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Multiple Airy beam generation by a digital micro mirror device

Zahra Abedi Kichi and Saeed Ghavami Sabouri

Open Access Open Access

Abstract

The Airy beam is the solution of Maxwell's wave equation and since this equation is linear, a superposition of Airy beams still remains the solution of the wave equation. In this paper, we propose a method for generating multiple Airy beams that includes a desirable number of up to 6 individual Airy beams with desirable acceleration properties. By introducing a decenter into the designed diffractive optical element (DOE) of an Airy beam the problem of generating dual airy beams patterns by an amplitude-based spatial light modulator is solved. By superimposing the designed DOEs of individual Airy beams and scaling them to the proper gray level range, the DOE of the multiple Airy beams is generated. Displaying this DOE on a digital micromirror device, multiple Airy beams are experimentally produced. The experimental studies of these beams show good agreement with the performed simulations.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Airy beams with parabolic accelerating trajectory are interesting solutions of maxwell’s wave equation. Unlike regular laser beams that propagate in a straight line, these accelerating waves follow a curved trajectory and the beam profile shifts laterally in the transverse plane along a parabolic path. In addition, peculiar properties of Airy beams such as self-healing [1], self-accelerating [2] and auto-focusing [3] have attracted more and more attentions. These waves are diffraction-free and remain invariant during propagation. Moreover, due to the self-healing property when defects occur in the beam, the beam reconstructs itself as it propagates. Furthermore, Airy beams are the only non-spreading and diffraction-free solutions of the paraxial wave equation in one transverse dimension [4]. On the other word, their intensity profile remains unchanged during propagation. These unique properties have attracted a great deal of interest and offer a wide range of applications, including dynamic imaging [5], routing surface plasmon polaritons [6], optical communications [7], nonlinear frequency conversion [8] and light-sheet microscopy [9]. Diffraction-free Airy beams have been generated by various methods. For example, an adaptive deformable mirror is used to generate Airy beams with adjustable acceleration [10]. Also, high quality Airy beams with strong deflection have been generated by using continuous cubic phase microplates [11,12]. In addition, integrated phase plates are used as compact optical devices to generate Airy beams [13]. A common method for generating multiple laser beams is to implement diffractive optical elements (DOE) on a spatial light modulator (SLM) [14]. In some applications such as laser machining [15], electron acceleration [16], plasma guidance [17], or microparticles manipulation [18], the propagation of two or more accelerating beams in close proximity to each other improves the performance of the application and provides additional degrees of freedom. For example, in particle manipulation with multiple laser beams, different types and sizes of particles can be studied simultaneously [19].

DOEs have the advantage of being multifunctional in laser beam shaping. Furthermore, SLMs provide the ability to control the shape of the laser beam in real time by dynamically displaying DOEs. For example, by taking advantage of this property of DOEs, Fresnel lenses with multiple focal lengths in both propagation and transverse directions have been fabricated [20,21]. SLMs are divided into two general categories of phase and amplitude modulation based on liquid crystal displays (LCD) [22] and digital micromirror devices (DMD) [23], respectively. Although Airy beams have been generated with phase-only LCDs, these instruments are limited by high cost and complicated fabrication. To solve this problem, amplitude modulation by DMD would be a suitable alternative. Moreover, the fill factor of a DMD can exceed 92.5% while providing a wide dynamic range of gray levels [24]. This means that the micromirrors in a DMD are densely packed as SLM pixels can lead to high resolution in displaying DOEs. However, converting the phase pattern of DOEs to an amplitude pattern can cause problems in the modulated beam. For example, displaying the real part of the cubic phase pattern for converting a Gaussian beam to an Airy beam on a DMD-based SLM, a double Airy beam will be generated. Due to this drawback, the power of the input Gaussian beam is split into two Airy beams [25,26].

In this work, we first optimized the DOE pattern for converting a Gaussian laser beam into a specified Airy beam by a DMD-based SLM. In this way, by adding a decenter to the designed DOE, the double Airy pattern is avoided and the input laser beam is directed into a single Airy beam. Then, according to the linear property of the wave equation, a superposition of Airy beams is proposed as the answer of the wave equation. Accordingly, by superimposing the real part of the cubic phases required to generate the individual Airy beams, laser beams consisting of up to 6 Airy beams with the desired acceleration properties are generated in both simulation and experiment. It is expected that these types of laser beams with their unique properties will be useful for various applications.

