Abstract

We relate local cell gap variations to visible defects in LCD panels using psychophysical methods of human eye perception of intensity variations. Our analysis is applicable to general shape of cell gap variation for any LCD mode. We use our method in an explicit example to determine the visible cell gap variation threshold for TN LCD panels.

© 2007 Optical Society of America

1. Introduction

Liquid crystal displays consist of a thin layer of liquid crystal molecules sandwiched between electrodes, polarizers, substrate glass and other components. The thickness of the liquid crystal layer is usually only a few microns and must be controlled to high accuracy for proper operation of the display and maintaining uniform output intensity. Cell gap variations that alter the thickness of the liquid crystal layer can result in intensity changes across the screen and might appear as visible defects. In this paper, we use psychophysical methods of human eye perception of intensity variation to study the conditions under which these intensity variations can become visible to human eye and appear as screen defects.

In section 2, we introduce the psychophysical methods of human eye perception of intensity variation. We lay out an explicit and general formalism and apply our methods to the case of periodic stimuli in section 3. We show that in the case of periodic stimuli, our formalism is consistent with and presents a justification for the well-known Campbell-Robson experiment on human-eye contrast perception. We establish the connection between the cell gap variation in a general LCD panel and the screen intensity variation in section 4. We then apply the methods of sections 2 and 3 to determine what kind of cell gap variation would result in visible defects on the screen. We next work out the explicit example of a TN LCD panel to determine the visible cell gap variation threshold for TN LCD panels.

2. Human eye and visibility of intensity variation

The human eye is a powerful yet imperfect optical system. Any visual stimulus passes through cornea, lens, and other optical elements in the eye and is finally detected by optic nerves, each element degrading the quality of the image. For example, the perceived image of a very narrow line spreads out into a thicker line quantified by a line spread function (LSF). In this section, we use the human eye contrast perception model presented in Ref. [1] to study the visibility of screen intensity variations. The results of this section are used in subsequent sections to investigate the relationship between local cell gap variations in LCD panels and LCD screen defects.

In this paper, we only consider local intensity variations in one dimension whose coordinate we label as x. The intensity variation is characterized by the real function I(x) and is a visual stimulus signal that passes through the human eye optical system. The human eye optical system is assumed to have a linear response to this signal [13] and can be characterized by a LSF Λ(x). The perceived image Ĩ(x) can be calculated using LSF according to

Ĩ=ʃ+dyI(y)Λ(xy).

The sensitivity function is usually defined as the Fourier transform of the LSF

s(u)=2ʃ+dxcos(2πux)Λ(x),

where we have assumed that Λ(x) and s(u) are both even and real functions and reduced the expression into a cosine transform. One can also easily show under these assumptions that

Λ(x)=2ʃ0+ducos(2πux)s(u).

We note that in this article, we always work in the positive spatial frequency domain. The reason for this will become clear later when we give an interpretation of the sensitivity function based on the experiment of Campbell and Robson [1, 2] on visible threshold of periodic stimuli. It is also interesting to write Eq. (1) in the frequency domain.

Ĩ(x)=2Re(ʃ0+duIu(u)s(u)e2πiux),

where Iu(u) is the Fourier transform of I(x).

We define the perceived local contrast of a visual stimulus as the difference between the maximum and minimum of the perceived local intensity variation divided by twice the average perceived background intensity level. This definition is motivated by Weber’s law which states that the change in a stimulus that will be just noticeable is a constant proportion of the original stimulus. For most cases where the perceived intensity variation is negligible compared to the perceived background intensity or the variation is periodic, the average perceived background intensity level is the same as the average of the maximum and minimum of the perceived local intensity variation. We can write

PC=ĨmaxĨmin2Ĩ0,

where Ĩ0 is the average perceived background intensity level. We next postulate that the perceived contrast is the determining factor for the visibility of a visual stimulus. We assume that the visible threshold of the perceived contrast is a fixed value PCth above which the visual stimuli become visible, independent of the shape and form of the stimulus [13].

In the next section, we apply the methods explained above and the concept of perceived contrast threshold to study periodic intensity variations of different spatial frequencies. Such gratings have been used in Campbell-Robson experiments [2] and we will see that the formalism we have presented in this section lead to a natural interpretation of the results of those experiments.

