Diffraction patterns from opaque planar objects simulated with Maggi-Rubinowicz method and angular spectrum theory

Thibault Vaillant de Guélis; Valery Shcherbakov; Alfons Schwarzenböck

doi:10.1364/OE.27.009372

1. Introduction

The phenomenon known as diffraction has been studied for almost four centuries. Most of these studies are based on the Huygens–Fresnel principle which states that every point of a wavefront may be considered as a source of secondary spherical wavelets, and the wavefront at any later instant may be regarded as the envelope of these wavelets [1]. This principle led to the prediction of the existence of a bright spot at the center of the shadow of an opaque disc: the “Poisson’s spot” [2].

The rigorous theory of diffraction is based on Maxwell’s equations; the effect of radiation incident on a perfectly conducting body may be interpreted in terms of the induced surface currents (see Chap. 11 in [1]). Such techniques can be considered as emission of waves by a specified fixed surface as a whole. At the same time, the diffraction pattern produced by an object with large dimensions and at large distances compared to the wavelength can be computed by the scalar diffraction theory resulting from Green’s theorem applied to the Helmholtz wave equation [2].

The Helmholtz–Kirchhoff diffraction integral, the Fresnel–Kirchhoff formula, the Rayleigh– Sommerfeld formulation, and the angular spectrum theory (AST) can be considered as different formulations of the general scalar diffraction theory [2,3]. The Maggi–Rubinowicz representation of the Helmholtz–Kirchhoff diffraction integral, hereafter the Maggi–Rubinowicz method (MRM), along with the generalization of [4] has the same domain of validity. The key peculiarity of the MRM consists in the fact that it is based on a boundary diffraction approach. The MRM can be adapted to different planar objects. However, the method is quite time consuming and the analytical parameterization of the contour has to be developed for each object.

Fast Fourier transform (FFT) can be used for numerical implementation of the AST. FFT-based algorithms are easy to implement and effective. That is why the AST is extensively employed in different domains, including simulations of diffraction patterns (e.g., see [5,6]). According to our computations, the AST-FFT modeling is several orders of magnitude faster than MRM-based calculations (all other things being the same). At the same time, the AST-FFT has shortcomings inherent in calculations employing 2D-FFT such as artifacts due to aliasing.

The first objective of this paper is to present simulation results obtained with the MRM for opaque planar objects (including tilted rectangles). A second objective is to compare diffraction patterns computed with the MRM and the AST-FFT in order to evaluate performance of a low-pass filter employed along with the AST-FFT calculations. The main novelty of this work is the fact that MRM is particularly well suited for evaluating anti-aliasing procedures applied to AST-FFT calculations. Throughout this work, it is assumed that the dimensions of opaque planar objects and the distance from an object to the corresponding diffraction pattern are much larger than the wavelength.

Section 2 is devoted to analytical formulations of the MRM and simulation results. Diffraction patterns obtained with the AST-FFT are compared to MRM results in Section 3.

2. Maggi–Rubinowicz method

Figure 1(a) schematically illustrates the geometric setup for the diffraction study of a monochromatic plane wave U(Q) of wavelength λ interacting with an opaque planar object. The contour Γ represents the boundary of the object. For simplicity sake, it is assumed that the wave is of unit amplitude. Within the accuracy of the Kirchhoff diffraction theory, the diffracted field U_d(Q) behind the object may be expressed as a superposition of the boundary wave U_b(Q) with the “geometrical wave” U_g(Q):

U_{d} (Q) = U_{g} (Q) + U_{b} (Q),

where

U_{g} (Q) = {\begin{array}{l} U (Q) & if Q \in direct beam \\ 0 & if Q \in geometrical shadow, \end{array}

and

U_{b} (Q) = \frac{1}{4 π} \oint_{Γ} e^{i k \vec{k} \cdot \vec{p} R} \frac{e^{i k S}}{S} \frac{(\vec{s} \land \vec{k}) \cdot \vec{l}}{1 - \vec{s} \cdot \vec{k}} d l .

