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Abstract
Starting from the wave equation with a non-zero space curvature, a generalized coordinate-independent expression for the evolution of a light beam on a curved space is derived. By defining the propagation axes, the expression reduces to integrable Green functions without an inevitable singular point. With a Gaussian incident field, the stationary status and refocusing effect of the light field on different shapes of curved surfaces are discussed. Different from a constant diffusion behavior in a flat space, the field experiences a periodical diffraction and refocusing spontaneously with no additional optical elements. To be more specific, we noticed that the laser field on a curved surface experiences a fractional Fourier transform, with a propagation angle to be the transform order. We hope our theoretical results can provide some references for the practical application in a curved surface space.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The characteristics of light beam propagation upon curved surface is now drawing increasing attention. It contains a vast range of applications towards versatile topics. In astrophysics, the Hanbury Brown and Twiss (HBT) measurement is a method to examine the angular size of stars, and it can be used to evaluate the curvature of surface [1,2]. Similarly, Wolf effect is developed on curved surface, which could also be a method of surface shape evaluating [3]. Surface Plasmon Polaritons (SPPs) are naturally surface bounded wave, for non-planer surface, its dynamics obey the generalized wave function as well [4]. The equivalence between the metric of surface and the index gradient of waveguide was discussed through conformal mapping, one subtle example is the comparison between Maxwell’s fisheye and the sphere [5–8]. For classical optics, the wave length is microcosmic, and it is acceptable to treat the light transmission on curved surface under the frame of geometrical optics [6,7,9–13]. However, with the increasing research demand for microstructures whose spatial size is of the same order with optical wavelength, the field must be treated by the means of wave optics. Wave optics on the curved space is nowadays a widely discussed topic, not only for its analogs of general relativity but also for its potential application in micro elements. To this end, Wang etc. have investigated the wave dynamics on toroidal surface by taking advantage of the principle of angular momentum conservation [14]. Batz etc. have investigated wave dynamics on constant Gaussian curvature surface with the method of an attempting solution [15–17]. The discussions about curved manifold are performed in various realms [18–23]. Recently, some experiments for optical field on curved manifolds were performed [24–26].
But until now, there is still no proper method for making a link between geometric optics and wave optics on curved surface. One may purse a differential geometric method for geometrical optics then turn to a Sturm-Liouville (S-L) eigenvalue problem for wave optics, separately. In traditional Euclidean space, a widely used method of linking these two concepts is the WKB approximation [27], which is a mathematical processing of neglecting over high order derivative term(s) of amplitude in wave equation. The WKB approximation is a treatment upon flat manifold originally, to expand its usage to generalized manifolds, some efforts should be made. Two inspiring examples are the assumptions of surface of revolution [15] and the radially symmetric media [20]. The basic method is to take the effect of space curvature into the derivative operator of wave function. However, these considerations would not be available near the poles of rotational symmetric system, limited with the undefinition of spherical or polar coordinate system at poles. Unfortunately, the currently used method allows non-geodesic lines to be the propagation axes, which is not tally with the facts and it is inevitable if adopting WKB approximation.
In this paper we conceived a new thought for this topic. It is to count in the effect of specific surface curvature after having solved the wave equation. A generalized WKB approximation is still utilized but before the definition of specific coordinate system. By defining the propagation axes, we derived integrable Green functions for light beams. It’s a concise and effective interpretation for the propagating field on curved surface. Our theory is confirmed by the flat space limit. The revolution of typical Gaussian beam is demonstrated thoroughly, a stationary state and the refocusing effect of light field are revealed. For a brand-new discovery, we found that a fractional Fourier transform can be perfectly performed on the surface of revolution without any additional optical elements, which may provide potential applications in signal processing. Finally, as an outlook, due to the complete description of optical phase in our method, the evolution of ultrashort pulses on curved surface can be readily further evaluated.
