Exact paraxial diffraction theory for polygonal apertures under Gaussian illumination

Riccardo Borghi

doi:10.1364/OSAC.3.000214

1. Introduction

Fresnel’s diffraction is a cornerstone of classical optics since more than two century, whereas Gaussian beams are the basic model to describe the light emitted by real laser sources. Recently, a “genuinely paraxial” revisitation of the well known Young-Maggi-Rubinowicz boundary diffraction wave (BDW) theory [1] has been developed [2–5] to deal with mainly plane wave sharp-edge diffraction. In [6], such a revisitation has been extended to the Miyamoto and Wolf theory [7] in order to include a Gaussian beam illumination rather than only the simpler plane-wave or spherical-wave ones. In doing so, it was conjectured that such extension can be obtained by introducing suitable complex angles within the mathematical representations of the two contributions to the diffracted wavefield predicted by the BDW theory: the so-called geometrical wavefield and the so-called BDW wavefield. In this way, two results were achieved at the same time:

(i) the reduction of the dimensionality of the Fresnel diffraction integral from two to one, which considerably reduces the computational time;
(ii) the automatic inclusion of impinging Gaussian beams, which are an incomparably more realistic model, with respect to plane or spherical waves, to describe the light emitted by real laser sources.

In the present paper it will be shown that, when a Gaussian beam is diffracted by a sharp-edge aperture having a polygonal arbitrary shape, the paraxial theory developed in [6] leads to an analytically exact representation of the diffracted wavefield. Surprisinlgy enough, the mathematical derivation of such exact representation turns out to be nearly straightforward, being it involved only elementary Calculus and Euclidean geometry.

It must be noticed how, although analytical descriptions for plane-wave Fraunhofer diffraction by polygonal apertures are available since many years [8–10], the same cannot be said when the diffracted field has to be estimated in the near field, apart from a few notable exceptions [11–14].

The knowledge of an exact, analytical representation of the solution of any problem, thus devoid of any numerical approximation, should be welcomed, regardless any possible practical implication or application. In any case, the availability of a complete knowledge of diffraction under an incomparably more realistic model of illumination than a plane wave should be viewed, even from a mere practical point of view, as an important achievement. Finally, what is contained in the present paper should also be viewed as a mean for further improving our theoretical understanding of diffraction, in the sense which will appear clearer to the reader on further going through the present work.

In what follows only the main ideas and results will be highlighted, with several mathematical steps and details having been dropped. We encourage the interested readers to go through the above quoted papers to get a more complete idea of the general context.

2. Theoretical analysis

The geometry of the problem is sketched in Fig. 1: a monochromatic, stigmatic Gaussian beam (wavenumber $k$, waist size $w_0$) orthogonally impinges on an opaque transverse plane placed at a distance $D$ from its waist plane having a sharp-edge aperture $\mathcal {A}$ arbitrarily shaped as a $N$-side polygon.

Fig. 1. Geometry of the problem.

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The general problem of finding the disturbance of the diffracted wavefield at the observation point $P \equiv (\boldsymbol {r};z)$ on a transverse plane at a distance $z\;>\;0$ from the aperture plane has been addressed in [6], whose main aspects will now be briefly resumed for the reader convenience. A key role is played by the following decomposition formula:

(1)$$\begin{array}{l} \displaystyle -\frac{\mathrm{i}\,k}{2\pi\,z} \int_{\boldsymbol{\rho}\in\mathcal{A}}\, \mathrm{d}^2\rho\, \exp\left(\frac{\mathrm{i}k}{2z}\,|\boldsymbol{r}-\boldsymbol{\rho}|^2\right)\,=\, \psi_G\,+\, \psi_\textrm{BDW} \end{array},$$

where the complex quantities $\psi _G$ and $\psi _\textrm {BDW}$ are both expressed via suitable one-dimensional contour integrals defined onto the aperture boundary $\Gamma =\partial \mathcal {A}$. In particular, on denoting the position of a typical point on $\Gamma$ by $Q$, let $Q=Q(t)$ be a suitable parametrization of $\Gamma$, with $t$ being a real parameter ranging within a given interval.

