Abstract

The problem of automated design of phase-shifting masks for enhanced-resolution optical lithography is examined. We propose a computationally viable algorithm for the rapid design of phase-shifting masks for arbitrary two-dimensional patterns. Our approach is based on the use of a class of optimal coherent approximations to partially coherent imaging systems described by the Hopkins model. These approximations lead to substantial computational and analytical benefits, and, in addition, the resultant approximation error can be quite small for imaging systems with coherence factor σ ≤ 0.5. These approximate models allow us to reduce the mask-design problem to the classical phase-retrieval problem in optics. A fast iterative algorithm, closely related to the Gerchberg–Saxton algorithm, is then applied to generate (suboptimal) phase-shifting masks. Analytical results related to practical requirements for phase-shifting masks are also presented. These results address questions related to the number of discrete phase levels required for arbitrary patterns and provide some insight into alternative strategies for the use of phase-shifting masks. A number of simulated phase-shifting mask-design examples are provided to illustrate the methods and ideas presented.

© 1994 Optical Society of America

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References

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  1. M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
    [CrossRef]
  2. Y. Liu, A. Zakhor, “Binary and phase-shifting mask design for optical lithography,”IEEE Trans. Semiconductor Manufacturing 5, 138–152 (1992).
    [CrossRef]
  3. C. Chang, C. D. Schaper, T. Kailath, “Computer-aided optimal design of phase-shifting masks,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1674, 65–72 (1992).
    [CrossRef]
  4. Y. Liu, A. Zakhor, “Computer aided phase-shift mask design with reduced complexity,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 477–493 (1992).
    [CrossRef]
  5. B. E. A. Saleh, K. M. Nashold, “Image construction: optimum amplitude and phase masks,” Appl. Opt. 24, 1432–1437 (1985).
    [CrossRef] [PubMed]
  6. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  7. Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
    [CrossRef]
  8. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,”J. Opt. Soc. Am. 7, 450–458 (1990).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  10. B. E. A. Saleh, M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansions,” Appl. Opt. 21, 2770–2777 (1982).
    [CrossRef] [PubMed]
  11. E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  12. A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 3, p. 127.
  13. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1: theory,” IEEE Trans. Med. Imaging MI-1(2), 81–94 (1982).
    [CrossRef]
  14. S. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2: applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (2), 95–101 (1982).
    [CrossRef]
  15. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  16. T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
    [CrossRef]

1992 (1)

Y. Liu, A. Zakhor, “Binary and phase-shifting mask design for optical lithography,”IEEE Trans. Semiconductor Manufacturing 5, 138–152 (1992).
[CrossRef]

1991 (1)

T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
[CrossRef]

1990 (2)

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,”J. Opt. Soc. Am. 7, 450–458 (1990).
[CrossRef]

1986 (1)

1985 (1)

1982 (4)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1: theory,” IEEE Trans. Med. Imaging MI-1(2), 81–94 (1982).
[CrossRef]

S. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2: applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (2), 95–101 (1982).
[CrossRef]

B. E. A. Saleh, M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansions,” Appl. Opt. 21, 2770–2777 (1982).
[CrossRef] [PubMed]

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
[CrossRef]

1981 (1)

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Balakrishnan, A. V.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 3, p. 127.

Bass, R.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Bradie, B.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Chang, C.

C. Chang, C. D. Schaper, T. Kailath, “Computer-aided optimal design of phase-shifting masks,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1674, 65–72 (1992).
[CrossRef]

Fienup, J. R.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,”J. Opt. Soc. Am. 7, 450–458 (1990).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

Fukuda, H.

T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Hasegawa, N.

T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
[CrossRef]

Kailath, T.

C. Chang, C. D. Schaper, T. Kailath, “Computer-aided optimal design of phase-shifting masks,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1674, 65–72 (1992).
[CrossRef]

Kowalczyk, A. M.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,”J. Opt. Soc. Am. 7, 450–458 (1990).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
[CrossRef]

Liu, Y.

Y. Liu, A. Zakhor, “Binary and phase-shifting mask design for optical lithography,”IEEE Trans. Semiconductor Manufacturing 5, 138–152 (1992).
[CrossRef]

Y. Liu, A. Zakhor, “Computer aided phase-shift mask design with reduced complexity,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 477–493 (1992).
[CrossRef]

Nashold, K. M.

