1 Introduction

Geometrical optics manifested by the rectilinear propagation of light rays in free space, was known to the ancient Greeks (400–300 BC), whereas the law of ray bending that takes place upon refraction through an interface, dates back to Snell (1591–1626). Fermat (1601–1665) put forward the hypothesis, known as Fermat’s principle, that a light ray from one point to another follows the path that takes the shortest time, from which rectilinear propagation in free space, the reflection law, and Snell’s law readily follow. In 1924 Rubinowicz [1] introduced the concept of boundary-diffracted rays, and in the 1950s Keller ([2], [3], [4]) developed the geometrical theory of diffraction. According to Keller’s theory, a diffracted ray satisfies Fermat’s principle of edge diffraction, which states that an edge-diffracted ray from a source point S to an observation point P is the curve that has stationary optical path length among all curves from S to P with one point on the edge.

Ray methods are important in diffraction theory for two completely different reasons. From a practical point of view they provide a tremendous saving in computing time because the integrands of diffraction integrals are rapidly oscillating functions. From a more fundamental point of view the transition from diffraction theory to ray theory is important because it provides a very valuable insight into the diffraction process. According to Huygen’s principle, an infinite number of secondary waves must be added to obtain the diffracted field. But ray theories show that only a few of these secondary waves contribute significantly. These particular secondary waves are obtained from the diffraction integral by applying the Method of Stationary Phase (MSP). By twisting George Orwell’s famous words, one may say that “All secondary waves are created equal, but some are more equal than others”. Those secondary waves that are “more equal” than the others, give rise to geometrical and diffracted rays.

The diffraction-integral approach to ray theory, which we discuss here, has the advantage that the use of the MSP automatically produces contributions from both geometrical and diffracted rays in a consistent and straightforward manner. To save space we restrict our attention to 3D diffraction by an aperture in a plane screen by use of the first Rayleigh-Sommerfeld diffraction formula combined with a Kirchhoff-type of approximation for the field in the aperture plane. Then the MSP for double integrals provides the transition from diffraction theory to ray theory. Geometrical rays are associated with interior stationary points, whereas diffracted rays are associated with critical points at the aperture boundary.

The paper is organized as follows. We begin by discussing conventional rays and aperture-plane Point-Spread-Function (PSF) rays in section 2, using the diffraction of a converging spherical wave through a circular aperture as an example. In section 3 we illustrate the use of aperture-plane PSF rays in connection with focusing through a plane interface between two isotropic media. In section 4 we study the diffraction of a plane wave by a circular aperture with particular emphasis on the axial caustic, the Poisson spot, and the axial interference pattern. In section 5 we study a Gaussian beam that is diffracted through a circular aperture and subsequently transmitted through a plane interface into a biaxially anisotropic medium. Finally, in section 6 we give some concluding remarks.

 

Fig. 1. Geometry for focusing through a circular aperture. The Geometrical Shadow Boundary (GSB) is described by rays passing through the focus and the aperture boundary.

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2 Conventional rays and aperture-plane Point-Spread-Function (PSF) rays

2.1 Conventional rays

To explain the difference between conventional rays and aperture-plane Point-Spread-Function (PSF) rays, we consider a scalar converging spherical wave with focus on the z axis that is incident upon a circular aperture A of radius a, centered at x = y = 0 in the plane z = 0 (Fig. 1). To determine the field in the half-space z > 0 we use the first Rayleigh-Sommerfeld diffraction formula combined with a Kirchhoff-type of approximation. The latter implies that the field in the aperture plane is taken to be equal to the incident field inside the aperture and equal to zero outside the aperture. Provided the observation distance R ₂ in Fig. 1 is much larger than the wavelength λ, the focused field is given by the diffraction integral (Eq. (11.29a) in [5] with u ₀(x,y) = 1)