3. Periodic stimuli and Campbell-Robson experiment

In order to show how we use the formalism in section 2 to draw conclusions on the visibility of stimulus, we consider the simplest yet the most important case of a periodic stimulus with periodicity u 1

I(x)=a1cos(2πu1x)+I0,

where I 0 is the constant background intensity and

C=a1I0

is the contrast. Eq. (4) can be used to evaluate the perceived intensity

Ĩ(x)=a1s(u1)cos(2πu1x)+I0s(0),

where I 0 s(0) = Ĩ0 is the perceived background intensity level. The perceived contrast is calculated using Eq. (5) where we get

PC=a1s(u1)I0s(0).

The contrast visibility threshold condition presented in section 2 can be written as

PCth=a1s(u1)I0s(0)|th.

It is interesting to note that PCth is the same as the visible threshold contrast of the periodic intensity variation in the low frequency (long wavelength) limit i.e.

u0PCth=a1I0=Cth.

Since s(u) and s(0) in this formalism always appear in the form of the ratio s(u)/s(0), we choose s(0) =1/ PCth or equivalently set s(0) = PCth =1. We can then use Eq. (10) to set up an experiment to determine s(u) for different values of u according to the familiar Campbell-Robson form [1, 2]

1s(u1)=a1I0th.

The interpretation of Eq. (12) is that in order to determine the sensitivity function s(u) or its Fourier transform, i.e. line spread function Λ(x), one can use the visible threshold of periodically modulated visual stimuli with different spatial frequencies. The Campbell-Robson [2] experiment is the broadly accepted method of determining the visible contrast threshold as a function of spatial frequency. A periodically varying intensity is created using gratings and the contrast is varied to determine the visible threshold as illustrated in Fig. 1.

 

Fig. 1. Sinusoidal grating with variable contrast and spatial frequency for Campbell-Robson experiment [2, 4].

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In this report, we use the measurements of the sensitivity function reported in Ref. [5]. The contrast sensitivity function that is presented in Ref. [5] is a slowly varying function of retinal illuminance level. The troland (Td) is the unit that is conventionally used for retinal illuminance level and is related to luminance (L in cd/m2 that is proportional to the LCD intensity transmission) and pupil area (Apupil in mm 2) according to

Td=L(cdm2)×Apupil(mm2).

The pupil diameter (dpupil in mm) itself scales with the background luminance and is given by [6]

dpupil=100.85580.000401[log(L)+8.6]3.

The results in Ref. [5] illustrate that there is little change in contrast sensitivity function from 90 to 900 Td retinal illuminance, corresponding to luminance levels of 12 and 264 cd/m2, respectively. We thus adopt the measurements of Ref. [5] for 900 Td and present a best fit in the form [1]

(ν)=K[exp(2παν)exp(2πβν)],

where

K,α,β761.868,0.01182degcycles,0.05897degcycles.

The sensitivity function presented in Eq. (15) is plotted in Fig. 2. The measured contrast threshold in Ref. [5] is reported against the angular frequency (ν in cycles per degree of the observation angle). We must translate the argument of Eq. (15) to spatial frequency. We can use

(πPu180)=s(u),

where P is the observation distance. In other words, we can write

s(u)=K[exp(2πau)exp(2πbu)],

where

ab=πP180αβ.

It is also easy to use Eq. (3) and show [1] that

Λ(x)=Kπ[aa2+x2bb2+x2].
 

Fig. 2. Sensitivity plotted as a function of angular frequency; see Eq. (15).

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We have so far provided a structured yet simple framework that can be used to determine the visible threshold of visual stimuli and can also be easily applied to intensity variation defects on LCD panels. For a more comprehensive study of human response to visual stimuli, we recommend the interested reader to look at the results of Watson [7] and references therein. There are alternative methods of determining the sensitivity function which are different from the standard Campbell-Robson method. For example, Ref. [8] determines the sensitivity function based on the human eye perception of the distortion of complex 2-dimensional pictures. The result is not far from the sensitivity function measured by reference [5] and the method can be potentially useful in ranking the severity of defects on LCD panels.

4. Local contrast stimuli

In the previous sections, we presented a method that can be used to determine whether or not a local intensity variation can be detected by human eye. The intensity variations on the screen of a LCD panel are of great importance since they can be regarded as defect if they can be seen by human eye. In this section, we study the relationship between local cell gap variation in LCD panels and screen defects. We use this relationship and the methods described in the previous sections to come up with practical bounds on the tolerance level for local cell gap variation that can cause visible defects on the screen.

Suppose z 0 is the nominal cell gap of the LCD panel and the local cell gap variation f(x) is parameterized as a function of the surface coordinate x. The cell gap z(x) is therefore given by

z(x)=z0f(x).

We can relate small local variations in cell gap to small changes in intensity transmission using a Taylor expansion

Izx=I0[1+κz0(zz0)]=I0(1κz0f(x)).

where I 0 is the nominal intensity and

κ=zIIz|z=z0.

