Abstract

We present a new numerical method to calculate the optimum lens transformation to implement on a monochromatic laser beam path, in order to maximize its coupling to the fundamental Gaussian mode of a resonator or to a single-mode optical fiber whose mode can be described as Gaussian to a good approximation. This method relies on a useful mathematical relation on Laguerre-Gauss modes of different waists and reduces in the end to numerically maximizing a polynomial that is a function of the state of the beam in a finite interval, thus being numerically very efficient. We show with a simple example that this method is particularly efficient against other common methods used in the laboratory when it comes to laser beams composed of a coherent superposition of higher-order Laguerre-Gauss modes, as it is the case for instance for beams traversing optical elements suffering from spherical aberration.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Shaping a laser beam for maximizing the power coupled to the less excited transverse mode of an optical resonator or to a singlemode optical fiber is an ubiquitous task in an optics laboratory. Within the paraxial approximation [1], and considering a given linear polarization of the light, the eigenmodes of an optical resonator with cylindrical symmetry can be described in the Laguerre-Gauss (LG) basis [1], where each mode is parametrized by a pair of integers $(p,l)$, with $p \in \mathbb {N}$ and $l \in \mathbb {Z}$. A LG basis is fully determined by the wavevector $\vec {k}$ of the light, the waist $w_0$ and its position $\vec {r}_0 = (x_0, y_0, z_0)$ in space (where we take $z$ as the beam propagation axis). For our target optical resonator, its geometry determine a natural $\vec {k}^{\textrm {T}}$, $w_0^{\textrm {T}}$ and $\vec {r}_0^{\textrm {T}}$, such that the less excited transverse mode of a resonator is the $(0,0)$ Gaussian mode in that specific LG basis [2]. In the case of a singlemode fiber, its propagation mode is also usually approximated by a $(0,0)$ Gaussian mode [3] of a LG basis with small loss of accuracy, and if the numerical aperture of the fiber is small enough, the paraxial approximation can still be considered valid for the propagation of the fiber mode into free space. For both cases, thus, the problem of maximizing the power coupled to a singlemode fiber or to an optical resonator reduces to find the correct set of lenses in order to assure a maximum of the superposition of the field of the incoming beam to the $(0,0)$ Gaussian mode of the LG basis determined by the physical system.

This maximization can be divided in two steps: First, we need to characterize the incoming beam, i.e., to express its state in a complete basis, for example on a LG basis. Second, we must be able to calculate the optimum set of lenses that will transform the state of the laser beam to the state that maximizes the projection on the target $(0,0)$ Gaussian mode. Quite often, the characterization of the laser beam in the laboratory is done under simplifying assumptions, such as to suppose that the beam is already in the $(0,0)$ mode of a LG basis, although of waist different than the waist of the target $(0,0)$ mode. This hypothesis is often oversimplifying, because it neglects the fact that higher-order Gaussian modes can be populated, which can be moreover difficult to identify through a simple measurement of the intensity profile [4]. To characterize the state of the laser beam without simplifying assumptions is a complicated task, but can be done by using, for example, holographic filters [5] or phase spatial light modulators (SLM) [6]. Now, these methods will find the full state of the laser beam in a predetermined LG basis, which is not necessarily the one that guarantee a maximum of the power in the $(0,0)$ mode. If we had a method to determine the waist $w_0^{\textrm {opt}}$ for this optimum basis, then we would just need to find the correct set of lenses that transforms a LG basis of waist $w_0^{\textrm {opt}}$ into the LG basis of waist $w_0^{\textrm {T}}$ determined by our target optical resonator, which is a trivial task of matrix optics [1]. In this paper, we deal with the task of finding for a specific laser beam the LG basis that maximizes the power into the $(0,0)$ mode, once we know its expression in any other LG basis. Although it is possible to do that fully numerically, we deduce in this paper several analytical expressions that reduces the numerical task to the maximization of a simple polynomial. Apart from maximizing the power coupled to an optical resonator, the relations shown here will surely find applicability in several other numerical problems related to Gaussian beams, such as, for example, estimating the coherence loss of light beams propagating in turbulent media [7].

This paper is organized as follows: In the second section, we define our problem mathematically. In the third section, we deduce a numerical efficient method to find the Gaussian basis that guarantee optimum projection in its $(0,0)$ Gaussian mode. Then we apply this method to a simple example, and compare it to other approximate methods, in the fourth section.

2. Definition of the problem

For completely determining a monochromatic $(0,0)$ Gaussian mode, we need to define its propagation direction (that we call the $z$ direction), the frequency $\omega$ of the mode, the waist size $w_0$, waist position $z_0$ and center $(x_0,y_0)$ in the transverse plane (that we use to define the $(0,0)$ position of the plane), such that the mode reads [1]

$$u_{(0,0)}^{(w_0,z_0)} (r,z)=\sqrt{\frac{2}{\pi}} \, \frac{1}{w(z)} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right) \exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \, \exp(i \psi^{(\textrm{G})}(z)) \; \; .$$

The half-width of the beam $w(z)$ is given by

$$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z-z_0}{z_{\mathrm{R}}} \right)}^2 } \ ,$$
with $z_{\mathrm {R}} = \frac {\pi w_0^2}{\lambda }$ the Rayleigh length of the beam, that describes the typical collimation distance for a beam of waist $w_0$, and $\lambda = 2 \pi c/\omega$ is the wavelength of light. The curvature radius of the wavefront is
$$R(z) = \left(z - z_0 \right) \left[{ 1+ {\left( \frac{z_{\mathrm{R}}}{z-z_0} \right)}^2 } \right] \,$$
and the Gouy phase $\psi ^{(\textrm {G})}(z)$
$$\psi^{(\textrm{G})}(z) = \arctan \left( \frac{z-z_0}{z_{\mathrm{R}}} \right) \ .$$

The mode $u_{(0,0)}^{(w_0,z_0)}$ is normalized in the metric defined by the inner product

$$\left( u_1, u_2 \right) = \int_0^{2\pi}d\phi \int_0^\infty r\,dr \, u_1^*(r,\phi,z)\, u_2 (r,\phi,z) \ ,$$
such that $\left ( u_{(0,0)}^{(w_0,z_0)} , u_{(0,0)}^{(w_0,z_0)} \right ) = 1$. This Gaussian mode is but the $(0,0)$ mode of a complete, orthonormal basis of LG modes [1], which are given, for same wavevector, waist and waist position, by
$$\begin{aligned}u_{(p,l)}^{w_0,z_0} (r,\phi,z)=\sqrt{\frac{2 p!}{\pi(p+|l|)!}} \, \frac{1}{w(z)}\left(\frac{ \sqrt{2} r}{w(z)}\right)^{\! |l|} &\exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|} \! \left(\frac{2r^2}{w^2(z)}\right) \times \\ \times &\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp({-}i l \phi) \, \exp(i \psi^{(\textrm{G})}_{(p,l)}(z)) \ , \end{aligned}$$
with $p \in \mathbb {N}$ and $l \in \mathbb {Z}$, and the generalized Laguerre polynomial $L_p^{|l|} (\xi )$ defined as
$$L_p^{l}(\xi) = \frac{x^{{-}l} e^{\xi}}{p!}\frac{d^p} {d \xi^p} \left(e^{-\xi} \xi^{n+l}\right) \ .$$