Then, on introducing the transverse vector $\boldsymbol {R}=\overrightarrow {PQ}$, quantities $\psi _G$ and $\psi _\textrm {BDW}$ can be evaluated through [6]

(2)$$\begin{array}{l} \displaystyle \psi_{G}\,=\,\frac{1}{2\pi}\,\oint_\Gamma\,\mathrm{d} t\, \dfrac{\boldsymbol{R}\times\dot{\boldsymbol{R}}} {\boldsymbol{R}\cdot\boldsymbol{R}} \end{array},$$

and

(3)$$\begin{array}{l} \displaystyle \psi_\textrm{BDW}\,=\,-\frac{1}{2\pi}\oint_\Gamma\,\mathrm{d} t\, \dfrac{\boldsymbol{R}\times\dot{\boldsymbol{R}}} {\boldsymbol{R}\cdot\boldsymbol{R}}\, \exp\left(\frac{\mathrm{i}k}{2z}\,\boldsymbol{R}\cdot\boldsymbol{R}\right) \end{array},$$

respectively. Here $\dot {\boldsymbol {R}}$ denotes the derivative of $\boldsymbol {R}$ with respect to the parameter $t$, while the cross product must be intended as the sole $z$-component, being both vectors $\boldsymbol {R}$ and $\dot {\boldsymbol {R}}$ purely transverse. In [6] it was conjectured the above integral representations to be valid, in principle, also for complex values of the Cartesian coordinates of the observation point $P$. Accordingly, the following recipe to retrieve the diffracted wavefield produced by the Gaussian beam in Fig. 1 was then proposed [6]:

(i) multiply all Cartesian coordinates of $P$, i.e., the position vector $(\boldsymbol {r};z)$, by the following dimensionless complex factor $(4)$$\begin{array}{l} \displaystyle \dfrac{1+\mathrm{i} D/L}{1+\mathrm{i} (z+D)/L} \end{array},$$$ with $L=kw^2_0/2$ being the incident Gaussian beam Rayleigh length;
(ii) evaluate both $\psi _G$ and $\psi _\textrm {BDW}$ for the above complex values of coordinates through Eqs. (2) and (3);
(iii) multiply $\psi _G+\psi _\textrm {BDW}$ by the wavefield obtained by letting the Gaussian beam to freely propagate from the waist plane up to the real observation point $P\equiv (\boldsymbol {r};z)$ to retrieve the diffracted wavefield.

That is all.

Now we are going to show how, when the aperture boundary $\Gamma$ is an arbitrarily shaped polygon, Eqs. (2) and (3) can be exactly expressed through analytical closed forms. The idea is quite simple: both $\psi _G$ and $\psi _\textrm {BDW}$ can be written as the sum of a finite number of contributions, each of them coming from one polygon side. In this way, the main problem reduces to evaluate Eqs. (2) and (3) for a typical segment, say $AB$, placed at the transverse observation plane, as sketched in Fig. 2.

Fig. 2. Parametrizations of the segment $AB$ for the evaluation of the contributions to the geometrical component $\psi _G(AB)$ (a) and of the BDW component $\psi _\textrm {BDW}(AB)$ (b) at the observation point $P$.

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We start with the contribution to the geometrical component, say $\psi _G(AB)$. To this aim, a natural parametrization of the segment is (see Fig. 2a)

(5)$$\begin{array}{l} \displaystyle \boldsymbol{R}(t)\,=\,\overrightarrow{PA}\,+\,\overrightarrow{AQ}\,=\,\overrightarrow{PA}\,+\,\overrightarrow{AB}\,t,\qquad\qquad t \in [0,1] \end{array},$$

so that $\dot {\boldsymbol {R}}\,=\,\overrightarrow {AB}$. On substituting from Eq. (5) into Eq. (2) we obtain at once

(6)$$\begin{array}{l} \displaystyle \psi_G(AB)\,=\,\dfrac 1{2\pi}\,\dfrac{\overrightarrow{PA}\times\overrightarrow{AB}}{\overline{AB}^2}\, \int^1_0\,\dfrac{\mathrm{d} t}{t^2\,+2\,\eta\,t\,+\,\chi^2} \end{array},$$

where

(7)$$\begin{array}{lcr} \displaystyle \eta\,=\,\dfrac{\overrightarrow{PA}\cdot \overrightarrow{AB}}{\overline{AB}^2}, & \chi^2\,=\,\dfrac{\overrightarrow{PA}\cdot \overrightarrow{PA}}{\overline{AB}^2}\,. \end{array}$$

