The transmission of structured light fields in uniaxial crystals employing the Laguerre-Gaussian mode spectrum

Fangqing Tang; Xiancong Lu; Xiancong Lu; Lixiang Chen; Lixiang Chen; Lixiang Chen

doi:10.1364/OE.27.028204

1. Introduction

The orbital angular momentum (OAM) of light is a different degree of freedom with respect to the spin angular momentum (SAM), which is related to the polarization of beam [1]. Allen et al. shows in 1992 that the Laguerre-Gaussian (LG) beam has a well-defined OAM [2]. Since then, the optical vortices have been a hot topic in the field of optics [3–7]. There are many ways to generate the OAM beams in experiments: passing Gauss beams through the spiral phase plate [8,9], loading phase hologram to the beams through a spatial light modulator (SLM) [10], the mode conversion method of cylindrical lens [11], the q-plate mode transformation method using the birefrigent effect of crystals [12,13], and the Fork grating method which can diffract Gauss beams into OAM beams of different orders [14].

Due to the orthogonality and infinity of OAM beams, they are widely used as the carriers of information. For example, the optical communication technology based on OAM has been greatly developed [15–19], since the first optical system for free-space OAM optical communication was built by Miles Padgett team in 2004 [20]. The concept of Digital spiral imaging (DSI) was proposed by Torner et al. [21], in which the information of objects in remote sensing is identified by analyzing the LG spectrum of scattered object. It is further demonstrated by Uribe-Patarroyo et al. that the correlated OAM states are also useful in the object identification [22]. Recently, the encrypted encoding and decoding of information using LG spectrum has been explored [23].

On the other hand, the propagating behavior of optical beam in anisotropic media is quite different from that in isotropic media or free space [24]. The most prominent phenomenon in anisotropic media is the spin-orbit coupling of angular momentum. Ciattoni and his collaborators have shown that, for a paraxial beam propagating along the optical axis of uniaxial crystal, the total angular momentum including SAM and OAM is conserved along the propagating direction [25], because of the rotational symmetry with respect to the optical axis. Meanwhile, the components of SAM and OAM can be exchanged each other upon propagating [26]. For the case of LG beam in uniaxial crystal, Ciattoni et al. derived the analytical expression of propagation when the azimuthal index of LG beam is zero, i.e., the circular symmetric LG beam [27]. Volyar et al. found a solution for the LG beams with real argument and azimuthal index $l = 1$ [28]. The propagating solutions of LG beams with arbitrary radial and azimuthal index were obtained in [29]. Besides, the propagation of nonparaxial beams in uniaxial crystals, whose waist is on the same order of the wavelength, are also well studied before [30–33].

In this paper, we use the angular-spectrum method and series expansion to obtain full expressions for arbitrary LG beam propagating in uniaxial crystals. The comparisons with results in [29] are performed. Based on this, we propose a scheme for the image transmission through the uniaxial crystals. The image is firstly decomposed into LG spectrum, and then each LG component, whose analytical solution is known, individually propagates through the uniaxial crystals. The final output image is the superposition of all OAM components. We will use the trefoil image as an example to demonstrate the effectiveness of our scheme.

2. Paraxial propagation of Laguerre-Gaussian beams In uniaxial crystal

The expressions for a paraxial Gaussian beam propagating along the optical axis of a uniaxial crystal are deduced in [24,26], using the angular-spectrum representation. We will follow the same steps to obtain the expressions for the standard Laguerre-Gaussian Beams. We consider an unbounded uniaxial crystal with ${n_o}$ and ${n_e}$ being ordinary and extraordinary refractive indices respectively, and choose the z axis to be the optical axis of the crystal. Assume that a right-hand circularly polarized beams bearing topological charge l is incident on the transverse surface of uniaxial crystal. The boundary field distribution at $z = 0$ is

