Abstract

A general theory for making an optical transform is presented using amplitude-phase holographic lenses (masks). It can be shown that an optical system composed of many amplitude-phase masks can do any given linear transform. A set of equations for determining the amplitude-phase distributions of masks is given. A feedback iterative approach to deal with these equations is also suggested. We show that any given optical transform can be achieved even with a single mask by increasing the number of sampling points in the mask. The relevant equations and the necessary conditions satisfied by the mask are also given. To free the fabrication of the mask from difficulty in technique, sometimes the information quantity carried by the single mask must be relaxed. The dual-mask system is discussed in detail. The general theory is demonstrated by examples for performing the four- and eight-sequence Walsh transforms in three different orders. The results agree well with the theory. Some experimental results and relevant applications are also reviewed briefly.

© 1986 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. J. R. Leger, S. H. Lee, “Optica Hoy Y Manana,” in Proceedings, Eleventh Congress of the International Commission for Optics, ICO-11, Madrid (11–17 Sept. 1979), p. 323.
  3. J. F. Walkup, “Space Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
    [CrossRef]
  4. Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).
  5. Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).
  6. Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (III) the Optimization Method and Related Problems,” Acta Phys. Sin. 27, 487 (1978).
  7. Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (IV) the Pattern Recognition and Projection Operator,” Acta Phys. Sin. 29, 153 (1980).
  8. G. Z. Yang, S. H. Pan, “A Scheme for Optical Realization of Walsh Transformation,” Acta Phys. Sin. 29, 1301 (1980).
  9. Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).
  10. J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).
  11. Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).
  12. C. H. Andrew et al., Computer Techniques in Image Processing (Academic, New York, 1970).
  13. W. Q. Li, Functional Analysis (Academic, Beijing, 1962).
  14. G. Z. Yang, “Theory of Optical Transformation by A Single Holographic Lens,” Acta Phys. Sin. 30, 1340 (1981).
  15. G. Z. Yang, B. Y. Gu, “On the Amplitude-phase Retrieval Problem in Optical System,” Acta Phys. Sin. 30, 410 (1981).
  16. B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

1984 (1)

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

1982 (2)

J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).

B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

1981 (2)

G. Z. Yang, “Theory of Optical Transformation by A Single Holographic Lens,” Acta Phys. Sin. 30, 1340 (1981).

G. Z. Yang, B. Y. Gu, “On the Amplitude-phase Retrieval Problem in Optical System,” Acta Phys. Sin. 30, 410 (1981).

1980 (4)

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (IV) the Pattern Recognition and Projection Operator,” Acta Phys. Sin. 29, 153 (1980).

G. Z. Yang, S. H. Pan, “A Scheme for Optical Realization of Walsh Transformation,” Acta Phys. Sin. 29, 1301 (1980).

Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).

J. F. Walkup, “Space Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

1978 (1)

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (III) the Optimization Method and Related Problems,” Acta Phys. Sin. 27, 487 (1978).

1976 (1)

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).

1975 (1)

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).

Andrew, C. H.

C. H. Andrew et al., Computer Techniques in Image Processing (Academic, New York, 1970).

Chen, Y. S.

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).

Dai, J. H.

J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).

Dong, B. Z.

B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

Goodman, J. W.

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

Gu, B. Y.

G. Z. Yang, B. Y. Gu, “On the Amplitude-phase Retrieval Problem in Optical System,” Acta Phys. Sin. 30, 410 (1981).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).

Huo, Y. P.

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (IV) the Pattern Recognition and Projection Operator,” Acta Phys. Sin. 29, 153 (1980).

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (III) the Optimization Method and Related Problems,” Acta Phys. Sin. 27, 487 (1978).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).

Jiang, C. S.

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

Lee, S. H.

J. R. Leger, S. H. Lee, “Optica Hoy Y Manana,” in Proceedings, Eleventh Congress of the International Commission for Optics, ICO-11, Madrid (11–17 Sept. 1979), p. 323.

Leger, J. R.

J. R. Leger, S. H. Lee, “Optica Hoy Y Manana,” in Proceedings, Eleventh Congress of the International Commission for Optics, ICO-11, Madrid (11–17 Sept. 1979), p. 323.

