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Abstract
Ray tracing is a fundamental geometric-optics issue which gives a single ray path but seldom presents the collective behavior of light. The optical field distribution usually involves the calculation of an electromagnetic field and is rarely discussed from the perspective of geometric optics. However, in this paper, we show for a concentric medium with spherically symmetric refractive index, how the relative angular distribution of refractive beams can be obtained from the pure classical geometric optics method. As a measurement of the distribution, we introduce the concept of the differential cross section (DCS), which can be calculated from the relation between aiming distance and deflecting the angle of the ray. We present a general method to solve this relation from both Snell’s law in a constant medium and the optical Binet equation (OBE) in a gradient-index (GRIN) medium. Even without observing the collective traces, the DCS can independently give a quantitative description for the deflected light density of concentric media at different directions. It may act as a reference index for the design of beam deflector.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As an ideal model in geometric optics, a beam, if reasonably ignoring its wave-like properties, can be always treated as a geometric curve regardless of what the background medium is. The refractive index field controls its trace by certain rules, and its direction represents the direction of light energy flux. Based on this theoretical framework, a variety of ray tracing techniques are proposed for calculating the ray path in a given refractive index field. For example, if the field is composed with several constant media, the ray trace can be simply obtained from the well-known Snell’s law [1], which can be further developed to such as ABCD matrix method [2–6]. For gradient-index (GRIN) media, the common ray tracing techniques including Fermat’s principle [7–9], optical Lagrange equation [10–13] and Hamilton equation [14–17] can be all used to calculate the ray path and analysis its geometric properties.
For whether constant or GRIN media, the existing ray tracing techniques mainly focus on the trajectory of a beam abstracted out as a zero-width curve. That is to say, the geometric property of a single ray is mostly concerned, but the collective behavior of light, if just seen from the trace, is still unclear. However, for some geometric-optics problems such as the caustics where the optical field has singularity [18], it is especially important to pay attention to the collective rays, since this singularity comes from a differentiable mappings performed by a family of rays, known as catastrophe theory [19,20]. This also involves the scattering problem of light and particles, such as rainbows in optics and atomic physics [19].
Usually, when we say the incident beam is “parallel”, its width is actually a finite number. Therefore, for those uniform beams with finite width, one can assume they are composed of many identical zero-width rays. As the refractive index field varies in the whole space, the emergent rays, in general, will be no more parallel but present an angular distribution. So we can naturally think that the field is strong at where the “number” of rays is large, and weak at where it is small. This viewpoint is also used to analysis the field near caustics [19]. In particular, if the parallel rays pass through a spherically symmetric field, the angular distribution will be also symmetric. To make clear that the relative density ofoptical field at every direction, calculatingthe deflecting angle is certainly necessary. But furthermore, it will be more explicit if a quantity is introduced to measure this angular distribution.
In this paper, we will develop a method to calculate the deflecting angle and angular distribution of optical field when a parallel beam is deflected by a concentric refractive index field. This method is purely based on the framework of classical geometric optics, which can be regarded as a result of short wavelength limit of wave optics [21]. In contrast with wave-optics method [1,21,22], it is relatively simple as it does not need to calculate the electromagnetic field, and thus it avoids the complicated description about the cylindrical wave field when performing the short wavelength approximation and analyzing the limit. Provided the ray trace exists analytical solution, the relative angular distribution can be also analytically solved. As a quantitative tool, the differential cross section (DCS), which is widely used in the collision and scattering of atomic electrons [23,24], is introduced in geometric-optics systems to describe the density of deflected rays in Sec. 2. In order to explain the physical significance of DCS, we will show detail procedures and common skills to calculate the DCS for several specific refractive index fields in Sec. 3.
2. Differential cross section of a concentric refractive index field
The concept of differential cross section (DCS) originates from the scattering problem of a central potential in mechanics [25]. When a particle comes into a central potential field from infinity, its motion will be deflected by the central force, known as scattering. The scattering angle depends on the distance from force center to incident line, called aiming distance. For a uniform particle flow, the incident particles with aiming distance varying over an infinitesimal range are distributed on an infinitesimal annulus and share a nearly same scattering angle, and they also cover an infinitesimal annulus area after scattering. For same amount of incident particles, if they are distributed over a relatively large space after scattering, the distribution should be relatively disperse, so the density of scattered particle should be small. Since the flow is uniform, the area of incident cross section is proportional to the number of particles. Therefore, the area of incident annulus divided by the solid angle covered by the annulus after scattering has the meaning of scattered particle density, and it is exactly the definition of DCS.
