1. Introduction

Laguerre-Gaussian (LG) beams are solutions to the paraxial wave equation in cylindrical coordinates, representing propagation-invariant fields that form a complete orthogonal basis. With quantized field components in the radial direction denoted by radial index $p\in \mathbb {N}$ and in azimuthal direction denoted by the azimuthal index $l\in \mathbb {Z}$, LG beams find diverse applications in optical communication [1–3], micro particle manipulation [4], quantum information [5,6], and other areas. Due to their azimuthal periodicity, LG beams serve as excellent prototypes of vortex beams carrying orbital angular momentum (OAM). Allen demonstrated that a LG beam with a helical phase $\exp (il\varphi )$ carries OAM of $l\hbar$ per photon [7]. Given their orthogonality, LG beams present promising channel candidates for mode division multiplexing technology.

In optical communication systems utilizing LG beams, the LG beams typically propagate coaxially along the axis of the optical system due to the rotational symmetry of both the LG beams and the optical system. Conventional orthogonality dictates that two coaxial LG beams are orthogonal to each other, forming the basis for mode division multiplexing using LG beams as channels. However, when an LG beam propagates off-axis, it is referred to as an unaligned LG beam and its rotational symmetry is disrupted [8]. The power of the unaligned LG beam spreads to neighboring aligned LG beams that remain coaxial with the optical system, resulting in non-zero power transfer and crosstalk between them. Previous studies have analyzed the impact of non-coaxiality on rotational Doppler effect [9,10], while various methods have been proposed to characterize and mitigate non-coaxiality effects such as minimizing root-mean-square of OAM spectra [11–14], compensating misalignment using fast steering mirrors placed at lens focal planes driven by quad-cell position sensitive detectors [15], deep-learning approaches [16], etc. Non-coaxiality can manifest in two forms: lateral displacement and tilt. Lateral displacement refers to parallel axes separated by a distance, whereas tilt implies an angle between two axes. Of particular interest is lateral displacement since it allows proper aperture capture due to alignment with the optical system’s axis. In this context, we designate lateral displacement as "non-coaxiality". Furthermore, non-coaxiality leads to crosstalk among LG beams; however, due to a lack of analytical expressions for this crosstalk phenomenon between LG beams, its properties remain unclear. The orthogonality between non-coaxial LG beams has not been fully elucidated, except for two specific scenarios. In the first scenario, when the axes of two LG beams are coaxial and their distance is zero, conventional orthogonality is observed, which has been extensively studied in earlier research. In the second scenario, where the distance between axes is infinite, LG beams are nearly orthogonal due to minimal overlap in geometric space. This implies that in an optical communication system with infinitely distant apertures, LG beams from different apertures do not interfere with each other. Obviously, this is denied by finite aperture of optical system in practical realisation. However, further investigation is required for the intermediate zone where there exists some overlap in geometric space; this complex orthogonal relationship is referred to as non-coaxial orthogonality within this context. Non-coaxial orthogonality holds significance for optical communication systems as it allows for reduced or negligible mutual interference between LG beams either within a single aperture or across different apertures when two apertures are brought closer together. It implies that one may pack more sub-space channels into one conventional space channel. Consequently, exploiting the non-coaxial orthogonality of LG beams may offer a novel mechanism to enhance throughput in optical communication systems. We know that everything has two sides, the multiple non-coaxial LG beams have many advantages, meanwhile, they may introduce somewhat complexity into the system.

This study primarily focuses on the intermediate zone and reveals the orthogonal relationship between two non-coaxial LG beams. We propose an analytical formula for a novel type of Hankel transform integral, incorporating an exponential function and two Laguerre polynomials, to derive the projection of non-coaxial LG beam onto an aligned one. The derived analytical expression unveils the non-coaxial orthogonality among any pair of LG beams within a set. This work also presents a potential scheme for compound space-space-mode multiplexing, wherein multiple LG beams can coexist in close proximity while propagating along distinct parallel axes.

