Evolving beam width

The Gaussian function has a 1/e² diameter (2w as used in the text) about 1.7 times the FWHM.

At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation:^[1]

$${\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}},}$$

where^[1]

$${\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}n}{\lambda }}}$$

is called the Rayleigh range as further discussed below, and ${\displaystyle n}$ is the refractive index of the medium.

The radius of the beam w(z), at any position z along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:^[7]

$${\displaystyle w(z)={\frac {{\text{FWHM}}(z)}{\sqrt {2\ln 2}}}.}$$

Wavefront curvature

The curvature of the wavefronts is largest at the Rayleigh distance, z = ±z_R, on either side of the waist, crossing zero at the waist itself. Beyond the Rayleigh distance, |z| > z_R, it again decreases in magnitude, approaching zero as z → ±∞. The curvature is often expressed in terms of its reciprocal, R, the radius of curvature; for a fundamental Gaussian beam the curvature at position z is given by:

$${\displaystyle {\frac {1}{R(z)}}={\frac {z}{z^{2}+z_{\mathrm {R} }^{2}}},}$$

so the radius of curvature R(z) is ^[1]

$${\displaystyle R(z)=z\left[{1+{\left({\frac {z_{\mathrm {R} }}{z}}\right)}^{2}}\right].}$$

Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.

Gouy phase

The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position z the Gouy phase of a fundamental Gaussian beam is given by^[1]

$${\displaystyle \psi (z)=\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right).}$$

Gouy phase.

The Gouy phase results in an increase in the apparent wavelength near the waist (z ≈ 0). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.

The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor.^[8] With e^iωt dependence, the Gouy phase changes from -π/2 to +π/2, while with e^-iωt dependence it changes from +π/2 to -π/2 along the axis.

For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to π radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.^[8]

Elliptical and astigmatic beams

Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for x and y and distinct definitions of the z = 0 point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range ±π/4 contributed by each dimension.

An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.

Beam waist

Gaussian beam width w(z) as a function of the distance z along the beam, which forms a hyperbola. w₀: beam waist; b: depth of focus; z_R: Rayleigh range; Θ: total angular spread

The shape of a Gaussian beam of a given wavelength λ is governed solely by one parameter, the beam waist w₀. This is a measure of the beam size at the point of its focus (z = 0 in the above equations) where the beam width w(z) (as defined above) is the smallest (and likewise where the intensity on-axis (r = 0) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range z_R and asymptotic beam divergence θ, as detailed below.

Rayleigh range and confocal parameter

The Rayleigh distance or Rayleigh range z_R is determined given a Gaussian beam's waist size:

$${\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}n}{\lambda }}.}$$

Here λ is the wavelength of the light, n is the index of refraction. At a distance from the waist equal to the Rayleigh range z_R, the width w of the beam is √2 larger than it is at the focus where w = w₀, the beam waist. That also implies that the on-axis (r = 0) intensity there is one half of the peak intensity (at z = 0). That point along the beam also happens to be where the wavefront curvature (1/R) is greatest.^[1]

The distance between the two points z = ±z_R is called the confocal parameter or depth of focus of the beam.^[9]

Beam divergence

Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where r = w(z). That is where the intensity has dropped to 1/e² of its on-axis value. Now, for z ≫ z_R the parameter w(z) increases linearly with z. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose r = w(z)) and the beam axis (r = 0) defines the divergence of the beam:

$${\displaystyle \theta =\lim _{z\to \infty }\arctan \left({\frac {w(z)}{z}}\right).}$$

In the paraxial case, as we have been considering, θ (in radians) is then approximately^[1]

$${\displaystyle \theta ={\frac {\lambda }{\pi nw_{0}}}}$$

where n is the refractive index of the medium the beam propagates through, and λ is the free-space wavelength. The total angular spread of the diverging beam, or apex angle of the above-described cone, is then given by

$${\displaystyle \Theta =2\theta \,.}$$

That cone then contains 86% of the Gaussian beam's total power.

Because the divergence is inversely proportional to the spot size, for a given wavelength λ, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to minimize the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section (w₀) at the waist (and thus a large diameter where it is launched, since w(z) is never less than w₀). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam.^[10] From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about 2λ/π.

Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size w₀. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as M² ("M squared"). The M² for a Gaussian beam is one. All real laser beams have M² values greater than one, although very high quality beams can have values very close to one.

