Abstract

We analyze the bulk elastic transformation of volume holograms as a general approach for three-dimensional pupil engineering. The physical relationship between transformation and the resulting point spread function is discussed by deriving the corresponding analytical expressions. For affine transformations, an analytical solution is directly possible. However, for nonaffine transformations, the analytical solution is not straightforward and we must turn to quasi-analytical solutions using the approximation of the stationary phase. Transformational volume holography offers richer design flexibility and real-time adjustment capabilities for imaging systems.

© 2014 Optical Society of America

1. INTRODUCTION

Volume holograms (VHs), given their three-dimensional (3D) nature, are inherently shift-variant and are able to provide richer opportunities for point spread function (PSF) design compared to conventional two-dimensional pupils. VHs have been widely utilized in various imaging applications as 3D pupils at the pupil planes of imaging systems [15]. However, since generally a VH is recorded by interference of two mutually coherent beams, the degrees of freedom in specifying the PSF are limited by those of the spatial-light modulators (SLMs) used to shape the two interfering wavefronts. To achieve more flexibility, we propose “transformational volume holography,” an analysis and design procedure for engineering the PSF by deforming the exterior of a hologram.

Previously, we have investigated the exterior deformations of the VH pupils using a superposition of point indenters [69]. However, not all PSFs are achievable by multiple point indenters. Here, we propose bulk elastic deformations besides point load, including compression, shearing, bending, twisting, etc. Through transformation, the system performance can in principle be tuned to fit more design criteria such as spectral composition of the PSF, anisotropic behavior, and so forth. This general approach is called transformational volume holography. It allows for real-time adjustment of an optical system, e.g., a microscope, for different imaging purposes. One can simply change the elastic deformations applied on the VHs to achieve various PSFs without having to replace the pupils. Here, we address only the forward problem of computing the PSF given the elastic deformation. However, the inverse problem, where given a desired PSF, we seek the deformation that produces it, is much more interesting but much more difficult to solve; hence, we have postponed it for future work.

Numerically, it is straightforward to compute the final PSF given a certain type of transformation. However, these computations require 3D integrations of highly oscillatory integrands. Hence, excessive sampling is required, especially in 3D. This poses unattainable demands on the CPU and extensive memory costs, while still allowing for the danger of numerical instabilities. Furthermore, a physically intuitive relationship between the transformation of the VH pupils and the resulting PSFs is lost. In this paper, we show how to find quasi-analytical expressions for the PSFs. In Section 2, we present the general architecture of volume holographic imaging systems. In Section 3, analysis of coordinate transformations as a result of elastic deformations is performed. In Section 4, we focus on the analytical solution of the PSFs under affine transformations. In Section 5, nonaffine transformations are investigated and the approximation of the stationary phase is utilized.

2. VOLUME HOLOGRAPHIC IMAGING SYSTEMS

In this paper, without loss of generality, we use the transmissive geometry of the volume holographic imaging system shown in Fig. 1. The hologram is recorded by two plane waves originating from point sources located at xs and xf. After recording, the hologram is probed by another plane wave, which is exactly the same as the reference beam used for recording (i.e., xp=xf). In this case, the hologram is perfectly Bragg-matched and the final optical field collected by the detector can be calculated as [2,3]

q(x)=dxdzEp(x,z)ϵ(x,z)s(x,z)·exp(i2πxxλf2)exp[i2π(1x22f22)zλ],
where Ep(x,z) is the probing field; ϵ(x,z) is the permittivity distribution of the recorded hologram; s(x,z)=rect(x/Lx)·rect(z/L) is the hologram support function (assuming a rectangular shape); f1 and f2 are the focal lengths of lenses before and after the VH, respectively; Lx and L are the lengths of the VH along the x and z axes, respectively; and λ is the wavelength. Note that contributions from y components are ignored because we assumed invariance along the y axis in most of the following calculations with the exception of twist in Section 5.C. We are able to analytically perform the integral of Eq. (1), which gives
q(x)=Lx·L·sinc[Lxλ(xsf1+xf2)]sinc[L2λ(xs2f12x2f22)].
It is composed of two “sinc” terms. Physically, the first sinc term corresponds to the finite lateral aperture of the hologram, and the second sinc term is a result of the non-negligible thickness of the hologram. The second term only appears when the pupil’s thickness should be considered (Bragg regime).

 

Fig. 1. System architecture for VH (top) recording and (bottom) probing processes.