If replacing the particle flow with a beam and the potential field with a refractive index field n=n(r), we can redefine the DCS in optics. When light goes through the concentric field with aiming distance b, its propagation direction will be deflected by an angle $\Theta $. Because the field is concentric, the deflecting angle is symmetric about the principal axis and only a function of aiming distance, $\Theta = \Theta (b)$. Consider an infinitesimal-width and annulus-shaped beam with uniform intensity comes from infinity, and its aiming distance is distributed at $[b,b + |{\textrm{d}b} |]$. After the refraction, the deflecting angle will be distributed at $[\Theta ,\Theta + |{\textrm{d}\Theta } |]$, as shown in Fig. 1. As the incident beam is restricted in an infinitesimal cross section with area $\textrm{d}\sigma = 2\pi b|{\textrm{d}b} |$, the emergence beam will also cover an infinitesimal space with solid angle $\textrm{d}\Omega = 2\pi \sin \Theta |{\textrm{d}\Theta } |$. Because the energy flux density of incident beam is uniformly distributed, the energy flux is proportional to the area of incident cross section $\textrm{d}\sigma$ and can act as a relative value for energy flux. To measure the density of refractive field, one can define the DCS ${\sigma _C}(\Theta )$ as the incident cross section per unit solid angle, namely
If a cylindrical uniform finite-width beam comes from infinity, ${\sigma _C}(\Theta )$ gives the angular distribution of relative light intensity at scattering angle $\Theta $. Once the relation $\Theta = \Theta (b)$ is known, the DCS can be obtained from Eq. (1).
Fig. 1. A parallel incident beam refracted by a concentric refractive index medium. All rays with aiming distance located at interval $[b,b + |{\textrm{d}b} |]$ will be finally refracted with deflecting angle located at interval $[\Theta ,\Theta + |{\textrm{d}\Theta } |]$.
Despite the definition of DCS in optics is formally consistent with that in mechanics, it is somewhat different. First, for a scattered particle in mechanics, its deflecting angle is not only related to the aiming distance, but also the initial velocity, which may affect its interaction time with the central force field. However, in geometric optics, there is no such a counterpart to “initial velocity” for an incident ray, and the so-called “time” in ray tracing procedure is also just a location parameter, not the physical time [14–16]. In fact, the light velocity is obviously not the counterpart to the particle’s velocity, because the light velocity only depends on the refractive index field (the same goes for the propagation time), while the initial velocity of particle can be independently set regardless of the force field. So the deflecting angle is purely determined by the aiming distance when the concentric field is given, which means the information of DCS at an arbitrary deflecting angle is totally implied in the refractive index field. Second, in the sense of ray trajectory, only the relative refractive index is significant. This also differs from mechanics, as multiplying the central force field by a constant will obviously change the trace of scattering, and consequently, change the DCS.
Now we give an explanation for DCS from the perspective of probability. Assume all the deflecting angles belongs to the interval $[{\Theta _1},{\Theta _2}]$, one can define the normalized differential cross section (NDCS) as
If the incident point of a zero-width beam presents a random fluctuation, so that it is uniformly distributed in a circle with center on the principal axis, the NDCS ${f_C}(\Theta )$ can be understood as the probability density of the emergence beam appearing at deflecting angle $\Theta $, that is, the probability of finding this beam at interval $[\Theta ,\Theta + |{\textrm{d}\Theta } |]$ is ${f_C}(\Theta )|{\textrm{d}\Theta } |$. If there are many incident rays coming from the infinity and are uniformly distributed in this circle, the field intensity at $\Theta $ will be proportional to ${f_C}(\Theta )$ (and also ${\sigma _C}(\Theta )$), as illustrated above.