(1)$${{\bf E}_ \bot }(r,\varphi ,0) = \exp (il\varphi ){E_ + }(r){{\bf e}_ + },$$

in which ${{\bf e}_ + }$ (${{\bf e}_ - }$) is the unit vector of right (left) handed circular polarization. The circular component of the optical field at a distance z obeys the following equation [26],

(2)$${{\bf E}_ \pm }(r,\phi ,z) = \exp (i{k_0}{n_o}z){A_ \pm }(r,\phi ,z){{\bf e}_ \pm },$$

where ${k_0} = {\omega \mathord{\left/ {\vphantom {\omega c}} \right.} c}$ is the wave number in vacuum and ${A_ \pm }$ are the slowly varying envelopes,

(3)$${A_ + }(r,\varphi ,z) = \exp (il\varphi )F_ + ^{(l)}(r,z),$$

(4)$${A_ - }(r,\varphi ,z) = \exp [i(l + 2)\varphi ]G_ + ^{(l + 2)}(r,z),$$

with

(5)$$F_ + ^{(l)}(r,z) = \pi \int\limits_0^{ + \infty } {dkk[\exp ( - \frac{{i{k^2}}}{{2{k_0}{n_o}}}z) + \exp ( - \frac{{i{k^2}{n_o}}}{{2{k_0}n_e^2}}z)]} {J_l}(kr)\tilde{E}_ + ^{(l)}(k),$$

(6)$$G_ + ^{(l + 2)}(r,z) = \pi \int\limits_0^{ + \infty } {dkk[\exp ( - \frac{{i{k^2}}}{{2{k_0}{n_o}}}z) - \exp ( - \frac{{i{k^2}{n_o}}}{{2{k_0}n_e^2}}z)]} {J_{l + 2}}(kr)\tilde{E}_ + ^{(l)}(k),$$

(7)$$\tilde{E}_ + ^{(l)}(k) = \frac{1}{{{{(2\pi )}^2}}}\int\limits_0^{ + \infty } {drr} {J_l}(kr)\int\limits_0^{2\pi } {d\varphi \exp ( - il\varphi )} {E_ + }(r,\varphi ,0).$$

Equations (2)–(7) form the whole propagating theory of vortex beam in uniaxial crystal. The exact solutions can be obtained once the integrals in Eqs. (5) and (6) can be performed analytically. For a standard right-hand circularly polarized LG beams, the boundary electric field distribution is

(8)$${\bf E}(r,\varphi ,0) = L{G^{pl}}{{\bf e}_{\bf + }} = \exp (il\varphi ){(\frac{{\sqrt 2 r}}{{{v_0}}})^{|l|}}\exp ( - \frac{{{r^2}}}{{{v_0}^2}})L_p^{|l|}(2\frac{{{r^2}}}{{v_0^2}}){E_0}{{\bf e}_{\bf + }},$$

where ${v_0}$ is the beams waist, $L_p^l$ is the Laguerre polynomial, and ${E_0} = {{\sqrt {{{2p!} \mathord{\left/ {\vphantom {{p!} {\pi (p + |l|)!}}} \right.} {\pi (p + |l|)!}}} } \mathord{\left/ {\vphantom {{\sqrt {{{p!} \mathord{\left/ {\vphantom {{p!} {\pi (p + |l|)!}}} \right.} {\pi (p + |l|)!}}} } {{v_0}}}} \right. } {{v_0}}}$ is a normalizing constant. Note that the value of l in Eq. (8) can be negative. Substituting Eq. (8) into Eq. (7), we have

(9)$$\tilde{E}_ + ^{(pl)}(k) = \frac{{{E_{0}(-1)^{p+(|l|-l) / 2}}}}{{2\pi }}{(\frac{{{v_0}}}{{\sqrt 2 }})^{|l|+ 2}}{k^{|l|}}\exp ( - \frac{{v_0^2{k^2}}}{4})L_p^{|l|}[\frac{{v_0^2{k^2}}}{2}].$$