The Gouy phase $\psi ^{(\textrm {G})}_{(p,l)}(z)$ for each mode now reads

$$\psi^{(\textrm{G})}_{(p,l)}(z) = \left(2p+|l|+1\right) \arctan \left( \frac{z-z_0}{z_{\mathrm{R}}} \right) = \left(2p+|l|+1\right) \psi^{(\textrm{G})}(z) \ .$$

This basis is complete for any beam in the paraxial approximation; this means that the most general state for a monochromatic light beam will be given, in the Dirac notation commonly used in quantum mechanics, by a density matrix that can be decomposed in this basis as

$$\rho = \sum_{p_1,l_1} \sum_{p_2,l_2} a_{p_1,l_1,p_2,l_2}^{w_0,z_0}\left| u_{(p_1,l_1)}^{w_0,z_0} {\bigg \rangle}{\bigg \langle} u_{(p_2,l_2)}^{w_0,z_0} \right| \ .$$

The particular case of a beam with full transverse spatial coherence can be represented by a pure state that is written as

$$\left| E (r,\phi,z) \right \rangle {=} \sum \limits _{l,p} b_{l,p}^{w_0,z_0} \left| u_{(l,p)}^{w_0,z_0} \right \rangle $$
or, alternatively, as the density matrix $\rho = \left | E (r,\phi ,z) \right \rangle \left \langle E (r,\phi ,z) \right |$. The transverse coherence of a laser beam depend on the characteristics of the laser resonator, and the quality of the optics elements it traverse during its propagation. A laser beam produced in a pure, coherent mode by i.e. a laser diode can be described as a coherent superposition of several LG modes after passage through optical elements with aberration or defects such as scratches on their surface. The propagation of a beam through turbid media, on the other hand, induce a loss of spatial coherence [7], such that a description in terms of a matrix density is needed. Also, an output beam from a laser resonator emitting in more than one transverse mode will not have full spatial coherence, since the modes do not have a definite phase relation among them, even if each mode has full spatial coherence [8], in which case it must also be described by a density matrix with reduced coherences. We finally note that, for the Eqs. (9) and (10) to express the electric field of a laser beam, we must have the $b_{l,p}^{w_0,z_0}$ dimensionally equivalent to an electric field and the $a_{p_1,l_1,p_2,l_2}^{w_0,z_0}$ equivalent to an electric field squared, since the basis of modes $u_{(l,p)}^{w_0,z_0}$ is normalized and thus with no dimension. The trace of the density matrix defined in Eq. (9) is thus proportional to the total power contained in the light, which means that it does not need to sum up to 1, as it is the case for the density matrix commonly used in Quantum Mechanics.

Finding the expression of a real light beam as Eq. (9) or (10) is feasible from an experimental point of view. It has been known for some time that the intensity measurement of the laser beam in one plane does not, in general, allow for the reconstruction of the full state of the beam [9,10]. But it is possible to implement a phase-retrieval algorithm [11] that converges iteratively to the correct beam state by comparing it to an intensity measurement in two planes, the focal plane and a plane in the infinity [9]. Alternatively, instead of dealing with numerical iterative methods, one can also implement a direct measurement of the modal decomposition of the beam by using hologram filters [5] or the phase modulation of the wavefront of the beam [6].

Once we have characterized the state of our beam, it is possible to act upon its transverse distribution of electric field by installing one or more lenses on its optical path. A set of perfect lenses (supposed free from aberration or diffraction effects) aligned with the propagation direction of the beam is a linear transformation that transform its waist $w_0$ and focus position $z_0$ [1,12], keeping the same amplitudes and relative phases for each $(l,p)$ mode of the LG decomposition. Considering this, we can divide in two parts the practical problem of maximizing the power in a specific $(0,0)$ Gaussian mode of waist $w_0^{\textrm {T}}$ . The first part consists in, for a given beam, to find the waist parameter $w_0'$ that guarantees maximum power in the $(0,0)$ mode, when decomposing the beam in the LG basis of waist $w_0'$. For a beam described by Eq. (9), the total power in the $(0,0)$ mode of waist $w_0'$ is given by the projection

$$f(w_0') = \textrm{Tr} \Biggl[ \rho \, \left| u_{(0,0)}^{w_0',z_0} {\bigg \rangle} {\bigg \langle} u_{(0,0)}^{w_0',z_0} \right|\Biggr]$$
which, in the particular case of a perfectly transverse coherent beam given by Eq. (10), can be cast in the form
$$f(w_0') = {\bigg|} {\bigg \langle} u_{(0,0)}^{w_0',z_0} \; \Big| \; E (r,\phi,z) {\bigg \rangle} {\bigg|}^2 \ ,$$
or alternatively still as Eq. (11), with $a_{p_1,0,p_2,0}^{w_0,z_0} = \left (b_{p_1,0}^{w_0,z_0}\right )^* b_{p_2,0}^{w_0,z_0}$. From here on, we will define the superposition function $f(w_0')$ of the laser beam in the $(0,0)$ mode of waist $w_0'$ only through the general expression of Eq. (11), valid for all beams, including fully coherent ones.

Once the optimal $w_0'$ is found (the one that maximizes the function $f(w_0')$), $w_0' \equiv W_0^{\textrm {opt}}$, the second part consists in installing a set of lenses in the beam path that transforms the LG basis of waist $W_0^{\textrm {opt}}$ into the LG basis of waist $w_0^{\textrm {T}}$, while keeping the same amplitude for each mode. But this second task can be easily accomplished by, for instance, a telescope composed of converging lenses of focal lengths $f_1$ and $f_2$ satisfying $f_1/f_2 = W_0^{\textrm {opt}}/w_0^{\textrm {T}}$, separated by a distance $f_1 + f_2$, with the first lens installed on the focal plane of the beam. So, the first part of the problem is what we will be dealing with in the rest of this paper. We note that we suppose, in this method, that the position of the focal plane of the beam is the same as the one of the target beam, i.e., $z_0 = z_0^{\textrm {T}}$. This is easy to do and verify experimentally, since the evolution of the second-order moment $W(z)$ of any beam verifies [13]

$$W^2(z) = W^2(z_0) \left(1 + \frac{(z-z_0)^2}{z_R^2}\right) \ ,$$
which means that, by measuring the evolution of the second-order moment of the beam, we willfind that it decreases monotonically before the focal plane and increases monotonically after the focal plane, being minimal at $z = z_0$.

3. Numerical method for choosing the best LG basis

All we need to deduce our method is the decomposition of the fundamental Gaussian mode with waist $w_0'$ in the LG basis of waist $w_0$:

$$u_{(0,0)} ^{w_0',z_0}(r,\phi,z) = \sqrt{1 - Q^2} \, \sum_{p} Q^p \, u_{(p,0)} ^{w_0,z_0}(r,\phi,z)\ , \ \textrm{with} \ Q = \tfrac{1- \eta^2}{1+ \eta^2} \ \textrm{and} \ \eta = \tfrac{w_0'}{w_0} \ .$$

This relation is verified in appendix A. Note that this relation is valid also for $w_0' < w_0$, which means that it is possible to create arbitrarily small beams by a combination of higher-order modes in a basis of finite $w_0$, at the cost of having $\sim 1/(2\eta ^2)$ populated modes for $\eta \ll 1$.

