Abstract

Assessment of the performance degradation caused by the mid-spatial frequency (MSF) structure on optical surfaces often relies on a perturbation method that dovetails with the familiar sequence of models based on geometrical and physical optics. In the case of imaging systems, the perturbative step yields estimates of wavefronts in the exit pupil which are, in turn, used to extract performance measures such as MTF, PSF, and Strehl ratio. To date, the validity of that perturbation appears to be poorly understood. We present methods to estimate the errors of this approach and thereby arrive at a rule of thumb for its accuracy: the error is approximately equal to the RMS of the MSF structure at its source multiplied by the square of the ratio between a particular Fresnel zone size and a characteristic length of the MSF structure.

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References

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  1. R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18(2), 182137 (1979).
    [Crossref]
  2. D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
    [Crossref]
  3. J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4825–4835 (2010).
  4. G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE 9525, 95251B (2015).
  5. R. N. Youngsworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198–2209 (2000).
    [Crossref]
  6. P. A. Sturrock, “Perturbation characteristic functions and their application to electron optics,” The Royal Society. 210–1101, 269–289 (1951).
  7. H. A. Buchdahl, “Perturbed characteristic functions, II second-order perturbation,” International Journal of Theoretical Physics. 24, 457–465 (1985).
    [Crossref]
  8. E. Garbusi and W. Osten, “Perturbation method in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A. 262538–2549 (2009).
    [Crossref]
  9. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 1733–55 (1966).
    [Crossref]
  10. H. A. Buchdahl, “Perturbed characteristic functions: application to the linearized gravitational field,” Aust. J. Phys. 32405–410 (1979).
    [Crossref]
  11. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A. 142837–2849 (1997).
    [Crossref]
  12. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9533–537 (1970).
    [Crossref] [PubMed]
  13. G.W. Forbes and B.D. Stone, “Restricted characteristic functions for general optical configurations,” J. Opt. Soc. Am. A 10, 1263–1269, (1993).
    [Crossref]

2015 (1)

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE 9525, 95251B (2015).

2010 (1)

2009 (1)

E. Garbusi and W. Osten, “Perturbation method in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A. 262538–2549 (2009).
[Crossref]

2000 (1)

1997 (1)

B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A. 142837–2849 (1997).
[Crossref]

1993 (1)

1985 (1)

H. A. Buchdahl, “Perturbed characteristic functions, II second-order perturbation,” International Journal of Theoretical Physics. 24, 457–465 (1985).
[Crossref]

1979 (2)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

H. A. Buchdahl, “Perturbed characteristic functions: application to the linearized gravitational field,” Aust. J. Phys. 32405–410 (1979).
[Crossref]

1970 (1)

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 1733–55 (1966).
[Crossref]

1951 (1)

P. A. Sturrock, “Perturbation characteristic functions and their application to electron optics,” The Royal Society. 210–1101, 269–289 (1951).

Aikens, D.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Buchdahl, H. A.

H. A. Buchdahl, “Perturbed characteristic functions, II second-order perturbation,” International Journal of Theoretical Physics. 24, 457–465 (1985).
[Crossref]

H. A. Buchdahl, “Perturbed characteristic functions: application to the linearized gravitational field,” Aust. J. Phys. 32405–410 (1979).
[Crossref]

Dallas, W.

DeGroote, J. E.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Forbes, G. W.

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE 9525, 95251B (2015).

Forbes, G.W.

Garbusi, E.

E. Garbusi and W. Osten, “Perturbation method in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A. 262538–2549 (2009).
[Crossref]

Hopkins, H. H.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 1733–55 (1966).
[Crossref]

Milster, T. D.

Noll, R. J.

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

Osten, W.

E. Garbusi and W. Osten, “Perturbation method in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A. 262538–2549 (2009).
[Crossref]

Rimmer, M.

Stone, B. D.

Stone, B.D.

Sturrock, P. A.

P. A. Sturrock, “Perturbation characteristic functions and their application to electron optics,” The Royal Society. 210–1101, 269–289 (1951).

Tamkin, J. M.

Tiziani, H. J.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 1733–55 (1966).
[Crossref]

Youngsworth, R. N.

Youngworth, R. N.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Appl. Opt. (3)

Aust. J. Phys. (1)

H. A. Buchdahl, “Perturbed characteristic functions: application to the linearized gravitational field,” Aust. J. Phys. 32405–410 (1979).
[Crossref]

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 1733–55 (1966).
[Crossref]

International Journal of Theoretical Physics. (1)

H. A. Buchdahl, “Perturbed characteristic functions, II second-order perturbation,” International Journal of Theoretical Physics. 24, 457–465 (1985).
[Crossref]

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

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

B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A. 142837–2849 (1997).
[Crossref]

E. Garbusi and W. Osten, “Perturbation method in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A. 262538–2549 (2009).
[Crossref]

Opt. Eng. (1)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE (1)

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE 9525, 95251B (2015).

