [CrossRef] [PubMed]
  5. B. J. Chang, “Grating-based interferometers,” Ph.D. dissertation, (University Michigan, 1974), University Microfilm 74-23-170.
  6. Y. S. Cheng, “Fringe formation in incoherent light with a two-grating interferometer,” Appl. Opt. 23, 3057-3059 (1984).
    [CrossRef] [PubMed]
  7. Y. S. Cheng, “Temporal coherence requirement in a symmetric-path grating interferometer,” Appl. Opt. 36, 800-804 (1997).
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    [CrossRef]
  13. H. Meiling, H. Meijer, V. Banine, R. Moors, R. Groeneveld, H.-J. Voorma, and U. MicKan, “First performance results of the ASML alpha demo tool,” Proc. SPIE 6151, 615108 (2006).
    [CrossRef]
  14. M. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
    [CrossRef]
  15. S. Heinbuch, M. Grisham, D. Martz, and J. J. Rocca, “Demonstration of a desk-top size high repetition rate soft x-ray laser,” Opt. Express 13, 4050-4055, (2005).
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  16. S. F. Horne, M. M. Besen, D. K. Smith, P. A. Blackborow, and R. DAgostino, “Application of a high-brightness electrodeless Z-pinch EUV source for metrology, inspection, and resist development,” Proc. SPIE 6151, 61510P (2006).
    [CrossRef]
  17. J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Eq. 3-74. We note that we have written the propagation phase in terms of angle and wavelength rather than in terms of spatial frequency as done by Goodman. It is for this reason we do not call call the propagation phase a transfer function. We also note that this phase is exact; the Fresnel approximation has not yet been made.
  18. J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Sec. 4.5.2.
  19. E. Hecht, Optics, third ed. (Addison-Wesley Longman, 1998), Eq. 10.61.
  20. Because the DOF dephasing term is odd in theta we need to use the full NA (2Δθ) to get the full dephasing for the DOF term. The tilt dephasing term is even in theta so only the half NA is required here.
  21. Here the small correction term does not contain a small parameter (g, γ, d) so we go to second order to maintain reasonable accuracy in the expansion.

2006 (3)

H. Meiling, H. Meijer, V. Banine, R. Moors, R. Groeneveld, H.-J. Voorma, and U. MicKan, “First performance results of the ASML alpha demo tool,” Proc. SPIE 6151, 615108 (2006).
[CrossRef]

M. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

S. F. Horne, M. M. Besen, D. K. Smith, P. A. Blackborow, and R. DAgostino, “Application of a high-brightness electrodeless Z-pinch EUV source for metrology, inspection, and resist development,” Proc. SPIE 6151, 61510P (2006).
[CrossRef]

2005 (1)

2000 (1)

1999 (1)

1997 (1)

1995 (1)

M. Wei, E. Gullikson, J. H. Underwood, T. K. Gustafson, and D. T. Attwood, “White-light spatial frequency multiplication using soft x-rays,” Proc. of SPIE 2516, 233-239 (1995).
[CrossRef]

1986 (1)

1985 (1)

1984 (1)

1977 (1)

E. N. Leith and B. J. Chang, “Image formation with an achromatic interferometer,” Opt. Commun. 23, 217-219 (1977).
[CrossRef]

1975 (1)

1973 (1)

1959 (1)

F. J. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227-230 (1959).
[CrossRef]

Appl. Opt. (8)

J. Sci. Instrum. (1)

F. J. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227-230 (1959).
[CrossRef]

Opt. Commun. (1)

E. N. Leith and B. J. Chang, “Image formation with an achromatic interferometer,” Opt. Commun. 23, 217-219 (1977).
[CrossRef]

Opt. Express (1)

Proc. of SPIE (1)

M. Wei, E. Gullikson, J. H. Underwood, T. K. Gustafson, and D. T. Attwood, “White-light spatial frequency multiplication using soft x-rays,” Proc. of SPIE 2516, 233-239 (1995).
[CrossRef]

Proc. SPIE (3)

H. Meiling, H. Meijer, V. Banine, R. Moors, R. Groeneveld, H.-J. Voorma, and U. MicKan, “First performance results of the ASML alpha demo tool,” Proc. SPIE 6151, 615108 (2006).
[CrossRef]

M. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

S. F. Horne, M. M. Besen, D. K. Smith, P. A. Blackborow, and R. DAgostino, “Application of a high-brightness electrodeless Z-pinch EUV source for metrology, inspection, and resist development,” Proc. SPIE 6151, 61510P (2006).
[CrossRef]

Other (6)

J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Eq. 3-74. We note that we have written the propagation phase in terms of angle and wavelength rather than in terms of spatial frequency as done by Goodman. It is for this reason we do not call call the propagation phase a transfer function. We also note that this phase is exact; the Fresnel approximation has not yet been made.