Despite its apparent simplicity, the evaluation of the integral in Eq. (6) requires some care since it has to be computed, for what it was said above, when $P$ attains complex values of its coordinates, say $(\boldsymbol {r}\,\exp (\mathrm {i}\phi );z\,\exp (\mathrm {i}\phi ))$.

First of all, let us recast the integral as follows:

(8)$$\begin{array}{lcr} \displaystyle \int^1_0\,\dfrac{\mathrm{d} t}{t^2\,+2\,\eta\,t\,+\,\chi^2}\,=\, \dfrac 1{2\sqrt{\eta^2-\chi^2}}\,\sum_{\sigma={\pm}}\,\sigma\, \int^1_0\,\dfrac{\mathrm{d} t}{t\,-\,\tau_\sigma} \end{array},$$

where $\tau _\pm$ denote the root of the algebraic equation $t^2+2\eta t+\chi ^2=0$, i.e.,

(9)$$\begin{array}{lcr} \displaystyle \tau_\pm\,=\,-\eta\,\pm\,\sqrt{\eta^2-\chi^2}\,. \end{array}$$

Then, on using the elementary integral

(10)$$\begin{array}{lcr} \displaystyle \int^1_0\,\dfrac{\mathrm{d} t}{t\,-\,\tau_\sigma}\,=\,\log\left(1\,-\,\dfrac 1{t_\sigma}\right) \end{array},$$

after simplifying and rearranging the following formula can be established:

(11)$$\begin{array}{l} \displaystyle \int^1_0\,\dfrac{\mathrm{d} t}{t^2\,+2\,\eta\,t\,+\,\chi^2} \displaystyle \,=\, \dfrac 1{2\sqrt{\eta^2\,-\,\chi^2}}\,\sum_{\sigma={\pm} 1}\,\sigma\,\log\left(1+\dfrac{\eta+\sigma\sqrt{\eta^2-\chi^2}}{\chi^2}\right) \end{array},$$

which, together with Eqs. (6) and (7), gives the exact representation of the geometrical wavefield contribution due to the segment $AB$. As a trivial check, it is an academic exercise to prove that, for real values of $\eta$ and $\chi$ (corresponding to plane-wave illumination), Eqs. (6)–(11) give at once

(12)$$\begin{array}{l} \displaystyle \psi_G(AB)\,=\,\dfrac {\beta\,-\,\alpha}{2\pi} \end{array},$$

where the sign of angles $\alpha$ and $\beta$ has to be considered positive for counterclockwise rotations (see Fig. 2b for the meaning of $\alpha$ and $\beta$).

The evaluation of the contribution to the BDW component of the diffracted wavefield coming from the segment $AB$, say $\psi _\textrm {BDW}(AB)$, is more cumbersome, since involves the use of a special function which is well known in statistics but still unknown in optics. To evaluate such contribution, the integration parameter into Eq. (3) has to be changed from $t$ (see Fig. 2a) to the polar angle $\varphi$ shown in Fig. 2b. Let $D$ denote the distance between $P$ and $AB$. On introducing the Cartesian axis $x$ whose origin coincides with $P$ and which is orthogonal to the $AB$ direction, the polar representation of the vector $\boldsymbol {R}$ is given by $R(\varphi )=D/\cos \varphi$, with $\varphi \in [\alpha ,\beta ]$. Accordingly, the contribution of $AB$ to the BDW wavefield can be recast as follows:

(13)$$\begin{array}{l} \displaystyle \psi_\textrm{BDW}(AB)\,=\, -\frac {1}{2\pi}\, \int_\alpha^\beta\, \exp\left(\frac {\mathrm{i} u/2}{\cos^2\varphi}\right)\,\mathrm{d}\varphi \end{array},$$

where the dimensionless parameter $u=kD^2/z$ has been introduced. Note that the distance $D$ in Fig. 2b can formally be expressed as