The Royal Society. (1)

P. A. Sturrock, “Perturbation characteristic functions and their application to electron optics,” The Royal Society. 210–1101, 269–289 (1951).

Other (1)

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 NRMSE plots for a sinusoidal MSF phase structure on a collimated beam, as a function of | z z M | / Z, for various values of h and C, with L = 2 cm. The lines indicated in the legend are the NRMSE between the perturbation model and numerically calculated exact fields. These NRMSE are compared with Eq. (22), which is shown as a thick black line.
Fig. 2
Fig. 2 The image space of an imaging system where the origin is placed at the image plane (blue). The image of the MSF phase (green) and the exit pupil (red) are located at zM and zP, respectively. Here, zM < zP < 0 but either or both could be positive. (a) We assume in this work that the lion’s share of the error in the perturbation model can be accounted for by just the step of carrying the MSF phase structure along the nominal rays (gray lines) from zM to zP. Note that ξ is the value of the x intercept at zM. The size of the exit pupil is L, and that of the beam footprint at zM is LM. (b) Definition of r1 as the half-width of the first Fresnel zone at zM, that is, as the height at which two circles, centered at the center of the exit pupil and the image point, respectively, and touching the axis at zM, have a separation of λ/2 in the Fresnel approximation.
Fig. 3
Fig. 3 (a) Cmax as a function of η and zM/zP given by Eq. (32) for L = 20 mm and λ = 632.8 nm. Note that the horizontal axis is scaled such that it is proportional to tan−1(2zM/zP − 1). (b) NRMSE for η = 0.05 as a function of r 1 2 / 2, for various values of C and h. The black line is the estimate from Eq. (29), and is seen to agree well with the numerically calculated values, shown as colored dots, particularly for small values of h.
Fig. 4
Fig. 4 Plots for an imaging system with η = 0.2, L = 20 mm, and a sinusoidal MSF structure. (a) NRMSE as a function of zM/zP for h = π/8 and various values of C given by the legend. The solid curves are the estimate from Eq. (29) and the dots are numerically calculated actual values. The green region indicates ϵφ/3. (b) The same information, plotted against r 1 2 / 2, where the single black line is the estimate. (c/d) show similar information to (a/b), but with a fixed value of C = 20 cycles and various values of h given by the legend. Note that in (c), unlike (a), the estimate is represented by a single black curve for all examples since they all have the same value of C.
Fig. 5
Fig. 5 The normalized RMSE plot, as a function of zM/zP, is shown in (a) for randomly generated MSF structures that possess a power-decay spectra, whose extent over the aperture is L = 2 cm. The solid curves are generated from Eq. (29), and the numerically calculated values are shown as dots whose color corresponds to the corresponding structure, shown in the subplot. Part (b) shows the same information but plotted against r 1 2 / 2, where the single black line is the estimate. The green region indicates ϵφ/3.

Equations (76)