J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Sec. 4.5.2.

E. Hecht, Optics, third ed. (Addison-Wesley Longman, 1998), Eq. 10.61.

Because the DOF dephasing term is odd in theta we need to use the full NA (2Δθ) to get the full dephasing for the DOF term. The tilt dephasing term is even in theta so only the half NA is required here.

Here the small correction term does not contain a small parameter (g, γ, d) so we go to second order to maintain reasonable accuracy in the expansion.

B. J. Chang, “Grating-based interferometers,” Ph.D. dissertation, (University Michigan, 1974), University Microfilm 74-23-170.

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Figures (5)

Fig. 1
Fig. 1

Two-grating interferometer. All diffraction gratings are assumed sinusodial. For instructive purposes, the illumination contains two distinct wavelengths, λ 1 and λ 2 , in a small NA of incidence angles centered on-axis. It is assumed the zero- order transmitted beam is blocked at the second grating so it is not shown. Darker shades indicate locations where the two distinct colors spatially overlap.

Fig. 2
Fig. 2

Computing the propagation phase of free space. This is a side-view schematic of the two-grating interferometer in nontilted (left) and tilted (right) configurations. The dashed line in the tilted case shows the ray that is used for the nontilted case. See Subsection 3A2 for an in-depth description.

Fig. 3
Fig. 3

Computing the grating phase. This is a side-view schematic of a plane wave propagating at angle θ striking a diffraction grating in nontilted (left) and tilted (right) configurations. For the tilted case to the right, the ray that would be used for the nontilted case is shown with dashed lines. See Subsection 3B for an in-depth description.

Fig. 4
Fig. 4

Two-grating interferometer nomenclature. This is a side-view schematic of the two-grating interferometer and describes the nomenclature used throughout this paper for distances, angles, regions, etc.

Fig. 5
Fig. 5

Plots of the C 1 T 1 product (left) and it’s derivative with respect to θ (right) for several values of η λ f 1 [ 0.05 , 0.25 ] with 45 ° < θ < 45 ° .

Tables (2)

Tables Icon

Table 1 Tracking the Propagation Phase

Tables Icon

Table 2 Tracking the Grating Phase

Equations (44)