2. Theory

In two transverse dimensions the scalar Helmholtz equation can be expressed in the paraxial approximation as follows

$$\nabla _T^2\psi + 2ik\frac{{\partial \psi }}{{\partial z}} = 0.$$
Where ∇2T is denotes the Laplacian in transverse coordinate and z and k are the propagation coordinate and wavenumber, respectively. The Airy function is a solution of Eq. 1 and can be written as
$$\psi (x,y,z) = Ai[a(x - \frac{{{a^3}{z^2}}}{{4{k^2}}})]\exp (\frac{{i{a^3}z}}{{2k}}(x - \frac{{{a^3}{z^2}}}{{6{k^2}}})).$$
Where Ai denotes the Airy function and α is the scaling parameter with unit 1/m. Due to the curved propagation path of the Airy beam, a deflection occurs in the plane perpendicular to the direction of propagation, which can be calculated by [27]
$$D = \sqrt 2 (\frac{{{a^3}{z^2}}}{{4{k^2}}})$$

As can be seen from Eq.3, the deflection of the Airy beam from the straight line is proportional to the square of the propagation length. To generate a particular Airy beam, a cubic phase is required for displaying on an SLM. However, since the DMD belongs to the category of amplitude SLMs, the real part of the cubic phase is selected for modulation of the SLM, which is illuminated by a Gaussian laser beam. Thus, the resulting DOE pattern for generating an Airy beam is given by [28]

$$\varphi (u,v) = \textrm {Re} [\exp (i\frac{{{{(2\pi b)}^3}}}{3}({(u - {u_0})^3} + {(v - {v_0})^3}))].$$

Here u, v, u0, and v0 are Cartesian coordinates in the SLM plane and the corresponding central adjustment parameters, respectively. Moreover, b is the spatial scaling factor in the SLM plane. It will be seen that the performance of the DMD in producing Airy beams can be improved by choosing the proper u0 and v0 and introducing a decenter into the cubic phase pattern. Since the Eq.1 is a linear partial differential equation, a superposition of Airy beams remains the answer to the Helmholtz equation. In other words, a general answer like the following still satisfies the wave equation.

$$\psi (x,y,z) = \sum\limits_j {Ai[{a_j}(x - \frac{{a_j^3{z^2}}}{{4{k^2}}})]\exp (\frac{{ia_j^3z}}{{2k}}(x - \frac{{a_j^3{z^2}}}{{6{k^2}}}))} .$$
Where ψ represents the field amplitude of a multiple Airy beam, while each component in the superposition may have the desired scaling factor αj. As shown in Eq.3, the amount of deflection and thus the path of each component in Eq.5 depends on αj. To generate a multiple Airy beam with SLM, the corresponding DOE pattern can be obtained by a Fourier transform (FT) from Eq.5. Since the FT of the Airy function is a cubic phase distribution, the phase pattern of a multiple Airy beam can be calculated as follows
$$\begin{array}{l} \sum\limits_{j = 1}^{j = N} {\int\limits_{ - \infty }^{ + \infty } {Ai(\frac{{(x - {x_{0j}})}}{{{b_j}}},\frac{{(y - {y_{0j}})}}{{{b_j}}})} \exp ( - 2\pi i(xu + yv))dxdy} = \\ \sum\limits_{j = 1}^{j = N} {b_j^2\exp (\frac{i}{3}{{(2\pi {b_j})}^3}({{(u - {u_{0j}})}^3} + {{(v - {v_{0j}})}^3}))} = \sum\limits_{j = 1}^{j = N} {{\varphi _j}} . \end{array}$$
Where x0j, y0j and u0j, v0j are the decentral coordinates for the jth component of Airy beams superposition and the corresponding DOE, respectively. Therefore, a superposition of two or more Airy beams can be generated with a phase pattern containing a summation of the individual DOE patterns.

