The hollow Gaussian beam propagation on curved surface based on matrix optics method
TThe hollow Gaussian beam propagation on curved surface based on matrix optics method
WEIFENG
DING, ZHAOYING
WANG Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] *[email protected]
Abstract:
In this paper, ABCD matrix is introduced to study the paraxial transmission of light on a constant gaussian curvature surface (CGCS), which is the first time to our knowledge. It is also proved that the curved surface can be used as the implementation of fractional Fourier transform, we further generalize that we can obtain the transfer matrix of an arbitrary surface with gently varying curvature by matrix optics. As a beam propagation example, based on the Collins integral, an analytical propagation formula for the hollow Gaussian beams (HGBs) on the CGCS is derived. The propagation characteristics of HGBs on one CGCS are illustrated graphically in detail, including the change of dark spots size and splitting rays. Besides its propagation periodicity and diffraction properties, a criterion for convergence and divergence of the spot size is proposed. The area of the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. In addition, we also study the special propagation properties of the hollow beam with a fractional order. Compared with HGBs in flat space, these novel characteristics of HGBs propagation on curved surface may further expand the application range of hollow beam.
1. Introduction
We know that under the general relativity theory, the gravitational field is described as space-time curvature, therefore, it is obvious that the study of the light beam propagation on curved space is meaningful for cosmology, such as the measurement of angular size of stars [1]. Moreover, the interaction between electromagnetic waves and curved space has a number of intriguing effects, such as Hawking radiation [2, 3] and Unruh effect [4]. On the other hand, although the condition of the curved space is difficult to implement in the laboratory, it appears many analogical theories and experiments of the curved space since the work by Unruh[5]. For instance, using Bose-Einstein condensation system to simulate the black hole Gibbons Hawking effect [6, 7]. Recently, the moving dielectric medium [8, 9] and the nonlinear Schrödinger Newton system[10, 11] are used to demonstrate the gravitational field caused by the curved space. Thus, the study of the transmission characteristics of light on curved space is not only a generalization of flat space, but also a reference for a large number of analogical experiments. Currently, two-dimensional surface is a common simplification model in the study of curved space, such as the two-dimensional Wolf effect of curved space [12], and other studies on geometric optics [13-15] and physical optics [16, 17] of two-dimensional manifolds. In our research, based on geometrical optics, ABCD matrix and Collins’ formula[18] are introduced to explore the light propagation characteristics on the curved surface under the paraxial approximation. The demonstration of optical transmission characteristics on curved surface is novel and widely applied. But in most cases, due to the complexity of curved surface algebra and differential geometry, one need a lot of numerical calculations, sometimes there are singularities, which greatly reduces the generality of the analytical expression and thus reduces its physical significance. In this paper, we shall first consider the beam propagation on a CGCS in paraxial approximation and then generalized to an arbitrary curved surface. The CGCS model, with an ssumption called surface of revolution, will be formed by the rotation of certain radius parameter ( ) h , which is the distance between one point on surface and the axis of symmetry. For a two-dimensional surface which is a sub-manifold imbedded into three-dimensional space, we define the expression of CGCS as ( ) ( ) cos h r h r = , where r and r are the parameters that describe the shape of surface. There is an entrance point and an exit point on this surface, the position , h h and transmission direction ', ' h h of the lights as shown in Fig. 1.(a)(b). According to Fermat’s principle on the surface [19], we know that light travels along geodesics, so from the geodesic equation, without considering the effect of refractive index, we can calculate the optical path under the coordinates we set. For the paraxial optical system with the propagation axis to be the direction, in our previous research, we deduced that the eikonal function of light rays between the incident point and the exit point had the following form[20] hr r rL L h h h h Or r r r = + − + + (1) Where is the angle of rotation between the incident point and the exit point, h is the arc length from the alternative point towards the maximum rotational circuit. L r = is used to describe the distance of the light along propagation axis.