(14)$$\begin{array}{l} \displaystyle D\,=\,\frac{\overrightarrow{PA}\,\times\,\overrightarrow{AB}}{\overline{AB}} \end{array},$$

where use has been made of $\overrightarrow {PB}\,=\,\overrightarrow {PA}\,+\,\overrightarrow {AB}$. To evaluate the integral in Eq. (13), it is then sufficient to make the variable change $\tau =\tan \varphi$, which yields

(15)$$\begin{array}{lc} \displaystyle \psi_\textrm{BDW}(AB)\,=\, T\left(\sqrt{-\mathrm{i}\,u},\tan\alpha\right)\,-\,T\left(\sqrt{-\mathrm{i}\,u},\tan\beta\right) \end{array},$$

where the symbol $T(a,\;b)$ denotes the so-called Owen T-function [15], defined as follows:

(16)$$\begin{array}{l} \displaystyle T(a,\;b)\,=\,\frac 1{2\pi}\, \int_0^{b}\,\dfrac{\mathrm{d}\tau}{1\,+\,\tau^2}\exp\left[-\dfrac {a^2}2\,(1+\tau)^2\right]\,. \end{array}$$

For real values of both parameters, the quantity $T(a,\;b)$ gives the probability that $\{X\;>\;a$ and $0\;<\;Y\;<\;b X\}$, with $X$ and $Y$ being two statistically independent normal random variables [15]. For our purposes it is important to note that Owen’s function $T(a,\;b)$ can be analytically continued to complex values of both arguments. A precious work about the main properties of the Owen function (including integral and series representations, connections with other special functions, evaluation of integrals and series containing the $T$-function) has recently been published [16]. Readers are encouraged to go through this work to familiarize with the Owen function. Analytical expressions of both $\tan \alpha$ and $\tan \beta$ in terms of the positions of points $A$, $B$, and $P$ can also be found through elementary geometry, which yields

(17)$$\left\{ \begin{array}{l} \displaystyle \tan\alpha\,=\,\dfrac{\overrightarrow{PA}\,\cdot\,\overrightarrow{AB}}{\overrightarrow{PA}\,\times\,\overrightarrow{AB}},\\ \\ \displaystyle \tan\beta\,=\,\dfrac{\overrightarrow{PB}\,\cdot\,\overrightarrow{AB}}{\overrightarrow{PA}\,\times\,\overrightarrow{AB}}\,. \end{array} \right.$$

Equations (6)–(11), together with Eqs. (14)–(17), are the main result of the present paper. Through them, paraxial diffraction of Gaussian beams by polygonal sharp-edges apertures finds an exact solution, as promised at the beginning.

3. Numerical results and discussion

In the rest of the paper a single but significant example of application of the theory previously exposed will be illustrated. Such example received a considerable attention, even recently [12,17–20]. Figure 3 depicts the geometry of the problem: a sharp-edge aperture shaped in the form of an equilateral triangle is orthogonally illuminated by a Gaussian beam whose mean propagation axis passes through the triangle centre. It is worth introducing a Cartesian reference frame $Oxyz$, in which the coordinates of the triangle vertices are $A \equiv (-\sqrt 3,-1)$, $B \equiv (\sqrt 3,-1)$, and $C \equiv (0,2)$, where for simplicity the length unit has been chosen in such way that each triangle side measures $2\sqrt 3$. Then, on denoting $(x,\;y)$ the Cartesian representation of the observation point $P$, Eqs. (7), (14), and (17) allow all parameters $\eta$, $\chi ^2$, $D$, $\tan \alpha$, and $\tan \beta$ to be easily evaluated for all triplets $(A,B,P)$, $(B,C,P)$, and $(C,A,P)$. For example, the parameters related to the triplet $(A,B,P)$ take on the following expressions:

(18)$$\left\{ \begin{array}{l} \displaystyle \eta\,=\,-\dfrac{3+x\sqrt 3}{6},\\ \\ \displaystyle \chi^2\,=\,\dfrac{x^2+2x\sqrt 3+y(2+y)+4}{12},\\ \\ \displaystyle D\,=\,1+y,\\ \\ \displaystyle \tan\alpha\,=\,-\dfrac{\sqrt 3+x}{1+y},\\ \\ \displaystyle \tan\beta\,=\,\dfrac{\sqrt 3-x}{1+y}\,. \end{array} \right.$$

Similar expressions hold for the other two triplets $(B,C,P)$ and $(C,A,P)$ but will not be given here. We start our analysis by exploring the structure of the geometrical component $\psi _G$. Before doing this, however, it is worth spending a few words in order to advice readers about the importance of what we are going to show.