2. Theorem

The Laguerre-Gaussian (LG) beams represent the eigen-solutions of the paraxial wave equation. In cylindrical coordinates, they can be expressed as follows [17]

The LG beams have conventional orthogonality as $\iint \psi _{p,l}^{w_0*}(r,\varphi,z)\psi _{p',l'}^{w_0}(r,\varphi,z)r\mathrm {d}r\mathrm {d}\varphi =\delta _{p,p'}\delta _{l,l'}$ on the transverse plane at $z$ [18,19]. Here * denotes complex conjugate and $\delta$ is Kronecker delta function. Such orthogonality holds true for two coaxial LG beams with the same waist. But for two non-coaxial LG beams shown in Fig. 1, the orthogonality has received limited attention despite its significance. This study addresses this gap by explicitly considering the complexity arising from laterally displaced Laguerre polynomials, which was previously avoided by restricting LG beams to zero radial index as per Vasnetsov’s work [8]. Herein, we derive the expression for non-coaxial orthogonality.

 

Fig. 1. Layout of two non-coaxial LG beams. $\left |p,l\right\rangle$ and $\left |p',l'\right\rangle$ denote the aligned LG beams. $\hat {T}(\Delta \vec {r})\left |p',l'\right\rangle$ indicates the laterally shifted LG beam $\left |p',l'\right\rangle$ by a displacement $\Delta \vec {r}$, resulting in the unaligned LG beam. Here, $\Delta \vec {r}=\Delta x\hat {e}_x+\Delta y\hat {e}_y$ and $(\eta,\theta )$ are the polar coordinates with $\Delta x=\eta \cos \theta$ and $\Delta y=\eta \sin \theta$.

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For convenience, we use ket $\left |p,l\right\rangle$ for LG beam $\psi _{p,l}^{w_0}(r,\varphi,z)$ with radial index $p$, azimuth index $l$ and waist radius $w_0$. The bra $\left\langle p,l\right |$ or $\psi _{p,l}^{w_0*}(r,\varphi,z)$ is complex conjugate of LG beam $\left |p,l\right\rangle$. The waists of LG beams lie on $xy$ plane and the axis of aligned LG beam $\left |p,l\right\rangle$ is $z$ axis, as shown in Fig. 1. The unaligned beam is denoted as an operation on ket vector, i.e., $\hat {T}(\Delta \vec {r})\left |p',l'\right\rangle$. The operator $\hat {T}(\Delta \vec {r})=\exp \left [-(i/\hbar )\Delta \vec {r}\cdot \hat {p}\right ]$ with $\Delta \vec {r}=\Delta x\hat {e}_x+\Delta y\hat {e}_y$ the displacement of the unaligned LG beam axis and $\hat {p}=-i\hbar (\hat {x}\partial _x+\hat {y}\partial _y)=-i\hbar \nabla _\bot$ the transverse linear momentum operator [20,21]. To demonstrate the orthogonality, the projection of non-coaxial LG beam on an aligned one is denoted as

The Fourier transform of $\psi _{p',l'}^{w_0}(x,y,0)$ is $(-i)^{2p'+|l'|}\psi _{p',l'}^{2/w_0}(k_x,k_y,0)$ with $(k_x,k_y)$ the transverse coordinates in spatial frequency domain. Because the right hand of Eq. (3) is $z-$ independent, we drop off $z$ and simplify the notation as $f_{p,l,p',l'}^{w_0}(\Delta x,\Delta y)$ in Cartesian coordinates or $f_{p,l,p',l'}^{w_0}(\eta,\theta )$ in cylindrical coordinates. The integral in last line of Eq. (3) is Hankel transform integral incorporating an exponential function and two Laguerre polynomials. The Hankel transform integral is the key of this work and depends on sign of $l\cdot l'$.

To our best knowledge, only when $l\cdot l'\le 0$ the integral in Eq. (3) has closed form in various handbooks of mathematics [23–26]. By using Equation 7.422-2 of Ref. [23] we can derive the closed form of integral in Eq. (3) and then the projection reads

For $l\cdot l'\ge 0$, the closed form of integral in Eq. (3) has not been reported till now. Following the new formula that we propose in the Appendix, we obtain the projection as

In Eqs. (4) and (5), Laguerre polynomials with negative upper index relates to that with positive upper index by [27]

