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Unified k-space theory of optical coherence tomography

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Abstract

We present a general theory of optical coherence tomography (OCT), which synthesizes the fundamental concepts and implementations of OCT under a common 3D $k$-space framework. At the heart of this analysis is the Fourier diffraction theorem, which relates the coherent interaction between a sample and plane wave to the Ewald sphere in the 3D $k$-space representation of the sample. While only the axial dimension of OCT is typically analyzed in $k$-space, we show that embracing a fully 3D $k$-space formalism allows explanation of nearly every fundamental physical phenomenon or property of OCT, including contrast mechanism, resolution, dispersion, aberration, limited depth of focus, and speckle. The theory also unifies diffraction tomography, confocal microscopy, point-scanning OCT, line-field OCT, full-field OCT, Bessel beam OCT, transillumination OCT, interferometric synthetic aperture microscopy (ISAM), and optical coherence refraction tomography (OCRT), among others. Our unified theory not only enables clear understanding of existing techniques but also suggests new research directions to continue advancing the field of OCT.

© 2021 Optical Society of America

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Supplementary Material (2)

NameDescription
Code 1       MATLAB scripts for generating Figures 6-10, 12, 14-17 and Visualization 1 in Unified k-space theory of optical coherence tomography.
Visualization 1       This visualization compares the OCT response to a single reflector to that to two reflectors whose separation is swept in time. When there is a single reflector, the filtered sample spectrum is a single frequency (a).

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Figures (18)

Figure 1.
Figure 1. Numerous forms and extensions of OCT have been developed over the years. This figure nonexhaustively shows some examples from the literature, including point-scanning OCT B-scans of human retina [31], full-field OCT en face images of human retina (reprinted with permission from [9]), line-field OCT B-scans of human retina (reprinted with permission from [15]), interferometric synthetic aperture microscopy of human skin (reprinted by permission from Macmillan Publishers Ltd.: Ahmad et al., Nat. Photonics 7, 444–448 (2013) [32]), and optical coherence refraction tomography of mouse vas deferens (reprinted by permission from Macmillan Publishers Ltd.: Zhou et al., Nat. Photonics 13, 794–802 (2019) [29]).
Figure 2.
Figure 2. (a) Conceptual schematic of a typical OCT system, based on a Michelson interferometer. A source with power spectrum $S(k)$ is split into two beams, which are reflected from a reference mirror and a multi-layer sample. The reflected light recombines and forms an interference pattern at the detector. (b) In conventional 1D theory of OCT, the axial and lateral resolutions are treated separately, where the former is determined by the source spectral width while the latter is determined by the beam-scanning optics in point-scanning OCT.
Figure 3.
Figure 3. Validity of the first Born approximation in OCT for thick samples. The scattered fields not only contribute to the detected backscattered signal but also may contribute to forward scattering. ${u_0}$ is the incident field (purple), and ${u_i}$ is a scattered field (red), where the subscript indexes which reflector(s) the field was scattered from. Ideally, if the cumulative RI variation is weak, the incident field is “regenerated” upon interaction with each reflector in the sample. If the RI variation is strong, then the incident beam becomes modified by forward scattering from earlier scattering events, resulting in multiple scattering.
Figure 4.
Figure 4. Incident plane wave, denoted by ${\textbf{k}_{\textbf{illum}}}$ , illuminates a sample that obeys the first Born approximation, which scatters light in potentially every direction (red circle, the Ewald sphere). As in Fig. 3, purple corresponds to incident fields, and red corresponds to scattered fields. Only the field contained within the solid angle covered by the objective lens, $U(x,y)\mathop \leftrightarrow \limits^{\cal F} U({k_x},{k_y})$ , is measured. In $k$ -space, the 2D measured field corresponds to the surface of the Ewald sphere. To obtain the $k$ -space coverage according to the illumination geometry, the origin-centered partial Ewald sphere is translated by subtracting out ${\textbf{k}_{\textbf{illum}}}$ .
Figure 5.
Figure 5. (a) FDT can be used to derive the transfer functions (TFs) and PSFs of holography (single illumination angle), (b) diffraction tomography (DT) in transmission (multi-angle illumination with fixed collection in transmission), (c) DT in reflection (multi-angle illumination with fixed collection in reflection), and (d) FF-OCT (single-angle illumination in reflection with multiple wavelengths).