The decomposition of $\left | u_{(0,0)}^{w_0',z_0} \right \rangle$ given by Eq. (14) now can be used to calculate the function $f(w_0')$. We have, starting from Eq. (11) (note that in the following $Q$ is a function of $w_0'$ through the definition of Eq. (14)),

$$\begin{aligned}&f(w_0') = \\ &= \left(1 - Q^2 \right) \textrm{Tr} \left[ \sum_{p_3,l_3} \sum_{p_4,l_4} a_{p_3,l_3,p_4,l_4}^{w_0,z_0}\left| u_{(p_3,l_3)}^{w_0,z_0} {\bigg \rangle}{\bigg \langle} u_{(p_4,l_4)}^{w_0,z_0} \right| \left(\sum_{p_1} Q^{p_1} \left| u_{(p_1,0)} ^{w_0,z_0} {\bigg \rangle} \right) \left( \sum_{p_2} Q^{p_2}{\bigg \langle} u_{(p_2,0)} ^{w_0,z_0} \right| \right) \right] = \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad= \left(1 - Q^2\right)\, \sum_{p_1} \sum_{p_2} Q^{p_1+p_2} \, a_{p_1,0,p_2,0}^{w_0,z_0} \ . \end{aligned}$$

This can be written in a polynomial form as simply

$$f(Q) = \sum_{p} C_p Q^p \ ,$$
with $C_0 = a_{0,0,0,0}^{w_0,z_0}$, $C_1 = a_{1,0,0,0}^{w_0,z_0} + a_{0,0,1,0}^{w_0,z_0}$ and
$$C_p = \sum_{t = 0}^{p} a_{t,0,p-t,0}^{w_0,z_0} \, - \, \sum_{t = 0}^{p-2} a_{t,0,p-2-t,0}^{w_0,z_0} \ , \ p \ge 2 \ .$$

This means that, in full generality, the optimal value of $w_0'$ must be found by a maximization of a polynomial on $Q(w_0')$ with infinite terms. Most often, though, if the basis in which we have characterized the laser beam (that is, the basis with waist $w_0$) is chosen judiciously (for example, taking as waist parameter the second-order moment of the beam at the focal plane), just a few modes are effectively populated, and the polynomial can be truncated. We note that, for $w_0' > 0$, the $Q$ parameter satisfies $-1 < Q < 1$, and thus the maximization of $f(Q)$ is to be done for $Q \in (-1,1)$.

4. Application to a coherent superposition of modes

We consider a specific coherent superposition of modes below, with equal power in each one of the $(0,0)$ and the $(1,0)$ modes, and a fixed relative phase $\theta$ between them:

$$E = \frac{u_{(0,0)}^{w_0,z_0} + e^{i \theta} u_{(1,0)}^{w_0,z_0}}{\sqrt{2}}$$

The total amount of power in the $(0,0)$ Gaussian mode of waist $w_0'$ is a function of the $Q (w_0')$ parameter through the polynomial of Eq. (16), which in this particular case become

$$f(Q) ={-}\frac{Q^4}{2} - \cos \theta \, Q^3 + \cos \theta \, Q + \frac{1}{2} \ .$$

This polynomial is exact and does not need to be truncated, since the beam has a finite quantity of modes populated. For a fixed $\theta$, the value of the polynomial can be maximized as a function of $Q$, and the optimal value $Q^{\textrm {opt}} (\theta )$ allows us to find the length parameter $W_0^{\textrm {(opt)}} (\theta )$ of the Gaussian basis that guarantees maximum power of the beam of Eq. (18) in the $(0,0)$ Gaussian mode.

It is instructive to compare the results we find from this exact method to other approximate ways to find a length parameter for the incoming laser beam. In the laboratory, usually we use simplifying assumptions in order to obtain an estimation of the best LG basis to describe a laser beam. If we assume that it is already in a pure $(0,0)$ mode of a LG basis, to find the length parameter of this basis we should just calculate the evolution of the second-order moment of the light, $W(z)$, defined for a, intensity profile of $I(r,\phi ,z)$ through the expression

$$W^2(z) = 2 \, \frac{\int_{0}^{\infty} r dr \int_{0}^{2 \pi} d\phi \, r^2 \, I(r,\phi,z)}{\int_{0}^{\infty} r dr \int_{0}^{2 \pi} d\phi \, I(r,\phi,z)} \ ,$$
which becomes, for the state of the light expressed in a basis with length scale $w_0$ as the density matrix $\rho$,
$$\begin{aligned}W^2 (z) &= 2 \,\frac{\textrm{Tr}\left[\rho \, r^2 \right]}{\textrm{Tr}\left[\rho \right]} = 2 \, \frac{\int_{\mathbb{R}^2} \, d \vec{r} \left\langle \vec{r} \right| \rho\, r^2 \left|\vec{r} \right\rangle}{\textrm{Tr}\left[\rho \right]} = \\ & = 2 \, \frac{\sum_{p_1,l_1} \sum_{p_2,l_2} a_{p_1,l_1,p_2,l_2}^{w_0,z_0} \int_{0}^{\infty} r dr \int_{0}^{2 \pi} d\phi \, r^2 \left(u_{(l_1,p_1)}^{w_0,z_0} (r,\phi,z) \right)^* u_{(l_2,p_2)}^{w_0,z_0}(r,\phi,z) } {\sum_{p,l} a_{p,l,p,l}^{w_0,z_0}} \ ,\end{aligned}$$
where $\vec {r}$ is the position in the plane $(r,\phi )$. It is an easy calculation to show that, indeed, for a pure Gaussian beam in the basis determined by $w_0$, the second-order moment at the focus position, $W_0 \equiv W (z_0)$, is equal to $w_0$, exactly the correct length parameter for the adapted LG basis in which to describe our laser beam. For a pure $(p,l)$ mode, on the other hand, $W (z_0) = \sqrt {2p+|l|+1} w_0$ [14]; this means that, for a beam mostly in a $(0,0)$ LG mode but with a small component of higher-order modes, $W_0$ can be a bad estimation for the length parameter of the LG basis that will maximize the amount of power in the $(0,0)$ mode. Based on the results of Ref. [15], we can imagine correcting this estimation by the propagation factor $\textrm {M}^2$ of the beam [4], through a new length parameter $W_0^{\textrm {(M)}} = W_0/\sqrt {\textrm {M}^2}$. The $\textrm {M}^2$ parameter is sometimes referred to in the literature as the beam quality, but this is often a source of confusion [4]. This parameter represents a normalized ratio between the second-order moment of the beam at the focal plane and its divergence, defined as
$$\textrm{M}^2 = \lim_{z \rightarrow \infty}\frac{\pi\, W(z_0)\, W(z)}{\lambda\, z} \ .$$

For a pure $(p,l)$ LG mode, $\textrm {M}^2 = 2p+|l|+1$ [14], and then indeed $W_0^{\textrm {(M)}} = w_0$. Next, we compare, for the electric field given in Eq. (18), the amount of power in the $(0,0)$ mode of the LG basis determined by each one of the three length parameters, $W_0^{\textrm {(opt)}} (\theta )$, $W_0 (\theta )$ and $W_0^{\textrm {(M)}} (\theta )$, that depend in general on the relative phase $\theta$ of the superposition. We show in Fig. 1(a) each one of the parameters as a function of $\theta$, while in Fig. 1(b) we show the amount of power in the $(0,0)$ Gaussian mode determined by them. The exact method shows a clear advantage for $\theta$ around $0$; in particular, for $\theta = 0$ the power available in a $(0,0)$ mode is maximum and equal to $27/32 = 0.84375$, a value that cannot be found by any one of the parameters discussed before; interestingly, we see that $W_0^{\textrm {(M)}}$ is a quite better estimation than $W_0$ for $\theta = 0$ (but still less efficient than $W_0^{\textrm {(opt)}}$), although to our knowledge it was never proposed in the literature as a length parameter to maximize the power coupling to a resonator. For $\pi /2 < \theta < 3 \pi /2$, on the other hand, $W_0^{\textrm {(M)}}$ is a worse estimation, and the naive estimation $W_0$ gives almost the best result achievable. All in all, only the exact parameter, found through the maximization of the polynomial of Eq. (19), guarantee the best possible result over all values of $\theta$.