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More generally, Eq. (1) can be written as

q(x,y)=dxEp(x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·x],
where we denote (x,y,z) as a vector x, including the previously neglected y component. It is actually a 3D Fourier transform [2].

3. TRANSFORMATION ANALYSIS

Bulk elastic deformations can be expressed as coordinate transforms [10,11]. Originally, the coordinates centered at the hologram are (x,y,z). After the deformations,

x(2)=fx(2)(x,y,z),y(2)=fy(2)(x,y,z),z(2)=fz(2)(x,y,z).
The linearized transformation matrix is
T=(fx(2)xfx(2)yfx(2)zfy(2)xfy(2)yfy(2)zfz(2)xfz(2)yfz(2)z).

To include this coordinate transformation in the VH analysis, we use the coordinates after the transformation for integration, as shown in Fig. 2. That is to say, the coordinates used for the probe beam and the Fourier transform should be transformed. In addition, the area (or volume) of each integration grid changes. Therefore, as a compensation, the Jacobian of the transformation matrix should be applied on dx, since mathematically, the Jacobian denotes the scaling of the grid area (or volume). This approach yields

q(x,y)=dx|T|Ep(T·x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·(T·x)].

 

Fig. 2. Analysis approach for including bulk transformations in volume holographic imaging systems. Left and right figures show coordinates used for calculating the integrals without and with the transformation, respectively.

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Alternatively, the same coordinates without transformation can be used for integration. The coordinates of permittivity distributions and hologram support functions would then be mapped back to the original coordinates; and due to the change of the hologram support function, the integration boundaries would be modified accordingly. Numerically, these two methods do not make much difference. However, the second method makes it difficult to find an analytical solution.

Note that we assume all transformations are small enough so that only material (permittivity or refractive index) redistribution is considered. For large transformations, anisotropy [11] becomes non-negligible and should be included. However, in transformational volume holography, a small transformation assumption suffices for most cases of interest because most large transformations result in significant Bragg-mismatch with the incoming probe beam so that the diffracted field becomes minimal.

4. AFFINE TRANSFORMATIONS

We first discuss some common affine transformations where exact analytical solutions are possible. Note that in this section, all transformations, as well as the VH itself, are assumed to be invariant along the y axis; thus, we only calculate along the xz plane. Without loss of generality, for all the numerical examples and discussions below, the following parameters are used for the volume holographic imaging system: wavelength λ=632nm, angle of the signal beam θs=8°, angle of the reference beam θf=8°, and size of the hologram Lx=3mm, L=0.3mm.

A. Hologram Shrinkage

First, we consider uniform hologram shrinkage [1215] and assume that the shrinkage occurs only along the z axis [see Fig. 3(a)]. The transformation can be expressed as

x(2)=x,z(2)=(1δ)z,
where δ is the compression ratio. The corresponding transformation matrix is
T=(1001δ)
with |T|=1δ. Substitute this into Eq. (6) and we can derive an analytical equation for the final field as
q(x)=Lx·sinc[Lxλ(xsf1+xf2)]·(1δ)L·sinc[L2λ(xs2f12x2f22)].
This result matches with the results presented in [6] using a perturbation theory approach. Note that, interestingly, the resulting PSF is simply a reduction of the intensity when compared with the undeformed case. This makes sense since shrinkage along z does not change the grating vector of the hologram, which is along the x direction. The hologram is still perfectly Bragg-matched. The lowered intensity is due to the reduced hologram area as a result of shrinkage. An example of the final PSF is demonstrated in Fig. 3(b), which is a scaled sinc function. The two sinc-function curves are shown separately as well in Figs. 3(c) and 3(d). Note that we are plotting the intensity, which is proportional to |q(x)|2.

 

Fig. 3. Shrinkage of the VH along (a) z and (e) x directions and the resulting PSFs when the compression ratio is (b) δ=0.1 and (f) δ=0.02, respectively. (c), (d) and (g), (h) Separation of two sinc functions in Eq. (9) and Eq. (11), which result in (b) and (f), respectively.