Li, W. Q.

W. Q. Li, Functional Analysis (Academic, Beijing, 1962).

Li, X. Y.

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).

Liu, J. L.

J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).

Pan, S. H.

G. Z. Yang, S. H. Pan, “A Scheme for Optical Realization of Walsh Transformation,” Acta Phys. Sin. 29, 1301 (1980).

Walkup, J. F.

J. F. Walkup, “Space Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

Wang, Y. T.

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).

Yang, G. Z.

B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

G. Z. Yang, “Theory of Optical Transformation by A Single Holographic Lens,” Acta Phys. Sin. 30, 1340 (1981).

G. Z. Yang, B. Y. Gu, “On the Amplitude-phase Retrieval Problem in Optical System,” Acta Phys. Sin. 30, 410 (1981).

G. Z. Yang, S. H. Pan, “A Scheme for Optical Realization of Walsh Transformation,” Acta Phys. Sin. 29, 1301 (1980).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).

Zhang, H. J.

J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).

Zhen, S. H.

B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

Acta Phys. Sin. (11)

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (I) the Analysis of Possibility,” Acta Phys. Sin. 24, 438 (1975).

Y. P. Huo, G. Z. Yang, B. Y. Gu, “Unitary Transformation and General Linear Transformation by An Optical Method (II) the Iterative Method of Solution,” Acta Phys. Sin. 25, 31 (1976).

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (III) the Optimization Method and Related Problems,” Acta Phys. Sin. 27, 487 (1978).

Y. P. Huo, “Unitary Transformation and General Linear Transformation by An Optical Method (IV) the Pattern Recognition and Projection Operator,” Acta Phys. Sin. 29, 153 (1980).

G. Z. Yang, S. H. Pan, “A Scheme for Optical Realization of Walsh Transformation,” Acta Phys. Sin. 29, 1301 (1980).

Y. S. Chen, Y. T. Wang, X. Y. Li, “The Optical Realization of Walsh Transformation with Coherent System,” Acta Phys. Sin. 29, 1307 (1980).

J. L. Liu, J. H. Dai, H. J. Zhang, “A Method to Achieve Optical Multi-characteristic Pattern Recognition,” Acta Phys. Sin. 31, 437 (1982).

Y. T. Wang, X. Y. Li, C. S. Jiang, Y. S. Chen, “Optical Realization of Coherent Walsh Transformation by a Single Element,” Acta Phys. Sin. 33, 1599 (1984).

G. Z. Yang, “Theory of Optical Transformation by A Single Holographic Lens,” Acta Phys. Sin. 30, 1340 (1981).

G. Z. Yang, B. Y. Gu, “On the Amplitude-phase Retrieval Problem in Optical System,” Acta Phys. Sin. 30, 410 (1981).

B. Z. Dong, S. H. Zhen, G. Z. Yang, “A Simulation Experiment of Phase Adjustment for A Phase-adjusted Focusing Laser Accelerator,” Acta Phys. Sin. 31, 895 (1982).

Opt. Eng. (1)

J. F. Walkup, “Space Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

Other (4)

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

J. R. Leger, S. H. Lee, “Optica Hoy Y Manana,” in Proceedings, Eleventh Congress of the International Commission for Optics, ICO-11, Madrid (11–17 Sept. 1979), p. 323.

C. H. Andrew et al., Computer Techniques in Image Processing (Academic, New York, 1970).

W. Q. Li, Functional Analysis (Academic, Beijing, 1962).

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Figures (7)

Fig. 1
Fig. 1

Schematic of a typical optical system for realizing a given general linear transform.

Fig. 2
Fig. 2

Schematic of a typical system composed of n planar masks for achieving a given LT.

Fig. 3
Fig. 3

Propagation process of lightwaves in free space.

Fig. 4
Fig. 4

Schematic of a single-mask system for realizing a given LT.