Figure 6.
Figure 6. Simulated (a) TFs and (b) and (c) PSFs for FF-OCT at ${\lambda _0} = 820\;{\rm nm} $ at a low NA (top row) and high NA (bottom row). The red and blue circles represent the Ewald spheres of the wavenumbers corresponding to the FWHM of the source spectrum. Similarly, the red and blue wedges correspond to the FWHM angular range of the focused Gaussian beam. In the low-NA case (top row), both the TF and PSF are approximately separable into their Gaussian axial and lateral components. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 7.
Figure 7. Example FF-OCT TFs with illumination directions at (a)  ${0^ \circ}$ , (b)  ${30^ \circ}$ , and (c)  ${60^ \circ}$ with respect to the ${-}{k_z}$ direction, for ${\lambda _0} = 820\;{\rm nm} $ , $\Delta \lambda = 300\;{\rm nm} $ , and ${\rm NA} = 0.05$ or 0.5 (first and second rows, respectively). Note that the TF not only shifts position with the illumination angle but also changes shape. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 8.
Figure 8. (a) Comparison of TFs for monochromatic holographic microscopy (Section 4.1), (b) FF-OCT (Section 4.3), (c) monochromatic reflective confocal microscopy (Section 4.6), and (d) point-scanning OCT (Section 4.5), for ${\lambda _0} = 820\;{\rm nm} $ , ${\rm NA} = 0.5$ , and, for the right column, $\Delta \lambda = 300\;{\rm nm} $ . The red curves are half-max contours. Note that confocal microscopy has axial sectioning, evident from the axial extent of its TF. Both confocal microscopy and point-scanning OCT obtain lateral resolution enhancement over their wide-field analogs. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 9.
Figure 9. Same comparison of TFs as in Fig. 8, except at a lower NA of 0.1 (the other parameters are the same, with ${\lambda _0} = 820\;{\rm nm} $ and $\Delta \lambda = 300\;{\rm nm} $ for the right column). In this regime, the (c) confocal microscopy TF has much less axial sectioning compared to the high-NA regime. Further, the (b) FF-OCT and (d) point-scanning OCT TFs are more separable between their axial and lateral components (i.e., better approximated by the product of orthogonal Gaussians). For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 10.
Figure 10. TF of LF-OCT is asymmetric in 3D $k$ -space and follows the curvature of a horn torus. For illustrative purposes, a discrete number of surfaces (5) are shown, corresponding to different wavenumbers. The projections onto the ${k_x}{k_z}$ and ${k_y}{k_z}$ planes correspond to the TFs of point-scanning and FF-OCT, respectively (for a LF-OCT system focusing in the $x$ dimension). The surfaces depicted are the resampling surfaces to obtain depth-invariant resolution in ISAM (Section 6). For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 11.
Figure 11. Comparison of three approaches for doubling the lateral frequency cutoff over (a) FF-OCT by increasing illumination $k$ -vector diversity, thus attaining the TF in the lower right panel of Fig. 8: (b) point-scanning OCT of a focused beam, (c) performing FF-OCT at multiple angles, and (d) FF-OCT with partially spatially coherent illumination via an extended source.
Figure 12.
Figure 12. (a) TF of transillumination OCT, which is similar to that of transmission confocal microscopy and DT. The different rescaled versions of the cross sections correspond to different wavenumbers within the broadband source. (b) TF of multi-angle transillumination OCT, which is obtained by rotating (a) about the ${k_x}$ - or ${k_z}$ -axis. The limiting volume is a sphere with a radius of $2{k_{{\max}}}{\rm NA}$ . For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 13.
Figure 13. Conventional OCT processing produces images with limited depths of focus, because they use 1D Fourier transforms along the wavenumber dimension. However, the measured information corresponds to non-planar manifolds in 3D $k$ -space. For FF-OCT, these manifolds are the Ewald spheres, as depicted in this figure. Thus, to obtain space-invariant lateral resolution, one needs to resample in 3D $k$ -space and perform a 3D Fourier transform.
Figure 14.