Inserting Eq. (9) into Eq. (5) and using the following integral formula [34]

\begin{array}{l} \int\limits_0^\infty {{x^{l + 1}}} \exp ( - \beta {x^2})L_p^l(\alpha {x^2}){J_l}(rx)dx\\ \quad \quad \quad = {2^{ - l - 1}}{\beta ^{ - l - p - 1}}{(\beta - \alpha )^p}{r^l}\exp ( - \frac{{{r^2}}}{{4\beta }})L_p^l[\frac{{\alpha {r^2}}}{{4\beta (\alpha - \beta )}}], \end{array}

it is straightforward to get the analytical expression for $F_ + ^{(l)}$, which is

(10)$$\begin{array}{l} F_ + ^{(pl)}(r,z) = F_{o + }^{(pl)}(r,z) + F_{e + }^{(pl)}(r,z),\\ F_{o + }^{(pl)}(r,z) = \frac{{{E_0}}}{2}{(\sqrt 2 r)^{|l|}}\frac{{{v_0}^{2p + |l|+ 2}{{[v_o^2(z)]}^{2p}}}}{{{{[{Q_o}(z)]}^{2p + |l|+ 1}}}}\exp ( - \frac{{{r^2}}}{{{Q_o}(z)}})L_p^{|l|}[\frac{{2{r^2}}}{{v_o^2(z)}}],\\ F_{e + }^{(pl)}(r,z) = \frac{{{E_0}}}{2}{(\sqrt 2 r)^{|l|}}\frac{{{v_0}^{2p + |l|+ 2}{{[v_e^2(z)]}^{2p}}}}{{{{[{Q_e}(z)]}^{2p + |l|+ 1}}}}\exp ( - \frac{{{r^2}}}{{{Q_e}(z)}})L_p^{|l|}[\frac{{2{r^2}}}{{v_e^2(z)}}], \end{array}$$

with

(11)$$\begin{array}{l} {v_o}(z) = {v_0}{[1 + {(\frac{{2z}}{{{k_0}{n_o}v_0^2}})^2}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}},\quad {Q_o}(z) = v_0^2 + i\frac{{2z}}{{{k_0}{n_o}}},\\ {v_e}(z) = {v_0}{[1 + {(\frac{{2{n_o}z}}{{{k_0}n_e^2v_0^2}})^2}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}},\quad {Q_e}(z) = v_0^2 + i\frac{{2{n_o}z}}{{{k_0}n_e^2}}. \end{array}$$

It can be seen that $F_ + ^{(pl)}$ consists of two parts, each of which is similar to the propagating solution of the LG beam in isotropic medium with proportional coefficients ${n_o}$ and ${{n_e^2} \mathord{\left/ {\vphantom {{n_e^2} {{n_o}}}} \right.} {{n_o}}}$, respectively. Note that for negative value of l, $F_ + ^{(l)} = F_ + ^{(|l|)}$.

In order to calculate the $G_ + ^{(l + 2)}$ in Eq. (6), we expand the Laguerre polynomial $L_p^l$ into power series and use the integral formula [34]

(12)$$\begin{array}{l} \int\limits_0^\infty {{x^\mu }{e^{ - \alpha {x^2}}}{J_v}(\beta x)} dx = \frac{{\Gamma (\frac{\mu }{2} + \frac{v}{2} + \frac{1}{2})}}{{\beta {\alpha ^{\frac{1}{2}\mu }}\Gamma (v + 1)}}\exp ( - \frac{{{\beta ^2}}}{{8\alpha }}){M_{\frac{1}{2}\mu ,\frac{1}{2}v}}(\frac{{{\beta ^2}}}{{4\alpha }}).\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\quad [{\mathop{\rm Re}\nolimits} \alpha > 0,{\mathop{\rm Re}\nolimits} (\mu + v) > - 1] \end{array}$$