S is the distance PQ, R is the distance OP,

k = \frac{2 π}{λ}

is the wavenumber, and k⃗, p⃗, s⃗, and l⃗ are the unit vectors shown in Fig. 1(a). Equations (1)–(3) are the Maggi–Rubinowicz representation of the Helmholtz–Kirchhoff diffraction integral [4]. Note that s⃗ has the opposite direction in [4]. Hence, the minus sign before s⃗ in the denominator of Eq. (1). In the numerator, the sign before s⃗ is unchanged because Eq. (3) is written for an opaque object (not for an aperture) and the Babinet’s principle is taken into account.

Fig. 1 (a) Scheme of an incident plane wave perpendicular to an opaque planar object of arbitrary shape producing a diffraction pattern at a distance Z. (b) Notations for opaque disc. (c) Notations for opaque rectangle.

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If the plane wave is perpendicular to the opaque planar object, then k⃗· p⃗ = 0 and (s⃗ Λ k⃗) · l⃗ = (k⃗ Λ l⃗) · s⃗ = n⃗ · s⃗ with n⃗ the inward unit vector normal to the contour Γ. Moreover, if the considered diffraction pattern at the distance Z is parallel to the opaque planar object, thus perpendicular to the plane wave, then $\vec{s} \cdot \vec{k} = \frac{Z}{S}$ . In this case, Eq. (3) can be simplified:

U_{b} (Q) = \frac{1}{4 π} \oint_{Γ} \frac{e^{i k S} \vec{n} \cdot \vec{s}}{S - Z} d l .

For any known shape, it is possible, using Eqs. (2) and (3) or Eqs. (2) and (4), to compute the corresponding diffraction patterns at any distance. Throughout this work, the z-axis is normal to the observation plane. In the following subsections we apply this theory to circular and rectangular planar objects.

2.1. Opaque disc

If the opaque planar object is a disc with radius R (Fig. 1(b)), then $\vec{n} \cdot \vec{s} = \frac{R - r cos α}{S}$ and integrating along the contour (dl = Rdα), Eq. (4) becomes

U_{b} (Q) = \frac{1}{4 π} \int_{0}^{2 π} \frac{e^{i k S} (R^{2} - R r cos α)}{S (S - Z)} d α,

with

S = \sqrt{R^{2} + Z^{2} + r^{2} - 2 R r cos α} .

The angle α in Eqs. (5) and (6) can be defined to start from the direction O′Q rather than from the direction Ox of the object plane due to the fact that the integrand of Eq. (5) is a periodic function with the period 2π. Equations (5) and (6) taken together lead to another form of Eq. (5) by straightforward algebraic operations

U_{b} (Q) = \frac{1}{4 π} \int_{0}^{2 π} e^{i k S} (1 + \frac{Z}{S} + \frac{R r cos α - r^{2}}{S (S - Z)}) d α .

Generally, S depends on α, so do all the terms of the integrand in Eq. (7). If point Q belongs to the axis OO′, r = 0 and the on-axis amplitude behind the disc is expressed as follows:

U_{b} (Q) = \frac{1}{2} e^{i k S} (1 + \frac{Z}{S}),

with

S = \sqrt{R^{2} + Z^{2}}

. Equation (8) is identical to Eq. (9) of the work by Lucke [7] where the on-axis amplitude was obtained using the Fresnel–Kirchhoff diffraction integral.

Figure 2(a) shows the diffraction pattern from an opaque disc for the Fresnel number N_F = R²/(Zλ) = 1. The diffraction pattern is presented in light intensity percentage of the incident light at the planar object. Figure axes ξ_x and ξ_y are x and y coordinates normalized by R. It is seen that the Poisson’s spot, i.e., a bright spot in the center of the disc shadow is well reproduced by the MRM. If we consider the outer limit of a 50 % intensity threshold (orange dashed line) as the edge of the disc shadow, we notice that the disc size is 30 % larger than the real disc object for this specific case when N_F = 1.