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2 U ( x , z ) + k 2 U ( x , z ) = 0 ,
| Φ | 2 1 = 1 i k 2 Φ .
Φ ( x , z ) = n = 0 Φ n ( x , z ) ( i k ) n .
| Φ 0 | 2 1 = 0 ,
Φ 0 Φ N = 1 2 ( 2 Φ N 1 + n = 1 N 1 Φ n Φ N n ) , N 1 .
x ( ξ , s ) = ξ + s W ( ξ ) , z ( ξ , s ) = z M + s χ ( ξ ) ,
Φ ¯ 0 ( ξ , s ) = W ( ξ ) + s ,
Φ ¯ 1 ( ξ , s ) = ln [ A ( ξ ) χ ( ξ ) Δ ( ξ , s ) ] + i ϕ ( ξ ) ,
Δ ( ξ , s ) ( x , z ) ( ξ , s ) = χ ( ξ ) + s W ( ξ ) χ ( ξ ) .
Φ 2 ¯ ( ξ , s ) = Ω 2 ¯ ( ξ , s ) + 1 2 [ ϕ 2 χ 2 i A 2 ( A 2 ϕ χ 3 ) ] ζ + i ϕ 2 χ 3 ( W χ 2 ) ζ 2 ,
ζ χ 2 ( ξ ) s Δ ( ξ , s ) = z z M 1 + ( z z M ) W ( ξ ) / χ 3 ( ξ ) .
d x = x ξ d ξ = [ 1 + ( z z M ) W ( ξ ) / χ 3 ( ξ ) ] d ξ ,
U P ¯ ( ξ , s ) = U ¯ ( ξ , s ) | ϕ ( ξ ) = 0 exp [ i ϕ ( ξ ) ] = A ( ξ ) χ ( ξ ) Δ ( ξ , s ) exp { i k [ W ( ξ ) + s ] i Ω 2 ¯ ( ξ , s ) k } exp [ i ϕ ( ξ ) ] ,
U ¯ ( ξ , s ) U P ¯ ( ξ , s ) exp { 1 2 k [ i ϕ 2 χ 2 + 1 A 2 ( A 2 ϕ χ 3 ) ] ζ + ϕ 2 k χ 3 ( W χ 3 ) ζ 2 } .
ϵ 2 ( z , z M ; ϕ ) a | U P ¯ [ ξ , ( z z M ) / χ ( ξ ) ] U ¯ [ ξ , ( z z M ) / χ ( ξ ) ] | 2 [ 1 + ( z z M ) W ( ξ ) / χ 3 ( ξ ) ] d ξ a | U ¯ [ ξ , ( z z M ) / χ ( ξ ) ] | 2 [ 1 + ( z z M ) W ( ξ ) / χ 3 ( ξ ) ] d ξ = a A 2 | 1 exp { 1 2 k [ i ϕ χ 3 + 1 A 2 ( A 2 ϕ χ 3 ) ] ζ ( z , ξ ) + O ( ξ 2 ) } | 2 a A 2 d ξ ,
Q A a Q A 2 d ξ a A 2 d ξ ,
ϵ 2 ( z , z M ; ϕ ) 1 4 k 2 { [ 1 A 2 ( A 2 ϕ χ 3 ) ] 2 + ( ϕ 2 χ 3 ) 2 } ζ 2 A .
ϵ 2 ( z , z M ; ϕ ) 1 4 k 2 χ 6 ζ 2 A ϕ 2 + ϕ 4 A .
φ 2 | U ( x , z ) | ϕ = 0 U ( x , z ) | 2 d x | U ( x , z ) | ϕ = 0 | 2 d x = | U ( x , z M ) | ϕ = 0 U ( x , z M ) | 2 d x | U ( x , z M ) | ϕ = 0 | 2 d x = | 1 exp ( i ϕ ) | 2 A = 4 sin 2 ( ϕ / 2 ) A ϕ 2 A ,
ϵ ( z , z M ; ϕ ) | z z M | 2 k ϕ 2 + ϕ 4 1 .
ϕ ( ξ ) = h sin ( 2 π κ ξ ) ,
ϵ ( z , z M ; ϕ ) | z z M | π λ h κ 2 1 2 + 3 h 2 8 | z z M | π λ h κ 2 2 ,
Z 1 π λ κ 2 φ h 1 2 + 3 h 2 8 1 π λ κ 2 ,
W ( ξ ) = z M 1 + ξ 2 z M 2 , A ( ξ ) = | z P | 1 / 2 ( z M 2 + ξ 2 ) 1 / 4 ,
Φ 0 ( x , z P ) = z P 1 + x 2 z P 2 ,
Φ 1 ( x , z P ) = ln [ | z P Φ 0 ( x , z P ) | ] + i π 4 [ sgn ( z P ) sgn ( z M ) ] + i ϕ ( x z M z P ) ,
| ζ | z = z P | = | z P z M z P / z M | = r 1 2 ( z P , z M ) λ ,
r 1 ( z P , z M ) λ | ( z M z P ) z M z P | .