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We then consider hologram shrinkage only along the x axis [see Fig. 3(c)]. Similar to the previous case, now the transformation matrix becomes

T=(1δ001)
with |T|=1δ. The final analytical equation of the output field is
q(x)=(1δ)Lx·sinc[Lxλ((1+δ)xsf1+(1δ)xf2)]·L·sinc[L2λ(xs2f12x2f22)].
In this case, the hologram is no longer Bragg-matched. The shrinkage does not contribute to the second “sinc” term because hologram thickness is unchanged. Shrinkage along x changes the lateral size as well as the grating vector of the hologram, thus δ shows up in the first “sinc” term. An example of the resulting PSF is shown in Fig. 3(d). The mainlobe is shifted because the shrinkage increases the grating vector of the hologram along x so that in order to make the best Bragg match, the diffracted beam should be at a different angle. The two sinc-function curves are shown separately as well in Figs. 3(g) and 3(h).

B. Elastic Affine Deformations

1. Axial Loading

For axial loading (compression) along the z axis, which tends to extend the hologram along the x axis,

T=(1+νδ001δ),
where ν is the Poisson’s ratio. We obtain
q(x)=(1+νδ)Lx·(1δ)L·sinc[Lxλ((1νδ)xsf1+(1+νδ)xf2)]·sinc[L2λ(xs2f12x2f22)].
This result is very similar to a combination of the two cases of shrinkage described above.

2. Rotation

For a hologram that is rotated counterclockwise by θ,

T=(cosθsinθsinθcosθ);
and
q(x)=Lx·L·sinc[Lxλ(2xsf1+cosθ(xf2xsf1)sinθ(xs22f12x22f22))]·sinc[Lλ(sinθ(xf2xsf1)+cosθ(xs22f12x22f22))].
Rotation mixes the lateral and axial “sinc” terms, which is an indication that rotation partially changes the hologram’s lateral structures into the axial direction, and vice versa. Investigating the rotation’s influence on the resulting PSF can be equivalently understood as the result of rotating the probe plane wave.

3. Shearing

For shearing of the hologram along the x direction,

T=(1α01),
where α is defined as the shearing ratio; and
q(x)=Lx·L·sinc[Lxλ(xf2+xsf1)]·sinc[Lλ((xs22f12x22f22)α(xsf1xf2))].
It is observed that the expression inside the first “sinc” term also appears in the second term with a coefficient of α. This is because shearing mixes part of the contributions from the x axis into the z direction. The same analysis applies to a shearing along the z axis, and the results show a mixture of the quadratic term from the axial thickness into the lateral “sinc” term. Detailed calculations are straightforward and not duplicated here.

5. NON-AFFINE TRANSFORMATIONS

A. Bending

For the first example of nonaffine deformations, we consider bending as shown in Fig. 4(a). Bending curves the hologram with a radius of curvature equal to the bending radius R [16]. Note that by the convention used here, a bending curved to the left [see Fig. 4(a)] results in a negative R since the radius extends to the left of the hologram. Radius R is positive when the bending curves to the right. We define the “bending ratio” γ=L/(2R) as a convenient measure of the amount of bending exerted on the hologram.

 

Fig. 4. (a) Bending of the VH. (b) Resulting PSFs at different bending ratios.

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The transformation matrix of bending deformation can be expressed as

|T|=(RzRcosx|R||R|Rsinx|R|Rz|R|sinx|R|cosx|R|),
where |T|=(zR)/R. The PSF at the detector is the result of the following double integral:
q(x)=dxdzrect(xLx)rect(zL)(zRR)·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)(|R|RzRsinx|R|)]·exp[i2πλ(xs22f12x22f22)·(R+(zR)cosxR)].
It can be observed that both integration variables x and z appear in the same exponential terms, which makes a direct integral solution impossible to obtain. Examples of resulting PSFs are illustrated in Fig. 4(b). It can be seen that for larger bending ratios, the PSF widens and flattens, and its peak also decreases.

The integral along z can be analytically performed without any approximation, and this reduces Eq. (19) into a single integral along x so that

q(x)=dxrect(xLx)[sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2]exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
where
ζ=1λ(xf2xsf1)|R|Rsinx|R|+1λ(xs22f12x22f22)cosxR.
Equation (20) can no longer be reduced analytically. Next, we show how the stationary phase method can be used as an approximation to find an analytical solution.