 

Fig. 1. (a) Estimations of the length parameter $w_0'$ of the Gaussian basis that maximizes the total power in the $(0,0)$ mode for the beam expressed in Eq. (18), as a function of the relative phase $\theta$ of the superposition. Continuous blue line: second-order moment of the beam at the focal plane, $W_0 (\theta )$. Short-dashed orange line: second-order moment at the focus, corrected by the $\textrm {M}^2$ parameter, $W_0^{\textrm {(M)}}(\theta )$. Long-dashed green line: Optimum value $W_0^{\textrm {(opt)}} (\theta )$ obtained by the maximization of the polynomial given by Eq. (19). (b) Relative power in the $(0,0)$ mode of the Gaussian basis determined by the $w_0'$ parameters expressed in fig. (a).

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In Fig. 2 we show the intensity profiles of the pure $(0,0)$ and the $(1,0)$ Gaussian modes of waist $w_0$, as well as of the coherent superposition of modes given by Eq. (18) with phases $\theta = 0$, $\pi /2$ and $\pi$. We see that the superposition of beams with $\theta = 0$ resembles a pure, although smaller, Gaussian beam, except for a faint ring outside the main lobe (that carries only $13.5~$% of its power); it is thus easy to understand that the projection on a Gaussian mode with smaller length scale ($W_0^{\textrm {(opt)}} = w_0/\sqrt {3}$ for $\theta = 0$) guarantee a fair amount of power in this mode. Moreover, as the phase of the faint ring is $\pi$-shifted with respect to the phase of the central lobe, as shown in the phase profile of Fig. 2(h), increasing the length parameter of the Gaussian quickly drops the total power projected on the $(0,0)$ mode (which has same phase on the whole wavefront), as the regions of the beam with opposite phase will contribute negatively to the projection. The estimation $W_0$ is, for this reason, specially bad for this beam, as it does not take into account the phase front of the beam, just its intensity distribution.

 

Fig. 2. (a-e) Intensity distribution at the focal plane for (a) the $(0,0)$ and (b) the $(1,0)$ Gaussian mode, and also for the linear combination of both modes given by Eq. (18), with (c) $\theta = 0$, (d) $\theta = \pi /2$ and (e) $\theta = \pi$. (f-j) Phase profile of each beam at the focal plane, respectively: (f) for the $(0,0)$ and (g) the $(1,0)$ Gaussian mode, as well as for the linear combination of both modes given by Eq. (18) with (h) $\theta = 0$, (i) $\theta = \pi /2$ and (j) $\theta = \pi$.

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Interestingly, the graphic of Fig. 1(b) shows that we can have the same coupling efficiency on a Gaussian $(0,0)$ mode for the $\theta = \pi$ superposition as we have for the $\theta = 0$ one (although for a higher waist, $W_0^{\textrm {(opt)}} = \sqrt {3} w_0$), which is moreover the best achievable efficiency for any of the beams described by Eq. (18), even if the intensity distribution shown in Fig. 2(e) is quite different from a Gaussian shape, getting to a perfect zero at its center. Again, the phase front of the beam (shown in Fig. 2(j)) has a direct influence on this result: Being constant over the whole beam, it allows for a positive contribution of all points in space. The phase wavefront is also responsible for the poor efficiency ($50$% maximum) achievable for the $\theta = \pi /2$ superposition, as its continuous variation from $0$ at $r = 0$ to $\pi /2$ at $r = \infty$ (see Fig. 2(i)) superposes poorly with the constant wavefront of a $(0,0)$ Gaussian mode.

5. Discussion and conclusion

In conclusion, we have used a relation between LG basis of different waists to obtain a polynomial that gives the value of the projection of a laser beam described in a LG basis on the $(0,0)$ mode of a LG basis of different waist. The numerical maximization of this polynomial on a finite interval allows to find the best possible coupling of this beam to a resonator, or to a single-mode fiber of small numerical aperture, whose singlemode can be well described by a Gaussian $(0,0)$ mode. The method developed here sheds light on the role of the wavefront of the light for the coupling to a $(0,0)$ mode, which is not directly accessible through intensity measurements of the beam but has great influence on the coupling efficiency. We have shown, in particular, that light beams with an intensity profile quite different from a Gaussian shape could have surprisingly high coupling efficiencies, guaranteed by the flatness of its wavefront.

As a final note, we should say something about the choice of the origin in the transverse plane for the initial choice of LG basis to describe our beam, which was overlooked during the treatment presented here. A beam presenting cylindrical symmetry (both in intensity and phase) has an obvious choice of center in the transverse plane: The center of mass of the intensity distribution. On the other hand, a beam without cylindrical symmetry has often in the center of mass still a good approximation for the choice of the transverse origin for the LG basis, although it is not anymore rigorously the best one for the applicability of this method. Of course, for a beam with small discrepancies from a cylindrically symmetric beam, the results presented here will still be approximately valid. The same can be said of an elliptical beam: Through a simple rescaling of one of its transverse dimensions by a telescope formed of cylindrical lenses, we can find a beam that will be quite suitable, if not rigorously suitable, to the method described in this paper.

Appendix A: Verifying the decomposition of the $(0,0)$ Gaussian mode in a basis with different waist, Eq. (14)

The decomposition of a $(0,0)$ Gaussian mode in a basis of different waist, Eq. (14), can be expanded as

$$\begin{aligned}&\sqrt{1 - Q^2} \, \sum_{p = 0}^{\infty} Q^p \, u_{(p,0)} ^{w_0,z_0}(r,\phi,z)\ = \\ & = \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w(z)} \, \sum_{p} Q^p \, \exp\! \left(\! -\frac{r^2}{w^2(z)} - i k \frac{r^2}{2 R(z)} \right) \, \exp\left((2p + 1) i \psi^{(\textrm{G})}(z) \right) \, L_p^{0} \! \left(\frac{2r^2}{w^2(z)}\right)\ . \end{aligned}$$

The generating function of the Laguerre polynomials $L_p^{0} (\xi )$ is [16]

$$\sum_{p = 0}^{\infty} t^p L_p^{0} (\xi) = \frac{1}{1-t} \exp{\left(-\frac{t \xi}{1-t}\right)} \ .$$