The numerical aperture of a Gaussian beam is defined to be NA = n sin θ, where n is the index of refraction of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by

$${\displaystyle z_{\mathrm {R} }={\frac {nw_{0}}{\mathrm {NA} }}.}$$

Power through an aperture

With a beam centered on an aperture, the power P passing through a circle of radius r in the transverse plane at position z is^[11]

$${\displaystyle P(r,z)=P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right],}$$

where

$${\displaystyle P_{0}={\frac {1}{2}}\pi I_{0}w_{0}^{2}}$$

is the total power transmitted by the beam.

For a circle of radius r = w(z), the fraction of power transmitted through the circle is

$${\displaystyle {\frac {P(z)}{P_{0}}}=1-e^{-2}\approx 0.865.}$$

Similarly, about 90% of the beam's power will flow through a circle of radius r = 1.07 × w(z), 95% through a circle of radius r = 1.224 × w(z), and 99% through a circle of radius r = 1.52 × w(z).^[11]

Peak intensity

The peak intensity at an axial distance z from the beam waist can be calculated as the limit of the enclosed power within a circle of radius r, divided by the area of the circle πr² as the circle shrinks:

$${\displaystyle I(0,z)=\lim _{r\to 0}{\frac {P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]}{\pi r^{2}}}.}$$

The limit can be evaluated using L'Hôpital's rule:

$${\displaystyle I(0,z)={\frac {P_{0}}{\pi }}\lim _{r\to 0}{\frac {\left[-(-2)(2r)e^{-2r^{2}/w^{2}(z)}\right]}{w^{2}(z)(2r)}}={2P_{0} \over \pi w^{2}(z)}.}$$

Lens equation

As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point ${\displaystyle (x,y)}$ of the gaussian beam as it travels through the lens.^[15] An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.^[16]

The exact solution to the above problem is expressed simply in terms of the magnification ${\displaystyle M}$

${\displaystyle {\begin{aligned}w_{0}'&=Mw_{0}\\[1.2ex](z_{0}'-f)&=M^{2}(z_{0}-f).\end{aligned}}}$

The magnification, which depends on ${\displaystyle w_{0}}$ and ${\displaystyle z_{0}}$, is given by

${\displaystyle M={\frac {M_{r}}{\sqrt {1+r^{2}}}}}$

where

${\displaystyle r={\frac {z_{R}}{z_{0}-f}},\quad M_{r}=\left|{\frac {f}{z_{0}-f}}\right|.}$

An equivalent expression for the beam position ${\displaystyle z_{0}'}$ is

${\displaystyle {\frac {1}{z_{0}+{\frac {z_{R}^{2}}{(z_{0}-f)}}}}+{\frac {1}{z_{0}'}}={\frac {1}{f}}.}$

This last expression makes clear that the ray optics thin lens equation is recovered in the limit that ${\displaystyle \left|\left({\tfrac {z_{R}}{z_{0}}}\right)\left({\tfrac {z_{R}}{z_{0}-f}}\right)\right|\ll 1}$. It can also be noted that if ${\displaystyle \left|z_{0}+{\frac {z_{R}^{2}}{z_{0}-f}}\right|\gg f}$ then the incoming beam is "well collimated" so that ${\displaystyle z_{0}'\approx f}$.

Beam focusing

In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot. Mathematically, this implies minimization of the magnification ${\displaystyle M}$. If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing ${\displaystyle z_{R}}$ and minimizing ${\displaystyle f}$. In this situation, it is justifiable to make the approximation ${\displaystyle z_{R}^{2}/(z_{0}-f)^{2}\gg 1}$, implying that ${\displaystyle M\approx f/z_{R}}$ and yielding the result ${\displaystyle w_{0}'\approx fw_{0}/z_{R}}$. This result is often presented in the form

${\displaystyle {\begin{aligned}2w_{0}'&\approx {\frac {4}{\pi }}\lambda F_{\#}\\[1.2ex]z_{0}'&\approx f\end{aligned}}}$

where

${\displaystyle F_{\#}={\frac {f}{2w_{0}}},}$

which is found after assuming that the medium has index of refraction ${\displaystyle n\approx 1}$ and substituting ${\displaystyle z_{R}=\pi w_{0}^{2}/\lambda }$. The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters ${\displaystyle 2w_{0}'}$ and ${\displaystyle 2w_{0}}$, rather than the waist radii ${\displaystyle w_{0}'}$ and ${\displaystyle w_{0}}$.

