Abstract

We investigated the pupil functions of specific-shaped apertures generated by Lambertian illumination with a rectangular light pipe to derive the corresponding optical transfer functions (OTFs) in aberration-free and defocused optical systems. The semianalytical results indicate that the curves of the OTF of the optical system vary with the form of the shaped apertures that are generated by illumination with different geometric structures of light pipes and light sources. It was found that the OTF values of even-peak frequencies decrease when the Lambertian light source decreases. If there are a total of n×n individual apertures within a pupil, then n near-periodical peaks will appear on the OTF curve. It is evident that the values of the OTF remain almost unchanged even when the lengths of the light pipes are different. Furthermore, the geometric structure of the light pipe does not affect the resolution limit of the optical system, and under the condition of a larger defocused coefficient ω20, the results of the defocused system can coincide with those of the aberration-free systems.

© 2006 Optical Society of America

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  2. W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
    [CrossRef]
  3. H. H. Hopkins, "Physics of the fiberoptics endoscope," in Endoscopy, G. Berci, ed. (Appleton-Century-Crofts, 1976), pp. 27-63.
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  9. E. M. Sabatke and J. H. Burge, "Basic principles in the optical design of imaging multiple aperture systems," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds., Proc. SPIE 4832, 236-248 (2002).
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    [CrossRef] [PubMed]
  13. H. H. Hopkins, "The frequency response of a defocused optical system," Proc. R. Soc. London, Ser. A 231, 143-153 (1955).
  14. W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III, pp. 2.1-2.53.
  15. M. S. Brennesholtz, "Light collection efficiency for light valve projection systems," in Projection Display II, M. H. Wu, ed., Proc. SPIE 2065, 71-79 (1996).
  16. E. H. Stupp and M. S. Brennesholtz, Projection Display (Wiley, 1999).
  17. E. Hecht, Optics (Addison-Wesley, 2002), pp. 543-544.
  18. MATHEMATICA version 4, Wolfram Research, Inc., 100 Trade Center Drive Champaign, Ill. 61820-7237.
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    [CrossRef]
  20. H. H. Hopkins, "Calculation of the aberration and image assessment for a general optical system," Opt. Acta 28, 667-714 (1981).
    [CrossRef]

2006 (1)

2000 (1)

1993 (1)

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

1989 (1)

1987 (2)

1986 (1)

1981 (1)

H. H. Hopkins, "Calculation of the aberration and image assessment for a general optical system," Opt. Acta 28, 667-714 (1981).
[CrossRef]

1971 (1)

1970 (1)

H. H. Hopkins and M. J. Yzuel, "The computation of diffraction patterns in the presence of aberrations," Opt. Acta 17, 157-182 (1970).
[CrossRef]

1955 (1)

H. H. Hopkins, "The frequency response of a defocused optical system," Proc. R. Soc. London, Ser. A 231, 143-153 (1955).

Barakat, R.

Berriel-Valdos, L. R.

Brennesholtz, M. S.

M. S. Brennesholtz, "Light collection efficiency for light valve projection systems," in Projection Display II, M. H. Wu, ed., Proc. SPIE 2065, 71-79 (1996).

E. H. Stupp and M. S. Brennesholtz, Projection Display (Wiley, 1999).

Burge, J. H.

E. M. Sabatke and J. H. Burge, "Basic principles in the optical design of imaging multiple aperture systems," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds., Proc. SPIE 4832, 236-248 (2002).

Cassarly, W.

W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III, pp. 2.1-2.53.

Chang, C. M.

Cheng, Y. K.

Chern, J. L.

Chung, C. S.

Dewa, P. G.

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Guha, A.

Hazra, L. N.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002), pp. 543-544.

Hillman, L.

T. Hough, J. Van Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Paper Series 940512 (Society of Automotive Engineers, 1994).

Hopkins, H. H.

C. S. Chung and H. H. Hopkins, "Influence of nonuniform amplitude on the optical transfer function," Appl. Opt. 28, 1244-1250 (1989).
[CrossRef] [PubMed]

H. H. Hopkins, "Calculation of the aberration and image assessment for a general optical system," Opt. Acta 28, 667-714 (1981).
[CrossRef]

H. H. Hopkins and M. J. Yzuel, "The computation of diffraction patterns in the presence of aberrations," Opt. Acta 17, 157-182 (1970).
[CrossRef]

H. H. Hopkins, "The frequency response of a defocused optical system," Proc. R. Soc. London, Ser. A 231, 143-153 (1955).

H. H. Hopkins, "Physics of the fiberoptics endoscope," in Endoscopy, G. Berci, ed. (Appleton-Century-Crofts, 1976), pp. 27-63.

Hough, T.

T. Hough, J. Van Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Paper Series 940512 (Society of Automotive Engineers, 1994).

Michaloski, P. F.

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Mino, M.

Montes, E. L.

Ojeda-Castaneda, J.