2. Wave function on a curved surface
A surface can always be treated as a hypersurface imbedded into a background manifold. Starting from pseudo-Riemannian space, the wave equation written with abstract indices gives [28],
where super- and sub-script obey the Einstein summation convention, that when an index variable appears twice in a single term and is not otherwise defined, it implies summation of that term over all the values of the index. The index $a$ goes through over four independent coordinates $({{q_1},{q_2},{q_3},{c^{ - 1}}t} )$. ${J_b} = ({c\rho ,{\textbf J}} )$ is the covariant electric current vector, where $\rho $ is the electric charge density and ${\textbf J}$ is the electric current vector. ${A_b}$ is the covariant vector potential. For two-dimensional surface, taking the separation with respect to the tangential part and the normal part $A({{q_1},{q_2},{q_3},t} )= {A^T}({{q_1},{q_2},t} ){A^N}({{q_3},t} )$ [29], in accompanying with the thin layer assumption, the wave function on two-dimensional sub-manifold writes: where H is the mean curvature while K is the Gaussian curvature. After the separation, the index a now runs over $({{q_1},{q_2},{c^{ - 1}}t} )$. For macroscopic radii of curvature, the influence of the second term on left of Eq. (2) can be neglected. By using the relationship between the field and vector potential and neglecting the polarization effects, the wave equation for propagating field on two-dimensional surface can be simplified as follows, with the index a now runs over $({{q_1},{q_2}} )$. Use the ansatz $E = C{e^{ikL}}$, substitute it back to Eq. (3) then we getUnder the WKB approximation which works on trivial manifolds, the second derivation term over amplitude should be neglected, since it’s a slowly varying term. Extract the real part and imagine part of Eq. (4), we obtain the following expressions.
Here we should emphasize, until now, in our analysis, we haven’t specially referred to the metric of surface, and coordinates $({{q_1},{q_2}} )$ remains undefined. In Eq. (5), it is always possible to define the first coordinate ${q_1} = L$ itself and the other coordinate perpendicular to it, writes ${\partial _{{q_2}}}L = 0$. The amplitude term C then satisfies
where g is the determinant of space metric upon the argument ${q_1}$ and ${q_2}$. ${q_1}$ or L in Eq. (5) was named the eikonal in flat space, we follow this note on curved surface. It is a key in this section to connect the optical phase with the definition of curved coordinate system. For a two-dimensional surface, this eikonal can be defined as the length of trajectory curve. Utilizing the principle of Green function, the distribution of propagation field will be derived as follows. All the ingredients in Eq. (7) now have a unambiguous physical correspondence.3. Evaluation of eikonal $L$
The space curvature could be resulted from various of elements, for example the shape of surface, the refractive index gradient, and gravity field, etc. Thanks to the equivalence in space metric between those circumstances, the evolution of field behaves similarly [30,31]. A common and useful type of surface is called surface of revolution, whose space metric writes
Drawn as sphere but without loss of generality, the sketch of surface and the corresponding coordinate system grids are defined in Fig. 1. The propagation trajectories are denoted as the red arrows, with A to be the incident point and B the exit point. $\theta $ is the angle of rotation, h is the arc length from the alternative point towards the maximum rotational circuit, and $\rho (h )$ is the radius of surface toward the rotation axis. For a specific surface with constant Gaussian curvature, the reliance between $\rho $ and h is given in Eq. (9),
where $1/r = {K^{1/2}}$ is the square root of Gaussian curvature and ${r_0}$ is the radius at the equator of surface of revolution. With this parametrization, the condition of constant Gaussian curvature is fulfilled. For $r = {r_0}$ one gets the sphere, for $r > {r_0}$ a spindle type, and for $r < {r_0}$ a bulge type surface. Once the space metric is given, the geodesic ray function or the optical trajectory writes
Fig. 1. Coordinate system definition of surface of revolution. Red arrowed lines indicate the vector of wave field.