Fig. 2 Diffraction pattern (a) from an opaque disc with radius R at distance Z such that the Fresnel number N_F = R²/(Zλ) = 1 and (b) from an opaque rectangle with width 2a = b and length 2b at distance Z such that the Fresnel number N_F = b²/(Zλ) = 1. Due to diffraction, a bright spot —called Poisson’s spot— appears in the center of the disc shadow. The orange dashed line in the disc shadow represents the outer limit of 50 % intensity threshold. The number of bright spots in the rectangle shadow (two spots here) varies with N_F.

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2.2. Opaque rectangle

If the opaque planar object is a rectangle with the width 2a and the length 2b (Fig. 1(c)), then integrating along the contour Γ of Eq. (4) yields:

\begin{array}{l} U_{b} (Q) = \frac{1}{4 π} (\int_{- b}^{b} \frac{e^{i k S} (y_{Q} - y_{P})}{S (S - Z)} d x_{P} \\ \begin{array}{l} + \int_{- a}^{a} \frac{e^{i k S} (x_{P} - x_{Q})}{S (S - Z)} d y_{P} \end{array} \\ \begin{array}{l} + \int_{- b}^{b} \frac{e^{i k S} (y_{P} - y_{Q})}{S (S - Z)} d x_{P} \end{array} \\ \begin{array}{l} + \int_{- a}^{a} \frac{e^{i k S} (x_{Q} - x_{P})}{S (S - Z)} d y_{P}), \end{array} \end{array}

with

S = \sqrt{{(x_{Q} - x_{P})}^{2} + {(y_{Q} - y_{P})}^{2} + Z^{2} .}

where y_P = −a in the first integral, x_P = b in the second integral, y_P = a in the third integral, and x_P = −b in the fourth integral.

We simulated a huge set of diffraction patterns from opaque rectangles of different sizes in a large range of distances Z for the wavelength of 0.783 μm. The patterns were computed for the screen of the size of 1000 × 1000 μm at the observation plane. The shorter and the longer sides of the rectangles were within the ranges 2a > 30λ and 2a < 2b < 400 μm.

In general, a diffraction pattern primarily depends on the Fresnel number N_F = b²/(Zλ). However, it also depends on the aspect ratio (length-to-width ratio). This is why the values of N_F should be considered as approximate estimates.

For N_F about 10 or higher, the patterns are quite close to the shape of the rectangle. For 2 < N_F < 10, the patterns are very complex and a number of small Poisson’s spots can be observed. For N_F < 2, the patterns are less complex and more smoothed. Figure 2(b) shows an example of the diffraction pattern for N_F = 1, i.e. the Fresnel diffraction regime. There are two large Poisson’s spots. Basically, the shape is an oval rather than a rectangle.

For N_F < 0.3, the patterns resemble those computed using the Fraunhofer approximation. It should be mentioned that patterns in the focal plane of a lens, i.e. for Z → ∞, are not considered in this work. The Fraunhofer diffraction regime was only considered for large but limited distances Z, i.e. N_F values were not smaller than 0.03. For example, if the Fraunhofer approximation is used (see [2], Chap. 4.4.1) to simulate diffraction from an opaque square of 100 × 100 μm at the distance of 2.5 cm and the Babinet’s principle is applied to the amplitude, the obtained intensity pattern has much in common with Fig. 3(a) (see Sect. 2.3).

Fig. 3 (a) Intensity diffraction pattern from the 100 × 200 μm opaque rectangle at the distance Z = 2.5 cm with the tilt angle θ = 60°. (b) Difference between (a) and the intensity diffraction pattern of the corresponding opaque square of 100 × 100 μm (see text). (c) As (a) but for the phase distribution. (d) As (b) but for the difference of the phase distributions.