Figure 14. Resampling with interferometric synthetic aperture microscopy (ISAM) realizes depth-invariant lateral resolution. (a) Comparison of OCT and ISAM for a single column of equally spaced point scatterers (left two columns) and multiple columns of scatterers (right two columns) at ${\rm NA} = 0.1 $ (top row) and ${\rm NA} = 0.3 $ (bottom row), simulated using ${\lambda _0} = 820\;{\rm nm} $ and $\Delta \lambda = 200\;{\rm nm} $ . Note that, away from the focus, the OCT responses to the beads are curved and the blurred spots do not superimpose incoherently. For simulation details and parameters, please see the code used to generate panel (a) of this figure (Code 1, Ref. [70]). (b) Experimental demonstration of ISAM on sub-resolution scatterers: left, OCT image without dispersion compensation; middle, OCT with dispersion compensation; right, ISAM. Adapted from [26].
Figure 15.
Figure 15. Angle compounding synthesizes incoherence and, therefore, reduces speckle. This figure compares responses of (a) an incoherent model, (b) an coherent model, and (c) an ${180^ \circ}$ angle compounded model to a pair of axially spaced scatterers (separation = ${d_z}$ ). The angle compounded result substantially reduces the coherent modulation artifacts, which would otherwise give rise to speckle. Zoom into the figure to avoid aliasing. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 16.
Figure 16. Speckle reduction via angle compounding over a limited angular range ( ${\pm}{15^ \circ}$ , ${\pm}{30^ \circ}$ , and ${\pm}{60^ \circ}$ ) with isotropic resolutions of (a) 2 µm and (b) 10 µm. For reference, the ${J_0}(2{k_0}{d_z})$ (Bessel) and the $\exp (- d_z^2/4/\sigma _z^2)$ (Gaussian) are plotted. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 17.
Figure 17. Simulation of incoherent averaging of 500 modulated wavefront patterns as a function of separation between the two scatterers for multiple angular ranges: (a)  ${\pm}{2^ \circ}$ , (b)  ${\pm}{30^ \circ}$ , (c)  ${\pm}{60^ \circ}$ , and (d)  ${\pm}{90^ \circ}$ . The gray regions capture 95% of the modulation patterns. The argument of the Bessel and cosine functions is $2{k_0}{d_z}$ . This simulation ignores the Gaussian prefactor. For simulation details and parameters, please see the code used to generate this figure (Code 1, Ref. [70]).
Figure 18.
Figure 18. Optical coherence refraction tomography (OCRT) obtains resolution enhancement and speckle reduction. OCRT uses intensity OCT images, which have (a), (b) anisotropic PSFs and TFs from multiple angles to reconstruct an image with isotropic resolution, (c), (d) limited by the original OCT axial resolution. The two samples shown are cross sections of (e), (f) a mouse femoral artery and (g), (h) a mouse bladder wall, whose (f), (h) OCRT reconstructions have higher resolution and reduced speckle compared to (e), (g) the intensity OCT image. Scale bars, 100 µm. Adapted from [29].

Equations (57)

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E ref ( k ) = E 0 ( k ) 2 exp ( j 2 k z ref ) .
E samp ( k ) = E 0 ( k ) 2 i = 1 N r i exp ( j 2 k z i ) .
I ( k ) = | E ref ( k ) + E samp ( k ) | 2 = S ( k ) 2 ( 1 + i = 1 N r i 2 + i j = 1 N r i r j cos ( 2 k ( z i z j ) ) + i = 1 N r i cos ( j 2 k ( z i z ref ) ) ) ,
F 1 { I x corr ( k ) } ( z ) γ ( z ) i = 1 N r i δ ( z ± 2 ( z i z ref ) ) ,
I ( z ref ) = 0 I x corr ( k ) d k = i = 1 N r i exp ( 2 σ k 2 ( z i z ref ) 2 ) cos ( 2 k 0 ( z i z ref ) ) ,
δ z = 2 ln ( 2 ) σ k = 2 ln ( 2 ) π λ 0 2 Δ λ 0.44 λ 0 2 Δ λ ,
δ xy 0.37 λ 0 N A ,
( 2 + k 0 2 n m 2 ) u ( r ) = V ( r ) u ( r ) ,
V ( r ) = k 0 2 ( n ( r ) 2 n m 2 )
u ( r ) u inc ( r ) + u sc ( r ) .