ϵ ( z P , z M ; ϕ ) r 1 2 ( z P , z M ) 4 π χ 6 A ϕ 2 + ϕ 4 A r 1 2 ( z P , z M ) 4 π χ 6 A ϕ 2 A ,
ϵ ( z P , z M ; ϕ ) r 1 2 ( z P , z M ) 2 φ ,
2 π ( ϕ 2 A ϕ 2 A ) 1 / 4 .
χ 6 A = 35 70 η 2 + 56 η 4 16 η 6 35 ( 1 η 2 ) 3 ,
C max 2 2 η L ( 1 η 2 ) 3 π λ | z M / z P z M / z P 1 | 35 35 70 η 2 + 56 η 4 16 η 6 .
ϕ ( ξ ) = m a m exp ( 2 π i κ m ξ ) ,
κ 4 ¯ m | a m | 2 κ m 4 m | a m | 2 .
U ( x , z P ) U P ( x , z P ) exp [ 1 2 k m = 1 M ϕ m ( x z m / z P ) χ 3 ( x z m / z P ) ζ m ] ,
ϵ T 2 m = 1 M ϵ 2 ( z P , z m ; ϕ m ) .
J r ξ = [ x ξ z ξ x s z s ] = [ 1 + s W ( ξ ) s W ( ξ ) W ( ξ ) χ ( ξ ) W ( ξ ) χ ( ξ ) ] = [ 1 + s W ( ξ ) s χ ( ξ ) W ( ξ ) χ ( ξ ) ] .
F [ x ( ξ ) , z ( ξ ) ] = F ¯ ( ξ , s ) = J 1 ξ F ¯ ( ξ , s ) ,
Φ 0 ¯ = ( 0 1 ) J = ( W , χ ) .
Φ 0 ¯ Φ N ¯ = ( 0 1 ) ξ Φ N ¯ = s Φ N ¯ .
Φ 0 Φ 1 = 1 2 2 Φ 0 .
r 2 Φ 0 ¯ = ( J 1 ξ ) Φ 0 ¯ = ( J 1 ξ ) [ ( 0 1 ) J ] = Tr ( J 1 s J ) = s ln ( Δ ) .
s Φ 1 ¯ = 1 2 s ln ( Δ ) ,
Φ 1 ¯ ( ξ , s ) = ln ( A χ Δ ) + i ϕ ( ξ ) .
Φ 0 Φ 2 = 1 2 2 exp ( Φ 1 ) exp ( Φ 1 ) .
Φ 2 ¯ s = 1 2 exp ( Φ 1 ¯ ) 2 exp ( Φ 1 ) ¯ = 1 2 1 A exp ( i ϕ ) Δ χ J i l 1 ξ l { J i j 1 ξ j [ A χ Δ exp ( i ϕ ) ] } ,
Φ 2 ¯ s = a i ϕ 2 A 2 χ ( 1 , W ) ξ ( A 2 χ Δ 2 ) + ϕ 2 i ϕ 2 Δ 2 = a i 2 A 2 χ ( A 2 χ ϕ Δ 2 ) + i χ ϕ Δ 3 + ϕ 2 2 Δ 2 ,
a = 1 2 A Δ χ ( J 1 ξ ) [ J 1 ξ ( A χ Δ ) ] .
0 s d s Δ 2 = s χ Δ , 0 s d s Δ 3 = ( χ + Δ ) s 2 χ 2 Δ 2 .
Φ 2 ¯ ( ξ , s ) = Φ 2 ¯ ( ξ , s ) i 2 A 2 χ ( A 2 ϕ s Δ ) + i χ ϕ ( χ + Δ ) s 2 χ 2 Δ 2 + ϕ 2 s 2 χ Δ = Φ 2 ¯ ( ξ , s ) i 2 A 2 ( A 2 ϕ Δ ) s Δ + i ϕ ( χ + Δ ) s 2 χ Δ 2 + ϕ 2 2 χ s Δ = Φ 2 ¯ ( ξ , s ) + 1 2 [ ϕ 2 χ i A 2 ( A 2 ϕ χ ) ] s Δ + i χ ϕ χ s Δ 2 ( Δ χ W s χ 2 ) + i ϕ 2 χ ( W χ ) s 2 Δ 2 = Φ 2 ¯ ( ξ , s ) + 1 2 [ ϕ 2 χ i A 2 ( A 2 ϕ χ ) + 2 i χ ϕ χ 2 ] s Δ + i ϕ 2 χ [ ( W χ ) 2 χ W χ 2 ] s 2 Δ 2 = Ω 2 ¯ ( ξ , s ) + 1 2 [ ϕ 2 χ 3 i A 2 ( A 2 ϕ χ 3 ) ] ζ + i ϕ 2 χ 2 ( W χ 3 ) ζ 2 ,
T ( p , p ) = f p p + δ 2 n p 2 δ 2 n p 2 δ 2 n ( p ) 2 ,
n λ ( 1 δ m δ i + 1 δ p δ m ) ( m ) 2 = n λ δ p δ i ( δ m δ i ) ( δ p δ m ) ( m ) 2
n λ δ p δ i ( δ m δ i ) ( δ p δ m ) ( m ) 2 = 1 λ n f 2 1 / δ p 1 / δ i ( 1 / δ m 1 / δ i ) ( 1 / δ p 1 / δ m ) ( m ) 2 = n λ δ p δ i ( δ m δ i ) ( δ p δ m ) ( δ m n f m ) 2 = n λ δ p δ i ( δ m δ i ) ( δ p δ m ) m 2 ,