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2.3. Tilted opaque rectangle

Diffraction calculations provide a basis for computer-generated holograms (e.g., see [8] and references therein). Fast algorithms are required for such calculations. That is why, the overwhelming majority of algorithms employ the AST-FFT modeling. Simulations of diffraction patterns from tilted planar objects are not an exception (e.g., see [5]). As mentioned above, the MRM is time consuming. However, it has the following advantages. It is a straightforward matter to develop an analytical parameterization of a tilted planar rectangle. Calculations can be performed on the equidistant sampling grid.

In this section, using the MRM, we treat the single-axis rotation around the y-axis (Fig. 1(c)). The diffraction patterns and the phase distributions from the tilted opaque planar rectangle of 2a × 2b = 100 × 200 μm at the distances of Z = 2.5 cm (N_F = 0.51) and Z = 0.5 cm (N_F = 2.55) are shown as a function of the tilt angle θ. The results for these two case studies can be seen in the video files ( Visualization 1, Visualization 2, Visualization 3, and Visualization 4). When the z-axis is normal to the rectangle, θ = 0°. The orthogonal projection of a 60° tilted rectangle on the observation plane is equal to a square of 100 × 100 μm.

It is instructive to compare diffraction properties of the tilted 100 × 200 μm rectangle when θ = 60° and the corresponding square (100 × 100 μm) when the z-axis is normal to the square. Figures 3 and 4 correspond to the distances Z = 2.5 cm and Z = 0.5 cm, respectively. Figures 3(a) and 4(a) show the intensity patterns and Figs. 3(c) and 4(c) the 2D distributions of phase at the observation plane. The figures were computed for the tilted 100 × 200 μm rectangle when θ = 60°. The respective figures computed for a 100 × 100 μm square are not shown because they look the same. Detailed analysis is needed to reveal differences between diffraction properties of the tilted rectangle and the square. The right-hand figures, i.e. Figs. 3(b), 3(d), 4(b) and 4(d), show the difference Δ = data_rectangle − data_square. For Z = 2.5 cm, the differences between the intensities Δ_I (Fig. 3(b)) and between the phase distributions Δ_P (Fig. 3(d)) are quite small. For Z = 0.5 cm, the differences Δ_I (Fig. 4(b)) and Δ_P (Fig. 4(d)) are well pronounced (different color scales compared to Figs. 3(b) and 3(d)). In our opinion, the differences Δ_I (Fig. 4(b)) between the intensity patterns are sufficient to distinguish the tilted rectangle from the square if the patterns will be used as holograms for reconstructing 3D object information.

Fig. 4 As Fig. 3 at Z = 0.5 cm.

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The important property of diffraction patterns computed with the MRM is their high quality. There are no artifacts such as aliasing. Figures 2–4 were obtained without any smoothing. If some applications need a higher resolution, this can be easily achieved by grid refinement. To complete this Sect. 2.3, we would like to point out that analytical parameterizations of laterally shifted or tilted laterally-shifted rectangles are straightforward as well. Diffraction patterns of the same quality for such planar objects can be obtained without extra effort.

3. Angular spectrum theory application

3.1. AST fundamental relationships

Other than the Maggi–Rubinowicz method, the angular spectrum theory (AST) can be used to compute the diffraction pattern from an illuminated opaque object. We recall the milestones of the AST following the textbooks [2,3]. Suppose that a monochromatic plane wave is propagating in the positive Z direction and the complex field across the plane Z = 0 is represented by U_i(x, y, 0). A diffracting structure is introduced in the plane Z = 0. The amplitude transmittance function t_A(x, y) is the ratio of the transmitted field amplitude U_t(x, y, 0) to the incident field amplitude U_i(x, y, 0) at each position (x, y) in the Z = 0 plane, with:

U_{t} (x, y, 0) = t_{A} (x, y) U_{i} (x, y, 0) .