B. Stationary Phase Method

The stationary phase method [1719] in the one-dimensional case is an approximation procedure for evaluating the following integrals:

I=Cf(x)exp[iΛϕ(x)]dx,
where Λ is a large constant parameter (Λ1), C is certain domain of integration, ϕ(x) is a fast-varying function over most of the range C, and f(x) is a slowly varying function compared with f(x). For most of the integral regions, ϕ(x) is rapidly varying therefore the integral result I is approximated as zero over these ranges. Main contributions are from some critical points where this assumption fails. One significant contribution is from the point of the stationary phase x0, where ϕ(x0)=0. At x=x0, ϕ(x) is no longer fast-oscillating and the contribution can be calculated by expanding ϕ(x) in a Taylor series up to the second-order derivative and substituting back into Eq. (22) [17,18]:
I=exp[iΛϕ(x0)]1Λ2π|ϕ(x0)|·exp[isign(ϕ(x0))·π4]f(x0).
Many other types of critical points will be discussed later. Final I is a sum of the contributions from all critical points along the integral domain.

The stationary phase has been widely used in electromagnetic scattering, diffraction, and radiation problems as a standard approach to solve the diffraction integrals [2022]. In this section, we propose that a similar approximation can be used to facilitate the integral of calculating PSFs of VH imaging systems under nonaffine deformations and help reach a quasi-analytical solution.

1. Integration Boundaries and Poles

We first note that for the integral of Eq. (20), the corresponding f(x) is a summation of three terms:

sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2.
It is beneficial to separate them and write Eq. (20) as a sum of three integrals. We refer to these three integrals as Part 1, Part 2, and Part 3. The following discussion focuses on Part 1, while Part 2 and Part 3 can be analyzed similarly. The comparison of contributions from all three components will also be illustrated later. Furthermore, the term sin(πLζ) is not slowly varying but it can be written as [exp(iπLζ)exp(iπLζ)]/(2i). By comparisons with the standard expression of the stationary phase method shown in Eq. (22), Part 1 can be rewritten as
f(x)=12iπζ,
exp[iΛϕ(x)]=[exp(iπLζ)exp(iπLζ)]·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
where Λ is defined here as 2π/λ, which satisfies Λ1.

Now, we are able to solve Part 1. We first realize that the term ζ expressed in Eq. (21) can be zero within the integral range resulting in a pole (one type of critical point) denoted as xp. Besides, rect(x/Lx) introduces two integration boundaries at a=Lx/2 and b=Lx/2. At these boundaries, oscillations from exponential terms stop abruptly; thus, the contributions from these points should be considered since the fast-varying expression does not cancel out with these sharp changes. Considering these two types of critical points (poles and integration boundaries) together, the final result is [17]

I=ejΛϕ(a)ϕ(a)jΛf(a)ejΛϕ(b)ϕ(b)jΛf(b){+icπexp[iΛϕ(xp)]ifϕ(xp)>0icπexp[iΛϕ(xp)]ifϕ(xp)<0,
where c=1/{[1/f(x)]|xp}.

2. Close Stationary Points

We plotted f(x) and exp[iΛϕ(x)] for the integral at different detector positions in Fig. 5. At x=3.2×104m [Figs. 5(c) and 5(d)], it can be observed that two stationary points appear within the integral range where ϕ(x)=0. Furthermore, these two stationary points are close to each other (within a few cycles of exponential oscillations), and they are also both close enough to the pole of f(x). That is to say, the contribution from each critical point should not be added independently; instead, all critical points need to be combined in order to find the total contribution. For detailed quantitative analysis on how close two critical points should be so that their individual contributions become dependent, see [21].

 

Fig. 5. Expressions of (left) f(x) and (right) real(exp[iΛϕ(x))]) for the one-dimensional integral of Eq. (20) at detector positions of (a), (b) x=0, (c), (d) x=3.2×104m, and (e), (f) x=3.1×104m. Note that only part of the integral range is shown. The bending ratio is γ=0.030 in these calculations.

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Now, we have in total five critical points that include one pole, two stationary points, and two integration boundaries. The following identical relationship can be used here to eliminate one integration boundary:

abf(x)exp[iΛϕ(x)]dx=af(x)exp[iΛϕ(x)]dxbf(x)exp[iΛϕ(x)]dx,
resulting in four critical points.

According to [17], the combined contributions from these four critical points can be expressed as

I=exp(iΛϕ0){2πif(xp)Afin(Λ23ξ,Λ13β,Λ13q)+πA0Λ13Φ(Λ23ξ,Λ13q)πB0iΛ23Φ(Λ23ξ,Λ13q)},
where Afin(p,β,q) is the incomplete Airy–Fresnel integral, Φ(p,q) is the incomplete Airy function, Φ(p,q) is the incomplete Airy function’s derivative, ξ=[(3/4)·(ϕ(x1)ϕ(x2))]2/3 is a representation of two stationary points at x1 and x2, β is a solution of ϕ(xp)=ϕ0+β3/3ξβ, which is an indication of the pole at xp, q is a solution of ϕ(a)=ϕ0+q3/3ξq, which represents the position of the integral boundary at x=a, and
A0=ξ1/4[f(x2)x2xp2|ϕ(x2)|+f(x1)x1xp2|ϕ(x1)|]2βf(xp)ξβ2,
B0=ξ1/4[f(x2)x2xp2|ϕ(x2)|f(x1)x1xp2|ϕ(x1)|]2f(xp)ξβ2.