Fig. 3. Geometry for Gaussian beam diffraction by equilateral triangular apertures.

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The BDW theory is not only an extremely useful computational tool but, as it was claimed at the end of Sec. 1, it could help in further improving our theoretical understanding of diffraction. The decomposition of the total diffracted wavefield as the superposition of the geometrical and of the BDW contributions, represented by Eq. (2) and Eq. (3), respectively, should be viewed as an added value with respect to the classical Fresnel diffraction theory, also from a purely theoretical point of view. In particular, as far as the “geometrical wavefield” contribution $\psi _G$ is concerned, a plane (or a spherical) wave impinging on a sharp-edge aperture will produce, within the geometrical optics limit $\lambda \to 0$, across any transverse observation plane a shadow having exactly the same shape of the diffracting aperture, apart from some scaling. Accordingly, the BDW contribution $\psi _\textrm {BDW}$ is nothing but a correction (which arises from the aperture edge) that must be added to the geometrical wavefield in order for the true paraxial diffracted field to be exactly retrieved. The binary nature of $\psi _G$ for plane wave illuminations can be immediately appreciated on writing Eq. (2) in terms of the angular variable $\varphi$, which gives

(19)$$\psi_G\,=\,\frac 1{2\pi}\,\oint_\Gamma\,\mathrm{d}\varphi,$$

and by noting that the above integral can take only the values $0$ or $2\pi$, depending on the position of the observation point $P$ with respect to $\Gamma$. However, under a Gaussian beam illumination it is no longer easy to grasp the structure of the geometrical wave field. A first answer was given by Otis in Ref. [21], where he proposed a simple mathematical recipe to calculate geometrical shadows produced by Gaussian beams impinging on arbitrarily shaped sharp-edge apertures. In particular, Otis claimed the Gaussian beam impinging on the aperture would produce a geometrical “Gaussian shadow” having exactly the aperture shape, scaled by the modulus of the factor in Eq. (4). Otis’ conjecture implies the geometrical field defined by Eq. (2), once evaluated at complex transverse positions $\boldsymbol {r}\,\exp (\mathrm {i}\phi )$, must necessarily satisfy the following relation:

(20)$$\begin{array}{l} \displaystyle \psi_G\,=\, \left\{ \begin{array}{lr} 1 & \boldsymbol{r} \in \mathcal{A},\\ & \\ 0 & \boldsymbol{r} \notin \mathcal{A}, \end{array} \right. \end{array}$$

irrespective the value of $\phi$. Equation (20) was rigorously proved for circular apertures, which share the axial symmetry with the impinging Gaussian beam [6]. In the same work, however, a numerical evidence which refutes the Otis thesis was obtained by analyzing the Gaussian beam diffraction produced by elliptic apertures. In [6] it was conjectured a different rule to establish the regions where it should be expected the integral in Eq. (20) to be identically unitary and the regions where the same integral is expected to vanish. According to such conjecture, it is expected $\psi _G\equiv 1$ at all observation points $P$ which are inside the circle centred at the Gaussian beam axis and inscribed within the geometrical projection of the aperture shape, while $\psi _G\equiv 0$ at all observation points $P$ which are outside the circumscribed circle [6]. At points inside the region between the two circles the geometrical wavefield $\psi _G$ displays a strong dependence on the phase $\phi$. At present a rigorous mathematical proof of our conjecture is not available, so that we believe the results contained in the present paper could encourage readers to utilize the log-based closed form expression of $\psi _G$ obtained for polygonal sharp-edge to further investigate such fascinating problem of the "Gaussian shadows", which will surely be the subject of future investigations. Here because, in the present paper, we intend to provide at least a further example of checking by using the aperture having the simplest geometry, i.e., the equilateral triangle.