Such an expansion to linear order is valid since the variation in the size of the cell gap that corresponds to the visible threshold is often much smaller than the cell gap (see Fig. 4). The parameter κ that is the normalized derivative of intensity with respect to cell gap determines the sensitivity of the transmission intensity of a given LCD panel to local cell gap variations and can be calculated based on the design geometry and electro-optical properties of a particular panel.

We can perform a Fourier transformation of the intensity transmission with respect to the x variable and write

τ(u)=I0δ(u)I0κz0ϕ(u),

where φ(u) is the Fourier transformation of f(x). The perceived intensity is calculated as

Ĩ(x)=I0s(0)I0κz0(x),

where

(x)=2Re(ʃ0+duϕ(u)s(u)e2πiux).

The perceived contrast in this case is defined as

PC=ĨmaxĨmin2I0=κ2z0(maxmin).

Applying the visible threshold condition described in the previous section, we arrive at

(maxmin)th=2z0κ.

On the left hand side of Eq. (28), we have an expression that depends on the size and shape of the local cell gap variation causing a visible defect on the LCD panel screen. On the right-hand side, we have an expression that is completely calculable based on the full knowledge of the design geometry and electro-optical properties of the LCD panel.

In order to fully understand the implications of Eq. (28), lets us examine a special case where the local cell gap variation can be approximated by an n-hump raised cosine

fxw=h2(1+cos(πxx0w)),

where h is non-zero only for x∊[−nw, nw] and x 0 = wmod(n +1). We refer to h as the height and to w as the width of the defect. The Fourier transformation of f(x;w) gives us

ϕuw=hsin(2πnuw)2πu(14u2w2),

from which we can calculate the perceived intensity variation

xw=xw,

where

Ωxw=2ʃ0+dusin(2πnuw)cos(2πux)s(u)2πu(14u2w2).

The relevant local maximum (minimum) of this function occurs at x = 0 for n odd (even). The other relevant extremum must be found numerically. We derive

hth=2z0κ[Ωmax(w)Ωmin(w)].

For the special case where the defect is periodic (n→∞), we will have

hth=2z0κs(12w).

So far, we have established specific relationships linking the cell-gap variation and the visibility of screen defects using Eq. (33) and Eq. (34). In the next section, we will present an explicit calculation of κ for a TN panel and use the results of Eq. (33) and Eq. (34) to come up with threshold curves for cell-gap variations that can lead to visible defects on the screen.

5. An explicit model for TN panels

The transmitted intensity of backlight (in the absence of an external voltage) in a normally-white TN-based LCD device is given by [9]

TTmax=1sin2(X)1+u2,

where

X=π21+u2,u=2zΔnλ,

z is the cell gap, Δn is the birefringence of LC material, and λ is the wavelength. We can use Eqs. (23), (35), and (36) to calculate κ.

κ=u02[2sin2(X0)X0sin(2X0)](1+u02)[cos2(X0)+u02],

where u 0 , X 0 are u, X in Eq. (36) evaluated at the nominal cell gap value z 0. We plot κ as a function of the u parameter in Fig. 3. We observe that the value of κ varies between 2 and 0 for the u parameter varying between 0 and 3 . The TN LCD panels are usually designed with a u value somewhat below u = √3 so a value of κ ≥ 0.2 is reasonable.

We can use the results in Eq. (33) and Eq. (34) to plot the visible height threshold of the cell gap variation as a function of the defect width parameter in Fig. 4 for a raised-cosine single-hump and periodic cell gap variation. We select a value of κ = 0.2 and an observation distance of p = 40cm in this example. The fact that the periodic feature happens to have a lower threshold than the single hump variation is related to the shape, the location of the peak of the sensitivity function in Fig. 2, and the observation distance P. It is evident from Eq. (34) that the minimum value for the curve related to the periodic cell gap variation occurs in the same location as the maximum of the sensitivity function in Fig. 2, i. e., 5.4 cycles/deg. This corresponds to a spatial frequency of 0.77 (1/mm) or an approximate defect width at minimum of 0.39 mm which is compatible with Fig. 4. Non-periodic features such as the single hump variation can have their corresponding curve lie below or above the curve corresponding to the periodic features in certain regions depending of the spatial frequency content of the feature. We observe that only a couple percentage local change in the value of the cell gap can lead to a visible defect on the screen of the LCD panel.

 

Fig. 3. κ is plotted as a function of the u parameter for a TN LCD device.