The final expression for $G_ + ^{(l + 2)}$ can be written as

(13)$$\begin{array}{l} G_ + ^{p(l + 2)}(r,z) = G_{o + }^{p(l + 2)}(r,z) - G_{e + }^{p(l + 2)}(r,z),\\ G_{o + }^{p(l + 2)}(r,z) = \frac{{{E_0}}}{{2r}}\exp ( - \frac{{{r^2}}}{{2{Q_o}(z)}}) \times \\ \quad \quad \quad \quad \quad \;\;\sum\limits_{m = 0}^p {\frac{{{K^{pl}}(m)}}{{Q_o^{(|l|+ 2m + 1)/2}(z)}}} {M_{(|l|+ 2m + 1)/2,\,|l + 2|/2}}(\frac{{{r^2}}}{{{Q_o}(z)}}),\\ G_{e + }^{p(l + 2)}(r,z) = \frac{{{E_0}}}{{2r}}\exp ( - \frac{{{r^2}}}{{2{Q_e}(z)}}) \times \\ \quad \quad \quad \quad \quad \;\;\sum\limits_{m = 0}^p {\frac{{{K^{pl}}(m)}}{{Q_e^{(|l|+ 2m + 1)/2}(z)}}} {M_{(|l|+ 2m + 1)/2,\,|l + 2|/2}}(\frac{{{r^2}}}{{{Q_e}(z)}}), \end{array}$$

in which ${K^{pl}}(m)$ is given by

(14)$${K^{pl}}(m) = {( - 1)^{m + p + \frac{{|l|- l}}{2}}}\left( {\begin{array}{{c}} {p + |l|}\\ {p - m} \end{array}} \right)\frac{{[(|l + 1|+ {\delta _{l, - 1}} + m]!}}{{2m!(|l + 2|)!}}{(\sqrt 2 {v_0})^{|l|+ 2m + 2}},$$

and ${M_{\lambda ,\mu }}(z)$ is the Whittaker functions. The ${\delta _{l, - 1}}$ in Eq. (14) equals 1 if $l = - 1$, otherwise it equals 0. For the special case of $l = 0$, Eq. (10) and Eq. (13) reduce to the results obtained by Ciattoni et al. in [27]. By doing some transformations, we also can recover the Eq. (27) and Eq. (30) in [29]. Especially for the case of $l = - 1$, the right-handed polarized light can be used to generate the cylindrically polarized fields by sharply focusing, as shown in [35,36].

Based on the expressions in Eq. (10) and (13), the propagating properties of any LG beams in uniaxial crystal can be obtained. We plot in Fig. 1 the optical intensity of a typical LG beam with indices $p = 3$ and $l = - 6$. The parameters used in Fig. 1 are chosen as follows: the wavelength $\lambda = 0.5\mu m$, the width of waist ${v_0} = 4.5\mu m$, refractive index ${n_o} = 1.658$, and ${n_e} = 1.486$. It can be seen that, as propagating away from the input plane, the left-handed component gradually appears which carries a topological vortex.

Fig. 1. The light intensity ${|{{E_ + }} |^2}$ (upper row) and ${|{{E_ - }} |^2}$ (lower row) of a LG beam ($p = 3,\;l = - 6$) at several propagating distances z in the uniaxial crystal. (a) and (e):$z = 0$, (b) and (f):$z = 100\mu m$, (c) and (g):$z = 200\mu m$, (d) and (h):$z = 300\mu m$.