Okano, Y.

Partlo, W. N.

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Sabatke, E. M.

E. M. Sabatke and J. H. Burge, "Basic principles in the optical design of imaging multiple aperture systems," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds., Proc. SPIE 4832, 236-248 (2002).

Shieh, H. P. D.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 279-281.

Stupp, E. H.

E. H. Stupp and M. S. Brennesholtz, Projection Display (Wiley, 1999).

Tompkins, P. J.

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Van Derlofske, J.

T. Hough, J. Van Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Paper Series 940512 (Society of Automotive Engineers, 1994).

Yzuel, M. J.

H. H. Hopkins and M. J. Yzuel, "The computation of diffraction patterns in the presence of aberrations," Opt. Acta 17, 157-182 (1970).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

H. H. Hopkins and M. J. Yzuel, "The computation of diffraction patterns in the presence of aberrations," Opt. Acta 17, 157-182 (1970).
[CrossRef]

H. H. Hopkins, "Calculation of the aberration and image assessment for a general optical system," Opt. Acta 28, 667-714 (1981).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, "The frequency response of a defocused optical system," Proc. R. Soc. London, Ser. A 231, 143-153 (1955).

Proc. SPIE (1)

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV limography using annular illumination," in Optical/Laser Microlithography VI, J. D. Cathbert, ed., Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Other (9)

H. H. Hopkins, "Physics of the fiberoptics endoscope," in Endoscopy, G. Berci, ed. (Appleton-Century-Crofts, 1976), pp. 27-63.

T. Hough, J. Van Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Paper Series 940512 (Society of Automotive Engineers, 1994).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 279-281.

E. M. Sabatke and J. H. Burge, "Basic principles in the optical design of imaging multiple aperture systems," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds., Proc. SPIE 4832, 236-248 (2002).

W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III, pp. 2.1-2.53.

M. S. Brennesholtz, "Light collection efficiency for light valve projection systems," in Projection Display II, M. H. Wu, ed., Proc. SPIE 2065, 71-79 (1996).

E. H. Stupp and M. S. Brennesholtz, Projection Display (Wiley, 1999).

E. Hecht, Optics (Addison-Wesley, 2002), pp. 543-544.

MATHEMATICA version 4, Wolfram Research, Inc., 100 Trade Center Drive Champaign, Ill. 61820-7237.

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of the optical projection system with light pipe and light valve to illustrate the relationship between pupil and field.

Fig. 2
Fig. 2

Schematic diagram and dimension of the light pipe and the Lambertian light source.

Fig. 3
Fig. 3

(a) Principle of the operation of a light pipe. (b) Image at the aperture stop in the optical system. The light pipe is made of parallel reflective sides with a rectangular cross section. The multiple reflections of the light source through the pipe can produce a spatial checkerboard-array-shaped light distribution that is the virtual image of the light at the entrance of the light pipe.

Fig. 4
Fig. 4

Chart of the luminous exitance for a uniform light source.

Fig. 5
Fig. 5

Geometry of a Lambertian light source radiating into the exit plane of a light pipe.

Fig. 6
Fig. 6

Illustration of a Lambertian light source radiating into the exit plane of the light pipe for the different virtual light spots on the entrance plane of the light pipe.

Fig. 7
Fig. 7

Total aperture functions on the normalized pupil in the condition of D = 20 , a = b = 2.5 , and L = 60 with (a) c = 0.5 , (b) c = 1.0 , (c) c = 1.5 , (d) c = 2.0 .

Fig. 8
Fig. 8

Total aperture functions on the normalized pupil in the condition of D = 20 , c = 2.0 , and L = 60 with (a) a = b = 2.5 , (b) a = b = 2.5 , (c) a = b = 5.0 , (d) a = b = 7.5 , (e) a = b = 10.0 .

Fig. 9
Fig. 9

Total aperture functions on the normalized pupil in the condition of D = 20 , a = b = 2.5 , and c = 2.0 with (a) L = 20 , (b) L = 30 , (c) L = 60 , (d) L = 120 .

Fig. 10
Fig. 10

OTFs in the aberration-free system with a clear aperture T 0 and specific apertures generated by a light pipe with a different geometric, structures with fixed a = b = 2.5 and fixed L = 60 and different conditions of c = 0.5 , 1.0, 1.5, and 2.0.

Fig. 11
Fig. 11

OTFs in an aberration-free system with a clear aperture T 0 and specific apertures generated by the different geometric structures of a light pipe with fixed c = 2.0 and fixed L = 60 and different conditions of a = b = 2.5 , 3.5, 5.0, 7.5, and 10.0.

Fig. 12
Fig. 12

OTFs in an aberration-free system with a clear aperture T 0 and specific apertures generated by the different geometric structures of a light pipe with fixed a = b = 2.5 and fixed c = 2.0 and different conditions of L = 20 , 30, 60, and 120.