The eikonal function L is the arc length between point A and $B$, with the help of Eq. (8) and Eq. (11), an analytical expression between variables $L,\theta $ and ${h_{1,2}}$ is derived and there is no need to retain an intermediate variable ${\varphi _{1,2}}$. The exact expression for eikonal L is complicated. However, For the paraxial optical system and for practical conditions of use, eikonal L could be expanded as a power series as given in Eq. (12).
The Eikonal function Eq. (14) is the main result of this section. Regardless of the reliance of eikonal definition on coordinate system, this calculation is flexible in conformal mapping on curved space. We are now able to describe several intuitive properties of the light propagation on two-dimensional curved surface.
We are now able to describe an intuitive property of the light propagation on two-dimensional curved surface. For flat surface limit when r approaches infinity, it’s able to define ${r_0}\theta = z$ as the propagation distance. ${r_0}\theta /r$ remains a small angle in this limit and the trigonometric functions are readily to expand in power series with the first order retained. Then the eikonal function Eq. (14) can be simplified as
which is the eikonal function under paraxial approximation in flat surface. Please note that, in the realm that the radius of curvature is much larger than the wavelength, in our discussion, it is the Gaussian curvature which determines the field propagation. In this regard, the radius at equator undergoes no restriction, namely, the field experience similar dynamics on flat surface and cylindrical surface.For a more complete description, the variation of amplitude term is also investigated. From Eq. (6), the amplitude term can also be defined simply from the determinate of space metric, or equivalently speaking, the Gaussian (intrinsic) curvature. After a little algebra, for surface with constant Gaussian curvature $K = 1/{r^2}$ defined in this paper, the amplitude term gives $C = {r^{ - 1/2}}{\sin ^{ - 1/2}}({L/r} )$. When choosing the equator to be the propagation axis, it can be naturally rewritten as $C = {r^{ - 1/2}}{\sin ^{ - 1/2}}({{r_0}\theta /r} )$.
4. Field propagation
We now consider a Gaussian incident field which is a most common type in laboratory case $E({{h_1}} )= {E_0}\textrm{exp} ({ - {h_1}^2/{\sigma_0}^2} )$, ${\sigma _0}$ is the initial beam size. On the basis of Eq. (7) and Eq. (14), an analytic expression of propagation field with Gaussian shape on curved surface is deduced
A special case of the initial beam size is ${\sigma _0} = {({2r/k} )^{1/2}}$. For this type of incidence, no matter how the surface shape with constant-Gaussian-curvature is, the beam size will keep a constant ${\sigma _\theta } = {\sigma _0}$ during the propagation and the amplitude of beams remains unaltered, as shown in Fig. 2. The stationary solution only depends on the relationship between the wave number and the Gaussian(intrinsic) curvature and have nothing to do with the extrinsic curvature, that how this surface is embedded into background manifold. The figures are plotted with a certain amount of transparency to show the revolution properties of field. On the other aspect of understanding, $k{\sigma _0}^2/2$ is the Rayleigh(diffraction) length of Gaussian beam, while r is the principal radius of surface, the propagation stabilization requires the equivalence between them. At the situation of stable transmission, the phase on axis varies as ${\phi _{axial}} = ({1/2r + k} ){r_0}\theta $, which principally explores a Gouy phase shift beyond an ordinary propagation phase. Different from a logarithm relationship on flat space, the Gouy phase now behaves a linear superposition over propagation distance.
Fig. 2. The stationary Gaussian light beam on surface of revolution. The initial beam size is ${\sigma _0} = {({2r/k} )^{1/2}}$. The relationship of equivalent radius of curvature r and radius at equator ${r_0}$ satisfies (a)$r = 0.5{r_0}$, (b)$r = 0.75{r_0}$, (c)$r = 1.5{r_0}$ and (d)$r = 2{r_0}$, respectively.
