Abstract

With the vector plane-wave spectrum and stationary phase method, a rigorous vector diffraction model of an aplanatic system when the polarized point source is at an arbitrary location on the optical axis is presented. The computer simulation is used to discuss in detail the effects of various angular semiapertures on the object and image sides on the resolution. Results show that angular semiapertures on the object and image sides have an obvious effect on the resolution and image fields, which indicates that the classical Wolf theory [Proc. R. Soc. London, Ser. A 253, 358 (1959) ] cannot be applied to the study of imaging properties of an aplanatic system when the point source is not located at infinity in the direction of the axis.

© 2006 Optical Society of America

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  1. P. R. T. Munro and P. Török, "Effect of detector size on optical resolution in phase contrast microscopes," Opt. Lett. 29, 623-625 (2004).
    [CrossRef] [PubMed]
  2. B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
    [CrossRef]
  3. F. van de Mast and A. Pirati, "ASML ArF leadership continues with TWINSCAN AT:1200B, a 0.85-NA production tool for 80-nm processing," Images (ASML's customer magazine, summer 2003), pp. 5-7, http://www.asml.com/doclib/productandservices/images/summer2003/asmllowbarsum03lowbarimages.pdf.
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  6. H. H. Hopkins, 'The Airy disc formula for systems of high relative aperture," Proc. Phys. Soc. London 55, 116-128 (1943).
    [CrossRef]
  7. J. P. McGuire, Jr., and R. A. Chipman, "Diffraction image formation in optical systems with polarization aberrations. I. Formulation and example," J. Opt. Soc. Am. A 7, 1614-1626 (1990).
    [CrossRef]
  8. Y. Unno, "Point-spread function for a rotationally symmetric birefringent lens," J. Opt. Soc. Am. A 19, 781-791 (2002).
    [CrossRef]
  9. E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
    [CrossRef]
  10. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
    [CrossRef]
  11. P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).
    [CrossRef] [PubMed]
  12. O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
    [CrossRef]
  13. O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: Confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
    [CrossRef]
  14. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000).
    [CrossRef] [PubMed]
  15. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
    [CrossRef]
  16. Y. Li, "Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory," J. Opt. Soc. Am. A 22, 68-76 (2005).
    [CrossRef]
  17. Y. Li, "Focal shifts in diffracted converging electromagnetic wave. II. Rayleigh theory," J. Opt. Soc. Am. A 22, 77-83 (2005).
    [CrossRef]
  18. C. J. R. Sheppard and Z. Hegedus, "Resolution for off-axis illumination," J. Opt. Soc. Am. A 15, 622-624 (1998).
    [CrossRef]
  19. J. J. Stamnes and H. Heier, "Scalar and electromagnetic diffraction point-spread functions," Appl. Opt. 37, 3612-3622 (1998).
    [CrossRef]
  20. H. Guo, J. Chen, and S. Zhuang, "Vector plane wave spectrum of an arbitrary polarized electromagnetic wave," Opt. Express 14, 2095-2100 (2006).
    [CrossRef] [PubMed]
  21. P. Varga and R. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
    [CrossRef]
  22. J. T. Foley and E. Wolf, "Wave-front spacing in the focal region of high-numerical-aperture systems," Opt. Lett. 30, 1312-1314 (2005).
    [CrossRef] [PubMed]

2006 (1)

2005 (3)

2004 (2)

P. R. T. Munro and P. Török, "Effect of detector size on optical resolution in phase contrast microscopes," Opt. Lett. 29, 623-625 (2004).
[CrossRef] [PubMed]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: Confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

2003 (1)

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

2002 (1)

2001 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

2000 (1)

1998 (3)

1997 (1)

1990 (1)

1959 (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

1943 (1)

H. H. Hopkins, 'The Airy disc formula for systems of high relative aperture," Proc. Phys. Soc. London 55, 116-128 (1943).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Brown, T. G.

Chen, J.

Chipman, R. A.

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Foley, J. T.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Goodman, J. W.

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

Guo, H.

Haeberlé, O.

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: Confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Hegedus, Z.

Heier, H.

Hendriks, B. H. W.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, 'The Airy disc formula for systems of high relative aperture," Proc. Phys. Soc. London 55, 116-128 (1943).
[CrossRef]

Leuchs, G

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Li, Y.

McGuire, J. P.

Munro, P. R. T.

Pirati, A.

F. van de Mast and A. Pirati, "ASML ArF leadership continues with TWINSCAN AT:1200B, a 0.85-NA production tool for 80-nm processing," Images (ASML's customer magazine, summer 2003), pp. 5-7, http://www.asml.com/doclib/productandservices/images/summer2003/asmllowbarsum03lowbarimages.pdf.