Fig. 13
Fig. 13

OTFs in a defocused system with (a) a clear aperture and (b) one specific aperture generated by the different geometric structures of a light pipe with a = b = 2.5 , c = 2.0 , and L = 60 for the different defocused coefficients ω 20 = 0 , λ π , 2 λ π , 3 λ π , 5 λ π , and 10 λ π .

Equations (23)

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B ( x , y ) = B 0 , x c 2 , y c 2 = [ H ( x + c 2 ) H ( x c 2 ) ] [ H ( y + c 2 ) H ( y c 2 ) ] = 0 , x > c 2 , y > c 2 ,
I Ω = I ( θ ) = I 0 cos θ , 90 ° θ 90 ° ,
cos θ = L L 2 + ( x 2 + y 2 ) = L R .
d F = I Ω d Ω = I 0 cos θ d A R 2 = I 0 L R d A R 2 = I 0 L R 3 d A = I 0 L [ L 2 + ( x 2 + y 2 ) ] 3 2 d x d y .
F 0 , 0 = b 2 b 2 a 2 b 2 I 0 L [ L 2 + ( x 2 + y 2 ) ] 3 2 d x d y .
F 0 , 1 = b 2 b 2 a 2 3 a 2 ρ I 0 L [ L 2 + ( x 2 + y 2 ) ] 3 2 d x d y ,
F 0 , 2 = b 2 b 2 3 a 2 5 a 2 ρ 2 I 0 L [ L 2 + ( x 2 + y 2 ) ] 3 2 d x d y ,
F m n = ( 2 n 1 ) b 2 ( 2 n + 1 ) b 2 ( 2 m 1 ) a 2 ( 2 m + 1 ) a 2 ρ m ρ n I 0 L [ L 2 + ( x 2 + y 2 ) ] 3 2 d x d y ,
m , n = 3 , 2 , 1 , 0 , 1 , 2 , 3 ,
L ( L 2 + ( x 2 + y 2 ) ) 3 2 = 1 L 2 ( 1 + ( x L ) 2 + ( y L ) 2 ) 3 2
1 L 2 ( 1 3 2 ( x L ) 2 3 2 ( y L ) 2 )
= 1 2 L 4 ( 2 L 2 3 x 2 3 y 2 ) .
F m n = I 0 2 L 4 ( 2 n 1 ) b 2 ( 2 n + 1 ) b 2 ( 2 m 1 ) b 2 ( 2 m + 1 ) a 2 ρ m ρ n ( 2 L 2 3 x 2 3 y 2 ) d x d y = ρ m ρ n I 0 2 L 4 ( a 3 b 4 a b 3 4 + 2 a b L 2 3 a 3 b m 2 3 a b 3 n 2 ) , m , n = 3 , 2 , 1 , 0 , 1 , 2 , 3 .
f ( x , y ) = T ( x , y ) exp i k ω 20 ( x 2 + y 2 ) , x 2 + y 2 1 , = 0 , x 2 + y 2 > 1 ,
T ( x , y ) = E ( x , y ) A δ ( x , y ) = E ( x , y ) m n ( F m , n A ) δ ( x 2 m a D ) δ ( y 2 n b D ) ,
m = Int [ D a 1 2 ] , n = Int [ D b 1 2 ] ,
D ( s ) = g ( s , 0 ) g ( 0 , 0 ) = f ( x + s 2 , y ) f * ( x + s 2 , y ) d x d y f ( x , y ) f * ( x , y ) d x d y ,
g ( s , 0 ) = [ 1 ( s 2 ) 2 ] 1 2 [ 1 ( s 2 ) 2 ] 1 2 [ ( 1 y 2 ) 1 2 s 2 ] [ ( 1 y 2 ) 1 2 s 2 ] [ T ( x , y ) ] 2 exp [ i K x ] d x d y ,
g ( 0 , 0 ) = 1 1 ( 1 y 2 ) 1 2 ( 1 y 2 ) 1 2 [ T ( x , y ) ] 2 d x d y ,
g ( s , 0 ) = [ 1 ( s 2 ) 2 ] 1 2 [ 1 ( s 2 ) 2 ] 1 2 [ ( 1 y 2 ) 1 2 s 2 ] [ ( 1 y 2 ) 1 2 s 2 ] [ T ( x , y ) ] 2 cos ( i K x ) d x d y .
g ( 0 , 0 ) = q = p p { ( 1 y 2 ) 1 2 ( 1 y 2 ) 1 2 [ T ( x , y ) ] 2 d x } Δ y ,
g ( s , 0 ) = q = p p { [ ( 1 y 2 ) 1 2 s 2 ] [ ( 1 y 2 ) 1 2 s 2 ] [ T ( x , y ) ] 2 cos ( i K x ) d x } Δ y ,
y = [ 1 ( s 2 ) 2 ] 1 2 p q , Δ y = [ 1 ( s 2 ) 2 ] 1 2 p .

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