U ~ ( k x , k y ) V ~ ( ( k x , k y , k 0 2 k x 2 k y 2 ) k illum ) ,
H FFOCT ( k x , k y , k z ) exp ( ( k r 2 k 0 cos ( k θ ) ) 2 8 σ k 2 cos 2 ( k θ ) ) exp ( 2 k θ 2 σ θ 2 ) ,
H FFOCT ( k x , k y , k z ) exp ( ( k z 2 k 0 ) 2 8 σ k 2 ) exp ( 2 k xy 2 σ k xy 2 ) ,
psf z ( z ) exp ( z 2 2 σ z 2 ) exp ( j 2 k 0 z ) ,
σ z = 1 2 σ k δ z FWHM = 2 2 ln ( 2 ) σ z = 2 ln ( 2 ) σ k = 2 ln ( 2 ) π λ 0 2 Δ λ 0.44 λ 0 2 Δ λ ,
psf xy ( x , y ) exp ( x 2 + y 2 2 σ xy 2 ) ,
σ xy = 2 σ k xy = 1 k 0 σ θ δ x y FWHM = 2 2 ln ( 2 ) σ xy = 2 ln ( 2 ) π λ 0 N A 0.44 λ 0 N A ,
H F F O C T ( k x , k y , k z ; k i l l u m ) exp ( ( k r 2 k 0 cos ( k θ ) ) 2 8 σ k 2 cos 2 ( k θ ) ) exp ( 2 k θ , 1 / 2 2 σ θ 2 ) ,
E ( k x , k y ) = E 0 exp ( k x 2 + k y 2 k 0 2 N A 2 ) ,
H O C T s c a n ( k x , k y , k z ) = k i , x 2 + k i , y 2 < k 0 2 E ( k i , x , k i , y ) H F F O C T × ( k x , k y , k z ; k i , x , k i , y , k 0 2 k i , x 2 k i , y 2 ) d k i , x d k i , y ,
E ( k x , k y ) = E 0 exp ( k x 2 k 0 2 N A 2 ) δ ( 0 , k y ) ,
δ ( x x 0 , z z 0 ) F exp ( j ( k x x 0 + k z z 0 ) ) .
H ( z z 0 ) F ( 1 π k z + δ ( k z ) ) exp ( j k z z 0 ) ,
r e c t ( ( z z 0 ) / w ) F s i n c ( w k z ) exp ( j k z z 0 ) .
c i r c ( ( x x 0 ) / w , ( z z 0 ) / w ) F j i n c ( w k x 2 + k z 2 ) exp ( j ( k x x 0 + k z z 0 ) ) ,
exp ( j k z z 0 ) k z exp ( j k z z 0 ) 2 k 0 .
s i n c ( w k z ) exp ( j k z z 0 ) sin ( w k z ) 2 w k 0 exp ( j k z z 0 ) = j 4 w k 0 ( exp ( j k z ( z 0 w ) exp ( j k z ( z 0 + w ) ) .
V ( r , k ) = k 2 k 0 2 V ( r ) exp ( j ϕ ( k ) ) ,
ϕ ( k ) = ϕ ( k 0 ) + d ϕ d k | k 0 ( k k 0 ) + 1 2 d 2 ϕ d k 2 | k 0 ( k k 0 ) 2 + 1 6 d 3 ϕ d k 3 | k 0 ( k k 0 ) 3 + .
n ( k ) = n 0 + C k ,
n g ( k ) = n ( k ) + k d n ( k ) d k = n 0 + 2 C k .
Δ k m = n ( k 2 ) k 2 n ( k 1 ) k 1 ,
Δ k m = Δ k ( n 0 + 2 C k 0 ) = Δ k n g ( k 0 ) ,
exp ( ( k r 2 k 0 cos ( k θ ) ) 2 8 σ k 2 cos 2 ( k θ ) ) = exp ( ( k r 2 2 k z k 0 ) 2 2 σ k 2 ) ,
k F F O C T , z = k r 2 2 k z = k x 2 + k y 2 + k z 2 2 k z
k x = k F F O C T , x , k y = k F F O C T , y , k z = k F F O C T , z ± k F F O C T , z 2 k x 2 k y 2 .
k x = k F F O C T , z k i l l u m , x k 0 , k y = k F F O C T , z k i l l u m , y k 0 , k z = k F F O C T , z k i l l u m , z k 0 ± k F F O C T , z 2 k x 2 k y 2 ,
k x = k O C T , x , k y = k O C T , y , k z = 4 k O C T , z 2 k x 2 k y 2 ,
k x = k L F O C T , x , k y = k L F O C T , y , k z = ( k L F O C T , z + k L F O C T , z 2 k y 2 ) 2 k x 2 ,
O C T ( x , y , z ) = | psf ( x , y , z ) V ( x , y , z ) | 2 | psf ( x , y , z ) | 2 | V ( x , y , z ) | 2 .