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3. The conversion of energy and spin-orbital coupling

In order to investigate the energy conversion between different components, we look at the energy of light field in the transverse plane, which is

(15)$${W_ \pm }(z) = \int\limits_0^\infty {drr} \int\limits_0^{2\pi } {d\varphi } |{A_ \pm }(r,\varphi ,z){|^2}.$$

For the right-hand polarized and normalized LG beam at the input plane ($z = 0$) as shown in Eq. (8), we have $W_ + ^{(l)}(0) = 1$ and $W_ - ^{(l)}(0) = 0$. By inserting Eq. (5) and Eq. (6) into Eq. (15) and after some algebra, we get

(16)$$W_ \pm ^{pl}(z) = \frac{1}{2}W_ + ^{pl}(0) \pm 4{\pi ^3}\int\limits_0^\infty {dkk\cos (\frac{{z\Delta }}{{2{k_0}{n_0}}}{k^2})} |\tilde{E}_ + ^{pl}(k){|^2},$$

where $\Delta = n_o^2/n_e^2 - 1$ describing the degree of anisotropy. After substituting Eq. (9) into Eq. (16) and performing the integral, the energy of two circular components can be written as

(17)$$\begin{array}{l} W_{_ \pm }^{pl}(z) = \frac{1}{2}W_{_+}^{pl}(0) \pm \frac{{(2p + |l|)!}}{{(p + |l|)!(p!)}} \times \\ \quad \quad \quad \;{\;_2}{F_1}( - p, - p - |l|; - 2p - |l|,1 + {\xi ^2}){\mathop{\rm Re}\nolimits} [{(\frac{1}{{1 + \xi i}})^{2p + |l|+ 1}}], \end{array}$$

in which $_2{F_1}$ is the hypergeometric function and $\xi = z\Delta /({k_0}{n_0}v_0^2)$. The energy of each components depends on the values of $\xi$, p, and l, and the total energy is conserved during propagation, i.e., ${W^{pl}}(z) = W_ - ^{pl}(z) + W_ + ^{pl}(z) = 1$. Based on Eq. (17), we can define the energy efficiency ${\eta _ \pm }\textrm{(z)}=W_ \pm ^{pl}(z)/W_{}^{pl}(z)$ as well as the spin angular momentum (SAM),

(18)$$\begin{array}{l} {S_z} = \frac{{W_ + ^{pl}(z) - W_ - ^{pl}(z)}}{{{W^{pl}}(0)}} = \frac{{(2p + |l|)!}}{{(p + |l|)!(p!)}} \times \\ \quad \;\;\quad \quad {\,_2}{F_1}( - p, - p - |l|; - 2p - |l|,1 + {\xi ^2}){\mathop{\rm Re}\nolimits} [{(\frac{1}{{1 + \xi i}})^{2p + |l|+ 1}}]. \end{array}$$

Equation (18) can recover the results in [29]. Note that there is a small mistake in the Eq. (31) of [29], in which the factor ${[({\sigma _o}\sigma _e^\ast )/(\sigma _o^\ast {\sigma _e})]^{2p + l}}$ is redundant and should be deleted. Although the qualitatively behavior of Eq. (31) in [29] is correct, the erroneous factor do result in quantitative difference, and in particularly the strange “kink” of ${S_z}$ at small z region.

The energy exchange between opposite circularly polarized components of LG beams with various radial and angular indices are plotted in Fig. 2 as a function of propagating distance $z/L$, with $L = {{{k_0}{n_0}v_0^2} \mathord{\left/ {\vphantom {{{k_0}{n_0}v_0^2} \Delta }} \right.} \Delta }$ being the characteristic length. All other parameters used in Fig. 2 are the same as those in Fig. 1. One can see that the larger are the indices l and p, the more strongly the energy oscillates, and also the more faster it converges to the saturated value of 0.5. The spin-orbital coupling shown in Fig. 2 can be understood as follows: the optical field inside the uniaxial crystal is a superposition of ordinary and extraordinary components, i.e. $F_ + ^{(pl)} = F_{ + o}^{(pl)} + F_{ + e}^{(pl)}$. The energy flux of the right-handed polarized field consists of three terms: $\int\!\!\!\int {|F_{ + o}^{(pl)}{|^2}dS}$, $\int\!\!\!\int {|F_{ + e}^{(pl)}{|^2}dS}$, and $\int\!\!\!\int {2{\mathop{\rm Re}\nolimits} (F_{ + o}^{(pl)}F_{ + e}^{(pl) \ast })dS}$. The first two terms are the energy flux of ordinary and extraordinary components, respectively, which are conservative during propagation. However, the third term denotes the interference between the ordinary and extraordinary components. This term will change the energy flux of right-hand field and is the key factor of spin-orbital coupling.