Before calculating the DCS, a problem is to get the relation $\Theta = \Theta (b)$. This can be categorized as two cases. One is constant medium, and we can just simply use Snell’s law to obtain the deflecting angle. The other is concentric GRIN medium, in which the ray trace satisfies the optical Binet equation (OBE) [26]
3. Examples
In the following examples, we will give some specific concentric refractive index fields to calculate the DCS and explain its significance as a contrast to the simulated ray trace. It is worth to mention again that the refractive index function is a relative value, that is, if multiplying $n(r)$ by a constant, the ray trace will not be changed. This can be clearly seen in Eq. (3). So we do not necessarily require $n(r) \ge 1$, and the condition can be relaxed to $n(r) > 0$.
3.1 Spherical constant medium
The refractive index field of a spherical constant medium can be expressed as
Assume beam comes from infinity, the general ray trace is shown in Fig. 2. ${\varphi _1}$ and ${\varphi _2}$ are incident and refractive angles when light goes into the sphere, and also refractive and incident angles when light goes out of the sphere. Snell’s law gives $\frac{{\sin {\varphi _1}}}{{\sin {\varphi _2}}} = n$, and from the geometry we see $\sin {\varphi _1} = \frac{b}{R}$. One can get the deflecting angle
Consider the geometric relations $\textrm{cos}{\varphi _1}\textrm{d}{\varphi _1} = \frac{{\textrm{d}b}}{R}$ and $\textrm{cos}{\varphi _1}\textrm{d}{\varphi _1} = \frac{{\textrm{d}b}}{R}$, the differential of Eq. (5) is
Equation (6) shows that $\frac{{\textrm{d}\Theta }}{{\textrm{d}b}} > 0$, so the deflecting angle will increase when the aiming distance increases. From Eq. (5) and Snell’s law, one can solve $\tan {\varphi _1} = \frac{{n\sin (\Theta /2)}}{{n\cos (\Theta /2) - 1}}$ and $\tan {\varphi _2} = \frac{{\sin (\Theta /2)}}{{n - \cos (\Theta /2)}}$. According to Eqs. (1) and (6), the DCS of spherical constant medium can be derived
From Eq. (7), if $n = 1$, the condition ${\sigma _C}(\Theta ) \ge 0$ gives $\Theta \equiv 0$, the ray will not be refracted. This is obvious because the refractive index is constant in the whole space. For $n > 1$, the deflecting angle should satisfy $\frac{1}{n} \le \cos \frac{\Theta }{2} \le 1$. Therefore, a large refractive index gives a large range of deflecting angle.
Fig. 2. Schematic of the refraction in a spherical constant medium with radius R. Ray comes from infinity with aiming distance b and incident angle ${\varphi _1}$.
We depict the above results in Fig. 3. In Fig. 3(a), the ray traces for n=2, 3 and 4 are respectively simulated. In Fig. 3(b), the relation of deflecting angle and aiming distance is calculated. It is seen that the deflecting angle cannot reach $\pi$, that is to say, the light cannot return at the reverse direction of original path. Actually, to ensure ${\sigma _C}(\Theta ) \ge 0$ in Eq. (7), $\Theta = \pi$ will result in $n = \infty$, so $\pi$ is a limiting deflecting angle at extremely high refractive index and impossible to reach.
Fig. 3. . (a) Ray traces of a parallel beam going through the spherical constant media with radius R, for n= 2, 3 and 4, from left to right. The contour plot shows the refractive index. (b) Dependence of deflecting angle on aiming distance. (c) Dependence of DCS on deflecting angle (angular distribution of optical field).
Figure 3(c) shows the relation between the DCS and deflecting angle. As seen, for all cases, the DCS reaches the maximum at $\Theta = 0$, and decreases with increase of $\Theta $. It means the optical field are mainly distributed near $\Theta = 0$, and get weaker at large deflecting angle. This can be definitely observed in Fig. 3(a), where the ray trajectories are more “compact” near $\Theta = 0$, and get “sparse” when $\Theta $ increases. On the other hand, when refractive index n increase, the DCS decreases as well. This is also shown in Fig. 3(a), where a smaller refractive index results in a more “compact” emergent trajectory. Therefore, the DCS can reasonably act as a quantity to measure the relative angular distribution of field intensity of refractive rays.