Because of the conventional orthogonality and completeness of a set of LG beams taking the whole radial index $p\in \mathbb {N}$ and azimuth index $l\in \mathbb {Z}$, any unaligned LG beam can be expanded in aligned LG beams as

3. Simulations and discussions

3.1 Projection of unaligned LG beam on aligned one and its radial nulls

According to Eq. (3), the projection $f_{p,l,p',l'}^{w_0}(\eta,\theta )$ of unaligned LG beam $\left |p',l'\right\rangle$ on aligned one $\left |p,l\right\rangle$ is a complex function and proportional to inverse Fourier transform of $\psi _{p,l}^{2/w_0*}(k_x,k_y,0)\cdot \psi _{p',l'}^{2/w_0}(k_x,k_y,0)$. Figures 2(a1) to (e1) show the transverse amplitude of the projections, which present many concentric dark rings. The point $(\Delta x,\Delta y)$ on the dark rings has zero value of the projection, which indicates that two non-coaxial LG beams are orthogonal to each other when the axes is separated by a distance $\eta =\sqrt {(\Delta x)^2+(\Delta y)^2}$. Then, no power is coupled from one LG beam into another one even they are spatially overlapped. The dark rings in the transverse amplitude patterns, i.e., radial nulls at $\eta >0$ in context, come from the radial phase dislocation in the phase pattern of the projection, e.g., Fig. 2(a3). The points outside the dark ring in Fig. 2(a1) to 2(e1) are also of significance. The presence of bright rings indicates the coupling of power from one LG beam to another. In more general scenarios, power coupling occurs between two non-coaxial LG beams leading to crosstalk between them. Interestingly, the relative power coupled into another LG beam remains unchanged as the LG beam propagates forward according to Eq. (3). The central null indicates the conventional coaxial orthogonality between LG beams because $\eta =0$, which occurs only when $p=p'$ and $l=l'$ resulting a projection 1; otherwise, it is 0.

 

Fig. 2. Projection $f_{p,l,p',l'}^{w_0}(\eta,\theta )$ of different unaligned LG beam $\left |p',l'\right\rangle$ on another aligned LG beam $\left |p,l\right\rangle$. (a) unaligned $\left |3,1\right\rangle$ on aligned $\left |1,2\right\rangle$, (b) unaligned $\left |2,0\right\rangle$ on aligned $\left |1,0\right\rangle$, (c) unaligned $\left |3,0\right\rangle$ on aligned $\left |2,1\right\rangle$, (d) unaligned $\left |2,-1\right\rangle$ on aligned $\left |2,1\right\rangle$, (e) unaligned $\left |3,-2\right\rangle$ on aligned $\left |2,1\right\rangle$. (a1) to (e1) amplitude patterns of the projections. (a2) to (e2) radial amplitude of the projections. (a3) phase pattern of the projection.

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The projection exhibits a helical phase, $\exp [i(l'-l)\theta ]$, with radial phase dislocations, as illustrated in Fig. 2(a3). The radial phase dislocation corresponds to the null rings in the amplitude pattern. A phase jump of $\pi$ occurs across the radial phase dislocation. Considering Eqs. (4) and (5), due to the consistent and uncomplicated transverse phase pattern during propagation, our focus will solely be on discussing the transverse amplitude pattern.

The number of radial nulls is directly related to the mode indices of two LG beams. Table 1 presents the count of dark rings or radial nulls for $f_{p,l,p',l'}^{w_0}(\eta,\theta )$ in four scenarios, exemplified by Fig. 2. The determination of the number of dark rings can be inferred from Eqs. (4) and (5). In case 1, when $l=l'=0$, the radial nulls arise from both equations: $L_p^{p'-p}(\xi )=0$ and $L_{p'}^{p-p'}(\xi )=0$ as per Eq. (4). If we have a situation where $p>p'$, then according to Eq. (6), we find that $L_p^{p'-p}(\xi )=(p'!/p!) (-\xi )^{p-p'}L_{p'}^{p-p'}(\xi )$. Therefore, $L_p^{p'-p}(\xi )=0$ and $L_{p'}^{p-p'}(\xi )=0$ share the same radial nulls. Obviously there are $p'$ dark rings according to the property of Laguerre polynomials. Similarly, there are $p$ dark rings when $p<p'$. Henceforth, the total number of dark rings can be determined as $\min (p,p')$. An example illustrating this case is depicted in Figs. 2(b1) and 2(b2) with $(p,l,p',l')=(1,0,2,0)$, resulting in one single dark ring.