Replacing in the equation above $t = Q \exp \left (2 i \psi ^{(\textrm {G})}(z) \right )$ and $\xi = \tfrac {2r^2}{w^2(z)}$, we arrive at

$$\begin{aligned}\sum_{p = 0}^{\infty} \left[Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right)\right]^p L_p^{0} \! &\left(\frac{2r^2}{w^2(z)}\right) = \\ &= \frac{1}{1-Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right)} \exp{\left(-\frac{Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right) \frac{2r^2}{w^2(z)}}{1\,-\,Q\exp\left(2 i \psi^{(\textrm{G})}(z) \right)}\right)} \ . \end{aligned}$$

Now, Eq. (23) can be rearranged to make appear the left side of Eq. (25):

$$\begin{aligned}&\sqrt{1 - Q^2} \, \sum_{p = 0}^{\infty} Q^p \, u_{(p,0)} ^{w_0,z_0}(r,\phi,z)\ = \\ &= \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w(z)} \, \exp\! \left(\! -\frac{r^2}{w^2(z)} - i k \frac{r^2}{2 R(z)} + i \psi^{(\textrm{G})}(z)\right) \sum_{p} \left[\left( Q \, \exp\left(2 i \psi^{(\textrm{G})}(z)\right) \right)^p \, L_p^{0} \! \left(\frac{2r^2}{w^2(z)}\right)\right] = \\ &\quad\quad\quad\quad = \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w(z)} \, \frac{\exp\! \left(\! -\frac{r^2}{w^2(z)} - i k \frac{r^2}{2 R(z)} + i \psi^{(\textrm{G})}(z) -\frac{Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right) \frac{2r^2}{w^2(z)}}{1\,-\,Q\exp\left(2 i \psi^{(\textrm{G})}(z) \right)}\right)}{1-Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right)} \ . \end{aligned}$$

Part of the last expression of the equation above expands as follows (renaming $\tfrac {z-z_0}{z_{\textrm {R}}} \equiv \zeta$):

$$\begin{array}{c}= \exp\! \left(\! -\frac{r^2}{w^2(z)} -\frac{Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right) \frac{2r^2}{w^2(z)}}{1\,-\,Q\exp\left(2 i \psi^{(\textrm{G})}(z) \right)}\right) = \exp\! \left(\! -\frac{r^2}{w^2(z)} \left[\frac{1 \,+ \, Q \exp\left(2 i \arctan \zeta \right)}{1\,-\,Q \exp\left(2 i \arctan \zeta \right)}\right] \right) = \\ = \exp\! \left(\! -\frac{r^2}{w^2(z)} \left[\frac{1 \,+ \, Q \cos\left(2 \arctan \zeta \right) \,+\, i Q \sin\left(2 \arctan \zeta \right)}{1\,-\, Q \cos\left(2 \arctan \zeta \right) \,-\, i Q \sin\left(2 \arctan \zeta \right)}\right] \right) = \\ = \exp\! \left(\! -\frac{r^2}{w^2(z)} \left[\frac{1 \,+ \, Q \frac{1-\zeta^2}{1+\zeta^2} \,+\, i Q \frac{2\zeta}{1+\zeta^2}}{1\,-\, Q \frac{1-\zeta^2}{1+\zeta^2} \,-\, i Q \frac{2\zeta}{1+\zeta^2}}\right] \right) = \exp\! \left(\! -\frac{r^2}{w^2(z)} \left[\frac{\left(1-Q^2\right) \left(1+\zeta^2\right) \,+\, 4 i Q \zeta}{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2}\right] \right). \end{array}$$

Replacing this result in Eq. (26), we arrive at

$$\begin{array}{c}\sqrt{1 - Q^2} \, \sum\limits_{p = 0}^{\infty} Q^p \, u_{(p,0)} ^{w_0,z_0}(r,\phi,z)\ = \\ = \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w(z)} \, \frac{\exp\! \left(i \psi^{(\textrm{G})}(z)\right) }{1-Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right)} \times \\ \times \, \exp\! \left(\! -\frac{r^2}{w^2(z)}\left[\frac{\left(1-Q^2\right) \left(1+\zeta^2\right)}{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2}\right] - i r^2 \left[\frac{k}{2 R(z)} + \frac{4 Q \zeta}{w^2(z) \left(\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2\right)}\right]\right) \ . \end{array}$$

To continue further, we first note that

$$\begin{aligned}&\quad w^2(z)\frac{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2}{\left(1-Q^2\right) \left(1+\zeta^2\right)} = w_0^2 \left(1+\zeta^2\right)\frac{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2}{\left(1-Q^2\right) \left(1+\zeta^2\right)} = \\ &w_0^2 \frac{\left(1-Q\right)^2}{\left(1-Q^2\right)} \left(1 + \frac{\left(1+Q\right)^2 \zeta^2}{\left(1-Q\right)^2}\right) = w_0^2 \frac{1-Q}{1+Q} \left(1 + \frac{(z-z_0)^2}{z_\textrm{R}^2\left(\frac{1-Q}{1+Q}\right)^2}\right) = w_0'^2 \left(1 + \frac{(z-z_0)^2}{z_\textrm{R}'^2}\right) = w'^2(z) \ , \end{aligned}$$
where, following the definition of $Q$ given in Eq. (14),
$$w_0^2 \, \frac{1-Q}{1+Q} = w_0^2 \, \eta^2 = w_0'^2$$
and
$$z_\textrm{R}^2\left(\frac{1-Q}{1+Q}\right)^2 = \frac{\pi^2 w_0^4}{\lambda^2}\, \eta^4 = \frac{\pi^2 w_0'^4}{\lambda^2} = z_\textrm{R}'^2\ .$$

For the imaginary part of the exponential of Eq. (28), we first compute the term proportional to $r^2$ (and we use the definition =$\zeta ' \equiv \tfrac {z-z_0}{z_{\textrm {R}}'}$):

$$\begin{array}{c}- i r^2 \left[\frac{k}{2 R(z)} + \frac{4 Q \zeta}{w^2(z) \left(\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2\right)}\right] = \\ ={-} i k r^2 \left[\frac{1}{2 (z - z_0) \left(1 + \frac{1}{\zeta^2}\right)} + \frac{4 Q \zeta}{k w_0^2 \left(1 + \zeta^2\right) \left(\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2\right)}\right] = \\ ={-} i \,\frac{k r^2}{2(z - z_0)} \left[\frac{\zeta^2}{1 + \zeta^2} + \frac{4 Q \zeta^2}{\left(1 + \zeta^2\right) \left(\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2\right)}\right] = \\ ={-} i \,\frac{k r^2}{2(z - z_0)} \,\frac{\zeta^2}{1 + \zeta^2} \,\frac{4 Q + \left(\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2\right)}{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2} ={-} i \,\frac{k r^2}{2(z - z_0)} \,\frac{\left(1+Q\right)^2 \zeta^2}{\left(1-Q\right)^2 + \left(1+Q\right)^2 \zeta^2} = \\ - i \,\frac{k r^2}{2(z - z_0)} \,\frac{1}{1+ \frac{\left(1-Q\right)^2}{\left(1+Q\right)^2 \zeta^2}} ={-} i \,\frac{k r^2}{2(z - z_0)} \,\frac{1}{1+ \frac{1}{\zeta'^2}} ={-} i \,\frac{k r^2}{2R'(z)} \ , \end{array}$$
with $R'(z) = (z - z_0) (1 + \tfrac {z_\textrm {R}^2}{(z - z_0)^2})$. Finally, we have to transform the expression
$$\sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w(z)} \frac{\exp\! \left(i \psi^{(\textrm{G})}(z)\right) }{1-Q \exp\left(2 i \psi^{(\textrm{G})}(z) \right)} = \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w_0 \sqrt{1+\zeta^2}} \frac{\exp\! \left(i \psi^{(\textrm{G})}(z)\right) }{1-Q \frac{1-\zeta^2}{1+\zeta^2}-\frac{2 i Q \zeta}{1 + \zeta^2}} \ .$$