Fig. 2. The normalized energy $W_ + ^{pl}(z)/W_ + ^{pl}(0)$ (solid lines) and $W_ - ^{pl}(z)/W_ + ^{pl}(0)$ (dash lines) for opposite circularly polarized components as a function of propagating distance.

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It should be pointed out that the total energy of the optical field equals to the addition of different components (either the Cartesian components or the circular components), due to the orthogonality of different components. Since the LG function is a complete and orthogonal basis in two-dimensional space, any right-handed polarized beam at the incident plane ($z = 0$) can be represented as a superposition of the LG beams,

(19)$${\bf E}(r,\varphi ,0) = {E_ + }(r,\varphi ,0){{\bf e}_{\bf + }} = \sum\limits_{pl} {{A_{pl}}L{G^{pl}}(r,\varphi ,0){{\bf e}_{\bf + }}} ,$$

where ${A_{pl}}$ is the weight of spectrum and can be calculated using,

(20)$${A_{pl}} = \int\limits_0^{2\pi } {\int\limits_0^\infty {{{[L{G^{pl}}(r,\varphi ,0)]}^\ast }} } {E_ + }(r,\varphi ,0)rdrd\varphi .$$

Using the Eq. (5), Eq. (6), and the following integral formula [34],

(21)$$\int_0^\infty r dr{J_l}({k_1}r){J_l}({k_2}r) = \frac{{{\delta _{{k_1}{k_2}}}}}{{{k_1}}},$$

one can prove that

(22)$$\begin{array}{l} \int_0^\infty r dr\int_0^{2\pi } {d\varphi } {[F_ + ^{pl}(r,z){\textrm{e}^{il\varphi }}]^\ast }F_ + ^{p^{\prime}l^{\prime}}(r,z){{\mathop{\rm e}\nolimits} ^{il^{\prime}\varphi }} = W_ + ^{pl}(z){\delta _{p^{\prime}p}}{\delta _{l^{\prime}l}},\\ \int_0^\infty r dr\int_0^{2\pi } {d\varphi } {[G_ + ^{p(l + 2)}(r,z){{\mathop{\rm e}\nolimits} ^{i(l + 2)\varphi }}]^ \ast }G_ + ^{p^{\prime}(l^{\prime} + 2)}(r,z){{\mathop{\rm e}\nolimits} ^{i(l^{\prime} + 2)\varphi }} = W_ - ^{pl}(z){\delta _{p^{\prime}p}}{\delta _{l^{\prime}l}}. \end{array}$$

That is, the $F_{^ + }^{pl}(r,z)$ and $G_{^ + }^{p(l + 2)}(r,z)$ are also orthogonal in the two-dimensional functional space. Therefore, we have

(23)$$\begin{array}{l} {W_ + }(z) = \int_0^\infty r dr\int_0^{2\pi } {d\varphi } |\sum\limits_{pl} {{A_{pl}}F_ + ^{pl}(r,z)} {|^2} = \sum\limits_{pl} {|{A_{pl}}{|^2}W_ + ^{pl}(z)} ,\\ {W_ - }(z) = \int_0^\infty r dr\int_0^{2\pi } {d\varphi } |\sum\limits_{pl} {{A_{pl}}G_ + ^{p(l + 2)}(r,z)} {|^2} = \sum\limits_{pl} {|{A_{pl}}{|^2}W_ - ^{pl}(z)} . \end{array}$$

We can use the Eq. (23) to investigate the image propagation in the uniaxial crystal, as will be discussed in the next section.