Table 1. Number of dark rings or radial nulls of $\left |f_{p,l,p',l'}^{w_0} (\eta,\theta )\right |$

View Table

For case 2, the radial nulls arise from the conditions $L_p^{p'-p} (\xi )=0$ and $L_{p'+|l'|}^{p+|l|-p'-|l'|}(\xi )=0$, where other condition in $l\cdot l'\ge 0$ hold. Similar to case 1, both functions $L_p^{p'-p}(\xi )$ and $L_{p'+|l'|}^{p+|l|-p'-|l'|}(\xi )$ have radial nulls with a number equal to $\min (p,p')$ and $\min (p+|l|,p'+|l'|)$, respectively. Therefore, the total number of radial nulls for function $\left |f_{p,l,p',l'}^{w_0}(\eta,\theta )\right |$ is given by $\min (p,p')+\min (p+|l|,p'+|l'|)$. Figure 2(a) illustrates an example with $(p,l,p',l')=(1,2,3,1)$, with the number of radial nulls is 4, which exhibits four radial nulls as clearly shown in Fig. 2(a2). When two LG beams represented by $\left |1,2\right\rangle$ and $\left |3,1\right\rangle$ are separated by a distance equal to four dark ring radii, they become orthogonal to each other. Similarly, examples for cases 3 and 4 can be observed in Figs. 2(d) and 2(e), respectively. It is noteworthy that cases 1 and 3 exhibit smooth nulls as depicted in Figs. 2(b2) and 2(d2), while cases 2 and 4 present sharp nulls as illustrated in Figs. 2(a2), 2(c2) and 2(e2), respectively. The presence of smooth nulls is particularly valuable due to their increased tolerance towards misalignment.

Considering the scenario where $p=p'$ and $l=l'$, it can be observed that $\psi _{p,l}^{2/w_0*}(k_x,k_y,0)\cdot \psi _{p',l'}^{2/w_0}(k_x,k_y,0)=|\psi _{p,l}^{2/w_0}(k_x,k_y,0)|^2$ is the transverse intensity of a LG beam in spatial frequency domain. By applying Eq. (3), its inverse Fourier transform is proportional to $f_{p,l,p,l}^{w_0}(\eta,\theta )$. The transverse amplitude exhibits $p$ dark rings when $l=0$ and $2p+|l|$ dark rings when $l\ne 0$ as well known. This conclusion aligns with cases 1 and 2 presented in Table 1. Moreover, from case 2 in Table 1, we find the unaligned fundamental mode beam $\left |0,0\right\rangle$ is not orthogonal to any aligned LG beam. From case 4 in Table 1, we also know unaligned LG beam $\left |0,l>0\right\rangle$ is not orthogonal to another aligned one $\left |0,l'<0\right\rangle$.