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Fig. 4. Visible height threshold of the cell gap variation (normalized to the cell gap) as a function of the width parameter. The results are plotted for a single hump and periodic features with the shape of a raised cosine.

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6. Summary

In this article, we applied psychophysical methods of human eye contrast perception to study the impact of cell gap variations on the screen intensity uniformity of LCD panels. Our results are rather general and can be applied to any LCD mode with arbitrary parameters. We show how one can conclude whether a particular cell gap variation will cause a visible defect on the screen of an LCD panel. We present an example of a TN LCD panel with cell gap variation in the form of a single hump or periodic raised cosine. For this particular example, we work out the details of the methods presented in this paper and derive the required height threshold for the cell gap variation to make a visible defect on the LCD panel screen.

Acknowledgments

We would like to thank William Wood, Kevin Sparks, Min Shen, Leslie Button, and Rakiba Chowdhury for valuable discussions.

References and links

1. F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969). [PubMed]  

2. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968). [PubMed]  

3. F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981). [CrossRef]   [PubMed]  

4. Izumi Ohzawa, Visual Neuroscience Laboratory, Osaka University, http://ohzawa-lab.bpe.es.osakau. ac.jp/ohzawa-lab/izumi/CSF/A_What_is_CSF.html.

5. F. L. van Ness and M. A. Bouman, “Spatial modulation transfer in the human eye,” J. Opt. Soc. Am. 57,401–406 (1967). [CrossRef]  

6. S. G. DeGroot and J. W. Gebhard, “Pupil size as determined by adapting luminance,” J. Opt. Soc. Am. 42,492–495 (1952). [CrossRef]  

7. A. Watson, “Visual detection of spatial contrast patterns: Evaluation of five simple models,” Opt. Express 6,12–33 (2000). [CrossRef]  

8. J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT–20,525–536 (1974). [CrossRef]  

9. P. Yeh and C. Gu, Optics of liquid crystal displays, (John Wiley & Sons, Inc., 1999).

References

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  1. F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
    [PubMed]
  2. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968).
    [PubMed]
  3. F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981).
    [Crossref] [PubMed]
  4. Izumi Ohzawa, Visual Neuroscience Laboratory, Osaka University, http://ohzawa-lab.bpe.es.osakau. ac.jp/ohzawa-lab/izumi/CSF/A_What_is_CSF.html.
  5. F. L. van Ness and M. A. Bouman, “Spatial modulation transfer in the human eye,” J. Opt. Soc. Am. 57,401–406 (1967).
    [Crossref]
  6. S. G. DeGroot and J. W. Gebhard, “Pupil size as determined by adapting luminance,” J. Opt. Soc. Am. 42,492–495 (1952).
    [Crossref]
  7. A. Watson, “Visual detection of spatial contrast patterns: Evaluation of five simple models,” Opt. Express 6,12–33 (2000).
    [Crossref]
  8. J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT– 20,525–536 (1974).
    [Crossref]
  9. P. Yeh and C. Gu, Optics of liquid crystal displays, (John Wiley & Sons, Inc., 1999).

2000 (1)

1981 (1)

F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981).
[Crossref] [PubMed]

1974 (1)

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT– 20,525–536 (1974).
[Crossref]

1969 (1)

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
[PubMed]

1968 (1)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968).
[PubMed]

1967 (1)

1952 (1)

Blommaert, F. J. J.

F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981).
[Crossref] [PubMed]

Bouman, M. A.

Campbell, F. W.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
[PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968).
[PubMed]

Carpenter, R. H. S.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
[PubMed]

DeGroot, S. G.

Gebhard, J. W.

Gu, C.

P. Yeh and C. Gu, Optics of liquid crystal displays, (John Wiley & Sons, Inc., 1999).

Levinson, J. Z.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
[PubMed]

Mannos, J. L.

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT– 20,525–536 (1974).
[Crossref]

Ohzawa, Izumi

Izumi Ohzawa, Visual Neuroscience Laboratory, Osaka University, http://ohzawa-lab.bpe.es.osakau. ac.jp/ohzawa-lab/izumi/CSF/A_What_is_CSF.html.

Robson, J. G.

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968).
[PubMed]

Roufs, J. A. J.

F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981).
[Crossref] [PubMed]

Sakrison, D. J.

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT– 20,525–536 (1974).
[Crossref]

van Ness, F. L.

Watson, A.

Yeh, P.