3.2 Coulomb-like field
Now we show a GRIN field in which the ray trace is similar to the trace of a charged particle scattering in a Coulomb field. Consider a special refractive index function
where ${\varepsilon _\infty } > 0$ is a constant representing the permittivity at infinity, and β is a proportionality coefficient with dimension of square root of length unit. We call Eq. (8) as Coulomb-likefield, because from Eq. (3) we have $\frac{1}{2}\varepsilon ^{\prime}(r) = \frac{1}{2}\frac{\textrm{d}}{{\textrm{d}r}}{n^2}(r) = \frac{{{\beta ^2}}}{{{r^2}}} = {\beta ^2}{u^2}$, which is similar with square inverse repulsive force field in mechanics [25]. For a GRIN field, the trace should be solved from OBE. Substituting Eq. (8) into Eq. (3), we haveThe solution of Eq. (9) is
As Eq. (9) is a second-order equation, there should be another integral constant added on θ. But if we properly choose the direction of polar axis, this constant can be set as zero. Equation (10) represents a hyperbolic trace with its focus at field center, and e > 1 is the eccentricity. Compared with the equilateral hyperbolic trace calculated under Cartesian coordinate [11], this is a more general results. We depict the general trace in Fig. 4. To observe the geometric relations clearly, we have rotated the system to make the incident line is horizontal.
Fig. 4. . Schematic of ray propagation in a concentric GRIN medium with refractive index function $n(r) = \sqrt {{\varepsilon _\infty } - \frac{{2{\beta ^2}}}{r}}$. The ray comes from infinity with aiming distance b.
Note that $r\sin \varphi = b$ and $n = \sqrt {{\varepsilon _\infty }}$ at $r = \infty$, according to Eq. (3), the Fermat’s invariant k is
From Eq. (9), the distance between the directrix and focus is $\frac{{{k^2}}}{{e{\beta ^2}}}$, from which one can get the relation between aiming distance and eccentricity through a straightforward calculation
According to Eqs. (11) and (12), we can rewrite the ray trace Eq. (8) as
As shown in Fig. 4, the deflecting angle should be the supplementary angle of the included angle between the two asymptotic lines, namely $\Theta = \pi - 2{\theta _0}$ and ${\theta _0} = \arccos (1/e)$, which gives
Equation (15) is similar to Rutherford formula [23,25] since the GRIN field has a similar form with Coulomb field, butit is just valid for those traces going through the spacewhere the refractive index is a positive real number in Eq. (8). Nevertheless, for aiming distance that is not too small, or deflecting angle that is not too large (most traces satisfy this condition), it is out of problem.
The above results are depicted in Fig. 5. The red curves in Fig. 5(a) are the ray traces for ${\varepsilon _\infty } =$ 5, 7.5 and 10. To observe clearly, we have rotated the system so that the principal axis is horizontal, which is in accordance with the geometry in Fig. 4. The incident lines (also asymptotic lines) of all the rays are parallel to the principal axis. As expected, when the incident line is far away from the field center, the ray presents a small deflection. This is also clearly seen in Fig. 4(b). Furthermore, a large ${\varepsilon _\infty }$ also leads to a small deflection, since the variation of GRIN field also gets flat at large ${\varepsilon _\infty }$ according to Eq. (8). Figure 5(c) shows the relation between DSC and deflecting angle. It is seen that the DSC drastically decreases with the increase of deflecting angle. Therefore, for most rays, the deflection is slight, as the refractive index is nearly a constant far from the center. Only ray with small aiming distance can realize a large deflection. This can be also qualitatively observed in Fig. 5(a), where the “number” of rays with small deflecting angles (they are nearly parallel) is far more than that with large deflecting angles.
Fig. 5. . (a) Ray tracesof a parallel incident beam going through a concentric field with refractive index function $n(r) = \sqrt {{\varepsilon _\infty } - \frac{{2{\beta ^2}}}{r}}$, for ${\varepsilon _\infty } =$ 5, 7.5 and 10, from left to right. The radius of circular areais set as $4{\beta ^2}$. The contour plot shows the refractive index. (b) Dependence of deflecting angle on aiming distance. (c) Dependence of DCS on deflecting angle (angular distribution of optical field).