where $k=\frac{2\pi }{\lambda }$ is the wave number, and

with

To obtain contributions from conventional geometrical and diffracted rays we assume that the observation point P ₂(x ₂, y ₂, z ₂) neither lies in the vicinity of the focus at P ₁(0, 0, z ₁) nor in the vicinity of the geometrical shadow boundary in Fig. 1. Then the stationary point P_s (x_s , y_s ) in the aperture plane z = 0, which is determined by the requirement that $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$ at (x_s ,y_s ), does not lie near the aperture boundary. A simple calculation shows that the stationary point P_s lies on the geometrical ray that passes through the focus and the observation point, as shown in Fig. 1. Consider first the case in which the stationary point lies inside the aperture, but not close to the aperture boundary. Then the contribution u_IS of the stationary point is given by the asymptotic formula (lowest-order term of Eq. (9.7a) in [5])

where

A straightforward calculation yields (Eq. (11.52a) in [5] with u ₀(x_s ,y_s ) = 1)

where R_jS (j = 1,2) is the value of R_j at the stationary point. Thus, as expected, the contribution from the interior stationary point is equal to the geometrical-optics field. We emphasize that this result is valid only at observation points far from both the focus and the shadow boundary. Note the well-known phase change of π on passage through the focus contained in (7).

The contribution u_IB from the aperture boundary to the integral in (1) can be obtained via a change to polar integration variables followed by integration by parts with respect to the radial variable. For observation points on the z axis this gives (Eq. (11.54b) with u ₀(a) = 1 in [5])

where

Thus, by combining the geometrical-optics result in (7) with the contribution from the boundary in (8), we get for the total field on the axis (Eq. (11.55a) in [5] with u ₀(0) = u ₀(a) = 1)

Note that the boundary contribution is comparable in strength to the geometrical-optics contribution in this case. The reason for this is that because of the symmetry all secondary waves from the boundary add in phase at any observation point on the z axis. This phenomenon is intimately connected with the existence of an axial caustic, which we discuss in more detail in section 4.

Note also that the result in (10) obtained by employing asymptotic techniques, turns out to be equal to the exact result obtained by performing the integration in (1) analytically without any approximations (Eq. (4.45a) in [5]).

In the paraxial approximation the result for the axial field in (10) readily reduces to the familiar result (Eq. (12.54a) in [5] multiplied by a factor -iλ to compensate for the fact that here the incident field is a converging spherical wave, whereas the result in Eq. (12.54a) pertains to an incident converging dipole wave)

where sinc(x) = sin(x)/x and

If the observation point lies off the z axis but well inside the lit cone in Fig. 1, there are only two critical points on the aperture boundary, namely those for which the distance R ₂ from the aperture boundary to the observation point is a maximum or a minimum. The secondary waves emitted by these two boundary points are associated with diffracted rays, and their contributions can be obtained using the MSP in a similar manner as explained in section 4 for a normally incident plane wave. The total field now consists of three terms, the geometrical-optics contribution given by (7), and the contributions u_IDRfoc from the two diffracted rays, given by

where

If the observation point lies well outside the lit cone in Fig. 1, the stationary point lies outside the aperture. Hence it does not contribute to the diffracted field, which now is determined entirely by the contributions from the two critical points on the aperture boundary given by (14).

2.2 Aperture-plane Point-Spread-Function (PSF) rays

To explain the concept of aperture-plane PSF rays we express (1) in the form (Eq. (4.13) in [5])

where

Here

with

Note that the only difference between (1) and (15) is that the approximation kR ₂ ≫ 1 was used to obtain the former result, whereas the latter result was obtained directly from the first Rayleigh-Sommerfeld formula without any approximations.

The physical interpretation of (15) follows by letting uⁱ (x, y, 0) = δ(x - x ₀)δ(y - y ₀), so that the field h(x ₂ - x, y ₂ - y, z ₂) becomes equal to the field at the observation point (x ₂,y ₂, z ₂) that is due to a point source at the point (x ₀, y ₀) in the aperture plane. Hence we call h(x ₂ - x, y ₂ - y, z ₂) the aperture-plane Point-Spread-Function (PSF). In (17) the aperture-plane PSF is expressed as an angular spectrum of plane waves. By applying a MSP formula analogous to that in (4) to the double integral in (17), we find that only one plane wave contributes, namely that which is directed from the point source at P(x, y, 0) in the aperture plane to the observation point P ₂(x ₂, y ₂, z ₂). The asymptotic contribution h_S (x ₂ - x, y ₂ - y, z ₂) obtained in this manner may be considered to be due to an aperture-plane PSF ray. It is given by