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

Schleipen, J. J. H. B.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

Sheppard, C. J. R.

Stallinga, S.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

Stamnes, J. J.

Török, P.

Török, R.

P. Varga and R. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Unno, Y.

van de Mast, F.

F. van de Mast and A. Pirati, "ASML ArF leadership continues with TWINSCAN AT:1200B, a 0.85-NA production tool for 80-nm processing," Images (ASML's customer magazine, summer 2003), pp. 5-7, http://www.asml.com/doclib/productandservices/images/summer2003/asmllowbarsum03lowbarimages.pdf.

van Houten, H.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

Varga, P.

P. Varga and R. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).
[CrossRef] [PubMed]

Wolf, E.

J. T. Foley and E. Wolf, "Wave-front spacing in the focal region of high-numerical-aperture systems," Opt. Lett. 30, 1312-1314 (2005).
[CrossRef] [PubMed]

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Youngworth, K. S.

Zhuang, S.

Appl. Opt. (2)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

P. Varga and R. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: Confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Rev. (1)

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA 0.85," Opt. Rev. 8, 211-213 (2001).
[CrossRef]

Proc. Phys. Soc. London (1)

H. H. Hopkins, 'The Airy disc formula for systems of high relative aperture," Proc. Phys. Soc. London 55, 116-128 (1943).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

Other (3)

F. van de Mast and A. Pirati, "ASML ArF leadership continues with TWINSCAN AT:1200B, a 0.85-NA production tool for 80-nm processing," Images (ASML's customer magazine, summer 2003), pp. 5-7, http://www.asml.com/doclib/productandservices/images/summer2003/asmllowbarsum03lowbarimages.pdf.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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

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Figures (3)

Fig. 1
Fig. 1

Geometry of vector diffraction in an aplanatic system for on-axis illumination when the polarized point source is located arbitrarily on the optical axis.

Fig. 2
Fig. 2

Relation between the resolution at the object plane and the angular semiaperture on the Φ o on the object side when the angular semiaperture Φ i on the image side is fixed [(a), (c) Φ o = 5 ° (b), (d) and Φ i = 50 ° ] and (a), (b) linearly polarized illumination in the x direction or (c), (d) circularly polarized illumination is used. Curves a, b, and c represent the resolution in the directions of the x axis, the y axis, and the CR, respectively.

Fig. 3
Fig. 3

Relation between the resolution at the image plane and the angular semiaperture Φ i on the image side when the angular semiaperture Φ o on the object side is fixed [(a), (c) Φ i = 5 ° (b), (d) and Φ i = 50 ° ] and (a), (b) linearly polarized illumination in the x direction (c), (d) or circularly polarized illumination is used. Curves a, b, and c represent the resolution in the directions of the x axis, the y axis, and the CR, respectively.

Equations (40)