3.2 Orthogonality of non-coaxial LG beams at a fixed distance

Figure 3 presents the radial amplitude of the projection, i.e., $\left |f_{p,l,p',l'}^{w_0}(\eta,\theta )\right |$, and LG mode spectrum corresponding to a radial null. The radial amplitude of the projection of an unaligned LG beam $\left |p',l'\right\rangle$ on to a set of aligned LG beams $\left |p,l\right\rangle$ is depicted in Fig. 3(a1) to (d1), illustrating the variation with respect to the distance between axes ranging from 0 to $8w_0$. Each subplot exhibits one or more common radial nulls as well as some non-common radial nulls. Non-common radial nulls indicate that an unaligned LG beam is orthogonal to only one aligned LG beam, whereas common radial nulls imply orthogonality between an unaligned LG beam and a set of aligned LG beams. For instance, the curves in Fig. 3(a1) exhibit a shared null point at $\eta =\sqrt {2}w_0$, indicating that an unaligned LG beam $\left |1,0\right\rangle$ is orthogonal to a set of aligned LG beams such as $\left |0,\pm 1\right\rangle$ and $\left |1,l\right\rangle$ with $|l|\le 6$ when the distance between their axes is $\eta =\sqrt {2}w_0$. This phenomenon can be explained by Eqs. (4) and (5). When $p'=1$ and $l'=0$, according to Eq. (4), the radial nulls arise from two equations: $L_p^{1-p}(\xi )=0$ and $L_1^{p+|l|-1}(\xi )=0$. The former solely depends on the radial index $p$ and represents the common radial nulls. Furthermore, for $p=1$, the solution yields either $\xi =1$ or $\eta =\sqrt {2}w_0$; thus all lines share a common radial null at $\eta =\sqrt {2}w_0$. Consequently, when the distance between axes is set as $\eta =\sqrt {2}w_0$, an unaligned LG beam $\left |1,0\right\rangle$ becomes orthogonal to LG beams $\left |1,l\in \mathbb {Z}\right\rangle$. The second equation relies on both radial and azimuthal indices of the aligned LG beams; only under special circumstances where either (a) $p+|l|=1$ or (b) $p=0$ while $l=\pm 1$ does it also yield common radial nulls at position $\eta =\sqrt {2}w_0$. Therefore, the unaligned LG beam $\left |1,0\right\rangle$ is orthogonal to a set of aligned LG beams, i.e., $\left |0,\pm 1\right\rangle$ and $\left |1,l\in \mathbb {Z}\right\rangle$, when the distance between axes is $\eta =\sqrt {2}w_0$. This observation is supported by Fig. 3(a2), which presents the LG mode spectrum when the LG beam $\left |1,0\right\rangle$ is laterally displaced by a distance $\eta =\sqrt {2}w_0$. The darkest squares in the first and second columns represent LG beams $\left |0,\pm 1\right\rangle$ and $\left |1,l\in \mathbb {Z}\right\rangle$ respectively with zero values. Furthermore, it confirms that an unaligned LG beam $\left |1,0\right\rangle$ remains orthogonal to aligned LG beams $\left |1,l\in \mathbb {Z}\right\rangle$ when their axes are separated by a distance of $\eta =\sqrt {2}w_0$. Other non-common radial nulls are of less interest and not discussed here.

 

Fig. 3. Radial amplitude of projections of different unaligned LG beams and corresponding LG mode spectra at different distances between axes. (a) unaligned $\left |1,0\right\rangle$, (b) unaligned $\left |1,2\right\rangle$, (c) unaligned $\left |1,2\right\rangle$ and (d) unaligned $\left |2,3\right\rangle$. (a2) $\eta =\sqrt {2}w_0$, (b2) $\eta =\sqrt {2}w_0$, (c2) $\eta =2w_0$ and (d2) $\eta =\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$. Here, $\eta$ denotes the distance between the axes of unaligned and aligned LG beams.

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Figures 3(b) and 3(c) also demonstrate that an unaligned LG beam is orthogonal to a set of LG beams at a specific distance between axes. However, there exists a slight distinction compared to the case when $l'>0$ as depicted in Figs. 3(b) and  3(c). For example, as illustrated in Fig. 3(b2), the unaligned LG beam $\left |1,2\right\rangle$ is not orthogonal to all aligned LG beams $\left |1,l<-1\right\rangle$ at an axes distance $\eta =\sqrt {2}w_0$. This discrepancy arises because both $L_p^{p'+|l'|-p}(\xi )$ and $L_{p'}^{p+|l|-p'}(\xi )$ do not possess common nulls independent of the azimuthal index $l$ for given values of $p'$ and $l'$ when the product $l\cdot l'<0$ according to Eq. (4).

Figure 3(d1) exhibits two common radial nulls. According to Eq. (5), the common nulls depend on $L_p^{2-p}(\xi )$, which has two solutions $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$ when $p=2$. Figure 3(d2) presents the LG mode spectrum when the distance between axes is $\eta =\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$. Furthermore, it also demonstrates that the unaligned LG beam exhibits orthogonality with a set of aligned LG beams positioned at an appropriate axial separation.