2. The matrix optical method of curved surface
In general, curved surfaces are worlds apart from linear systems, but under the paraxial approximation and ignore the higher-order small terms, the expression of its function is the homogeneous expression of h and h . So, in order to better describe the propagation characteristics of light, we introduce the method of matrix optics which is the first time that matrix optics has been introduced into a curved surface system to our knowledge. We're going to start with CGCS and then generalize to arbitrary surfaces. The transformation relationship between the input and the output beams satisfies ' ' h hA Bh hC D = . (2) For a general eikonal function L r r L Ur Vr r Wr = + − + . the ray transmission angle can be regarded as the gradient of the eikonal function, thus the value of ABCD can be derived as ' /2 /' /
Lh U A Bh V BLh W D Bh = = = = = . (3) By substituting the eikonal expression Eq. (1) into Eq. (3), the ABCD matrix of the CGCS can be written as the follows. cos sin( ) 1 sin cos r rrA B r rM C D r rr r r = = − . (4) Fig. 1 (a) Definition of the surface coordinates and parameters. The blue line represents a light ray on the surface. (b) HGB propagation diagram on the surface, which shows the simulation of the light intensity distribution when n=3 and 𝑟 ∕ 𝑟 = 2 . (c) An arbitrary surface can be thought of as a combination of CGCS The determinant of this matrix is 1( ( ) det 1 M = ), which means the effect of the change in refractive index is not considered. Furthermore, the output light field ( ) E h can be obtained after the input light field ( )
E h passing through the CGCS system, which satisfies the Collins formula as shown in Eq. (5): ( ) ( ) ( )exp 2 + d .2 2 ikL ik ikE h e E h Ah h h Dh hB B − = − − (5) After presenting the ABCD matrix and the Collins formula, we find that the field expression of output beam has some similarities with the following general expression of the fractional Fourier transform (FFT): u ( ) u ( ) ( )exp 2 dtan sin e e x x x xx F x u x i hf f − += = − , (6) Where is the fractional order parameter, e f is the scaled parameter. Then we can easily get the correspondence: e f r = , / r r = . That means, we can control the fractional order parameter of the FFT by controlling the transmission distance (described here in terms of angle ). In a word, the CGCS can also be used as a fractional Fourier transformer, just as done by the gradient refractive index medium[21, 22] systems and Lohmann’s lens systems[23]. After we introduce the FFT, the Collins formula on the curved surface is more succinctly expressed as ( ) exp( ) [ ( )]2 rr ikE h ikr F E hB − = . (7) We have shown that the transmission of light over CGCS is equivalent to the fractional Fourier transform system. Then, due to its convolution property, we can consider the general multi-process optical transmission, which can also be clearly explained from the perspective of transformation matrix. A significant advantage of matrix is that it can deal with the combination of optical devices, that is, the transformation matrix is simply multiplied in order, so our study on the surface of constant Gaussian curvature can be extended to arbitrary curved surface. To ake the matrix more general, we can rewrite our Eq. (4) by using the transmission distance z r = and the Gaussian curvature r − = − . ( ) ( )( ) ( ) z zM z z z − − −= − − − − . (8) Research on arbitrary curved surface of the matrix, we identified the z axis is usually the geodesic of surface, and then we use the paraxial approximation. For a general curved surface, is not constant, but varies with the change of z , we remember to ( ) z . According to the idea of the element method, as Fig. 1 (c), the light transmission is divided into paragraphs, so the total can be subdivided into such small enough for a short period of them can be seen as part of CGCS. The total equivalent matrix is going to be the product of all of these matrices: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) m mmmn m n n m n m mm z z z zn nzzM z z M z n z z z zz n n → = → = − − − = = − − − − (9) The symbol of m n = means multiplying matrices in reverse order, the Eq. (9) expresses that the transmission distance is divided into n segments, and then the matrix of each segment is multiplied in reverse order and the limit is taken. And very cleverly, using the trigonometric properties and some approximation, we can say that Eq. (9) can be converted to the sum form: ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) cos sin, lim sin cos1cos sinsin n nm mnm mmmn n n nm m mm m mz zz zzz z z z znn nzM z z z z z z zn n nz dz z dzz dz = ==→ = = = − − − − − − − − − − − − − ( ) cos zz z dz − (10) Where ( ) ( ) / zz z dz z z = − from and says the average gaussian curvature. Eq. (10) is worth noting that the results need to be in surface when gaussian curvature change is not too big, will no longer be established or approximate conditions, is determined by the geometric properties of the surface itself, if we need to implement some optical transformation, the needed surface transmission system can be constructed using the Eq.(10). The matrix contains all the information of light transformation, which is a function of the transmission distance. Although different optical devices differ greatly in style, if the transmission matrix is the same, then the effect on the near-axis optical transformation is equivalent. . Propagation of hollow Gaussian beams on curved surfaces Next, in this paper, the introduction of ABCD matrix will be used to study a special case – the propagation of hollow Gaussian beam on CGCS. An analysis of a hollow Gaussian beam is a good starting point. We can not only visually see the propagation properties of the split light, but also study its hollowness, a property common to many other hollow beams (including vortex beams). In general, we set the electric field of the initial hollow Gaussian beam simply as follows[24]
211 1 2 ( ) n h hE h e − = . (11) In which, n represents the order of the HGB and the initial transverse spot size is h w n = . Using the method described in the first part, the Eq. (8) is substituted into the Collins’ formula, then an analytic solution is derived:
22 0 n niDkh ikrB k iAkE iB Bh kn HF n B AB − −+ = − + + − − . (12) Where ( ) x denotes to the gamma function.