Fig. 5
Fig. 5

Plots of the amplitude and phase distributions of the mask vs the sequence of sampling points for realizing an eight-sequence naturally ordered Walsh transform: (a) amplitude distribution, (b) phase distribution. (N1 = N2 = 8, N = 64, l0 = 20 cm, l1 = 12.5 cm, a0 = a1 = a2 = 0.225 cm, λ = 0.6328 μm.)

Fig. 6
Fig. 6

Schematic of a dual-mask system for achieving a LT.

Fig. 7
Fig. 7

Amplitude distribution (solid curve) and phase distribution (dashed curve) of the masks as a function of the sequence of sampling points for realizing a four-sequence naturally ordered Walsh transform in the dual-mask system: (a) mask H1, (b) mask H2. (N1 = N2 = 4, N = 8, l0 = 100 cm; l1 = 30 cm, l2 = 100 cm, a0 = a1 = a2 = a3 = 0.225 cm, λ = 0.6328 μm.)

Tables (2)

Tables Icon

Table I Relevant Parameters in a Single-Mask System for Realizing Four- and Eight-Sequence WTs in the Three Different Orders

Tables Icon

Table II Corresponding Matrix Elements Between G and G for a Four-Sequence Naturally Ordered Walsh Transform

Equations (64)

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f ( x , y ) = G ˜ ( x , y ; x , y ) f ( x , y ) d x d y .
f = G ˜ f .
G ˜ = G n G n - 1 G n - 2 G 3 G 2 G 1 .
H m ( x m ) = ρ m ( x m ) exp [ - i ϕ m ( x m ) ] ,             m = 1 , 2 , 3 , n .
G ˜ = G 0 ( l n ) H n H 2 G 0 ( l 1 ) H 1 G 0 ( l 0 ) ,
G 0 ( x , x ; l ) = [ ( 1 + cos θ ) / i 2 λ r ] exp ( i k r ) ,
D 2 = G - G ˜ 2 = ( 1 / N 2 ) i , j = 1 N G i j - G ˜ i j 2 .
D 1 2 = G - G 0 ( l 1 ) H 1 G 0 ( l 0 ) 2 .
D 2 2 = G - G 0 H 2 G 0 H 1 ( 1 ) G 0 2 .
D 3 2 = G - G 0 H 2 G 0 H 1 G 0 2 .
G ˜ = P 1 H k P 2 ,
P 1 = G 0 H n G 0 G 0 H k + 1 G 0
P 2 = G 0 H k - 1 G 0 G 0 H 1 G 0 .
Re j = 1 N A l j B j l ( H k ) j j exp [ i ( ϕ k ) l l ] = Re { C l l exp [ i ( ϕ k ) l l ] } ,
Im j = 1 N A l j B j l ( H k ) j j exp [ i ( ϕ k ) l l ] = Im { C l l exp [ i ( ϕ k ) l l ] } ,
j = 1 N A l j B j l ( H k ) j j = C l l ,             l = 1 , 2 , 3 N ,
A = P 1 + P 1 = G 0 + H k + 1 + G 0 + G 0 + H n + G 0 + G 0 H n G 0 × G 0 H k + 1 G 0 ,
B = P 2 P 2 + = G 0 H k - 1 G 0 G 0 H 1 G 0 G 0 + H 1 + G 0 + × G 0 + H k - 1 + G 0 + ,
C = P 1 + G P 2 + = G 0 + H k + 1 + G 0 + G 0 + H n + G 0 + G G 0 + H 1 + G 0 + × G 0 + H k - 1 + G 0 + ,
( ρ k ) j j = 1 ,             j = 1 , 2 , 3 , N .