It is worth noting that in the simulation and experiment, the beam reflected from the SLM was Fourier transformed through a spherical lens with a phase profile of exp(-π/λf(u2 + v2)). Where f is the focal length of the FT lens. Furthermore, the scaling parameters in the DOE plane, bj and the screen plane, αj are related to gather with αj =1/(bjλf). Therefore, the deflection of each individual Airy beam attending in the superposition Eq.6 can be tailored by adjusting bj. On the other hand, SLMs offer the possibility of integrating multiple functions into one DOE pattern [29]. Accordingly, to design a DOE pattern to produce a multiple Airy beam with the desired specifications, we have proposed a method based on the normalized superposition of the individual real parts of the cubic phase patterns with specified acceleration properties. Therefore, the resulting pattern containing a number of N individual DOE patterns of specified Airy beam can be expressed as

$$\Phi (u,v) = \left[ {\frac{1}{N}\sum\limits_{j = 1}^{j = N} {{\varphi_j}(u,v)} } \right] \times 255,$$
where φj is the DOE pattern of the jth Airy beam and Φ is the resulting multiple Airy DOE pattern. According to Eq.7, the superposition pattern, Φ contains pixels with gray levels from 0 to 255, suitable for display on amplitude based SLM. To perform the simulation of single and multiple Airy beams according to the proposed method, the Fresnel diffraction integral for obtaining the generated beam pattern at the location of the observation screen is solved numerically. The Fresnel diffraction integral is given by
$$\begin{array}{l} E(x,y) = \frac{{\exp (ikz)\exp (i\frac{k}{{2z}}({x^2} + {y^2}))}}{{i\lambda z}} \times \\ \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\Phi (u,v)\exp (i\frac{k}{{2z}}({u^2} + {v^2}))\exp ( - i\frac{{2\pi }}{{\lambda z}}(xu + yv))} } dudv. \end{array}$$

By multiplying the DOE pattern by the phase pattern of the FT lens, the intensity pattern of the Airy beam at a desired distance z can be determined.

3. Experiment and simulation

A schematic diagram of the experimental setup for the generation single and multiple Airy beams by DMD is shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic representation of the generation of single and multiple Airy beams by DMD.

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The DOE patterns are displayed on a DMD-based SLM (Texas Instruments XR-325) with a resolution of 800 × 600 and an active area of 8.4mm×11.2 mm. A He:Ne laser with a wavelength of 632.8 nm is used as the laser source in the experimental setup. A couple of lenses are used to diverges the laser beam to cover the entire area of the DMD chip and the FT lens has a focal length of 30 cm. The intensity profile of the generated Airy beam is recorded by the CCD camera (DMK23U274) with a resolution of 1600×1200 and a pixel size of 4.4µm×4.4µm. Generation of the Airy beam by amplitude modulation SLM needs to relate pure phase to an amplitude pattern. As described previously, we select the real part of the phase pattern for display on the DMD. To investigate the effects of the amplitude pattern DOE on the intensity distribution of the Airy beam, the Fresnel diffraction integral to obtain the generated beam profile at the place of the CCD is solved numerically. The resolution and dimensions used in the simulation are the same as for the DMD and CCD presented. In addition, the intensity profile of the Airy beams is recorded in the laboratory. The simulation and experimental results are shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Effect of the DOE design method for amplitude-based SLM on the intensity profile of the generated Airy beam. For the simulation and the recorded images, the distance to the DMD is z = 30 cm.

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The size of the designed DOEs is 8.4 mm × 8.4 mm to cover the maximum attainable square on the DMD area. As can be seen in the first row of Fig. 2 when comparing the experimental and simulated results, there is an asymmetry in the recorded intensity. The physical reason for this is the rectangular shape of the DMD, which results in a small elongation of the screen width while the designed DOE are displayed on the DMD chip. Moreover, the simulations and the recorded images were performed at a distance of z = 30 cm from the DMD, so the recorded patterns are shifted due to the acceleration properties of the Airy beams. To see more details a selected part of the designed DOE is presented in Fig. 2. As can be seen in the first row of Fig. 2, displaying the real part of the cubic phase on the DMD creates a double pattern of Airy beams is consisted in both the simulation and the experiment. This phenomenon is due to the selection of the real part of the phase and has already been observed in similar experimental setups [25,26,30]. To avoid splitting the input laser beam to the DMD into double Airy beams, half of the DOE pattern around the symmetry axis is eliminated. As can be seen from the second row of Fig. 2, only one Airy beam in generated. However, the noisy pattern in the background of the intensity profile is clearly visible. The disadvantage of this method for eliminating the second airy beam is the omission of half of the DMD pixels attending in the DOE displaying. To overcome to this problem, a quarter of the cubic phase pattern is selected for display on the DMD. This is done by setting u0j and v0j in Eq. (6) equal to half the DMD size (i.e., 4.2 mm) for all individual DOEs. However, according to Eq. (6), this decentering causes a constant displacement for all generated airy beams. This technique not only eliminates the generation of duplicate Airy beams, but also suppresses the aforementioned noisy behavior and converts the input laser power incident on the DMD into one Airy beam.