4. The Transmission of structured optical field in uniaxial crystals

4.1 Numerical LG spectral decomposition

Based on the analytical solutions of LG beam in Eq. (10) and Eq. (13), we propose a scheme for the transmission of structured optical field through the uniaxial crystals. The optical field is firstly decomposed into a series of LG spectra at the input plane, and then let each LG component individually propagate through the crystal. To demonstrate this scheme, we look at a typical example of trefoil image, which is shown in Fig. 3(a). In order to obtain the OAM spectrum numerically, we separate the trefoil image into $100 \times 100$ points of pixels, and set the distance between two adjacent pixels to be $1\mu m$. The gray value of each point is defined as the intensity of electric field. Without loss of generality, we assume the whole image is right-handed polarized. The OAM spectrum $|{A_{pl}}{|^2}$ can be calculated by the following formula,

(24)$${A_{pl}} = \sum\limits_{{n_x}} {\sum\limits_{{n_y}} {{{[L{G^{pl}}(h{n_x},h{n_y})]}^\ast }} } E(h{n_x},h{n_y}),$$

where h is the physical distance between pixels ($1\mu m$), $E(h{n_x},h{n_y})$ is the intensity of electric field (i.e., the gray level), ${n_x}$ and ${n_y}$ are integers representing the pixel spacing of each point relative to the center of image. The Eq. (24) is actually a discrete version of Eq. (20).

Fig. 3. The trefoil image with size $100 \times 100$ pixels (a) and its LG spectra $|{A_{pl}}{|^2}$ (b). The lower left panel (c) is the recovered image superposed the finite LG spectra in panel (b).

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The calculated OAM spectra of the trefoil images are shown in Fig. 3(b), where the index l ranges from −15 to 15 and p ranges from 0 to 5. Note that the input trefoil image only contains the intensity information and the phase is uniform in the lateral plane, i.e., the input light field carries zero OAM. Taking into account the feature size of the trefoil, we choose the beam waist of LG mode ${v_0}$ to be $4.5\mu m$, which will minimize the distribution of the spectra. Due to the ${C_3}$ symmetry of the image, its spectrum is negligible if l does not equal to the multiple of 3, therefore only the multiple of 3 are shown in Fig. 3(b), in which the total spectrum is normalized ${\sum\limits_{pl} {|{A_{pl}}|} ^2} = 1$. Figure 3(c) is the superposed image using the finite spectrum in Fig. 3(b). One can see that it is very similar to the original image in Fig. 3(a). This implies that the spectral ranges of l and p and the beam waist of LG mode we used are reasonable.

4.2 The evolution of OAM spectrum and field intensity in a uniaxial crystal

After decomposing the trefoil image into a serial of LG spectrum at the input plane, we let each LG component propagates through the uniaxial crystals individually. Due to the orthogonality of $F_{^ + }^{pl}(r,z)$ and $G_{^ + }^{p(l + 2)}(r,z)$ as shown in Eq. (22), the optical field of trefoil image propagating inside the uniaxial crystal can be written as follows,

(25)$$\begin{array}{l} {E_ + }(r,\varphi ,z) = \sum\limits_{pl} {A_{pl}^{}F_ + ^{pl}(r,z)\exp (il\varphi )} \exp (i{k_0}{n_o}z),\\ {E_ - }(r,\varphi ,z) = \sum\limits_{pl} {A_{pl}^{}G_ + ^{p(l + 2)}(r,z)\exp [i(l + 2)\varphi ]} \exp (i{k_0}{n_o}z). \end{array}$$

That is, the optical field is a superposition of each LG components in according to their spectral weight ${A_{pl}}$ respectively. We depict in Fig. 4 the optical field of right-handed polarized trefoil image propagating at several typical values of distance. The parameters of uniaxial crystal used in Fig. 4 are the same as those in Fig. 1, and $L$ again is the characteristic length as before. One can see that the left-handed circular component develops from zero when propagating into the uniaxial crystal. The optical field changed rapidly for small value of z, while its shape tends to fixed at large distance. This agrees with the saturated behavior of energy of LG beam in Fig. 2. Noted that, at large propagating distance, the shape of the right-handed optical field is similar to the Fourier transform of original trefoil image at $z = 0$.