Hermite-Gaussian modes

Twelve Hermite-Gaussian modes

It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called Hermite-Gaussian modes, any of which are given by the product of a factor in x and a factor in y. Such a solution is possible due to the separability in x and y in the paraxial Helmholtz equation as written in Cartesian coordinates.^[19] Thus given a mode of order (l, m) referring to the x and y directions, the electric field amplitude at x, y, z may be given by:

$${\displaystyle E(x,y,z)=u_{l}(x,z)\,u_{m}(y,z)\,\exp(-ikz),}$$

where the factors for the x and y dependence are each given by:

$${\displaystyle u_{J}(x,z)=\left({\frac {\sqrt {2/\pi }}{2^{J}\,J!\;w_{0}}}\right)^{\!\!1/2}\!\!\left({\frac {{q}_{0}}{{q}(z)}}\right)^{\!\!1/2}\!\!\left(-{\frac {{q}^{\ast }(z)}{{q}(z)}}\right)^{\!\!J/2}\!\!H_{J}\!\left({\frac {{\sqrt {2}}x}{w(z)}}\right)\,\exp \left(\!-i{\frac {kx^{2}}{2{q}(z)}}\right),}$$

where we have employed the complex beam parameter q(z) (as defined above) for a beam of waist w₀ at z from the focus. In this form, the first factor is just a normalizing constant to make the set of u_J orthonormal. The second factor is an additional normalization dependent on z which compensates for the expansion of the spatial extent of the mode according to w(z)/w₀ (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders J.

The final two factors account for the spatial variation over x (or y). The fourth factor is the Hermite polynomial of order J ("physicists' form", i.e. H₁(x) = 2x), while the fifth accounts for the Gaussian amplitude fall-off exp(−x²/w(z)²), although this isn't obvious using the complex q in the exponent. Expansion of that exponential also produces a phase factor in x which accounts for the wavefront curvature (1/R(z)) at z along the beam.

Hermite-Gaussian modes are typically designated "TEM_lm"; the fundamental Gaussian beam may thus be referred to as TEM₀₀ (where TEM is transverse electro-magnetic). Multiplying u_l(x, z) and u_m(y, z) to get the 2-D mode profile, and removing the normalization so that the leading factor is just called E₀, we can write the (l, m) mode in the more accessible form:

$${\displaystyle {\begin{aligned}E_{l,m}(x,y,z)={}&E_{0}{\frac {w_{0}}{w(z)}}\,H_{l}\!{\Bigg (}{\frac {{\sqrt {2}}\,x}{w(z)}}{\Bigg )}\,H_{m}\!{\Bigg (}{\frac {{\sqrt {2}}\,y}{w(z)}}{\Bigg )}\times {}\\&\exp \left({-{\frac {x^{2}+y^{2}}{w^{2}(z)}}}\right)\exp \left({-i{\frac {k(x^{2}+y^{2})}{2R(z)}}}\right)\times {}\\&\exp {\big (}i\psi (z){\big )}\exp(-ikz).\end{aligned}}}$$

In this form, the parameter w₀, as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at z = 0. Given that w₀, w(z) and R(z) have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with l = m = 0 we obtain the fundamental Gaussian beam described earlier (since H₀ = 1). The only specific difference in the x and y profiles at any z are due to the Hermite polynomial factors for the order numbers l and m. However, there is a change in the evolution of the modes' Gouy phase over z:

$${\displaystyle \psi (z)=(N+1)\,\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right),}$$

where the combined order of the mode N is defined as N = l + m. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by ±π/2 radians over all of z (and only by ±π/4 radians between ±z_R), this is increased by the factor N + 1 for the higher order modes.^[8]

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

Laguerre-Gaussian modes

Intensity profiles of the first 12 Laguerre-Gaussian modes.