U ( x , z P ) = 1 λ U ˜ ( α ) exp { i k [ α x + 1 α 2 ( z P z M ) ] } d α ,
U ˜ ( α ) = U ( ξ , z M ) exp ( i k α ξ ) d ξ .
U ˜ ( α ) = a ( ξ ) exp [ i k γ ( ξ ) ] d ξ = exp [ i k γ ( ξ 0 ) ] ( a ( ξ 0 + τ ) exp { i k [ γ ( ξ + τ ) γ ( ξ 0 ) γ ( ξ 0 ) τ 2 2 ] } ) × exp [ i k γ ( ξ 0 ) τ 2 2 ] d τ ,
τ 2 n exp ( i k 2 τ 2 β ) d τ = k ( n + 1 / 2 ) 2 π | β | ( 2 n 1 ) ! ! | β | n exp [ i π 4 sgn ( β ) ] ,
U ( x , 0 ; x ) = x | K ^ z P a ^ P K ^ z P z M M ^ M K ^ z M | x ,
U P ( x , 0 ; x ) = x | K ^ z P a ^ P M ^ P K ^ z P | x = x | K ^ z P a ^ P M ^ P K ^ z P z M | x .
ε I 2 ( x ) = N 1 | U P ( x , 0 ; x ) U ( x , 0 , x ) | 2 d x = N 1 x | K ^ z M T ^ I a ^ P 2 T ^ I K ^ z M | x ,
U ( x , 0 ; x ) = x | K ^ z M M ^ M K ^ z M z P a ^ P K ^ z P | x ,
U P ( x , 0 ; x ) = x | K ^ z p M ^ P a ^ P K ^ z P | x = x | K ^ z M K ^ z M z P M ^ P a ^ P K ^ z P | x ,
ϵ II 2 ( x ) = N 1 x | K ^ z P a ^ P T ^ II T ^ II a ^ P K ^ z P | x ,
ϵ I 2 ¯ = 1 N ϵ I 2 d x 1 Tr ( K ^ z M T ^ I a ^ P 2 T ^ I K ^ z M ) = 1 Tr ( a ^ P 2 T ^ I T ^ I ) ,
ϵ II 2 ¯ = 1 N ϵ II 2 d x 1 Tr ( K ^ z M a ^ P T ^ II T ^ II a ^ P K ^ z P ) = 1 Tr ( a ^ P 2 T ^ II T ^ II ) ,
U i ( ξ , z M ) = 1 i λ ( z M z P ) U i ( x , z P ) exp [ i π ( ξ x ) 2 λ ( z M z P ) ] d x .
U ii ( x , z P ) 1 i λ ( z M z P ) U i ( ξ , z M ) [ 1 + i h sin ( 2 π κ ξ ) ] exp [ i π ( x ξ ) 2 λ ( z M z P ) ] d ξ = U i ( z , z P ) + h 2 exp ( i π κ x κ ) [ U i ( x + x κ , z P ) exp ( i 2 π κ x ) U i ( x x κ , z P ) exp ( i 2 π κ x ) ] ,
U P ( x , z P ) U 0 exp ( i π x 2 λ z P ) rect ( x L ) [ 1 + i h sin ( 2 π κ z M z P x ) ] .
ϵ j 2 h 2 4 L | σ = ± σ exp ( σ i 2 π C L x ) [ r σ , j ( x ) exp ( i π C L x κ ) rect ( x L ) ] | 2 d x ,
ϵ I 2 φ 2 4 [ 1 sinc ( 2 π C ) ] sin 2 ( π κ 2 r 1 2 2 ) ,
ϵ II 2 φ 2 2 { 1 + κ 2 r 1 2 C C cos ( π κ 2 r 1 2 ) + 1 + 2 cos ( π κ 2 r 1 2 ) π C sin 2 ( π κ 2 r 1 2 2 ) sin [ 2 π ( C κ 2 r 1 2 ) ] } ,
C Max [ 1 , π κ 2 r 1 2 2 sin 2 ( π κ 2 r 1 2 2 ) ] .
ϵ I 2 ¯ / φ 2 4 sin 2 ( π κ 2 r 1 2 / 2 )
ϵ II 2 ¯ / φ 2 4 sin 2 ( π κ 2 r 1 2 / 2 ) + 2 κ 2 r 1 2 C cos ( π κ 2 r 1 2 . )
ϵ I 2 m | a m   2 | 4 sin 2 ( π κ m 2 r 1 2 2 ) φ 2 Σ m | a m | 2 4 sin 2 ( π κ m 2 r 1 2 / 2 ) Σ m | a m | 2 ,

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