Fig. 4. The intensity of optical field ${|{{E_ + }} |^2}$ (upper row) and ${|{{E_ - }} |^2}$ (lower row) of a right-hand polarized trefoil image propagating in the uniaxial crystals. The propagating distance are chosen as follows, (a) and (e): $z = 0$, (b) and (f): $z = 0.1L$, (c) and (g): $z = 0.5L$, (d) and (h): $z = L$.

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Due to the importance of OAM in the optical communication, we look at the evolution of OAM spectra in the crystal. Due to the orthogonality of propagating modes as discussed in Section 3, the spectra of circular components inside the crystal can be calculated by $I_ \pm ^{pl}(z) = |{A_{pl}}{|^2}W_ \pm ^{pl}(z)$. We plot the OAM spectra of optical field in Fig. 5, where all parameters used are exactly the same as those in Fig. 4. One can see that the spectrum of left-handed component appears when z is larger than 0. At large distance, the spectra of right-handed components are half of those at $z = 0$. Figure 4(d) and (h) show that the spectral distribution of right-handed component is similar to that of left-handed component, but with a shift of l value by 2.

Fig. 5. The evolution of OAM spectra $I_ \pm ^{pl}(z)$ of a right-hand polarized trefoil image propagating in the uniaxial crystals. The upper (lower) row is the spectra of ${E_ + }$ (${E_ - }$). The observation planes are chosen as follows, (a) and (e): $z = 0$, (b) and (f): $z = 0.1L$, (c) and (g): $z = 0.5L$, (d) and (h): $z = L$. The radial index p ranges from 0 to 5 and angular index l ranges from −9 to 9.

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5. Conclusions

The expression of arbitrary order LG beams propagating along the optical axis in uniaxial crystals is derived using the angular-spectrum method. When studying the energy conversion of LG beams with different mode index, we find that the higher the mode index is, the faster the energy conversion curve oscillates and converges to a saturated value. We also prove that different propagating modes ($F_ + ^{(pl)}$ and $G_ - ^{p(l + 2)}$) inside the uniaxial crystals are orthogonal. Based on this observation, we propose a scheme for the transmission of structured optical field through the uniaxial crystals. The optical field is firstly decomposed into LG spectrum, and then each LG component individually propagates through the uniaxial crystals. The final output optical field is the superposition of all OAM components. Using this scheme, the propagating characters of a trefoil image in uniaxial crystal are obtained. For a small propagating distance, the optical field changes rapidly and different OAM modes converse heavily. For the large propagating distance, the distribution of spectra tends to a saturated value. Our scheme not only provides a universal way for the propagation of structured optical field in uniaxial crystal, but also may find great applications in the techniques such as DSI, or encrypted encoding and decoding of information using LG spectrum.

Funding

Program for New Century Excellent Talents in University (NCET-13-0495); Fundamental Research Funds for the Central Universities (20720180015, 20720190057); National Natural Science Foundation of China (11104233, 91636109); Fujian Province Funds for Distinguished Young Scientists (2015J06002).

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The transmission of structured light fields in uniaxial crystals employing the Laguerre-Gaussian mode spectrum

Abstract

1. Introduction

2. Paraxial propagation of Laguerre-Gaussian beams In uniaxial crystal

3. The conversion of energy and spin-orbital coupling

4. The Transmission of structured optical field in uniaxial crystals

4.1 Numerical LG spectral decomposition

4.2 The evolution of OAM spectrum and field intensity in a uniaxial crystal

5. Conclusions

Funding

References

Cited By

Figures (5)

Equations (26)

Optics Express