Let’s calculate the modulus and the phase of the complex expression above. For the modulus,

$$\begin{aligned}\left| \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{1}{w_0 \sqrt{1+\zeta^2}} \frac{\exp\! \left(i \psi^{(\textrm{G})}(z)\right) }{1-Q \frac{1-\zeta^2}{1+\zeta^2}-\frac{2 i Q \zeta}{1 + \zeta^2}}\right| = \\ = \sqrt{\frac{2(1 - Q^2)}{\pi}} \, \frac{\sqrt{1+\zeta^2}}{w_0} \frac{1}{\sqrt{\left(1 + \zeta^2-Q \left(1-\zeta^2\right)\right)^2+\left(2 Q \zeta\right)^2}} = \\ = \sqrt{\frac{2}{\pi}} \frac{1}{w_0}\sqrt{\frac{(1 - Q^2)(1+\zeta^2)}{\left(1+\zeta^2\right) \left(\left(1-Q\right)^2+ \left(1+Q\right)^2 \zeta^2 \right)}} = \sqrt{\frac{2}{\pi}} \frac{1}{w_0\sqrt{\frac{1-Q}{1+Q}}\sqrt{1 + \frac{\left(1+Q\right)^2}{\left(1-Q\right)^2}\zeta^2 }} = \\ = \sqrt{\frac{2}{\pi}} \frac{1}{w_0' \sqrt{1 + \zeta'^2}} = \sqrt{\frac{2}{\pi}} \frac{1}{w'(z)} \ . \end{aligned}$$

For calculating the complex phase of the expression (33), we note that it must have the same phase as the expression

$$\begin{aligned}\frac{\exp\! \left(i \psi^{(\textrm{G})}(z)\right) }{1-Q \frac{1-\zeta^2}{1+\zeta^2}-\frac{2 i Q \zeta}{1 + \zeta^2}} &= \frac{\frac{1-\zeta^2}{\sqrt{1+\zeta^2}}+\frac{i \zeta}{\sqrt{1 + \zeta^2}}}{1-Q \frac{1-\zeta^2}{1+\zeta^2}-\frac{2 i Q \zeta}{1 + \zeta^2}} = \\ &= \sqrt{1+\zeta^2} \frac{(1+i\zeta)\left(1+\zeta^2-Q \left(1-\zeta^2\right)+ 2 i Q \zeta \right)}{\left(1 + \zeta^2-Q \left(1-\zeta^2\right)\right)^2+\left(2 Q \zeta\right)^2} = \\ &\quad\quad = \left(1+\zeta^2\right)^{3/2} \frac{1-Q + i (1+ Q) \zeta}{\left(1 + \zeta^2-Q \left(1-\zeta^2\right)\right)^2+\left(2 Q \zeta\right)^2} \ . \end{aligned}$$

Since the denominator of the last fraction is real, the complex phase of the number is equal to

$$\arctan{\frac{(1+ Q) \, \zeta}{1-Q}} = \arctan{\frac{z-z_0}{z_\textrm{R}'}} \ ,$$
which leads us to the result
$$\begin{aligned}\sqrt{1 - Q^2} \, &\sum_{p = 0}^{\infty} Q^p \, u_{(p,0)} ^{w_0,z_0}(r,\phi,z)\ = \\ &= \sqrt{\frac{2}{\pi}} \, \frac{1}{w'(z)} \, \exp\! \left(\! -\frac{r^2}{w'^2(z)} - i k \frac{r^2}{2 R'(z)} \right) \, \exp\left(i \arctan{\frac{z-z_0}{z_\textrm{R}'}} \right) = u_{(0,0)} ^{w_0',z_0}(r,\phi,z) \ . \end{aligned}$$

Funding

Fundação de Amparo à Pesquisa do Estado de São Paulo (2013/04162-5, 2018/23873-3).

Disclosures

The authors declare no conflicts of interest.

References

1. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef]  

2. J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of optical engineering (Marcel Dekker, Inc., 2003), pp. 999–1013.

3. J. A. Buck, Fundamentals of optical fibers (John Wiley & Sons, 2004).

4. A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).

5. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using cgh-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009). [CrossRef]  

6. D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37(13), 2478–2480 (2012). [CrossRef]  

7. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005). [CrossRef]  

8. E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1(5), 541–546 (1984). [CrossRef]  

9. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005). [CrossRef]  

10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef]  

11. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

12. M. A. Porras, J. Alda, and E. Bernabeu, “Complex beam parameter and abcd law for non-gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992). [CrossRef]  

13. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27(5), 1146–1148 (1991). [CrossRef]  

14. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998). [CrossRef]  

15. C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20(25), 27866–27873 (2012). [CrossRef]  

16. M. L. Boas, Mathematical methods in the physical sciences (John Wiley & Sons, 2006).

References

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  1. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
    [Crossref]
  2. J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of optical engineering (Marcel Dekker, Inc., 2003), pp. 999–1013.
  3. J. A. Buck, Fundamentals of optical fibers (John Wiley & Sons, 2004).
  4. A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).
  5. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using cgh-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009).
    [Crossref]
  6. D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37(13), 2478–2480 (2012).
    [Crossref]
  7. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005).
    [Crossref]
  8. E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1(5), 541–546 (1984).
    [Crossref]
  9. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
    [Crossref]
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [Crossref]
  11. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  12. M. A. Porras, J. Alda, and E. Bernabeu, “Complex beam parameter and abcd law for non-gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992).
    [Crossref]
  13. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27(5), 1146–1148 (1991).
    [Crossref]
  14. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
    [Crossref]
  15. C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20(25), 27866–27873 (2012).
    [Crossref]
  16. M. L. Boas, Mathematical methods in the physical sciences (John Wiley & Sons, 2006).

2012 (2)

2009 (1)

2005 (2)

Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005).
[Crossref]

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

1998 (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[Crossref]

1992 (1)

1991 (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27(5), 1146–1148 (1991).
[Crossref]

1984 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1966 (1)

Abouraddy, A. F.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

Agarwal, G. S.

Alda, J.

M. A. Porras, J. Alda, and E. Bernabeu, “Complex beam parameter and abcd law for non-gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992).
[Crossref]

J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of optical engineering (Marcel Dekker, Inc., 2003), pp. 999–1013.

Bernabeu, E.

Boas, M. L.

M. L. Boas, Mathematical methods in the physical sciences (John Wiley & Sons, 2006).

Buck, J. A.

J. A. Buck, Fundamentals of optical fibers (John Wiley & Sons, 2004).

Davidson, F. M.

Dikmelik, Y.