1( , , )
HF a b x represents the Cumor confluence hypergeometric function and the first few terms of the expansion are (1 ) (1 )(2 )1( , , ) 1 [ ]2 (1 ) 6 (1 )(2 ) ax a a x a a a xHF a b x O xb b b b b b + + += + + + ++ + + (13) In order to facilitate the discussion of the propagation properties of hollow beam, the curved surface is flattened into a plane surface and a two-period image can be drawn, as shown in Fig. 2(a). Its horizontal coordinate is the angle of revolution for transmission and its vertical coordinate is h . Figure 2 (a) is the light intensity flattened maps of Gaussian beam and hollow Gaussian beam during the propagation on the curved surface. We explore that the hollow Gaussian beam can keep the hollow characteristic in the near field, but becomes no longer “hollow” in the far field, the area of dark region depends on n. For comparison, the propagation of hollow beams in a flat space is presented in Fig. 2(b). In the following, we focus on the beam divergence and convergence properties during its transmission. Before that, the study of Gaussian beam (i.e., n = ) can serve as a reference for us, the transverse waist width of the Gaussian beam is calculated as follows:
3, 0 / h n w k = = , (14) The change of waist width equals to
2, 0 0 0 2 h n dw r r rd r r k = = − . (15) From Eq.(15), we explore that the derivative is a periodic function with period / r r = , and the first half of the period is positive or negative depending on the parameter d r k = , which is called the divergence coefficient in this paper. This coefficient is closely related to the original beam size and the shape of surface. When d , the beam converges and then diverges, when d , the beam first diverges and then converges after emission, and in the case of d = , the variation of Gaussian beam spot size vanishes. The last wo transmission features are shown in the first raw plot of Fig. 2 (a). But as the order n increases, it does not work that way. The second-order moment is used to describe the overall mean spot width:
222 2 22 2 20, 222 2 h n d h E dh n r nw r nE dh −− + − = = + − − (16) So, we can use the change in average waist width as a criterion for overall convergence and divergence. According to Eq.(16), now the divergence coefficient changed from d to (8 1) / (16 1) d n n − − . Fig. 2. Propagation of the n-order HGBs on curved surfaces(a) and flat surfaces(b). On the curved surfaces, we calculated the propagation of a HGB of different order n ( n = ). Each plot holds two periods. As observed, with the increase of n , the maximum number of splitting rays increased in the process of transmission. Transverse comparison shows that d significantly affects the dark spot size. What we can see is that the surface accelerates the near-field and far-field conversion of light transmission, i.e., the fast conversion between the Fresnel diffraction region and the Fraunhofer diffraction region within half a transmission period. So, this method of creating far-field conditions over short distances can be considered for the manufacture of optical elements. In the case of Gaussian beam, the light intensity distribution along h direction has only one single peak, while for the propagation process of HGB, the transverse light intensity develops from bimodal image to multiple peaks. The maximum number of peaks is equal to n + , which can also be proved analytically by Eq.(12). In order to obtain the transverse positions of each peak, we can calculate the extreme value of light intensity during the half period based on Eq.(12), we demonstrate that their positions satisfy the following expression. ( )( ) ( ) ( ) ( )
22 2 222 20 0 ! 0! ! 2 ! 2 1 ! i jn ni j nh kh r n i n j i j += = − = − − + . (17) From Eq.(17), we can see that, except for the zero point at infinity, h has (4 1)th n + order at ost, that is, there are n + solutions corresponding to n + extremum points of light intensity, including n + peaks and n valleys, alternately. Considering the Gaussian beam, when the light diverges, the intensity on the propagation axis becomes weaker, and vice versa, but for the hollow light, the situation is more complicated. Its light intensity on the propagation axial is as follows: cot(2 1)!' 2 ( 1)! sin nnd dnh rn rE rn r − −−= + − = − . (18) According to Eq. (18), we divide the intensity shapes on the propagation axis into two types as shown in Fig. 3: A type and B type. There is only one maximum-intensity between two minimum-intensity positions for A type, but there are two maximum-intensity positions for B type. It is shown that the maximum intensity of type A and the second minimum intensity of type B are both at the half period position with / 2 r r = . The longitudinal length of the dark spot on the propagation axis w is defined as the distance from the center of the dark spot to half of the maximum light intensity (type A), or to the first maximum intensity (type B), as shown in Fig. 3. Fig. 3. Two types of light intensity distribution. The transverse coordinate represents the angular distance on the propagation axis, the longitudinal coordinate represents the light intensity. A-type (a) and B-type (b) are unimodal and bimodal structures within one period, respectively.