It is demonstrated in Fig. 3 that a properly laterally displaced LG beam is orthogonal to a set of aligned LG beams. Furthermore, it is observed that any two LG beams within the set are mutually orthogonal at a fixed distance between their axes. As an illustrative example, we consider a specific set of LG beams $\left |0,-2\right\rangle$, $\left |1,-1\right\rangle$ and $\left |2,l\ge 0\right\rangle$ with the fixed axis distance denoted as $\eta =\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$. The corresponding LG mode spectra are depicted in Fig. 4 where the green squares represent the marked LG beams within the set. It can be readily observed that any pair of these LG beams exhibit orthogonality when separated by this fixed distance. We provide a proof for this conclusion based on the following steps.

 

Fig. 4. LG mode spectra of unaligned LG beams selected from a predefined set consist of LG beams $\left |0,-2\right\rangle$, $\left |1,-1\right\rangle$ and $\left |2,l'\ge 0\right\rangle$ (denoted by green squares). (a) unaligned $\left |0,-2\right\rangle$, (b) unaligned $\left |1,-1\right\rangle$, (c) unaligned $\left |2,0\right\rangle$, (d) unaligned $\left |2,1\right\rangle$, (e) unaligned $\left |2,2\right\rangle$, (f) unaligned $\left |2,3\right\rangle$. The distance between axes of two LG beams is set as $\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$.

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For LG modes with $p'=2$ and $l'\ge 0$, as shown in Fig. 4(c), if $l\ge 0$, the radial nulls arise from the conditions $L_p^{2-p}(\xi )=0$ and $L_{2+|l'|}^{p+|l|-2-|l'|}(\xi )=0$, according to Eq. (5). Notably, the equation $L_p^{2-p}(\xi )=0$ is independent of azimuthal indices such as $l$ and $l'$. Thus, this equation provides a suitable distance between axes for a given value of $p$. For instance, when $p=2$, the distances between axes are determined by either $\xi =2\pm \sqrt {2}$ or $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$. In Fig. 4, only LG mode spectra with a distance $\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$ between axes are depicted. Consequently, any two LG beams represented by $\left |2,l\ge 0\right\rangle$ and $\left |2,l'\ge 0\right\rangle$ exhibit orthogonality.

Next, we consider LG beams $\left |p,l<0\right\rangle$ and demonstrate their orthogonality to $\left |2,l'\ge 0\right\rangle$ when the distance between axes is given by $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$. Since $l\cdot l'\le 0$ (as per Eq. (4)), the radial nulls of projection arise from the conditions $L_p^{2+|l'|-p}(\xi )=0$ and $L_2^{p+|l|-2}(\xi )=0$. Notably, the second equation is independent of $l'$, implying potential orthogonality between LG beam $\left |p,l<0\right\rangle$ and all LG beams $\left |2,l'\ge 0\right\rangle$. Obviously $p+|l|=2$ is required to make $L_2^{p+|l|-2}(\xi )=L_2(\xi )$ which gives common nulls at $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$. Consequently, both LG modes $\left |0,-2\right\rangle$ and $\left |1,-1\right\rangle$ exhibit orthogonality with respect to all LG beams $\left |2,l'\ge 0\right\rangle$ when the distance between axes is given by $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$.

The remaining task is to demonstrate the orthogonality between LG modes $\left |0,-2\right\rangle$ and $\left |1,-1\right\rangle$. Since $l\cdot l'>0$, the radial nulls arise from $L_p^{p'-p}(\xi )=0$ and $L_{p'+|l'|}^{p+|l|-p'-|l'|}(\xi )=0$ according to Eq. (4). If the radial nulls satisfy $L_2(\xi )=0$, the first equation requires $p=p'=2$ which is not applicable in this case. The second equation requires $p+|l|=p'+|l'|=2$, thus establishing that LG modes $\left |0,-2\right\rangle$ and $\left |1,-1\right\rangle$ are orthogonal to each other when the distance between axes is $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$.

In conclusion, a set of LG beams comprising $\left |0,-2\right\rangle$, $\left |1,-1\right\rangle$ and $\left |2,l\ge 0\right\rangle$ is found to exhibit orthogonality when laterally displaced by $\eta =\sqrt {2}\cdot \sqrt {2\pm \sqrt {2}}w_0$. The LG beams are depicted as squares in Fig. 4. This generalization can be extended to other sets of LG beams.