Beam profiles which are circularly symmetric (or lasers with cavities that are cylindrically symmetric) are often best solved using the Laguerre-Gaussian modal decomposition.^[3] These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index p ≥ 0 and the azimuthal index l which can be positive or negative (or zero):^[20]

$${\displaystyle {\begin{aligned}u(r,\phi ,z)={}&C_{lp}^{LG}{\frac {1}{w(z)}}\left({\frac {r{\sqrt {2}}}{w(z)}}\right)^{\!|l|}\exp \!\left(\!-{\frac {r^{2}}{w^{2}(z)}}\right)L_{p}^{|l|}\!\left({\frac {2r^{2}}{w^{2}(z)}}\right)\times {}\\&\exp \!\left(\!-ik{\frac {r^{2}}{2R(z)}}\right)\exp(-il\phi )\,\exp(i\psi (z)),\end{aligned}}}$$

where L_p^l are the generalized Laguerre polynomials. C^LG
_lp is a required normalization constant:

$${\displaystyle C_{lp}^{LG}={\sqrt {\frac {2p!}{\pi (p+|l|)!}}}\Rightarrow \int _{0}^{2\pi }d\phi \int _{0}^{\infty }rdr|u(r,\phi ,z)|^{2}=1}$$

.

w(z) and R(z) have the same definitions as above. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor N + 1:

$${\displaystyle \psi (z)=(N+1)\,\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right),}$$

where in this case the combined mode number N = |l| + 2p. As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in r but now multiplied by a Laguerre polynomial. The effect of the rotational mode number l, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor exp(−ilφ), in which the beam profile is advanced (or retarded) by l complete 2π phases in one rotation around the beam (in φ). This is an example of an optical vortex of topological charge l, and can be associated with the orbital angular momentum of light in that mode.

Ince-Gaussian modes

Transverse amplitude profile of the lowest order even Ince-Gaussian modes.

In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by^[4]

$${\displaystyle u_{\varepsilon }\left(\xi ,\eta ,z\right)={\frac {w_{0}}{w\left(z\right)}}\mathrm {C} _{p}^{m}\left(i\xi ,\varepsilon \right)\mathrm {C} _{p}^{m}\left(\eta ,\varepsilon \right)\exp \left[-ik{\frac {r^{2}}{2q\left(z\right)}}-\left(p+1\right)\zeta \left(z\right)\right],}$$

where ξ and η are the radial and angular elliptic coordinates defined by

$${\displaystyle {\begin{aligned}x&={\sqrt {\varepsilon /2}}\;w(z)\cosh \xi \cos \eta ,\\y&={\sqrt {\varepsilon /2}}\;w(z)\sinh \xi \sin \eta .\end{aligned}}}$$

C^m
_p(η, ε) are the even Ince polynomials of order p and degree m where ε is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for ε = ∞ and ε = 0 respectively.^[4]

Hypergeometric-Gaussian modes

There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.

These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate ρ = r/w₀ and the normalized longitudinal coordinate Ζ = z/z_R as follows:^[21]

$${\displaystyle {\begin{aligned}u_{{\mathsf {p}}m}(\rho ,\phi ,\mathrm {Z} ){}={}&{\sqrt {\frac {2^{{\mathsf {p}}+|m|+1}}{\pi \Gamma ({\mathsf {p}}+|m|+1)}}}\;{\frac {\Gamma \left({\frac {\mathsf {p}}{2}}+|m|+1\right)}{\Gamma (|m|+1)}}\,i^{|m|+1}\times {}\\&\mathrm {Z} ^{\frac {\mathsf {p}}{2}}\,(\mathrm {Z} +i)^{-\left({\frac {\mathsf {p}}{2}}+|m|+1\right)}\,\rho ^{|m|}\times {}\\&\exp \left(-{\frac {i\rho ^{2}}{\mathrm {Z} +i}}\right)\,e^{im\phi }\,{}_{1}F_{1}\left(-{\frac {\mathsf {p}}{2}},|m|+1;{\frac {\rho ^{2}}{\mathrm {Z} (\mathrm {Z} +i)}}\right)\end{aligned}}}$$

where the rotational index m is an integer, and ${\displaystyle {\mathsf {p}}\geq -|m|}$ is real-valued, Γ(x) is the gamma function and ₁F₁(a, b; x) is a confluent hypergeometric function.

Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes,^[21] and the modified Laguerre–Gaussian modes.

The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist (z = 0):

$${\displaystyle u(\rho ,\phi ,0)\propto \rho ^{{\mathsf {p}}+|m|}e^{-\rho ^{2}+im\phi }.}$$