Park, D.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Pati, Y. C.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Peckerar, M. C.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Rabbani, M.

Rhee, K.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Saleh, B. E. A.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schaper, C. D.

C. Chang, C. D. Schaper, T. Kailath, “Computer-aided optimal design of phase-shifting masks,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1674, 65–72 (1992).
[CrossRef]

Sezan, S. I.

S. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2: applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (2), 95–101 (1982).
[CrossRef]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
[CrossRef]

Stark, H.

S. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2: applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (2), 95–101 (1982).
[CrossRef]

Teolis, A.

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Terasawa, T.

T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
[CrossRef]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
[CrossRef]

Wackerman, C. C.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1: theory,” IEEE Trans. Med. Imaging MI-1(2), 81–94 (1982).
[CrossRef]

Wolf, E.

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1: theory,” IEEE Trans. Med. Imaging MI-1(2), 81–94 (1982).
[CrossRef]

Zakhor, A.

Y. Liu, A. Zakhor, “Binary and phase-shifting mask design for optical lithography,”IEEE Trans. Semiconductor Manufacturing 5, 138–152 (1992).
[CrossRef]

Y. Liu, A. Zakhor, “Computer aided phase-shift mask design with reduced complexity,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 477–493 (1992).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Electron Devices (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,”IEEE Trans. Electron Devices, ED-29, 1828–1836 (1982).
[CrossRef]

IEEE Trans. Med. Imaging (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1: theory,” IEEE Trans. Med. Imaging MI-1(2), 81–94 (1982).
[CrossRef]

S. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2: applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (2), 95–101 (1982).
[CrossRef]

IEEE Trans. Semiconductor Manufacturing (1)

Y. Liu, A. Zakhor, “Binary and phase-shifting mask design for optical lithography,”IEEE Trans. Semiconductor Manufacturing 5, 138–152 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,”J. Opt. Soc. Am. 7, 450–458 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Vac. Sci. Technol. B (1)

Y. C. Pati, A. Teolis, D. Park, K. Rhee, R. Bass, B. Bradie, M. C. Peckerar, “An error measure for dose correction in e-beam nanolithography,”J. Vac. Sci. Technol. B 8, 1882–1885 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Terasawa, N. Hasegawa, H. Fukuda, “Imaging characteristics of multi-phase-shifting and halftone-phase-shifting masks,” Jpn. J. Appl. Phys. 30, 2991–2997 (1991).
[CrossRef]

Opt. Commun. (1)

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

C. Chang, C. D. Schaper, T. Kailath, “Computer-aided optimal design of phase-shifting masks,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1674, 65–72 (1992).
[CrossRef]

Y. Liu, A. Zakhor, “Computer aided phase-shift mask design with reduced complexity,” in Optical/Laser Microlithography V, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 477–493 (1992).
[CrossRef]

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 3, p. 127.

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Figures (14)

Fig. 1
Fig. 1

Optical imaging of a double-pulse pattern with (a) a conventional transmission mask and (b) a phase-shifting mask.

Fig. 2
Fig. 2

Percentage error in the OCA versus coherence factor σ, for both a first-order and a zero-order coherent approximation to a partially coherent imaging system. (Here we define percentage error by normalizing the L2 norm of the approximation error with respect to the norm of W.)

Fig. 3
Fig. 3

Squared magnitude of the Fourier transform of the dominant eigenfunction of A for coherence factors σ = 0, σ = 0.3, and σ = 0.5.

Fig. 4
Fig. 4

Empirically determined curve demonstrating the nonlinearity in the etching process.

Fig. 5
Fig. 5

Partitioning of the image plane.

Fig. 6
Fig. 6

Alternating-projection loop for the phase-shifting mask design. H, complex image; Q, (linear) filter defined by the approximate model.

Fig. 7
Fig. 7

Adjacency in Manhattan patterns.

Fig. 8
Fig. 8

(a) Metal pattern requiring four phases; all spacings and feature sizes are minimum (= δ). (b) The pattern is split so that only two phases are required for each layer.