j = 1 N Im ( A l j B j l exp { i [ ( ϕ k ) l l - ( ϕ k ) j j ] } ) = Im { C l l exp [ i ( ϕ k ) l l ] } ,             l = 1 , 2 , 3 , N ,
( H k ) l l = C l l / C l l ,             l = 1 , 2 , 3 N .
exp [ - i ( ϕ k ) j j ] = 1 ,             j = 1 , 2 , 3 N .
j = 1 N ( ρ k ) j j Re ( A l j B j l ) = Re ( C l l ) ,             l = 1 , 2 , 3 N .
G ˜ = G 0 ( l 1 ) H G 0 ( l 0 ) .
l = 1 N A k l B l k H l l = C k k ,             k = 1 , 2 , 3 , N .
A = G 0 + ( l 1 ) G 0 ( l 1 ) ,
B = G 0 ( l 0 ) G 0 + ( l 0 ) ,
C = G 0 + ( l 1 ) G G 0 + ( l 0 ) ,
G 0 ( l 0 ) p q = K 0 ( ( 1 + ( 1 + S 0 ) 1 / 2 ) / ( 2 ( 1 + S 0 ) ) ) 1 / 2 × exp ( i 2 π l 0 ( ( 1 + S 0 ) 1 / 2 - 1 ) / λ ) , if S 0 0.1
= K 0 ( ( 1 + ( 1 + S 0 ) 1 / 2 ) / ( 2 ( 1 + S 0 ) ) ) 1 / 2 × exp ( i 2 π l 0 ( S 0 / 2 - S 0 2 / 8 + S 0 3 / 16 - 5 S 0 4 / 128 + 7 S 0 5 / 256 ) / λ ) , if S 0 < 0.1 ,
K 0 = 1 i λ l 0 exp ( i 2 π l 0 / λ ) ,
S 0 = ( p a 1 / N l 0 - q a 0 / N 1 l 0 ) 2 ,
G 0 ( l 0 ) p q = K 0 exp ( i π l 0 S 0 / λ ) .
A k l = s = 1 N 2 G 0 * ( l 1 ) s k G 0 ( l 1 ) s l = K 1 2 exp [ - i π ( k 2 - l 2 ) N 2 L 1 ] · s = 1 N 2 exp [ i 2 π s ( k - l ) N 2 N ( L 1 L 1 ) 1 / 2 ] = K 1 2 exp [ - i π ( k 2 - l 2 ) N 2 L 1 ] exp [ i π ( N 2 + 1 ) ( k - l ) N 2 N ( L 1 L 1 ) 1 / 2 ] · sin { π ( k - l ) / [ N ( L 1 L 1 ) 1 / 2 ] } / sin { π ( k - l ) / [ N 2 ( L 1 L 1 ) 1 / 2 ] } ,
B l k = p = 1 N 1 G 0 ( l 0 ) l p G 0 * ( l 0 ) k p = K 0 2 exp [ - i π ( k 2 - l 2 ) N 2 L 0 ] · p = 1 N 1 exp [ i 2 π p ( k - l ) N N 1 ( L 0 L 0 ) 1 / 2 ] = K 0 2 exp [ - i π ( k 2 - l 2 ) N 2 L 0 ] × exp [ i π ( N 1 + 1 ) ( k - 1 ) N N 1 ( L 0 L 0 ) 1 / 2 ] sin { π ( k - l ) / [ N ( L 0 L 0 ) 1 / 2 ] } × ( sin { π ( k - l ) / [ N N 1 ( L 0 L 0 ) 1 / 2 ] } ) - 1 ,
L 0 = l 0 λ / a 0 2 ,             L 0 = l 0 λ / a 1 2 ,             L 1 = l 1 λ / a 1 2 ,             L 1 = l 1 λ / a 2 2 ,             K 0 2 = 1 / λ l 0 ,             K 1 2 = 1 / λ l 1 .
N ( L 1 L 1 ) 1 / 2 = 1 , N 1 ( L 0 L 0 ) 1 / 2 = 1 , N = N 1 × N 2 , }             or             N 2 ( L 1 L 1 ) 1 / 2 = 1 , N ( L 0 L 0 ) 1 / 2 = 1 , N = N 1 × N 2 . }
A k l B l k = K 0 2 K 1 2 N 1 N 2 δ k , l ,             k , l = 1 , 2 , 3 N ,
H k k = C k k / K 0 2 K 1 2 N 1 N 2 ,             k = 1 , 2 , 3 N ,
C k k = p = 1 N 2 q = 1 N 1 G 0 * ( l 1 ) p k G p q G 0 * ( l 0 ) k q .