It is worth mentioning that the quality of the generated Airy beams is lower compared to the Airy beams generated with phase-based SLMs. The physical reason for this is the phase nature of the designed DOE. As described in Eq. (4), the DOE to generate an Airy beam consists of a cubic phase, and to match this phase pattern to the amplitude-based SLM, we used the real part of it. Moreover, the degradation of the quality of the Airy beams when using an amplitude only SLM [23] compared to the Airy beams generated by a phase-based SLM is obvious [31]. The propagation and acceleration behavior of the generated Airy beam based on the display of a quarter of the real part of the cubic phase pattern on the DMD is recorded by varying the distance from the back focal point (BFP) of the FT lens and shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Experimentally recorded intensity of the generated Airy beam by displaying a quarter of the real part of the cubic phase pattern on the DMD. Horizontal and vertical yellow dashed lines cross in the middle of the main lobe of the generated Airy beam at z = 1 cm. The parameter b1 is assumed to be 350 m-1. The z indicates the distance from the BFP of the FT lens in the experimental setup.

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As can be seen in the Fig. 3, the main lobe of the airy beam is moving across the off-diagonal line of the indicated coordinate. As expected from Eq. (3), the deflection of the Airy beam increases at a lower rate when the distance from the BFP of the lens FT is short, and the deflection becomes more pronounced as z increases. To get a better insight into the propagation characteristics of the generated Airy beam, the deflection of the beam from the straight propagation direction is measured and compared with the simulation results and illustrated in Fig. 4.

 figure: Fig. 4.

Fig. 4. Deflection of the Airy beam shown in Fig. 3, simulation and experiment. The scaling parameter in the DMD plane is supposed b1 = 300.

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As evident from the Fig. 4, the experimental results agree well with the simulation. As expected from Eq.3, the value of deflection is proportional to the square of the distance from the BFL of the FT lens. However, there are some deviations from the simulation. This may be due to the non-uniformity of the main lobe of the generated Airy beam and thus errors in determining the center of the main lobe. It is worth mentioning that the extraction of the position of the main lobe of the Airy beam is done by finding the pixel with the maximum intensity in the image captured by the CCD. In addition, the DMD pixels are completely closed in the simulation, but in the real DMD there is a gap between two consecutive micromirrors. Moreover, the residual field curvature perturbations of the laser beam incident on the DMD are the other source of the difference between simulation and experiment.

By producing DOE patterns for N = 2 and N = 3 in Eq.7 and displaying on the DMD double and triple Airy beams are generated and the profile intensity is recorded for different distances from the BFL of the FT lens. The designed DOEs and the recorded intensities are illustrated in Fig. 5.

 figure: Fig. 5.

Fig. 5. The profile intensity of the generated multiple Airy beams by propagation from the BFP of the FT lens for a) N = 2 and b) N = 3 and the generated DOE patterns for c) N = 2 and d) N = 3. The color bar indicates the gray levels. Horizontal and vertical yellow dashed lines cross in the middle of the main lobe of the generated Airy beam at z = 1 cm.

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As can be seen from Fig. 5, the proposed DOE design method successfully generates multiple Airy beams. Examining the intensity profile when propagating from the BFP of the FT lens, the acceleration behavior is evident from the Fig. 5(a) and (b). Also, as can be found in Fig. 5(c) and (d), the DOE pattern becomes more complicated when the number of Airy beams in the superposition of Eq. (7) is increased. Therefore, for efficient displaying of DOE of multiple Airy beams, a high-resolution SLM is preferable. Looking more closely at Fig. 5(a) and (b), we see that the main lobes of the intensity profile of the multiple Airy beams are completely separated after the propagation length of about 12 cm from the BFP of the FT lens. Furthermore, for distance of more than 22 cm a significant fraction of the intensity is concentrated in the main lobes of the double- and triple- Airy beams. This property of the multiple Airy beams makes them suitable for trapping or accelerating of multi particles. It should be noted that the experiment is repeated several times for different sets of scaling parameters bj and the generated intensity pattern is recorded. The separation of the Airy main lobes was evident for all selected parameters. The separation ratio of the Airy beams is strongly dependent on the bj parameters. To consider the acceleration behavior of the generated multiple Airy beams in more detail, the deflection of the main lobes from the primary direction is shown in Fig. 6.