For type A, w has no analytic solution and satisfies the following equation: cot 2 12sin n nd d r wr nr wr − − − − + = + , (19) But based on Eq.(18), We can simply obtain the size of the dark spot of type B by calculating the maximum value of the light intensity as: d20 d r nw nr = + − , . (20) Here, we mainly focus on the beam propagation of type B. With the increase of r , the longitudinal size of the dark spot becomes smaller, but tends to a constant value r w nk = , which is consistent with the result of flat space[24]. It is obvious that the quare of the longitudinal widths is linear to n in flat space, while on curved surface, its hollow region will keep longer, and it will grow faster than flat space as n grows. However, the transverse size of our dark spot is consistent with that of the flat space, which indicates that our treatment of the surface follows such a phenomenon: the paraxial approximation ignores the surface property of light perpendicular to the propagation axis, but focuses on the surface property along the propagation axis. Fig. 4. Fractional order beam transmission images on the curved surfaces. The intensity of half-integer-order beams on the axis remain dark during its propagation, other fractional orders can cause asymmetry of transverse light intensity distribution.
All the above work is based on the fact that 𝑛 is an integer, but n can also be a fraction. Next, we focus on the optical transmission characteristics when n is a fraction. Figure 4 shows the examples when the value of 𝑛 is fraction and half-integer. As we can see, the most intuitive one is that when n is half integer, HGB remains hollow intensity along the propagation axis all the time, which is as same as the beam propagation on flat space. In addition, for other general fractions, the beam fringes in adjacent periods are distorted, causing periodic changes in the axial light intensity. Here, we can use the statistical skewness to describe the overall deviation of the light field distribution in two adjacent periods. For convenience, we can take the half period light field distribution as the skew analysis, and the results are presented in Fig. 5. Fig. 5. Skewness changes with respect to n . The parameter m denotes to an arbitrary integer. Here, we take the position of half period within two adjacent periods as an index to describe the overall skewness. We find that, with the change of n , the skewness oscillates and reaches 0 when n is an integer and a half integer. Moreover, the variation of skewness is complementary. When n is an arbitrary number, we can also get a more general magnitude of the light intensity on the axis. The light intensity on the propagation axial is as follows: ( ) cos 1 2 cot2' sin nnd dh n n rrE rr − −= + + = . (21) From Eq. (21), it can be seen that, due to the existence of the factor cos n , the light intensity on the propagation axis oscillates with n , and when n is half integer, the light intensity on the propagation axis is zero everywhere.
4. Conclusion
In conclusion, we have introduced a ABCD matrix of the CGCS under paraxial approximation, and find one of its application as a fractional Fourier transform system. Then, we extend the results to arbitrary surfaces by using the properties of ABCD matrix. As the focus of the whole manuscript, we derived an analytical propagation formula for HGBs on CGCS by using the Collins integral and plotted the transmission properties of HGB. We found the periodic divergence and convergence of light and the light is split into some rays in the process of transmission. The longitudinal dark area depends on the order n, the original beam size and the shape of curved surface, which is totally different from that of flat space. The transverse dark size is totally consistent with that of flat space. Moreover, we extend the optical transmission to a more general case where n is a fraction. The ABCD matrix has been shown to be an ideal and convenient model with which to describe beam propagation on CGCS. In addition, our matrix processing method is not limited to the propagation on the selected axis. Since the misaligned optical system can be further described by using a tensor matrix, in the future, we can also deal with the cases of eccentricity or rotation of HGB on curved surface[25]. Disclosures . The authors declare no conflicts of interest.
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