Fig. 9
Fig. 9

Phase-shifting mask design for two parallel 0.25-μm lines. (a) Phase-shifting mask: black areas are opaque; phase values are indicated in the legend. (b) Image produced with a conventional transmission mask. (c) Image produced with the designed phase-shifting mask. The ideal image is outlined by dark lines. (d) Contour plot of the image produced with the designed phase-shifting mask. The ideal image is outlined by dark lines. Images (c) and (d) display intensity levels greater than 0.3 times the maximum intensity.

Fig. 10
Fig. 10

Phase-shifting mask design for a 0.25-μm × 0.25-μm contact hole. (a) Phase-shifting mask: black areas are opaque; phase values are indicated in the legend. (b) Image produced with a conventional transmission mask. (c) Image produced with the designed phase-shifting mask. The ideal image is outlined by dark lines. (d) Contour plot of the image produced with the designed phase-shifting mask and the ideal image. Images (b)–(d) display intensity levels greater than 0.3 times the maximum intensity.

Fig. 11
Fig. 11

Phase-shifting mask design for a U-shaped pattern. (a), (b) Image of designed phase-shifting mask with continuously varying phase and amplitude. (c), (d) Image produced with a conventional transmission mask. Images (b) and (d) display intensity levels greater than 0.3 times the maximum intensity.

Fig. 12
Fig. 12

Double-exposure phase-shifting mask design for a U-shaped pattern. (a), (b) The pair of designed two-phase phase-shifting masks. Black areas are opaque; phase values are indicated in the legend. (c) Image obtained by successive imaging of the pair of two-phase masks. Dark lines outline the ideal image. (d) Contour plot of the image obtained by successive imaging of the pair of two-phase masks. Dark lines outline the ideal image. Image (d) displays intensity levels greater than 0.3 times the maximum intensity.

Fig. 13
Fig. 13

Single-exposure four-phase phase-shifting mask design for a U-shaped pattern. (a) Phase-shifting mask. Phase values are indicated by shading defined in the legend. (b) Contour plot of the image of the four-phase mask. Image (b) displays intensity levels greater than 0.3 times the maximum intensity.

Fig. 14
Fig. 14

Manhattan-pattern map that cannot be colored with less than four distinct colors.

Equations (36)