For propagation in free space we can evaluate the aperture-plane PSF integral in (17) analytically by use of Weyl’s plane-wave expansion of a spherical wave. Thus we obtain ([6], Eq. (4.15) in [5])

from which it follows that h_S (x ₂ - x, y ₂ - y, z ₂) is approximately equal to h_Exact (x ₂ - x, y ₂ - y, z ₂) when kR ₂ ≫ 1. Thus, for propagation in free space the asymptotic result in (20) is superfluous. But for wave propagation in inhomogeneous media, the aperture-plane PSF integral can not be evaluated analytically, whereas a very accurate approximation to it can be obtained using the MSP for double integrals [7 – 17], as shown in section 3.

In conclusion, the important difference between conventional rays and aperture-plane PSF rays is that a conventional ray starts at the source of the incident field, whereas an aperture-plane PSF ray starts in the aperture plane z = 0. Thus, an aperture-plane PSF ray is a conventional ray that connects an integration point P(x, y, 0) in the aperture plane with the observation point P ₂(x ₂, y ₂, z ₂).

The incident field in the aperture plane may be obtained using conventional rays. In the case of focusing the important point is that rays from the source to the aperture plane do not suffer from coalescence of stationary points, whereas rays from the source to the focal region do indeed suffer from such problems. Another important point is that when the medium between the aperture plane z = 0 and the observation point is not homogeneous, a significant saving of computing time can be obtained by using aperture-plane PSF rays, as illustrated in section 3.

3 Focusing through a plane interface

Consider the same focusing geometry as in Fig. 1, except that at the plane z = z ₀ < z ₁ there is an interface with a medium with a different refractive index. Thus, the wave number in the half-space z < z ₀ of the incident field is k ₁, and the wave number in the half-space z > z ₀ of the transmitted field is k ₂. The exact solution to the problem of reflection and refraction of an electromagnetic field at plane interface between two different media can be expressed in terms of a vectorial aperture-plane PSF, which appears in the form of an angular spectrum of plane waves. If the field in the plane z = 0 is generated by an aperture current that produces a convergent spherical wave polarized in the x direction, the exact solution for the transmitted electric field is given by ([7], Eq. (16.30b) with A(x′, y′) = - 1/2R ₁ and ϕ(x′, y′) = R ₁ and Eqs. (16.31h, j, l, m) and (16.32e, g) in [5])

where $R_1=\sqrt{x^{\prime 2}+y^{\prime 2}+z_1^2}$, and where the vectorial aperture-plane PSF is given by

Here

with $k_{\mathit{zj}}=\sqrt{k_j^2-k_t^2}$, $k_t^2$ = $k_x^2$ + $k_y^2$, k ^t = k_x ê_x + k_y ê_y + k _z2 ê_z, and with T^TE and T^TM being the Fresnel transmission coefficients given by Eqs. (15.40b, d) in [5].

The physical interpretation of this result is as follows. Each point (x′ , y′) in the aperture plane z = 0 emits a secondary diverging spherical wave which is weighted by the aperture field exp(-ik _i R ₁)/R ₁. This secondary wave produces an angular spectrum of plane waves (given by (23)) which are incident on the plane interface at z = z ₀, and each of these plane waves is multiplied by the transmission coefficient T^TE or T^TM on refraction at the interface.

By applying the MSP for double integrals to the aperture-plane PSF integral in (23), we obtain a reduction in (22) from a quadruple integration to a double integration, and consequently, a significant reduction in the computing time at a negligible reduction of accuracy ([12], [13]).

In accordance with the physical interpretation given above, one finds by use of the MSP that the main contribution to the field at an observation point (x, y, z) in the second medium due to a point source at (x′, y′,0) in the aperture plane is provided by that particular plane wave emitted by the source at (x′, y′, 0), which after refraction through the interface is directed towards (x, y, z). Thus, through the use of aperture-plane PSF rays we obtain both a valuable insight into the diffraction process and a significant reduction in the computing time.

4 Plane-wave diffraction by a circular aperture

Let a plane wave be normally incident upon a circular aperture in the plane z = 0 (Fig. 2). We let kR ₂ ≫ 1 so that the diffracted field is given by (15) with uⁱ (x, y, 0) = 1 and with the aperture-plane PSF given by (20). Further, we introduce polar co-ordinates and integration variables given by

and we express the integral over the aperture area A as the integral over the entire aperture plane minus the integral over the area outside the aperture area to obtain

where

with

 

Fig. 2. Diffraction of a plane wave that is normaly incident upon a circular aperture.