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ϕ ( z , θ , λ ) = 2 π λ z cos θ ,
z 1 ( x ) = z 1 + x tan g .
ϕ ( z , θ , λ ) = 2 π λ ( z x tan θ 1 + x tan θ 2 ) cos θ ,
ϕ front ( x ) = 2 π λ x ( sin θ in + tan g cos θ in ) ,
ϕ rear ( x ) = 2 π λ x ( sin θ out + tan g cos θ out ) ,
ϕ g ( x ) = ϕ rear ϕ front = 2 π λ x [ sin θ out sin θ in ] + 2 π λ x tan g [ cos θ out cos θ in ] .
θ out = arcsin [ sin ( θ in + g ) + m λ f ] g ,
I ( x ) = | E T + E B | 2 = | E T | 2 + | E B | 2 + E T E B * + E T * E B = 2 + 2 cos ( ϕ ( x , θ , λ , f 1 , f 2 , z 1 , z 2 , g , w ) ) ,
θ T 1 = arcsin [ sin θ + λ f 1 ] , θ B 1 = arcsin [ sin θ + λ f 1 ] , θ T 2 = arcsin [ sin ( θ T 1 + g ) λ f 2 ] g , θ B 2 = arcsin [ sin ( θ B 1 + g ) + λ f 2 ] g .
ϕ = 2 π λ z 1 C 1 + 2 π λ ( z 2 + x tan w ) C 2 + 2 π λ x S 2 ,
C 1 ( λ , θ , f 1 ) = cos θ T 1 cos θ B 1 , C 2 ( λ , θ , f 1 ) = cos θ T 2 cos θ B 2 , S 2 ( λ , θ , f 1 ) = sin θ T 2 sin θ B 2 .
arcsin ( α + δ ) arcsin α + δ ( 1 α 2 ) 1 / 2 ,
θ out arcsin [ sin ( θ in + g ) + m λ f g ( 1 [ sin ( θ in + g ) + m λ f ] 2 ) 1 / 2 ] .
sin ( θ in + g ) sin θ in + g cos θ in ,
θ out arcsin [ sin θ in + m λ f ] ,
f = f + g m λ [ cos θ in ( 1 [ sin θ in + g cos θ in + m λ f ] 2 ) 1 / 2 ] ,
f 2 = 2 f 1 ( 1 + γ ) g λ C 1 .
θ T 1 = arcsin [ sin θ + λ f 1 ] , θ B 1 = arcsin [ sin θ λ f 1 ] , θ T 2 arcsin [ sin θ B 1 2 λ f 1 γ + g C 1 ] , θ B 2 arcsin [ sin θ T 1 + 2 λ f 1 γ g C 1 ] .
cos [ arcsin ( sin α + δ ) ] cos α δ tan α δ 2 2 cos 3 α ,
C 2 C 1 + ( 2 λ f 1 γ g C 1 ) T 1 ,
T 1 ( λ , θ , f 1 ) = tan θ T 1 + tan θ B 1 .
ϕ 2 π λ [ d C 1 + z 1 ( 2 λ f 1 γ g C 1 ) T 1 ] θ , λ     dependent     fringe     shift 2 π x ( 2 f 1 ) ( 1 + 2 γ ) desired     modulation + 2 π λ x [ 2 g C 1 + tan w ( C 1 + 2 λ f 1 γ T 1 g C 1 T 1 ) ] . unwanted     modulation
ϕ 2 π λ d C 1 + 2 π z 1 2 f 1 γ T 1 2 π λ z 1 g C 1 T 1 2 π x ( 2 f 1 ) ( 1 + 2 γ ) ,
DOF Δ λ = cos 4 θ 0 2 λ ¯ Δ λ f 1 3 tan θ 0 ,
DOF Δ θ = cos 2 θ 0 2 f 1 Δ θ .
Δ λ λ ¯ Δ θ cos 2 θ 0 ( λ ¯ f 1 ) 2 tan θ 0 .
tan [ arcsin ( sin α + δ ) ] tan α + δ cos 3 α + 3 δ 2 sin α 2 cos 5 α .
C 1 2 λ f 1 tan θ , T 1 2 tan θ + 3 λ 2 f 1 2 sin θ cos 5 θ ,
g 2 d z 1 Δ θ = 1 f 1 z 1 ( Δ θ ) 2 .
( C 1 T 1 ) θ 8 λ f 1 sin θ cos 3 θ 12 λ 3 f 1 3 sin θ cos 7 θ ( cos 2 θ + 3 sin 2 θ ) , C 1 θ 2 λ f 1 cos 2 θ ,
z 1 g Δ θ [ ( C 1 T 1 ) θ ] θ = θ 0 d Δ θ [ C 1 θ ] θ = θ 0 .
g d 4 z 1 tan θ 0 = cos 3 θ 0 8 f 1 z 1 Δ θ sin θ 0 .
ϕ 2 π λ d C 1 + 2 π z 1 2 f 1 γ N 2 π λ z 1 g C 1 T 1 2 π x ( 2 f 1 ) ( 1 + 2 γ ) ,
N ( θ , λ , θ 0 , λ ¯ ) = [ 1 λ ¯ T 1 ( θ 0 , λ ¯ ) C 1 ( θ , λ ) λ T 1 ( θ , λ ) C 1 ( θ 0 , λ ¯ ) ] T 1 ( θ , λ ) ,
N ( θ , λ ) θ 0 = 0 3 λ f 1 C 1 2 ( 1 cos 4 θ ) 3 λ f 1 C 1 θ 2 ,
γ d 6 λ 2 f 1 2 z 1 ( Δ θ ) 2 = 1 24 λ 2 z 1 ( f 1 Δ θ ) 3 .
f = f + g m λ { cos θ in ( 1 [ sin θ in + g cos θ in + m λ f ] 2 ) 1 / 2 } ,
f 2 T = 2 f 1 ( 1 + γ ) g λ { cos θ T 1 ( 1 [ sin θ B 1 + g cos θ T 1 2 λ f 1 γ ) ] 2 ) 1 / 2 } ,
f 2 T 2 f 1 ( 1 + γ ) g λ [ cos θ T 1 cos θ B 1 ]
C 1 = cos θ T 1 cos θ B 1 ,
C 1 2 λ f 1 tan θ .
C 1 θ 2 λ f 1 cos 2 θ .
d 2 π λ Δ θ [ C 1 θ ] θ = θ 0 = π ,
2 d = DOF Δ θ = cos 2 θ 0 2 f 1 Δ θ ,

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