I ( x , z ) = | psf ( x d x 2 , z d z 2 ) + psf ( x + d x 2 , z + d z 2 ) | 2 = | psf ( x d x 2 , z d z 2 ) | 2 + | psf ( x + d x 2 , z + d z 2 ) | 2 + 2 | psf ( x d x 2 , z d z 2 ) | | psf ( x + d x 2 , z + d z 2 ) | cos ( 2 k 0 d z ) ,
I ( x , z ) = | n = 1 N r n psf ( x d x n , z d z n ) | 2 = n = 1 N | r n psf ( x d x n , z d z n ) | 2 + N N n m 2 r n r m | p s f ( x d x n , z d z n ) | × | p s f ( x d x m , z d z m ) | cos ( 2 k 0 ( d z n d z m ) ) ,
A ~ ( k ) = H z ( k ) ( cos ( 2 k ( z d z 2 ) ) + cos ( 2 k ( z d z 2 ) ) ) = 2 H z ( k ) cos ( k d z ) cos ( 2 k z ) , H z ( k ) = exp ( ( k z 2 k 0 ) 2 8 σ k 2 ) ,
psf θ ( x , z ) = exp ( ( x cos ( θ ) + z sin ( θ ) ) 2 2 σ x 2 ) exp ( ( x sin ( θ ) z cos ( θ ) ) 2 2 σ z 2 ) × exp ( j 2 k 0 ( x sin ( θ ) z cos ( θ ) ) ,
I θ ( x = 0 , z = 0 ) = | psf θ ( 0 , d z 2 ) + psf θ ( 0 , d z 2 ) | 2 = 2 exp ( d z 2 4 ( cos 2 ( θ ) σ z 2 + sin 2 ( θ ) σ x 2 ) ) ( 1 + cos ( 2 k 0 d z cos ( θ ) ) ) .
S ( d z ) = 1 π π / 2 π / 2 exp ( d z 2 4 ( cos 2 ( θ ) σ z 2 + sin 2 ( θ ) σ x 2 ) ) cos ( 2 k 0 d z cos ( θ ) ) d θ = exp ( d z 2 4 σ z 2 ) 1 π π / 2 π / 2 exp ( d z 2 4 ( 1 σ x 2 1 σ z 2 ) sin 2 ( θ ) ) cos ( 2 k 0 d z cos ( θ ) ) d θ exp ( d z 2 4 σ z 2 ) 1 π π / 2 π / 2 cos ( 2 k 0 d z cos ( θ ) ) d θ = exp ( d z 2 4 σ z 2 ) J 0 ( 2 k 0 d z ) ,
psf ϕ mod ( x , z ) = Θ psf θ ( x , z ) exp ( j ϕ ( θ ) ) d θ ,
I ϕ ( x = 0 , z = 0 ) = | psf ϕ mod ( 0 , d z 2 ) + psf ϕ mod ( 0 , d z 2 ) | 2 = | Θ ( psf θ ( 0 , d z 2 ) + psf θ ( 0 , d z 2 ) ) exp ( j ϕ ( θ ) ) d θ | 2 = Θ Θ ( psf α ( 0 , d z 2 ) + psf α ( 0 , d z 2 ) ) × ( psf β ( 0 , d z 2 ) + psf β ( 0 , d z 2 ) ) exp ( j ( ϕ ( α ) ϕ ( β ) ) ) d α d β 2 exp ( d z 2 4 σ z 2 ) Θ Θ [ cos ( k 0 d z ( cos ( α ) cos ( β ) ) ) + cos ( k 0 d z ( cos ( α ) + cos ( β ) ) ) ] exp ( j ( ϕ ( α ) ϕ ( β ) ) ) d α d β ,
S mod ( d z ) = 2 exp ( d z 2 4 σ z 2 ) Φ Θ Θ cos ( k 0 d z ( cos ( α ) + cos ( β ) ) ) × exp ( j ( ϕ ( α ) ϕ ( β ) ) ) P ( ϕ ) d α d β d ϕ = 2 exp ( d z 2 4 σ z 2 ) Φ Θ Θ [ cos ( k 0 d z cos ( α ) ) cos ( k 0 d z cos ( β ) ) sin ( k 0 d z cos ( α ) ) sin ( k 0 d z cos ( β ) ) ] exp ( j ( ϕ ( α ) ϕ ( β ) ) ) P ( ϕ ) d α d β d ϕ ,
S mod ( d z ) = 2 exp ( d z 2 4 σ z 2 ) Θ Θ [ cos ( k 0 d z cos ( α ) ) cos ( k 0 d z cos ( β ) ) sin ( k 0 d z cos ( α ) ) sin ( k 0 d z cos ( β ) ) ] [ Φ exp ( j ( ϕ ( α ) ϕ ( β ) ) ) d ϕ ] d α d β .