3. Close Quasi-Stationary Points

In other positions, e.g., x=3.1×104m [see Figs. 5(e) and 5(f)], the first derivative of ϕ(x) does not go to zero, which means that there are no stationary points. However, in this case, another type of critical point, the quasi-stationary point, becomes important when these two points are close to each other. Quasi-stationary points are virtual stationary points at “imaginary” positions xq1 and xq2 where ϕ(x)=0. For example, if ϕ(x)=x3+3x, it has quasi-stationary points at xq1=1i and xq2=1i.

In this case, we have four critical points, one pole, two quasi-stationary points, and one integration boundary. The total contributions can be expressed as [17]

I=exp(iΛϕ0)·{2πif(x0)Afin[Λ23(2ϕ)13α,Λ13β,Λ13q]+2π(2ϕ)13f0Λ13Φ[Λ23(2ϕ)13α,Λ13q]2πi(2ϕ)23f1Λ23Φ[Λ23(2ϕ)13α,Λ13q]},
where α=min(ϕ(x)) is an indication of two quasi-stationary points, f0=f(0), and f1=f(0).

After applying Eqs. (29) and (32), the resulting PSF is shown in Fig. 6(b). It can be clearly observed that the analytical result matches that of the full numerical solution. It is interesting to see that the contributions only from the poles [Eq. (27)] outline the envelope of the PSF [see Fig. 6(a)].

 

Fig. 6. PSF at the detector for a bent hologram calculated using (a) Eq. (27) (including the contributions only from poles) and (b) the stationary phase approximation combining four critical points, which is compared with the full numerical solution.

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Similar analyses can be applied to Part 2 and Part 3 of Eq. (24). The total PSF is plotted in Fig. 7. We notice that the major contribution to the final PSF is from Part 1. This is because we assume a small bending so that |R|/L1 and |R·ζ|1 [see Eq. (20)]. Calculation of Part 1 suffices if the required accuracy is not high.

 

Fig. 7. PSFs calculated from Parts 1–3 and combined values.

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C. Twisting

As another example of nonaffine deformation, we consider twisting. Twisting, which is different from all cases discussed above, does not have an invariant axis therefore all three coordinate axes should be considered. In this section, we consider twisting deformation along the optical axis (z axis). Instead of a rectangular-shaped VH, it is more convenient for the analysis to use a VH shaped as a cylinder (with the cylinder axis coincident with both the optical axis of propagation and the direction of the torque vector). A cylindrical hologram also allows for easy application of uniform twisting torque on the hologram.

For twisting, the rotating angles at different z planes are [16]

θ(z)=zL/2θm,
where θm is the maximum twisting angle on each side. Typical PSFs at different θm values are plotted in Fig. 8. It can be observed that with increasing θm, the PSF extends along the y axis with its peak reduced. This can be intuitively explained as shown in Fig. 9. The probe beam was diffracted by platelets located at different z positions with grating vectors rotated at different angles, each contributing to a spot at different positions of the detector. Adding coherently the contributions from all platelets will result in the extended PSF.

 

Fig. 8. PSFs for twisting at different maximum twisting angles: (a) θm=0° (without deformation), (b) θm=0.5°, and (c) θm=1°. Note that in order to show sidelobes clearly, q(x,y) is plotted instead of the actual intensity, which is proportional to |q(x,y)|2.

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Fig. 9. Intuitive explanation of the PSF shape after twisting.