3.3 Spherical reflector
The above two examples are about the light refraction. However, if we treat the reflection as a special refraction, the concept of DSC also goes for a concentric reflection system. Here we can present a simple but meaningful example.
Consider a spherical reflector with radius R, as shown in Fig. 6. The parallel beam comes from left with aiming distance b and incident angle $\varphi$, and is reflected at the interface of the sphere. From the geometry, $b = R\sin \varphi$, and the deflecting angle can be expressed as
Fig. 6. . General trace for a ray reflected by a spherical reflector. The ray comes from infinity with aiming distance b and incident angle $\varphi$.
Equation (16) gives $\frac{{\textrm{d}\Theta }}{{\textrm{d}b}} ={-} \frac{2}{{\sqrt {{R^2} - {b^2}} }} ={-} \frac{2}{{R\sin (\Theta /2)}} < 0$, so the deflecting angle decreases when aiming distance increases. From Eq. (1), one can get the DCS of spherical reflector
What is interesting in this result is, despite the deflecting angle $\Theta $ depends on the incident point, the DCS is irrelevant to $\Theta $, which means the deflection is isotropic. That is, the reflected rays will be uniformly distributed at every direction.
We depict the ray traces and the variation of deflecting angle versus aiming distance in Fig. 7(a) and Fig. 7(b). To distinguish the incident and reflected rays, they are respectively marked as blue and red. The reflected ray traces and the variation of deflecting angle are very simple. Nevertheless, the presented traces in Fig. 7(a) are helpful to observe that the angular density of reflected rays are uniform, as predicted in Eq. (17).
Fig. 7. (a)Ray traces of a parallel beam reflected by a spherical reflector with radius R. (b) Dependence of deflecting angle on aiming distance.
We can make a further discussion for Eq. (17). As known, in paraxial region, the focal length of a convex mirror f equals half radius of the sphere. Therefore, for paraxial rays, Eq. (17) generates a useful relation between the focal length and the DCS
Actually, Eq. (18) gives an optical method to find the focus of reflection surface. Assume we can measure the angular distribution of reflective paraxial rays by some means, one can numerically calculate the DCS and thus get the focus of a reflection surface. It should be noted that this method is not only restricted to spherical surface, but also applicable for an arbitrary curved surface, just need to understand R as the curvature radius in Eq. (18). For those systems which obeys the paraxial optical theory [17,28], it is useful when one needs to check whether the surface is spherical (only need to check whether the energy flux is constant with the variation of angle), or measure the curvature of the surface. Inversely, focal length can also give the DCS at corresponding deflecting angle.
The above examples indicate that, the ray trace simulations in Figs. 3(a), 5(a) and 7(a) can only roughly present the density of refractive (or reflective) rays. If we want to obtain a quantitative index about the relative angular distribution of emergent field, the DCS is a decent tool. It can be directly calculated just from the relation between deflecting angle and aiming distance, and does not need to depict the ray traces first. As seen in Eqs. (7), (15) and (17), the density of emergent rays at deflecting angle $\Theta $ is clearly presented.
4. Conclusion
In conclusion, we have introduced the concept of DCS in a geometric optical system to describe the angular distribution of a parallel incident beam deflected by a concentric refractive index field. By using Snell’s law or OBE in a concentric constant or GRIN medium, we give a systematic procedure to solve the ray trace and its defecting angle, which provides the prerequisite for calculating the DCS. The information about relative density of deflected rays at an arbitrary direction can be completely obtained from the DCS, and it facilitates the analysis about the collective behavior of deflected rays in a concentric field. Our theory is applicable for light refraction and reflection problems in geometric optical systems with spherically symmetric refractive index field [12,13,26], If one can measure the angular distribution of optical field and find the collective ray traces to match this distribution. It is also feasible to consider the inverse problem [11,26,29] to solve the concentric field. Moreover, the DSC may provide a reference index for the design of beam deflector [30–32].
Funding
Scientific Research and Technology Development Program of Guangxi (AD23026159); Natural Science Foundation of Guangxi Zhuang Autonomous Region (2022GXNSFBA035529); Doctoral Foundation of Guangxi University of Science and Technology (21Z18); National Natural Science Foundation of China (11864032).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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