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The first term u_IS in (27) we recognize as the geometrical-optics contribution. In (29) we may integrate by parts with respect to ρ to obtain the contribution due to the aperture boundary as

where

Next, we obtain the contribution due to diffracted rays by applying the MSP for single integrals to the integral in (31). For r ≠ 0 the phase function f(ϕ) in (31) has two stationary points, one for ϕ = β and another for ϕ = β + π. These correspond to the minimum and maximum values of R ₂. The asymptotic result is (lowest-order term of Eq. (8.8a) in [5])

where

Note that this result is valid only if the observation point is sufficiently far away from the axis. At all off-axis observation points the phase function in (31) has two stationary points. But if the observation point moves onto the axis, the phase function becomes independent of ϕ, and we have an axial caustic ([18], section 11.1.7 in [5]). In the limit as r → 0 all points on the aperture boundary become stationary because the secondary waves they emit are in phase at any axial observation point.

On the axis we obtain from (27) and (31)

which in the paraxial approximation reduces to

Thus, the axial intensity is proportional to sin²(ka ²/4z ₂). The axial intensity distribution results from interference between the geometrical-optics wave and the boundary-diffracted waves. Because of the symmetry all secondary waves emitted by the aperture boundary are in phase, as noted above, and therefore the boundary-diffracted field is just as strong as the geometrical-optics field, giving full contrast of the axial interference pattern, i.e. axial zeros.

If we replace the aperture in Fig. 2 by an opaque disk, the diffracted field is given by -u_ID where u_ID is given in (29). On the axis the field then becomes

Thus, the intensity on the axis behind an opaque diskis given by I = cos²(θ), where θ is the angle subtended by the disk at the axial observation point. This is the famous Poisson spot. In 1818 Poisson observed that Fresnel’s wave theory of light predicted a bright spot at the center of the shadow of a small circular disk, and he used this to dispute Fresnel’s theory. But when Arago later performed the experiment, he found the prediction to be correct. This bright spot is caused by the axial caustic, i.e. by the infinite number of diffracted rays issued by the boundary whose contributions are identical.

Asymptotic techniques analogous to the ones presented in this section, have been used to obtain fast and accurate evaluations of high-aperture diffractive lenses [19] and to analyze imaging in the presence of a sinusoidal phase modulation [20].

5 Transmission of a truncated Gaussian beam into a biaxial crystal

Consider a Gaussian beam that is polarized in the x direction and has its main propagation direction in the z direction. The beam waist is σ ₀ in the plane z = 0, and in this plane there is a circular aperture of radius a. After passing through this aperture the diffracted beam is transmitted through a plane interface at z = z ₀, which separates the isotropic medium of the incident beam from a biaxial crystal with one of its principal axes along the interface normal. The principal indices of refraction of the crystal are n ₁, n ₂, and n ₃, and the index of refraction of the isotropic medium is n ⁽¹⁾. To simplify matter, we take the distance z ₀ from the aperture plane to the interface to be so small that we may neglect depolarization effects on transmission through the interface. Then a scalar theory is adequate, and the transmitted field can be expressed as [17]

where

This result was obtained from an exact solution of the problem of reflection and refraction at a plane interface between an isotropic and a biaxially anisotropic medium [14]. The exact solution is expressed in terms of an aperture-plane PSF, which in turn is given as an angular spectrum of plane waves. This aperture-plane PSF can not be evaluated analytically, but by applying the MSP for double integrals to the angular-spectrum integral, we obtain a very accurate result and a substantial saving in computing time. Thus, the aperture-plane PSF ray approach was applied to obtain the result in (38).

We wish to compare the result in (38) with that obtained when the second medium is isotropic with index of refraction n. Therefore we let n ₁ = n ₂ = n ₃ = n to obtain from (38)

where $n_r=\frac{n}{n^{\left(1\right)}}$ and

On the axis the integration in (43) can be performed analytically and the result is

where $\overline{u}=k^{\left(1\right)}\frac{a^2}{\overline{Z}}$.