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In order to derive an analytical solution, we start with the transformation matrix of twisting,

T=(cos(2zLθm)sin(2zLθm)0sin(2zLθm)cos(2zLθm)0001),
where |T|=1, which indicates that twisting does not change the volume of each integral voxel. According to Eq. (6), the resulting PSF is
q(x,y)=dxdydzcirc(x2+y2Rc)rect(zL)·exp[j2πxs(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[jπxs2zλf12]exp[j2π2xsxλf1]·exp[j2πx(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[j2πy(sin(2zLθm)x+cos(2zLθm)y)λf1]·exp[j2π(x2+y2)z2f22λ],
where Rc is the radius of the cylinder. Integration along x and y can be derived exactly, reducing into
q(x,y)=dzrect(zL)πRc2jinc(Rcu2+v2)·exp[iπxs2zλf12]exp[i2πx2+y22f22zλ],
where
u=1λ[xscos(2zLθm)f1+2xsf1+xcos(2zLθm)f2+ysin(2zLθm)f2],
v=1λ[+xssin(2zLθm)f1xsin(2zLθm)f2+ycos(2zLθm)f2],
jinc(x)=J1(2πx)πx,
where J1 is the Bessel function of the first kind [23].

Without deformation, i.e., θm=0°, the PSF can be calculated as

q(x,y)=πRc2·jinc[Rcλ(xsf1+xf2)2+(yf2)2]·L·sinc[Lλ(xs22f12x2+y22f22)].
This matches with the PSF result shown in Fig. 8(a). However, with twisting, because the integral variable z exists inside the jinc function, an exact analytical solution for Eq. (36) is not possible. A similar procedure for incorporating the stationary phase approximation like the one used in the bending transformation above should be used.

First of all, u2+v2 can be rewritten as

u2+v2=1λ(xsf1+xf2)2+(yf2)2+1λ2xsyf1f2(xsf1+xf2)2+(yf2)22zLθm,
assuming that (2zθm/L)1. To apply the stationary phase, the jinc function can be approximated as [24]
jinc(x){cos(πx)if|x|<0.32681π|πx|3cos(|2πx|3π4)if|x|0.3268.

With Eqs. (41) and (42), the original 1D integral [Eq. (36)] has been reduced and expressed in the standard form of the stationary phase method [Eq. (22)]. Using the same analysis discussed in Section 5.B, a quasi-analytical solution can be derived. The resulting PSF at the detector is illustrated in Fig. 10, together with the difference for the result from the full numerical approach. The analytical result is in agreement with the numerical result.

 

Fig. 10. (a) PSF calculated using stationary phase approximation, and (b) its difference to the result calculated using full numerical approach (Fig. 8(b)). Maximum twisting angle is θm=0.5°.

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6. CONCLUSION

In conclusion, we proposed a quasi-analytical method for calculating the PSF in transformational volume holography under bulk elastic deformations. For affine transformations, analytical equations of the final PSFs can be directly derived. For nonaffine transformations, we showed how to use the stationary phase to derive quasi-analytical expressions for the PSFs. Transformational volume holography provides more design flexibility for imaging systems. Additionally, the analytical solution not only reduces the computing costs, but it also enhances the physical intuition between the deformation and resulting PSF. The approach proposed here is general and can be applied to other types of bulk transformations or their superpositions.

ACKNOWLEDGMENTS

We are grateful to Zhengyun Zhang for useful discussions. Funding support was from Singapore’s National Research Foundation through the Singapore–MIT Alliance for Research and Technology (SMART) Centre.

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7. H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

8. H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

9. H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).

10. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef]  

11. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]  

12. G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent 5,978,112 (November 2, 1999).

13. J. T. Gallo and C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994). [CrossRef]  

14. P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).

15. D. H. R. Vilkomerson and D. Bostwick, “Some effects of emulsion shrinkage on a holograms image space,” Appl. Opt. 6, 1270–1272 (1967). [CrossRef]  

16. R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).

17. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

18. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).

19. J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982). [CrossRef]  

20. V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

21. O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001). [CrossRef]  

22. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).

23. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

References

  • View by:
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  • |

  1. H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).
  2. G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
    [CrossRef]
  3. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
    [CrossRef]
  4. Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
    [CrossRef]
  5. H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
    [CrossRef]
  6. K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
    [CrossRef]
  7. H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.
  8. H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.
  9. H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).
  10. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  11. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef]
  12. G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).
  13. J. T. Gallo and C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994).
    [CrossRef]
  14. P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).
  15. D. H. R. Vilkomerson and D. Bostwick, “Some effects of emulsion shrinkage on a holograms image space,” Appl. Opt. 6, 1270–1272 (1967).
    [CrossRef]
  16. R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).
  17. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).
  18. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).
  19. J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
    [CrossRef]
  20. V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).
  21. O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
    [CrossRef]
  22. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).
  23. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

2013 (1)

2011 (1)

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

2006 (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

2005 (1)

2004 (1)

2001 (1)

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

1999 (1)

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

1995 (1)

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

1994 (1)

1982 (1)

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

1967 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Barbastathis, G.