S mod deterministic ( d z ) 2 exp ( d z 2 4 σ z 2 ) Θ Θ [ cos ( k 0 d z cos ( α ) ) cos ( k 0 d z cos ( β ) ) sin ( k 0 d z cos ( α ) ) sin ( k 0 d z cos ( β ) ) ] d α d β = 2 exp ( d z 2 4 σ z 2 ) [ ( Θ cos ( k 0 d z cos ( θ ) ) d θ ) 2 ( Θ sin ( k 0 d z cos ( θ ) ) d θ ) 2 ] 2 exp ( d z 2 4 σ z 2 ) [ J 0 ( k 0 d z ) 2 H 0 ( k 0 d z ) 2 ] ,
S mod stochastic ( d z ) 2 exp ( d z 2 4 σ z 2 ) Θ Θ [ cos ( k 0 d z cos ( α ) ) cos ( k 0 d z cos ( β ) ) sin ( k 0 d z cos ( α ) ) sin ( k 0 d z cos ( β ) ) ] [ Φ δ ( α ϕ ) δ ( β ϕ ) d ϕ ] d α d β = 2 exp ( d z 2 4 σ z 2 ) Φ [ ( Θ cos ( k 0 d z cos ( θ ) ) δ ( θ ϕ ) d θ ) 2 ( Θ sin ( k 0 d z cos ( θ ) ) δ ( θ ϕ ) d θ ) 2 ] d ϕ = 2 exp ( d z 2 4 σ z 2 ) Φ [ cos 2 ( k 0 d z cos ( ϕ ) ) sin 2 ( k 0 d z cos ( ϕ ) ) ] d ϕ = 2 exp ( d z 2 4 σ z 2 ) Φ cos ( 2 k 0 d z cos ( ϕ ) ) d ϕ exp ( d z 2 4 σ z 2 ) J 0 ( 2 k 0 d z ) ,
V ( x , z ) = n = 1 N r n δ ( x x n , z z n ) ,
V ~ ( k x , k z ) = n = 1 N r n exp ( j ( k x x n + k z z n ) ) .
psf lpf ( x , z ) = | psf ( x , z ) | 2 exp ( x 2 σ xy 2 z 2 σ z 2 ) F × exp ( 1 4 ( σ xy 2 k x 2 + σ z 2 k z 2 ) ) H lpf ( k x , k z ) ,
I ~ ( k x , k z ) H lpf ( k x , k z ) × [ n = 1 N r n 2 exp ( j ( k x d x n + k z d z n ) ) + N N n m [ ( d z n d z m ) 2 4 σ z 2 2 r n r m cos ( 2 k 0 ( d z n d z m ) ) × exp ( j 1 2 ( k x ( d x n + d x m ) + k z ( d z n + d z m ) ) ) × exp ( ( d x n d x m ) 2 4 σ xy 2 ( d z n d z m ) 2 4 σ z 2 ) ] ] .
psf OCRT ( x , z ) exp ( x 2 + z 2 σ z 2 ) , H OCRT ( k x , k z ) exp ( σ z 2 4 ( k x 2 + k z 2 ) ) ,

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