Figure 4 shows the geometry: the observation point $P$ belongs to the $x$-axis. For a given value of $x$, the geometrical wavefield $\psi _G$ is then computed, via Eqs. (2) and (3), at $x\,\exp (\mathrm {i}\phi )$, with $\phi \in [-\pi ,\pi ]$. The result is shown in Fig. 5: it turns out that $\psi _G=1$ within the white region, $\psi _G=0$ within the black region, and $\psi _G=1/2$ within the grey region. From a look to Fig. 5 it is clear that Eq. (20) is not satisfied, except when the observation point is inside the inscribed circle $\gamma _m$ ($0\le x \le 1$), for which $\psi _G\equiv 1$, or when it is outside the circumscribed circle $\gamma _M$ ($x \ge 1$), for which $\psi _G\equiv 0$. Several others numerical trials, made at different observation points but not shown here, gave results in perfect agreement with the conjecture given in [6].

Fig. 4. Geometry for checking the conjecture in [6].

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Fig. 5. Two-dimensional map of the geometrical wavefield $\psi _G$ for the triangular geometry in Fig. 4, numerically evaluated at complex transverse observation points $(x\exp (\mathrm {i}\phi ),0)$.

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Before concluding this section, in order to offer at least one example of computation of $\psi _\textrm {BDW}$, in Fig. 6 two-dimensional maps of the optical intensity produced by the diffraction of a Gaussian beam with $\lambda =0.5$ $\mu$m whose waist and aperture planes coincide ($D=0$), are shown at the observation plane $z=100$ mm. The radius of the circumscribed circle $\gamma _M$ is set to 1 mm. To put into evidence the influence of the Gaussian nature of the illumination, different values of the spot-size $w_0$ have been chosen, and precisely $w_0=500$ $\mu$m (a), $w_0=1$ mm (b), $w_0=2$ mm (c), and $w_0=10$ mm (d). The last value of $w_0$ has been chosen in order for Fig. 6d to be visually compared to Fig. 5a of [12], which was generated by using, in place of our collimated Gaussian beam, a plane wave. In fact, since the chosen value of the spot size is one order of magnitude greater than the triangle side length, it should be expected the Gaussian beam to act approximately like a plane wave. This is confirmed by the excellent visual agreement between Fig. 6d and Fig. 5a.

Fig. 6. Two-dimensional maps of the optical intensity produced by the diffraction of a Gaussian beam with $\lambda =0.5$ $\mu$m and $D=0$ at the observation plane $z=100$ mm. The radius of the circumscribed circle $\gamma _M$ is set to 1 mm. The spot-size has been set to $w_0=500$ $\mu$m (a), $w_0=1$ mm (b), $w_0=2$ mm (c), and $w_0=10$ mm (d).

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4. Summary and conclusions

Sharp-edge diffraction by Gaussian beams has been recently reformulated within the paraxial framework via a suitable revisitation of the BDW theory [6]. Based on this, in the present paper an analytically exact mathematical representation of the field diffracted by arbitrary polygonal sharp-edge apertures under Gaussian beam illumination has been obtained. No further approximations beyond paraxial diffraction have here been invoked. Surprisingly enough, the whole mathematical derivation of such representation turned out to be nearly straightforward, with only elementary Calculus as well as Euclidean geometry being involved. We believe our achievement could help to further improve the theoretical understanding of paraxial diffraction, in particolar about the conjecture conceived in [6] concerning the binary nature of the geometrical wavefield $\psi _G$ under Gaussian illumination. We believe the results contained in the present paper could encourage readers, for instance, to utilize the log-based closed form expression of $\psi _G$ here obtained to investigate the fascinating “Gaussian shadow” problem. Moreover, the analytical and complete knowledge of the diffracted field under such a realistic model of illumination should also be acknowledged as an effective computational tool of considerable usefulness for a wide range of future applications.

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Exact paraxial diffraction theory for polygonal apertures under Gaussian illumination

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical results and discussion

4. Summary and conclusions

References

Cited By

Figures (6)

Equations (20)

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