H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
[CrossRef]

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
[CrossRef]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
[CrossRef]

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

Battey, D. E.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Borovikov, V. A.

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

Bostwick, D.

Brady, D. J.

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

Cátedra, M. F.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Cheng, H.

H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).

Conde, O. M.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Cooke, J. C.

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

Cuingnet, T.

Gallo, J. T.

Gao, H.

H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
[CrossRef]

H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Hariharan, P.

P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).

Hibbeler, R. C.

R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).

Kamm, R. D.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Kinber, B. Y.

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

Li, Z.

Liu, W.

Luo, Y.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Mok, F.

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

Oh, S. B.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Owen, H.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Pelletier, M. J.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Pérez, J.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Psaltis, D.

K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
[CrossRef]

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Shih, T.

Sinha, A.

Slater, J. B.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Stuart, J. S.

Sun, W.

Tian, K.

Verber, C. M.

Vilkomerson, D. H. R.

Watson, J. M.

Zervantonakis, I. K.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

IMA J. Appl. Math. (1)

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

J. Biomed. Opt. (1)

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

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

Opt. Express (1)

Proc. IEEE (1)

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

Proc. SPIE (1)

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Other (12)

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).

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Figures (10)

Fig. 1.
Fig. 1.

System architecture for VH (top) recording and (bottom) probing processes.

Fig. 2.
Fig. 2.

Analysis approach for including bulk transformations in volume holographic imaging systems. Left and right figures show coordinates used for calculating the integrals without and with the transformation, respectively.

Fig. 3.
Fig. 3.

Shrinkage of the VH along (a) z and (e) x directions and the resulting PSFs when the compression ratio is (b) δ=0.1 and (f) δ=0.02, respectively. (c), (d) and (g), (h) Separation of two sinc functions in Eq. (9) and Eq. (11), which result in (b) and (f), respectively.

Fig. 4.
Fig. 4.

(a) Bending of the VH. (b) Resulting PSFs at different bending ratios.

Fig. 5.
Fig. 5.

Expressions of (left) f(x) and (right) real(exp[iΛϕ(x))]) for the one-dimensional integral of Eq. (20) at detector positions of (a), (b) x=0, (c), (d) x=3.2×104m, and (e), (f) x=3.1×104m. Note that only part of the integral range is shown. The bending ratio is γ=0.030 in these calculations.

Fig. 6.
Fig. 6.

PSF at the detector for a bent hologram calculated using (a) Eq. (27) (including the contributions only from poles) and (b) the stationary phase approximation combining four critical points, which is compared with the full numerical solution.

Fig. 7.
Fig. 7.

PSFs calculated from Parts 1–3 and combined values.

Fig. 8.
Fig. 8.

PSFs for twisting at different maximum twisting angles: (a) θm=0° (without deformation), (b) θm=0.5°, and (c) θm=1°. Note that in order to show sidelobes clearly, q(x,y) is plotted instead of the actual intensity, which is proportional to |q(x,y)|2.

Fig. 9.
Fig. 9.

Intuitive explanation of the PSF shape after twisting.

Fig. 10.
Fig. 10.

(a) PSF calculated using stationary phase approximation, and (b) its difference to the result calculated using full numerical approach (Fig. 8(b)). Maximum twisting angle is θm=0.5°.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