3.3 Compound space-space-mode multiplexing utilizing non-coaxial orthogonality

Conventional space-mode multiplexing is achieved through the deployment of densely packed space channels (SC) containing multiple orthogonal mode channels (MC), as illustrated in Fig. 5(a). The MCs are formed by a set of coaxial orthogonal mode beams such as LG beams. Each SC has a sufficiently large aperture $D$ to prevent inter-channel crosstalk between adjacent SCs, similar to traditional MIMO systems [30,31].

 

Fig. 5. Conventional space-mode multiplexing and compound space-space-mode multiplexing. (a) Conventional space-mode multiplexing employing densely packed space channels (SC), each containing multiple orthogonal mode channels (MC) in an aperture of diameter $D$. (b) Compound space-space-mode multiplexing packing three sub-space channels (i.e., SSC#1, SSC#2 and SSC#3) into a single space channel. The axes of three SSCs intersect at the vertices of the white triangle and are equidistant ($\eta$) from each other. (c) Compound space-space-mode multiplexing packing seven sub-space channels into a single space channel. In each SSC, numerous coaxial LG beams serve as modal channels.

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By exploiting the non-coaxial orthogonality revealed herein, it becomes possible to condense three sets of LG beams into a single spatial channel, as depicted in Fig. 5(b). These three sets of LG beams constitute distinct sub-space channels (SSC), denoted as SSC#1, SSC#2, and SSC#3. Within each SSC, numerous LG beams form mode channels such as MC#1 and MC#2, as shown in Fig. 5(c). The selection of LG beams is made from a predefined set, for example a set comprising $\left |0,-2\right\rangle$, $\left |1,-1\right\rangle$ and $\left |2,l\ge 0\right\rangle$. Furthermore, the separation between adjacent SSCs remains constant at $\eta =\sqrt {2}\cdot \sqrt {2-\sqrt {2}}w_0$. Owing to the non-coaxial orthogonality property exhibited by LG beams, those belonging to different SSCs are orthogonal to one another; similarly within an individual SCC due to conventional coaxial orthogonality. Consequently, both intra-SSC and inter-SSC LG beam interactions can be theoretically avoided. As illustrated in Fig. 5(b), during propagation through these parallel SSCs the distance between their axes remains unaltered. This compound space-space-mode multiplexing scheme incorporates two layers of spatial multiplexing (SC and SSC) along with one layer of mode multiplexing (MC), leveraging the non-coaxial orthogonality among any pair of LG beams within a given set.

A dense layout of SSCs, as shown in Fig. 5(c), is expected. Every three groups of beams, such as SSC#0, SSC#1 and SSC#2, located at the vertices of an equilateral triangle with a side length $\eta$ are orthogonal to each other due to non-coaxial orthogonality. However, it is challenging to ensure strict orthogonality between two farther SSCs, such as SSC#1 and SSC#3 separated by $\sqrt {3}\eta$ or SSC#1 and SSC#4 separated by $2\eta$. Fortunately, the smooth radial nulls of $\left |f_{p,l,p',l'}^{w_0}(\eta,\theta )\right |$ discussed in Section 3.1 provide more tolerance to misalignments. Therefore, by appropriately selecting LG modes, e.g., $\left |2,3\right\rangle$, $\left |2,6\right\rangle$ and $\left |2,7\right\rangle$, we can achieve low crosstalk when the axes of two LG modes are separated by $\sqrt {3}\eta$ and $2\eta$.

4. Conclusions

In this study, we present a mathematical derivation of the expression for the projection of an unaligned Laguerre-Gaussian (LG) beam onto an aligned one. The resulting projection exhibits helical phase and rotationally symmetric amplitude with concentric null rings, where the number of null rings is closely related to the indices of both LG beams. By analyzing these nulls in the projection, we reveal the existence of non-coaxial orthogonality between LG beams with appropriate axial separation. Furthermore, our analysis demonstrates that this non-coaxial orthogonality extends to any pair of LG beams selected from a set thereof. This novel phenomenon suggests new possibilities for compound space-space-mode multiplexing, which involves two layers of spatial channels and one layer of mode channel. Importantly, this unique non-coaxial orthogonality has potential implications for enhancing throughput in optical communication systems.