 figure: Fig. 6.

Fig. 6. The deflection of the main lobes of the generated multiple Airy beams for a) N = 2 with b1 = 300 m-1 and b2 = 350 m-1 and b) N = 3 with b1 = 330 m-1, b2 = 350 m-1 and b3 = 370 m-1.

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As can be seen in Fig. 6, each lobe of the multiple Airy beams is individually accelerated according to its scaling factor. At distances greater than 15 cm from the BFP of the FT lens the measured deflation of the center of the main lobes from the primary propagation path agrees with the simulation. However, at near distances there is a discrepancy between simulation and experiment. As described earlier, the main reason for this is the uncertainty in determining the center of the main lobes. As evident from Fig. 5 near the BFP, where the lobes of the airy beam overlap, it is difficult to find the center of each lobe. Since the measurement of the position of the main lobes is based on the search for the pixel with the maximum intensity in the image and the pixel size of the CCD (4.4µm×4.4µm) is small in relation to the area of the main lobes, the non-uniform intensity distribution of the main lobes, especially between z = 1 cm and z = 12 cm, leads to a significant error and deviation from the simulation results.

The presented method can be extended to generate multiple Airy beams with more participating single Airy beams. For z = 32 cm distance from the BFP of the FT lens, where the main lobes of the Airy beams contain most of the intensity and are clearly distinguishable (see Fig. 5(a) and (b)), the profile of the intensity for N = 4 to 6 is illustrated in Fig. 7. As can be seen in Fig. 7, the main lobes of the multiple Airy beams up to N = 6 are fully distinguishable. However, when the number of individual Airy beams is increased, the distribution of intensity within each lobe becomes uneven and stretching occurs. This effect is due to the limited resolution of the DMD and the inability to display the DOE pattern in the experimental set up, especially in the lateral regions where the lines of the DOE pattern become narrow (see Fig. 5(c) and (d)). In addition, both the experiment and the simulation were performed for N = 7 and above, and no distinct main lobes number equal with N were evident and the maximum number distinguishable main Airy lobes was 6. Comparing the simulation and experimental results, we can see that the unequal intensities of the main lobes are predicted and the intensity distributions become more unbalanced as the number of N increases. This problem also occurs in the experiment, but the simulation results are not compatible in terms of the intensity of the main lobes.

 figure: Fig. 7.

Fig. 7. The generated multiple Airy beam at a distance of z = 32 cm from the BFP of FT lens for a) N = 4 with scaling factors b1 = 360 m-1, b2 = 330 m-1, b3 = 300 m-1, b4 = 280 m-1, b) N = 5 with scaling factors b1 = 280 m-1, b2 = 300 m-1, b3 = 330 m-1, b4 = 370 m-1, b4 = 420 m-1 and c) for N = 6 with scaling factors b1 = 480 m-1, b2 = 400 m-1, b3 = 360 m-1, b4 = 340 m-1, b5 = 300 m-1, b6 = 280 m-1. The color bar shows the normalized intensity.

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The reason could be that a perfect DMD with densely packed micromirrors was included in the simulation. In addition, the wavefront of the incident laser beam in the DMD area is assumed to be perfect and flat, which is not consistent with the real experimental setup.

4. Conclusion

In this paper, a method for efficient generation of Airy beams by an amplitude-based SLM is presented. It is shown that adding a specific decenter to the designed DOE eliminates the double Airy beam pattern and directs the input laser power to one Airy beam. By superimposing the individual Airy beam DOE patterns with certain acceleration characteristics, the DOE of the multiple Airy beams is generated. Accordingly, the designed DOE is displayed on a DMD and multiple Airy beams with the desired number of N = 2 to 6 are generated. The experimental and simulation results show a good agreement and indicate that the acceleration behavior of each Airy beams participating in the multiple Airy beams depend on the scaling factor. The generated multiple Airy beams can help to improve the performance of applications such as particles manipulation and acceleration.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Schematic representation of the generation of single and multiple Airy beams by DMD.