q(x)=dxdzEp(x,z)ϵ(x,z)s(x,z)·exp(i2πxxλf2)exp[i2π(1x22f22)zλ],
q(x)=Lx·L·sinc[Lxλ(xsf1+xf2)]sinc[L2λ(xs2f12x2f22)].
q(x,y)=dxEp(x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·x],
x(2)=fx(2)(x,y,z),y(2)=fy(2)(x,y,z),z(2)=fz(2)(x,y,z).
T=(fx(2)xfx(2)yfx(2)zfy(2)xfy(2)yfy(2)zfz(2)xfz(2)yfz(2)z).
q(x,y)=dx|T|Ep(T·x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·(T·x)].
x(2)=x,z(2)=(1δ)z,
T=(1001δ)
q(x)=Lx·sinc[Lxλ(xsf1+xf2)]·(1δ)L·sinc[L2λ(xs2f12x2f22)].
T=(1δ001)
q(x)=(1δ)Lx·sinc[Lxλ((1+δ)xsf1+(1δ)xf2)]·L·sinc[L2λ(xs2f12x2f22)].
T=(1+νδ001δ),
q(x)=(1+νδ)Lx·(1δ)L·sinc[Lxλ((1νδ)xsf1+(1+νδ)xf2)]·sinc[L2λ(xs2f12x2f22)].
T=(cosθsinθsinθcosθ);
q(x)=Lx·L·sinc[Lxλ(2xsf1+cosθ(xf2xsf1)sinθ(xs22f12x22f22))]·sinc[Lλ(sinθ(xf2xsf1)+cosθ(xs22f12x22f22))].
T=(1α01),
q(x)=Lx·L·sinc[Lxλ(xf2+xsf1)]·sinc[Lλ((xs22f12x22f22)α(xsf1xf2))].
|T|=(RzRcosx|R||R|Rsinx|R|Rz|R|sinx|R|cosx|R|),
q(x)=dxdzrect(xLx)rect(zL)(zRR)·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)(|R|RzRsinx|R|)]·exp[i2πλ(xs22f12x22f22)·(R+(zR)cosxR)].
q(x)=dxrect(xLx)[sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2]exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
ζ=1λ(xf2xsf1)|R|Rsinx|R|+1λ(xs22f12x22f22)cosxR.
I=Cf(x)exp[iΛϕ(x)]dx,
I=exp[iΛϕ(x0)]1Λ2π|ϕ(x0)|·exp[isign(ϕ(x0))·π4]f(x0).
sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2.
f(x)=12iπζ,
exp[iΛϕ(x)]=[exp(iπLζ)exp(iπLζ)]·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
I=ejΛϕ(a)ϕ(a)jΛf(a)ejΛϕ(b)ϕ(b)jΛf(b){+icπexp[iΛϕ(xp)]ifϕ(xp)>0icπexp[iΛϕ(xp)]ifϕ(xp)<0,
abf(x)exp[iΛϕ(x)]dx=af(x)exp[iΛϕ(x)]dxbf(x)exp[iΛϕ(x)]dx,
I=exp(iΛϕ0){2πif(xp)Afin(Λ23ξ,Λ13β,Λ13q)+πA0Λ13Φ(Λ23ξ,Λ13q)πB0iΛ23Φ(Λ23ξ,Λ13q)},
A0=ξ1/4[f(x2)x2xp2|ϕ(x2)|+f(x1)x1xp2|ϕ(x1)|]2βf(xp)ξβ2,
B0=ξ1/4[f(x2)x2xp2|ϕ(x2)|f(x1)x1xp2|ϕ(x1)|]2f(xp)ξβ2.
I=exp(iΛϕ0)·{2πif(x0)Afin[Λ23(2ϕ)13α,Λ13β,Λ13q]+2π(2ϕ)13f0Λ13Φ[Λ23(2ϕ)13α,Λ13q]2πi(2ϕ)23f1Λ23Φ[Λ23(2ϕ)13α,Λ13q]},
θ(z)=zL/2θm,
T=(cos(2zLθm)sin(2zLθm)0sin(2zLθm)cos(2zLθm)0001),
q(x,y)=dxdydzcirc(x2+y2Rc)rect(zL)·exp[j2πxs(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[jπxs2zλf12]exp[j2π2xsxλf1]·exp[j2πx(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[j2πy(sin(2zLθm)x+cos(2zLθm)y)λf1]·exp[j2π(x2+y2)z2f22λ],
q(x,y)=dzrect(zL)πRc2jinc(Rcu2+v2)·exp[iπxs2zλf12]exp[i2πx2+y22f22zλ],
u=1λ[xscos(2zLθm)f1+2xsf1+xcos(2zLθm)f2+ysin(2zLθm)f2],
v=1λ[+xssin(2zLθm)f1xsin(2zLθm)f2+ycos(2zLθm)f2],
jinc(x)=J1(2πx)πx,
q(x,y)=πRc2·jinc[Rcλ(xsf1+xf2)2+(yf2)2]·L·sinc[Lλ(xs22f12x2+y22f22)].
u2+v2=1λ(xsf1+xf2)2+(yf2)2+1λ2xsyf1f2(xsf1+xf2)2+(yf2)22zLθm,
jinc(x){cos(πx)if|x|<0.32681π|πx|3cos(|2πx|3π4)if|x|0.3268.

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