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G = T ( F ) .
F L { F L 2 ( R 2 ) F ( x , y ) { 0 , 1 , - 1 } } .
G ( x , y ) = [ T ( F ) ] ( x , y ) = R 4 F ( ξ 1 , ξ 2 ) J 0 ( ξ 1 , ξ 2 , η 1 , η 2 ) F * ( η 1 , η 2 ) × K ( x , y , ξ 1 , ξ 2 ) K * ( x , y , η 1 , η 2 ) d ξ 1 d ξ 2 d η 1 d η 2 ,
G = F * K 2 ,
( G * H ) ( x , y ) = G ( ξ 1 , ξ 2 ) H ( x - ξ 1 , y - ξ 2 ) d ξ 1 d ξ 2 .
G = F 2 * K 2 .
G d ( x , y ) = [ T ( F ) ] ( x , y ) .
G ( x ) = R R F ( η 1 ) J 0 ( η 1 - η 2 ) F * ( η 2 ) K ( x - η 1 ) × K * ( x - η 2 ) d η 1 d η 2 .
J 0 ( η ) = J 0 ^ ( ω ) exp ( i ω η ) d ω .
G ( x ) = J 0 ^ ( ω ) F ( η 1 ) F * ( η 2 ) exp [ i ω ( η 1 - η 2 ) ] × K ( x - η 1 ) K * ( x - η 2 ) d η 1 d η 2 d ω = J 0 ^ ( ω ) F * K ω 2 d ω ,             K ω ( η ) = exp ( i ω η ) K ( η ) .
G ( x ) = m a m F * K m 2 ,
G ( x ) = [ T ( F ) ] ( x ) = F ( η 1 ) F * ( η 2 ) W ( x - η 1 , x - η 2 ) d η 1 d η 2 ,
W ( η 1 , η 2 ) = J 0 ( η 2 - η 1 ) K ( η 1 ) K * ( η 2 ) .
W ( η 1 , η 2 ) = Q ( η 1 ) Q * ( η 2 ) + R ( η 1 , η 2 ) .
G ( x ) = F ( η 1 ) F * ( η 2 ) [ Q * ( x - η 2 ) Q ( x - η 1 ) + R ( x - η 1 , x - η 2 ) ] d η 1 d η 2 = ( F * Q ) ( x ) 2 + R ( F ) ( x ) .
[ A ( x ) f ] ( η 1 ) = W ( x - η 1 , x - η 2 ) f ( η 2 ) d η 2 .
G ( x ) = [ T ( F ) ] ( x ) = A ( x ) F , F ,
W ( η 1 , η 2 ) = k = 1 λ k ϕ k ( η 1 ) ϕ k * ( η 2 ) ,
( T n F ) ( x ) = k = 1 n A k ( x ) F , F = k = 1 n λ k ( F * ϕ k ) ( x ) 2             for n = 1 , 2 , ,
W k ( x - η 1 , x - η 2 ) = λ k ϕ k ( x - η 1 ) ϕ ( x - η 2 ) .
W - W ˜ 2 ,
sup x G ( x ) - G ˜ ( x ) F 2 2 ,
S 1 = { H L 2 ( R 2 ) supp ( H ^ ) S Ω } ,
G - G d 2 = G - G d 2 .
S 2 1 = { H L 2 ( R 2 ) H 2 = G ˜ d } .
W ( G d ) = { G L 2 ( R 2 ) G 0 , [ G ( x , y ) χ - for ( x , y ) D - G ( x , y ) χ + for ( x , y ) D + ] } .
[ P W ( G ) ] ( x , y ) = { χ + if ( x , y ) D + and G ( x , y ) < χ + χ - if ( x , y ) D - and G ( x , y ) < χ - G ( x , y ) else .
S 2 2 = { H L 2 ( R 2 ) H 2 W ( G d ) } .
T ( F ˜ ) - P 2 1 T ( F ˜ ) T ( F ) - P 2 1 T ( F )         for any F L 2 ( R 2 ) .
T ( F ˜ ) - P 2 2 T ( F ˜ ) T ( F ) - P 2 2 T ( F )             for any F L 2 ( R 2 ) ,
( P 1 P 2 P n ) N f 0 f j = 1 n C j             as N .
Projection onto S 1 : P 1 ^ f ( ω 1 , ω 2 ) = { f ^ ( ω 1 , ω 2 ) if ( ω 1 , ω 2 ) Ω 0 if ( ω 1 , ω 2 ) Ω , Projection onto S 2 ( 1 ) : P 2 1 f = ( G ˜ d ) 1 / 2 exp ( i ϕ f ) , Projection onto S 2 ( 2 ) : P 2 2 f = ( P W f ) 1 / 2 exp ( i ϕ f ) ,
T ( F ) = G
G = T ( F 1 ) + T ( F 2 ) .
G ( x ) - G ˜ ( x ) = A ( x ) F , F - A ˜ ( x ) F , F = [ A ( x ) - A ˜ ( x ) ] F , F ( A ( x ) - A ˜ ( x ) ) F 2 F 2             ( by Cauchy - Schwarz ) F 2 2 .
G ( x ) = [ T ( F ) ] ( x ) = A ( x ) F , F = A ( x ) Re F + i A ( x ) Im F , Re F + i Im F = A ( x ) Re F , Re F + A ( x ) Im F , Im F + i A ( x ) Im F , Re F - i A ( x ) Re F , Im F = [ T ( Re F ) ] ( x ) + [ T ( Im F ) ] ( x ) + i ( A ( x ) Im F , Re F - A ( x ) Re F , Im F ) = [ T ( Re F ) ] ( x ) + [ T ( Im F ) ] ( x ) + i ( Im F , A ( x ) Re F - A ( x ) Re F , Im F ) [ since A ( x ) is self - adjoint ] = [ T ( Re F ) ] ( x ) + [ T ( Im F ) ] ( x ) - 2 Im Im F , A ( x ) Re F = A ( x ) Re F , Re F + A ( x ) Im F , Im F ( since Im F , A ( x ) Re F is real ) = [ T ( F 1 ) ] ( x ) + [ T ( F 2 ) ] ( x ) .

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