In this work, the amplitude transmittance function t_A(x, y) is defined as follows:

t_{A} (x, y) = {\begin{array}{l} 1 & if outside the opaque object \\ 0 & if in the opaque object . \end{array}

In the Fourier domain, the angular spectrum of U_t(x, y, 0) is:

A (f_{x}, f_{y}, 0) = \iint_{- \infty}^{+ \infty} U_{t} (x, y, 0) e^{- i 2 π (f_{x} x + f_{y} y)} d x d y .

The angular spectrum of the wave field U(x, y, z) at Z = z is given by a solution of the differential equation which represents the Helmholtz equation in the Fourier domain [3]. The solution can be written as:

A (f_{x}, f_{y}, z) = A (f_{x}, f_{y}, 0) H (f_{x}, f_{y}),

where

H (f_{x}, f_{y}) = e^{i z \sqrt{k^{2} - 4 π^{2} (f_{x}^{2} + f_{y}^{2})}}

under the condition of homogeneous waves

(k^{2} > 4 π^{2} (f_{x}^{2} + f_{y}^{2}))

, which is the case in this study. Applying the inverse Fourier transform gives the resulting wave field at the distance Z = z:

U (x, y, z) = \iint_{- \infty}^{+ \infty} A (f_{x}, f_{y}, z) e^{- i 2 π (f_{x} x + f_{y} y)} d f_{x} d f_{y} .

Note that despite the apparent differences of their approaches, “the angular spectrum approach and the first Rayleigh–Sommerfeld solution yield identical predictions of diffracted fields” (see [2], p. 61). A theoretical demonstration of the interrelationship between the AST and the general Rayleigh–Sommerfeld diffraction formula can be found in the work by Harvey [9].

3.2. Comparison of AST and MRM simulations

Fast Fourier transform (FFT) algorithms [10] allow very fast computations of discrete Fourier transform. In this study, we implement AST-FFT to compute diffraction patterns produced from planar opaque objects. The resulting wave field can be computed by two approaches. The first one is direct: the amplitude transmittance function t_A(x, y) corresponds to an opaque planar object and Eqs. (11)–(15) can be employed. The second approach is based on Babinet’s principle: the resulting wave field U(x, y, z) is computed for the corresponding aperture and the complex field U_i(x, y, z) is subtracted (amplitude and phase), not only the intensity (amplitude squared) (e.g., see [1], Section 8.3). Both approaches give the same results for the spatial distributions of the amplitude and of the phase when the AST-FFT modeling is used.

Figure 5(a) shows the diffraction pattern from the opaque planar rectangle 2a × 2b = 200 × 400 μm at the distance of 2.5 cm (N_F = 2.04) computed with the AST-FFT. Detailed analysis of the pattern (e.g., see the magnified image in the inset) reveals some artifacts, i.e. aliasing noise, appearing as very close horizontal and vertical lines. When looking at the intensity profiles along the y-axis (Fig. 5(b)) and x-axis (Fig. 5(c)), these artifacts correspond to sharp peaks on the intensity profiles (red curves) compared to the smooth profiles from Maggi–Rubinowicz method simulations (blue curves). It is well known that aliasing noise is common to algorithms using the FFT (see e.g, [8,11,12] and references therein). A number of anti-aliasing procedures can be found in the literature. The effectiveness of anti-aliasing procedures has to be evaluated by FFT-free means. In our opinion, the MRM is particularly well suited for such an evaluation.

Fig. 5 (a) Diffraction image of a 200 × 400 μm opaque rectangle at Z = 2.5 cm computed with the angular spectrum theory. (b) Intensity profile along the y-axis of this image (red) compared to the one obtained with the Maggi-Rubinowicz method (blue). (c) As (b) for the x-axis. (d,e,f) As (a,b,c) with low-pass filter. The root mean square deviation (RMSD), expressed in light intensity percentage, between the MRM and the AST images is given for both comparisons.