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Fig. 2.
Fig. 2. Effect of the DOE design method for amplitude-based SLM on the intensity profile of the generated Airy beam. For the simulation and the recorded images, the distance to the DMD is z = 30 cm.

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Fig. 3.
Fig. 3. Experimentally recorded intensity of the generated Airy beam by displaying a quarter of the real part of the cubic phase pattern on the DMD. Horizontal and vertical yellow dashed lines cross in the middle of the main lobe of the generated Airy beam at z = 1 cm. The parameter b1 is assumed to be 350 m-1. The z indicates the distance from the BFP of the FT lens in the experimental setup.

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Fig. 4.
Fig. 4. Deflection of the Airy beam shown in Fig. 3, simulation and experiment. The scaling parameter in the DMD plane is supposed b1 = 300.

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Fig. 5.
Fig. 5. The profile intensity of the generated multiple Airy beams by propagation from the BFP of the FT lens for a) N = 2 and b) N = 3 and the generated DOE patterns for c) N = 2 and d) N = 3. The color bar indicates the gray levels. Horizontal and vertical yellow dashed lines cross in the middle of the main lobe of the generated Airy beam at z = 1 cm.

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Fig. 6.
Fig. 6. The deflection of the main lobes of the generated multiple Airy beams for a) N = 2 with b1 = 300 m-1 and b2 = 350 m-1 and b) N = 3 with b1 = 330 m-1, b2 = 350 m-1 and b3 = 370 m-1.

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Fig. 7.
Fig. 7. The generated multiple Airy beam at a distance of z = 32 cm from the BFP of FT lens for a) N = 4 with scaling factors b1 = 360 m-1, b2 = 330 m-1, b3 = 300 m-1, b4 = 280 m-1, b) N = 5 with scaling factors b1 = 280 m-1, b2 = 300 m-1, b3 = 330 m-1, b4 = 370 m-1, b4 = 420 m-1 and c) for N = 6 with scaling factors b1 = 480 m-1, b2 = 400 m-1, b3 = 360 m-1, b4 = 340 m-1, b5 = 300 m-1, b6 = 280 m-1. The color bar shows the normalized intensity.

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Equations (8)

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∇ T 2 ψ + 2 i k ∂ ψ ∂ z = 0.
ψ ( x , y , z ) = A i [ a ( x − a 3 z 2 4 k 2 ) ] exp ⁡ ( i a 3 z 2 k ( x − a 3 z 2 6 k 2 ) ) .
D = 2 ( a 3 z 2 4 k 2 )
φ ( u , v ) = Re [ exp ⁡ ( i ( 2 π b ) 3 3 ( ( u − u 0 ) 3 + ( v − v 0 ) 3 ) ) ] .
ψ ( x , y , z ) = ∑ j A i [ a j ( x − a j 3 z 2 4 k 2 ) ] exp ⁡ ( i a j 3 z 2 k ( x − a j 3 z 2 6 k 2 ) ) .
∑ j = 1 j = N ∫ − ∞ + ∞ A i ( ( x − x 0 j ) b j , ( y − y 0 j ) b j ) exp ⁡ ( − 2 π i ( x u + y v ) ) d x d y = ∑ j = 1 j = N b j 2 exp ⁡ ( i 3 ( 2 π b j ) 3 ( ( u − u 0 j ) 3 + ( v − v 0 j ) 3 ) ) = ∑ j = 1 j = N φ j .
Φ ( u , v ) = [ 1 N ∑ j = 1 j = N φ j ( u , v ) ] × 255 ,
E ( x , y ) = exp ⁡ ( i k z ) exp ⁡ ( i k 2 z ( x 2 + y 2 ) ) i λ z × ∫ − ∞ + ∞ ∫ − ∞ + ∞ Φ ( u , v ) exp ⁡ ( i k 2 z ( u 2 + v 2 ) ) exp ⁡ ( − i 2 π λ z ( x u + y v ) ) d u d v .
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