Duparré, M.

Fienup, J. R.

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

Flamm, D.

Forbes, A.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

Kaiser, T.

Kogelnik, H.

Li, T.

Naidoo, D.

Ngcobo, S.

Porras, M. A.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schröter, S.

Schulze, C.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27(5), 1146–1148 (1991).
[Crossref]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).

Wolf, E.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27(5), 1146–1148 (1991).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. Lett. (1)

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref]

Other (4)

J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of optical engineering (Marcel Dekker, Inc., 2003), pp. 999–1013.

J. A. Buck, Fundamentals of optical fibers (John Wiley & Sons, 2004).

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).

M. L. Boas, Mathematical methods in the physical sciences (John Wiley & Sons, 2006).

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Figures (2)

Fig. 1.
Fig. 1. (a) Estimations of the length parameter $w_0'$ of the Gaussian basis that maximizes the total power in the $(0,0)$ mode for the beam expressed in Eq. (18), as a function of the relative phase $\theta$ of the superposition. Continuous blue line: second-order moment of the beam at the focal plane, $W_0 (\theta )$. Short-dashed orange line: second-order moment at the focus, corrected by the $\textrm {M}^2$ parameter, $W_0^{\textrm {(M)}}(\theta )$. Long-dashed green line: Optimum value $W_0^{\textrm {(opt)}} (\theta )$ obtained by the maximization of the polynomial given by Eq. (19). (b) Relative power in the $(0,0)$ mode of the Gaussian basis determined by the $w_0'$ parameters expressed in fig. (a).
Fig. 2.
Fig. 2. (a-e) Intensity distribution at the focal plane for (a) the $(0,0)$ and (b) the $(1,0)$ Gaussian mode, and also for the linear combination of both modes given by Eq. (18), with (c) $\theta = 0$, (d) $\theta = \pi /2$ and (e) $\theta = \pi$. (f-j) Phase profile of each beam at the focal plane, respectively: (f) for the $(0,0)$ and (g) the $(1,0)$ Gaussian mode, as well as for the linear combination of both modes given by Eq. (18) with (h) $\theta = 0$, (i) $\theta = \pi /2$ and (j) $\theta = \pi$.

Equations (37)