l = 1 N G 0 ( l 1 ) k l H l l G 0 ( l 0 ) l p = G k p ,             k = 1 , 2 , 3 N 2 ,             p = 1 , 2 , 3 N 1 .
G ˜ = P 1 H 2 P 2 ,
P 1 = G 0 ( l 2 )             and             P 2 = G 0 ( l 1 ) H 1 ( 0 ) G 0 ( l 0 ) .
A = P 1 + P 1 = G 0 + ( l 2 ) G 0 ( l 2 ) , B = P 2 P 2 + = G 0 ( l 1 ) H 1 ( 0 ) G 0 ( l 0 ) G 0 + ( l 0 ) H 1 ( 0 ) + G 0 + ( l 1 ) C = P 1 + G P 2 + = G 0 + ( l 2 ) G G 0 + ( l 0 ) H 1 ( 0 ) + G 0 + ( l 1 ) .
l = 1 N A k l B l k ( H 2 ) l l = C k k ,             k = 1 , 2 , 3 N .
G ˜ = P 1 H 1 P 2 , P 1 = G 0 ( l 2 ) H 2 ( 1 ) G 0 ( l 1 ) , P 2 = G 0 ( l 0 ) , } A = P 1 + P 1 ,             B = P 2 P 2 + ,             C = P 1 + G P 2 + .
l = 1 N A k l B l k ( H 1 ) l l = C k k ,             k = 1 , 2 , 3 N .
G ˜ = G 0 H n H k + 1 G 0 H k G 0 H k - 1 G 0 H 2 G 0 H 1 G 0 .
G ˜ = P 1 H k P 2 ,
P 1 = G 0 G n G 0 H k + 1 G 0 ,
P 2 = G 0 H k - 1 G 0 H 2 G 0 H 1 G 0 .
D 2 = G - G ˜ 2 = G - P 1 H k P 2 2 = [ T r ( G + G ) + Δ 2 ] / N 2 = [ T r ( G + G ) + T r ( B H k + A H k ) - 2 Re ( C H k + ) ] / N 2 ,
A = P 1 + P 1 ,             B = P 2 P 2 + ,             C = P 1 + G P 2 + ,
Δ 2 = T r ( B H k + A H k ) - 2 Re [ T r ( C H k + ) ] .
Δ 2 = i , j B j i H i i + A i j H j j - 2 Re i C i i H i + .
Δ 2 ρ k k = i , j { δ i , k ρ i j A i j B j i exp [ i ( ϕ i i - ϕ j j ) ] + δ j , k ρ i i A i j B j i · exp [ i ( ϕ i i - ϕ j j ) ] } - 2 Re [ C k k exp ( i ϕ k k ) ] = i ρ i i 2 Re { A k i B i k exp [ i ( ϕ k k - ϕ i i ) ] } - 2 Re [ C k k exp ( i ϕ k k ) ] = 0 ,             k = 1 , 2 , 3 N ,
Re i = 1 N A k i B i k exp ( i ϕ k k ) H i i = Re [ C k k exp ( i ϕ k k ) ] ,             k = 1 , 2 , 3 N .
2 Δ 2 ρ k k 2 = 2 Re ( A k k B k k ) 0 ,
A k k = i ( P 1 * ) i k ( P 1 ) i k = i P 1 i k 2 0 ;
Δ 2 ϕ k k = i ρ k k exp ( i ϕ k k ) [ j ρ j j A k j B j k exp ( - i ϕ j j ) - C k k ] + c . c . = - 2 ρ k k Im { [ j ρ j j A k j B j k exp ( - i ϕ j j ) - C k k ] · exp ( i ϕ k k ) } = 0 ,
Im j A k j B j k H j j exp ( i ϕ k k ) = Im [ C k k exp ( i ϕ k k ) ] ,             k = 1 , 2 , 3 N .
2 Δ 2 ϕ k k 2 = - ρ k k exp ( i ϕ k k ) [ j ρ j j A k j B j k exp ( - i ϕ j j ) - C k k ] + c . c . + 2 ρ k k 2 A k k B k k = 2 ρ k k 2 A k k B k k 0.
j = 1 N A k j B j k H j j = C k k ,             k = 1 , 2 , 3 N ,

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