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It is seen that the aliasing noise belongs to the high-frequency domain in the case of Figs. 5(a)– 5(c). A cut-off low-pass filter, which is a very simple anti-aliasing procedure, has been tested on the AST-FFT patterns. The cut-off frequency (f_c = 0.06 μm⁻¹) was chosen using the root mean square deviation (RMSD) between the MRM and the AST-FFT patterns. When the low-pass filter was applied, the AST-FFT provided the diffraction pattern shown in Fig. 5(d). The pattern now appears smooth (see the magnified image in the inset). Nevertheless, there are smaller differences remaining between the MRM and the filtered AST-FFT data (see the profiles in Figs. 5(e) and 5(f)). Still, it can be seen that the AST-FFT profiles are not excessively smoothed.

4. Conclusions

The simulations performed in this study demonstrate that the Maggi–Rubinowicz method is a flexible tool to compute diffraction patterns from opaque planar objects. The MRM has the same domain of validity as the angular spectrum theory and the first Rayleigh–Sommerfeld solution and can provide valuable data within a large range of Fresnel-number values. An important feature of the MRM is the high quality of simulated diffraction patterns (both intensity and phase 2D distributions). The patterns are artifact-free and the resolution can be increased just by grid refinement. The MRM has two major shortcomings. First, the method is time consuming (according to our estimations, it is about three orders of magnitude slower than the AST). And second, the analytical parameterization of the contour has to be developed for each object. Direct computing of the Rayleigh–Sommerfeld integral should be free of issues inherent in FFT calculations. Compared to such an approach, the MRM has the advantage that a surface integral is replaced by a line integral. Generally, that leads to more efficient calculations.

We adapted the MRM to planar rectangles. It follows from our simulations that the intensity diffraction patterns are quite close to the shape of the rectangle for the Fresnel number N_F = 10 or higher. For 2 < N_F < 10, the patterns are very complex and a number of small Poisson’s spots can be observed. For N_F < 2, the patterns are less complex and more smoothed and their shape is an oval rather than a rectangle. For N_F < 0.3, the patterns resemble those computed using the Fraunhofer approximation.

We compared diffraction patterns from a tilted 100 × 200 μm opaque rectangle when tilt angle θ = 60° and from a 100 × 100 μm opaque square when tilt angle θ = 0°, which is equal to the projection of the rectangle on the observation plane. The patterns (both intensity and phase 2D distributions) look very similar. Still, the detailed analysis revealed some differences, which should be sufficient to distinguish between the tilted rectangle and the non-tilted square if the patterns might be used as holograms for reconstructing 3D object information.

We simulated the diffraction pattern from the opaque planar rectangle (N_F = 2.04) using AST-FFT and compared it with MRM results. This comparison demonstrates that the Maggi– Rubinowicz method is particularly well suited for evaluating anti-aliasing procedures applied to AST-FFT calculations. This fact is the major novelty of this work.

Funding

Centre national d’études spatiales (CNES).

References

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5. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. 20, 1755–1762 (2003). [CrossRef]

6. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009). [CrossRef] [PubMed]

7. R. L. Lucke, “Rayleigh–Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006). [CrossRef]

8. T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016). [CrossRef]

9. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979). [CrossRef]

10. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965). [CrossRef]

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Name	Description
Visualization 1	Intensity diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 0.5 cm from the object as a function of tilt angle.
Visualization 2	Phase diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 0.5 cm from the object as a function of tilt angle.
Visualization 3	Intensity diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 2.5 cm from the object as a function of tilt angle.
Visualization 4	Phase diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 2.5 cm from the object as a function of tilt angle.

Diffraction patterns from opaque planar objects simulated with Maggi-Rubinowicz method and angular spectrum theory

Abstract

1. Introduction

2. Maggi–Rubinowicz method

2.1. Opaque disc

2.2. Opaque rectangle

2.3. Tilted opaque rectangle

3. Angular spectrum theory application

3.1. AST fundamental relationships

3.2. Comparison of AST and MRM simulations

4. Conclusions

Funding

References

Supplementary Material (4)

Cited By

Figures (5)

Equations (15)

Optics Express