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u ( 0 , 0 ) ( w 0 , z 0 ) ( r , z ) = 2 π 1 w ( z ) exp ( r 2 w 2 ( z ) ) exp ( i k r 2 2 R ( z ) ) exp ( i ψ ( G ) ( z ) ) .
w ( z ) = w 0 1 + ( z z 0 z R ) 2   ,
R ( z ) = ( z z 0 ) [ 1 + ( z R z z 0 ) 2 ]
ψ ( G ) ( z ) = arctan ( z z 0 z R )   .
( u 1 , u 2 ) = 0 2 π d ϕ 0 r d r u 1 ( r , ϕ , z ) u 2 ( r , ϕ , z )   ,
u ( p , l ) w 0 , z 0 ( r , ϕ , z ) = 2 p ! π ( p + | l | ) ! 1 w ( z ) ( 2 r w ( z ) ) | l | exp ( r 2 w 2 ( z ) ) L p | l | ( 2 r 2 w 2 ( z ) ) × × exp ( i k r 2 2 R ( z ) ) exp ( i l ϕ ) exp ( i ψ ( p , l ) ( G ) ( z ) )   ,
L p l ( ξ ) = x l e ξ p ! d p d ξ p ( e ξ ξ n + l )   .
ψ ( p , l ) ( G ) ( z ) = ( 2 p + | l | + 1 ) arctan ( z z 0 z R ) = ( 2 p + | l | + 1 ) ψ ( G ) ( z )   .
ρ = p 1 , l 1 p 2 , l 2 a p 1 , l 1 , p 2 , l 2 w 0 , z 0 | u ( p 1 , l 1 ) w 0 , z 0 u ( p 2 , l 2 ) w 0 , z 0 |   .
| E ( r , ϕ , z ) = l , p b l , p w 0 , z 0 | u ( l , p ) w 0 , z 0
f ( w 0 ) = Tr [ ρ | u ( 0 , 0 ) w 0 , z 0 u ( 0 , 0 ) w 0 , z 0 | ]
f ( w 0 ) = | u ( 0 , 0 ) w 0 , z 0 | E ( r , ϕ , z ) | 2   ,
W 2 ( z ) = W 2 ( z 0 ) ( 1 + ( z z 0 ) 2 z R 2 )   ,
u ( 0 , 0 ) w 0 , z 0 ( r , ϕ , z ) = 1 Q 2 p Q p u ( p , 0 ) w 0 , z 0 ( r , ϕ , z )   ,   with   Q = 1 η 2 1 + η 2   and   η = w 0 w 0   .
f ( w 0 ) = = ( 1 Q 2 ) Tr [ p 3 , l 3 p 4 , l 4 a p 3 , l 3 , p 4 , l 4 w 0 , z 0 | u ( p 3 , l 3 ) w 0 , z 0 u ( p 4 , l 4 ) w 0 , z 0 | ( p 1 Q p 1 | u ( p 1 , 0 ) w 0 , z 0 ) ( p 2 Q p 2 u ( p 2 , 0 ) w 0 , z 0 | ) ] = = ( 1 Q 2 ) p 1 p 2 Q p 1 + p 2 a p 1 , 0 , p 2 , 0 w 0 , z 0   .
f ( Q ) = p C p Q p   ,
C p = t = 0 p a t , 0 , p t , 0 w 0 , z 0 t = 0 p 2 a t , 0 , p 2 t , 0 w 0 , z 0   ,   p 2   .
E = u ( 0 , 0 ) w 0 , z 0 + e i θ u ( 1 , 0 ) w 0 , z 0 2
f ( Q ) = Q 4 2 cos θ Q 3 + cos θ Q + 1 2   .
W 2 ( z ) = 2 0 r d r 0 2 π d ϕ r 2 I ( r , ϕ , z ) 0 r d r 0 2 π d ϕ I ( r , ϕ , z )   ,
W 2 ( z ) = 2 Tr [ ρ r 2 ] Tr [ ρ ] = 2 R 2 d r r | ρ r 2 | r Tr [ ρ ] = = 2 p 1 , l 1 p 2 , l 2 a p 1 , l 1 , p 2 , l 2 w 0 , z 0 0 r d r 0 2 π d ϕ r 2 ( u ( l 1 , p 1 ) w 0 , z 0 ( r , ϕ , z ) ) u ( l 2 , p 2 ) w 0 , z 0 ( r , ϕ , z ) p , l a p , l , p , l w 0 , z 0   ,
M 2 = lim z π W ( z 0 ) W ( z ) λ z   .
1 Q 2 p = 0 Q p u ( p , 0 ) w 0 , z 0 ( r , ϕ , z )   = = 2 ( 1 Q 2 ) π 1 w ( z ) p Q p exp ( r 2 w 2 ( z ) i k r 2 2 R ( z ) ) exp ( ( 2 p + 1 ) i ψ ( G ) ( z ) ) L p 0 ( 2 r 2 w 2 ( z ) )   .
p = 0 t p L p 0 ( ξ ) = 1 1 t exp ( t ξ 1 t )   .
p = 0 [ Q exp ( 2 i ψ ( G ) ( z ) ) ] p L p 0 ( 2 r 2 w 2 ( z ) ) = = 1 1 Q exp ( 2 i ψ ( G ) ( z ) ) exp ( Q exp ( 2 i ψ ( G ) ( z ) ) 2 r 2 w 2 ( z ) 1 Q exp ( 2 i ψ ( G ) ( z ) ) )   .
1 Q 2 p = 0 Q p u ( p , 0 ) w 0 , z 0 ( r , ϕ , z )   = = 2 ( 1 Q 2 ) π 1 w ( z ) exp ( r 2 w 2 ( z ) i k r 2 2 R ( z ) + i ψ ( G ) ( z ) ) p [ ( Q exp ( 2 i ψ ( G ) ( z ) ) ) p L p 0 ( 2 r 2 w 2 ( z ) ) ] = = 2 ( 1 Q 2 ) π 1 w ( z ) exp ( r 2 w 2 ( z ) i k r 2 2 R ( z ) + i ψ ( G ) ( z ) Q exp ( 2 i ψ ( G ) ( z ) ) 2 r 2 w 2 ( z ) 1 Q exp ( 2 i ψ ( G ) ( z ) ) ) 1 Q exp ( 2 i ψ ( G ) ( z ) )   .
= exp ( r 2 w 2 ( z ) Q exp ( 2 i ψ ( G ) ( z ) ) 2 r 2 w 2 ( z ) 1 Q exp ( 2 i ψ ( G ) ( z ) ) ) = exp ( r 2 w 2 ( z ) [ 1 + Q exp ( 2 i arctan ζ ) 1 Q exp ( 2 i arctan ζ ) ] ) = = exp ( r 2 w 2 ( z ) [ 1 + Q cos ( 2 arctan ζ ) + i Q sin ( 2 arctan ζ ) 1 Q cos ( 2 arctan ζ ) i Q sin ( 2 arctan ζ ) ] ) = = exp ( r 2 w 2 ( z ) [ 1 + Q 1 ζ 2 1 + ζ 2 + i Q 2 ζ 1 + ζ 2 1 Q 1 ζ 2 1 + ζ 2 i Q 2 ζ 1 + ζ 2 ] ) = exp ( r 2 w 2 ( z ) [ ( 1 Q 2 ) ( 1 + ζ 2 ) + 4 i Q ζ ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ] ) .
1 Q 2 p = 0 Q p u ( p , 0 ) w 0 , z 0 ( r , ϕ , z )   = = 2 ( 1 Q 2 ) π 1 w ( z ) exp ( i ψ ( G ) ( z ) ) 1 Q exp ( 2 i ψ ( G ) ( z ) ) × × exp ( r 2 w 2 ( z ) [ ( 1 Q 2 ) ( 1 + ζ 2 ) ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ] i r 2 [ k 2 R ( z ) + 4 Q ζ w 2 ( z ) ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) ] )   .
w 2 ( z ) ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ( 1 Q 2 ) ( 1 + ζ 2 ) = w 0 2 ( 1 + ζ 2 ) ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ( 1 Q 2 ) ( 1 + ζ 2 ) = w 0 2 ( 1 Q ) 2 ( 1 Q 2 ) ( 1 + ( 1 + Q ) 2 ζ 2 ( 1 Q ) 2 ) = w 0 2 1 Q 1 + Q ( 1 + ( z z 0 ) 2 z R 2 ( 1 Q 1 + Q ) 2 ) = w 0 2 ( 1 + ( z z 0 ) 2 z R 2 ) = w 2 ( z )   ,
w 0 2 1 Q 1 + Q = w 0 2 η 2 = w 0 2
z R 2 ( 1 Q 1 + Q ) 2 = π 2 w 0 4 λ 2 η 4 = π 2 w 0 4 λ 2 = z R 2   .
i r 2 [ k 2 R ( z ) + 4 Q ζ w 2 ( z ) ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) ] = = i k r 2 [ 1 2 ( z z 0 ) ( 1 + 1 ζ 2 ) + 4 Q ζ k w 0 2 ( 1 + ζ 2 ) ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) ] = = i k r 2 2 ( z z 0 ) [ ζ 2 1 + ζ 2 + 4 Q ζ 2 ( 1 + ζ 2 ) ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) ] = = i k r 2 2 ( z z 0 ) ζ 2 1 + ζ 2 4 Q + ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 = i k r 2 2 ( z z 0 ) ( 1 + Q ) 2 ζ 2 ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 = i k r 2 2 ( z z 0 ) 1 1 + ( 1 Q ) 2 ( 1 + Q ) 2 ζ 2 = i k r 2 2 ( z z 0 ) 1 1 + 1 ζ 2 = i k r 2 2 R ( z )   ,
2 ( 1 Q 2 ) π 1 w ( z ) exp ( i ψ ( G ) ( z ) ) 1 Q exp ( 2 i ψ ( G ) ( z ) ) = 2 ( 1 Q 2 ) π 1 w 0 1 + ζ 2 exp ( i ψ ( G ) ( z ) ) 1 Q 1 ζ 2 1 + ζ 2 2 i Q ζ 1 + ζ 2   .
| 2 ( 1 Q 2 ) π 1 w 0 1 + ζ 2 exp ( i ψ ( G ) ( z ) ) 1 Q 1 ζ 2 1 + ζ 2 2 i Q ζ 1 + ζ 2 | = = 2 ( 1 Q 2 ) π 1 + ζ 2 w 0 1 ( 1 + ζ 2 Q ( 1 ζ 2 ) ) 2 + ( 2 Q ζ ) 2 = = 2 π 1 w 0 ( 1 Q 2 ) ( 1 + ζ 2 ) ( 1 + ζ 2 ) ( ( 1 Q ) 2 + ( 1 + Q ) 2 ζ 2 ) = 2 π 1 w 0 1 Q 1 + Q 1 + ( 1 + Q ) 2 ( 1 Q ) 2 ζ 2 = = 2 π 1 w 0 1 + ζ 2 = 2 π 1 w ( z )   .
exp ( i ψ ( G ) ( z ) ) 1 Q 1 ζ 2 1 + ζ 2 2 i Q ζ 1 + ζ 2 = 1 ζ 2 1 + ζ 2 + i ζ 1 + ζ 2 1 Q 1 ζ 2 1 + ζ 2 2 i Q ζ 1 + ζ 2 = = 1 + ζ 2 ( 1 + i ζ ) ( 1 + ζ 2 Q ( 1 ζ 2 ) + 2 i Q ζ ) ( 1 + ζ 2 Q ( 1 ζ 2 ) ) 2 + ( 2 Q ζ ) 2 = = ( 1 + ζ 2 ) 3 / 2 1 Q + i ( 1 + Q ) ζ ( 1 + ζ 2 Q ( 1 ζ 2 ) ) 2 + ( 2 Q ζ ) 2   .
arctan ( 1 + Q ) ζ 1 Q = arctan z z 0 z R   ,
1 Q 2 p = 0 Q p u ( p , 0 ) w 0 , z 0 ( r , ϕ , z )   = = 2 π 1 w ( z ) exp ( r 2 w 2 ( z ) i k r 2 2 R ( z ) ) exp ( i arctan z z 0 z R ) = u ( 0 , 0 ) w 0 , z 0 ( r , ϕ , z )   .

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