Another special case is the beam propagation on sphere (Fig. 3). The sphere is a kind of surface with perfect isotropy, and this characteristic is inherited by the light field on it. No matter how the incident field is, it will return to the initial status after one circle of traveling. This corresponds to the fact that the spherical harmonic function is a complete set on sphere, any type of field could be expanded as a linear superposition of it. The refocusing effect is clearly revealed in Fig. 3 that the field diffuses and focuses spontaneously, and this property holds for alternative incident field. From this refocusing effect, we find that the eigenvalue condition is contained naturally in the geometric of curved surface. As proved in geometrical optics [11] and now in wave optics as a major progress, the surface of revolution which satisfies $r = m{r_0}$ will always show its translational symmetry after going $m/2$ circles. Two typical stationary solutions with limited circles are plotted in Fig. 4(a) for $r = 0.5{r_0}$ and Fig. 4(b) for $r = 1.5{r_0}$, where the field undergoes a retrieval $2/m = 4$ and $2/m = 4/3$ times per circle, respectively. The self-interference effect is neglected in our simulation.
Fig. 3. The propagations of non-stationary light beam on sphere. The initial incident positions are noted in red, with incident field as a Gaussian shape whose initial beam size (a)${\sigma _0} = 0.2\sqrt {2r/k} $ and (b)${\sigma _0} = 5\sqrt {2r/k} $.
Fig. 4. Two typical examples of stationary solutions on the surface of revolution, (a)$r = 0.5{r_0}$ and (b)$r = 1.5{r_0}$. It shows that the translational symmetric circuit number are (a)$m = 0.5$ and (b)$m = 1.5$, respectively, indicating that the field undergoes a retrieval (a)$2/m = 4$ and (b)$2/m = 4/3$ times per circle.
A more subtle feature of the curved surface is its possibility of performing a fractional Fourier transformation. The emergent field $E({{h_2}} )$ with an incidence $E({{h_1}} )$ after a propagation angle $\theta $ gives
On the other hand, the definition of fractional Fourier transformation is [33]
By means of the substitution ${x^\prime } = {h_1}{(k/r)^{1/2}}$, $x = {h_2}{(k/r)^{1/2}}$ and $\alpha = {r_0}\theta /r$, in spite of a constant coefficient and an axial propagation phase, they are essentially the same thing. When $\alpha = \pi /2$ and $- \pi /2$, Eq. (18) regains the usual Fourier transforms, and it corresponds to a propagation distance $L ={-} \pi r/2$ and $\pi r/2$ on curved surface. As a close connection between the fractional Fourier transform and a parabolic index profile microlens [34], Wigner rotation [35,36], etc., we now find a new correspondence, the curved surface. As an example, the diffraction pattern of light field with a plane incidence is plotted in Fig. 5, the broken red line indicates an imaginary diagram. The revolution properties of transformation and diffraction pattern on curved surface are clearly revealed. This transform property may provide potential application in signal processing. By blocking-up certain areas which corresponds to the cur-off action in classical time-frequency analysis, the curved surface can play a role as an spatial optical filter. This processing occurs at the transform between lateral distribution and lateral spatial spectrum, and it is actually a two-dimensional spatial wave filter.
5. Conclusion
We focused on the light beam propagation on two-dimensional curved surface. In this paper, we strictly reviewed the wave equation with non- Euclidean metric elements, then performed the paraxial approximation before giving specific metric tensor. It is evident that on curved manifold, the light field could still be represented as phase and amplitude term, separately. No need to define an additional concept “proper length” on curved surface, the complete description of light field is derived. The major advantage of our method over previously used ones is that both the illumination evolution and the interference properties can be described. When assuming an infinite diffraction length, the light degenerates to the realm of geometrical optics.
To be more visualized, we gave some plots of field on different types of surface. Thanks to the complete description for Green function, we are able to talk about the periodicity and diffraction properties. Finally, we discovered an intriguing feature that laser field on curved surface experiences a fractional Fourier transform. This may provide more possibilities in laser optics and measurements in condensed matter and astronomical physics.
Funding
National Key Research and Development Program of China (2017YFC0601602).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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