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h o sin θ o = h i sin θ i .
d o sin θ o = d i sin θ i .
E o ( A ) 2 d S o = E i ( B ) 2 d S i ,
d S o cos θ o = d S i cos θ i .
E ( r ) = λ 2 e TM ( s ) E ̃ TM ( s ) exp [ j k ( s x x + s y y + s z z ) ] d s x d s y λ 2 e TE ( s ) E ̃ TE ( s ) exp [ j k ( s x x + s y y + s z z ) ] d s x d s y ,
E ̃ TM ( s ) = s z 1 ( s x 2 + s y 2 ) 1 2 ( s x E x + s y E y ) exp [ j k ( s x x + s y y + s z z ) ] d x d y ,
E ̃ TE ( s ) = ( s x 2 + s y 2 ) 1 2 ( s y E x s x E y ) exp [ j k ( s x x + s y y + s z z ) ] d x d y ,
e TM ( s ) = ( s x 2 + s y 2 ) 1 2 [ s z ( x s x + y s y ) z ( s x 2 + s y 2 ) ] ,
e TE ( s ) = ( s x 2 + s y 2 ) 1 2 ( x s y y s x ) ,
E ( r ) = λ 2 E ̃ ( s ) exp { j k [ ( s x x + s y y + s z z ) ] } d s x d s y ,
E ̃ ( s ) = [ x E x + y E y z s z 1 ( s x E x + s y E y ) ] exp [ j k ( s x x + s y y ) ] d x d y ,
E o ( A ) = ( λ d o ) 1 s o z E ̃ o ( s o ) exp [ j ( k d o π 2 ) ] ,
E o ( A ) = C e A exp [ j ( k d o π 2 ) ] ,
e A = s o z [ x e o x + y e o y z s o z 1 ( s o x e o x + s o y e o y ) ] ,
E i ( B ) = d i 1 a ( s i x , s i y ) exp ( j k d i ) ,
e i = { [ e TE ( s o ) e o ] e TE ( s i ) + [ e TM ( s o ) e o ] e TM ( s i ) } ,
e i = e A 1 ( s o x 2 + s o y 2 ) 1 2 ( s i x 2 + s i y 2 ) 1 2 × { ( s o y s o z e o x s o x s o z e o y ) ( x s i y y s i x ) + ( s o x e o x + s o y e o y ) [ x s i x s i z + y s i y s i z z ( s i x 2 + s i y 2 ) ] } .
e i = e A 1 { [ ( cos θ o sin 2 φ + cos θ i cos 2 φ ) e o x + ( cos θ i cos θ o ) sin φ cos φ e o y ] + y [ ( cos θ i cos θ o ) sin φ cos φ e o x + ( cos θ o cos 2 φ + cos θ i sin 2 φ ) e o y ] + z sin θ i ( cos φ e o x + sin φ e o y ) } .
E i ( B ) = e i C e A cos 1 2 θ o cos 1 2 θ i exp ( j k d i ) .
E i ( B ) = ( λ d i ) 1 s i z E ̃ i ( s i ) exp [ j ( k d i + π 2 ) ] ,
E ¯ i ( s i ) = C λ d i cos 1 2 θ o cos 1 2 θ i exp ( j π 2 ) { x [ ( cos θ o sin 2 φ + cos θ i cos 2 φ ) e o x + ( cos θ i cos θ o ) sin φ cos φ e o y ] + y [ ( cos θ i cos θ o ) sin φ cos φ e a x + ( cos θ o cos 2 φ + cos θ i sin 2 φ ) e o y ] + z sin θ i ( cos φ e o x + sin φ e o y ) } .
H ̃ i ( s i ) = s i × E ̃ i ( s i ) ,
H ̃ i ( s i ) = C λ d i cos 1 2 θ o cos 1 2 θ i exp ( j π 2 ) × { x [ ( 1 cos θ o cos θ i ) sin φ cos φ e o x + ( sin 2 φ + cos θ o cos θ i cos 2 φ ) e o y ] + y [ ( cos 2 φ + cos θ o cos θ i sin 2 φ ) e o x + ( 1 cos θ o cos θ i ) sin φ cos φ e o y ] + z cos θ o sin θ i ( sin φ e o x cos φ e o y ) } .
E ( r ) = λ 2 0 Φ i sin θ i cos θ i d θ i 0 2 π E ̃ i ( s i ) exp [ j k ( s i x x + s i y y + s i z z ) ] d φ ,
H ( r ) = λ 2 0 Φ i sin θ i cos θ i d θ i 0 2 π H ̃ i ( s i ) exp [ j k ( s i x x + s i y y + s i z z ) ] d φ ,
0 2 π cos n φ exp [ j ρ cos ( φ β ) ] d φ = 2 π j n J n ( ρ ) cos n β ,
0 2 π sin n φ exp [ j ρ cos ( φ β ) ] d φ = 2 π j n J n ( ρ ) sin n β ,
E x = j π λ 1 d i C [ ( A 0 + A 2 cos 2 β ) e o x + A 2 sin 2 β e o y ] ,
E y = j π λ 1 d i C [ A 2 sin 2 β e o x + ( A 0 A 2 cos 2 β ) e o y ] ,
E z = 2 π λ 1 d i C A 1 ( cos β e o x + sin β e o y ) ,
H x = j π λ 1 d i C [ B 2 sin 2 β e o x ( B 0 + B 2 cos 2 β ) e o y ] ,
H y = j π λ 1 d i C [ ( B 0 B 2 cos 2 β ) e o x B 2 sin 2 β e o y ] ,
H z = 2 π λ 1 d i C B 1 ( sin β e o x cos β e o y ) ,
A 0 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i ( cos θ o + cos θ i ) J 0 ( ) exp ( ) d θ i ,
A 1 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin 2 θ i J 1 ( ) exp ( ) d θ i ,
A 2 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i ( cos θ o cos θ i ) J 2 ( ) exp ( ) d θ i ,
B 0 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i ( 1 + cos θ o cos θ i ) J 0 ( ) exp ( ) d θ i ,
B 1 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin 2 θ i J 1 ( ) exp ( ) d θ i ,
B 2 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i ( 1 cos θ o cos θ i ) J 2 ( ) exp ( ) d θ i ,
R = n sin Φ 0.61 λ ,

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