 

Fig. 3. Axial intensities of the transmitted beam. (a) Computed from Eq. (34) with an aperture radius of a = 5 mm (solid curve) and a = 2 mm (dashed curve). (b) Computed from Eq. (34) (solid curve) and from Eq. (39) (dashed curve) for a = 2 mm.

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Fig. 3 (a) shows axial intensities of the transmitted beam. The biaxial crystal is KIO ₃ with principal refractive indices of n ₁ = 1.700, n ₂ = 1.828, and n ₃ = 1.832. The interface with the crystal is taken to be at a distance of z ₀ = 1.0 mm behind the aperture plane z = 0. The beam waist in the aperture plane is taken to be σ ₀ = 1.0 mm, and the wavelength of the incident beam is taken to be λ⁽¹⁾ = 0.633 μm. The solid curve in Fig. 3 (a) is computed from (38) with an aperture radius of a = 5 mm. Since a ≫ σ ₀, the field at the aperture boundary is so small that it gives negligible diffraction effects. Thus, the solid curve in Fig. 3 (a) shows the axial intensity of a non-truncated transmitted Gaussian beam. The dashed curve in Fig. 3 (a) is computed from the same equation as the solid curve but for an aperture radius of a = 2 mm. Now we see strong interference effects due to the boundary-diffracted waves.

The solid curve in Fig. 3 (b) shows the axial intensity computed from (38) for an aperture radius of a = 2 mm, whereas the dashed curve shows the axial intensity computed from (44). Note that according to (44), the axial intensity maxima of a truncated Gaussian beam that is transmitted into an isotropic medium, decay monotonically with z, as illustrated in the dashed curve in Fig. 3 (b). However, for a truncated Gausian beam that is transmitted into a biaxially anisotropic medium, the axial intensity maxima, shown by the solid curve in Fig. 3 (b), increase with z in the region z ≤ 4 m. To explain this unexpected behavior let us first consider the case in which the incident field is a plane wave propagating along the z axis and the second medium is isotropic. Then at every observation point on the z axis all boundary-diffracted waves are in phase, so that they interfere constructively to give a diffracted field of the same strength as the geometrical-optics field. Hence we get axial intensity zeros, as shown by (36). For an incident Gaussian beam all boundary-diffracted waves are still in phase on the axis, but the field amplitude at the aperture boundary is now exp(-a ² ${\sigma }_0^2$). Thus, the diffracted intensity no longer has axial zeroes, as shown by the dashed curve in Fig. 3 (b). But the axial maxima decrease monotonically with z because each edge diffracted wave decays as 1/R, where $R=\sqrt{a^2+z^2}$. Also when the second medium is biaxially anisotropic, each boundary-diffracted wave decays as 1/R, but now the various boundary-diffracted waves are not in phase on the z axis because the phase velocity depends on the direction of propagation. However, as z increases the boundary-diffracted waves get more in phase on the axis, and this explains the growth of the maxima of the on-axis intensity in the region z ≤ 4 m in Fig. 3 (b).

6 Conclusions

We have reviewed some aspects concerning the connection between waves and rays, based on a diffraction-integral approach to diffraction and wave propagation combined with the application of the Method of Stationary Phase (MSP). As discussed in the paper, the advantage of this approach is that it produces both geometrical and diffracted rays in a consistent and straightforward manner. To save space we have restricted our discussion to 3D diffraction by an aperture in a plane screen, in which case the MSP for double integrals is appropriate.

To simplify matters we have largely considered cases in which the non-uniform MSP for double integrals apply. This means that we have avoided to discuss complications that arise in caustic regions, where we have coalescence of interior stationary points and associated diffraction catastrophes ([21], [22], sections 8.2.2, 8.2.3, 9.2.3 in [5]). Also, we have avoided to discuss complications that arise near shadow boundaries, where we have coalescence between interior stationary points and critical points at the boundary. In such cases uniform asymptotic methods are required to get quantitative descriptions of the diffraction phenomena (see e.g. sections 8.2 and 9.2 in [5]). Alternatively, as discussed in the paper, we may use aperture-plane PSF rays to avoid complications in caustic regions due to the coalescence of stationary points.