P. Yeh and C. Gu, Optics of liquid crystal displays, (John Wiley & Sons, Inc., 1999).

IEEE Trans. Inform. Theory (1)

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inform. Theory IT– 20,525–536 (1974).
[Crossref]

J. Opt. Soc. Am. (2)

J. Physiol. (2)

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson , “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 204,283–298 (1969).
[PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197,551–566 (1968).
[PubMed]

Opt. Express (1)

Vision Res. (1)

F. J. J. Blommaert and J. A. J. Roufs, “The foveal point spread function as a determinant for detail vision,” Vision Res. 21,1223–1233 (1981).
[Crossref] [PubMed]

Other (2)

Izumi Ohzawa, Visual Neuroscience Laboratory, Osaka University, http://ohzawa-lab.bpe.es.osakau. ac.jp/ohzawa-lab/izumi/CSF/A_What_is_CSF.html.

P. Yeh and C. Gu, Optics of liquid crystal displays, (John Wiley & Sons, Inc., 1999).

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Figures (4)

Fig. 1.
Fig. 1.

Sinusoidal grating with variable contrast and spatial frequency for Campbell-Robson experiment [2, 4].

Fig. 2.
Fig. 2.

Sensitivity plotted as a function of angular frequency; see Eq. (15).

Fig. 3.
Fig. 3.

κ is plotted as a function of the u parameter for a TN LCD device.

Fig. 4.
Fig. 4.

Visible height threshold of the cell gap variation (normalized to the cell gap) as a function of the width parameter. The results are plotted for a single hump and periodic features with the shape of a raised cosine.

Equations (37)

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Ĩ = ʃ + dyI ( y ) Λ ( x y ) .
s ( u ) = 2 ʃ + dx cos ( 2 πux ) Λ ( x ) ,
Λ ( x ) = 2 ʃ 0 + du cos ( 2 πux ) s ( u ) .
Ĩ ( x ) = 2 Re ( ʃ 0 + du I u ( u ) s ( u ) e 2 πiux ) ,
PC = Ĩ max Ĩ min 2 Ĩ 0 ,
I ( x ) = a 1 cos ( 2 π u 1 x ) + I 0 ,
C = a 1 I 0
Ĩ ( x ) = a 1 s ( u 1 ) cos ( 2 π u 1 x ) + I 0 s ( 0 ) ,
PC = a 1 s ( u 1 ) I 0 s ( 0 ) .
P C th = a 1 s ( u 1 ) I 0 s ( 0 ) | th .
u 0 P C th = a 1 I 0 = C th .
1 s ( u 1 ) = a 1 I 0 th .
Td = L ( cd m 2 ) × A pupil ( mm 2 ) .
d pupil = 10 0.8558 0.000401 [ log ( L ) + 8.6 ] 3 .
( ν ) = K [ exp ( 2 παν ) exp ( 2 πβν ) ] ,
K , α , β 761.868 , 0.01182 deg cycles , 0.05897 deg cycles .
( πPu 180 ) = s ( u ) ,
s ( u ) = K [ exp ( 2 πau ) exp ( 2 πbu ) ] ,
a b = πP 180 α β .
Λ ( x ) = K π [ a a 2 + x 2 b b 2 + x 2 ] .
z ( x ) = z 0 f ( x ) .
I z x = I 0 [ 1 + κ z 0 ( z z 0 ) ] = I 0 ( 1 κ z 0 f ( x ) ) .
κ = z I I z | z = z 0 .
τ ( u ) = I 0 δ ( u ) I 0 κ z 0 ϕ ( u ) ,
Ĩ ( x ) = I 0 s ( 0 ) I 0 κ z 0 ( x ) ,
( x ) = 2 Re ( ʃ 0 + du ϕ ( u ) s ( u ) e 2 πiux ) .
PC = Ĩ max Ĩ min 2 I 0 = κ 2 z 0 ( max min ) .
( max min ) th = 2 z 0 κ .
f x w = h 2 ( 1 + cos ( π x x 0 w ) ) ,
ϕ u w = h sin ( 2 πnuw ) 2 πu ( 1 4 u 2 w 2 ) ,
x w = x w ,
Ω x w = 2 ʃ 0 + du sin ( 2 πnuw ) cos ( 2 πux ) s ( u ) 2 πu ( 1 4 u 2 w 2 ) .
h th = 2 z 0 κ [ Ω max ( w ) Ω min ( w ) ] .
h th = 2 z 0 κs ( 1 2 w ) .
T T max = 1 sin 2 ( X ) 1 + u 2 ,
X = π 2 1 + u 2 , u = 2 z Δ n λ ,
κ = u 0 2 [ 2 sin 2 ( X 0 ) X 0 sin ( 2 X 0 ) ] ( 1 + u 0 2 ) [ cos 2 ( X 0 ) + u 0 2 ] ,

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