Gaussian beam

In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or "TEM00 mode" of the laser's optical resonator. When refracted by a lens, a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread model in laser optics.



The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the Helmholtz equation. The solution, in the form of a Gaussian function, represents the complex amplitude of the electric field, which propagates along with the corresponding magnetic field as an electromagnetic wave in the beam.

Mathematical form
For a Gaussian beam, the complex electric field amplitude is given by


 * $$E(r,z) = E_0 \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w^2(z)}\right) \exp \left( -ikz -ik \frac{r^2}{2R(z)} +i \zeta(z) \right)\, $$

where
 * $$r$$ is the radial distance from the center axis of the beam,
 * $$z$$ is the axial distance from the beam's narrowest point (the "waist"),
 * $$i$$ is the imaginary unit (for which $$i^2 = -1$$),
 * $$ k = { 2 \pi \over   \lambda  } $$ is the wave number (in radians per meter),
 * $$E_0 = |E(0,0)|$$,
 * $$w(z)$$ is the radius at which the field amplitude and intensity drop to 1/e and 1/e2 of their axial values, respectively, and
 * $$w_0 = w(0)$$ is the waist size (described in more detail below).

The functions $$w(z)$$, $$R(z)$$, and $$\zeta(z)$$ are parameters of the beam, which we define below.

The corresponding time-averaged intensity (or irradiance) distribution is


 * $$I(r,z) = { |E(r,z)|^2  \over  2 \eta   }  = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w^2(z)} \right)\, $$

where $$I_0 = I(0,0)$$ is the intensity at the center of the beam at its waist. The constant $$\eta \,$$ is the characteristic impedance of the medium in which the beam is propagating. For free space, $$ \eta = \eta_0 \approx 377 \ \mathrm{\Omega} $$.

Beam parameters
The geometry and behavior of a Gaussian beam are governed by a set of beam parameters, which are defined in the following sections.

Beam width or "spot size"


For a Gaussian beam propagating in free space, the spot size w(z) will be at a minimum value w0 at one place along the beam axis, known as the beam waist. For a beam of wavelength λ at a distance z along the beam from the beam waist, the variation of the spot size is given by


 * $$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_R} \right)}^2 } \ . $$

where the origin of the z-axis is defined, without loss of generality, to coincide with the beam waist, and where


 * $$z_R = \frac{\pi w_0^2}{\lambda}$$

is called the Rayleigh range.

Rayleigh range and confocal parameter
At a distance from the waist equal to the Rayleigh range zR, the width w of the beam is


 * $$ w(\pm z_R) = w_0 \sqrt{2} \, $$

The distance between these two points is called the confocal parameter or depth of focus of the beam:


 * $$b = 2 z_R = \frac{2 \pi w_0^2}{\lambda}\ .$$

Radius of curvature
R(z) is the radius of curvature of the wavefronts comprising the beam. Its value as a function of position is


 * $$R(z) = z \left[{ 1+ {\left( \frac{z_R}{z} \right)}^2 } \right] \ . $$

Beam divergence
The parameter $$w(z)$$ approaches a straight line for $$z \gg z_R$$. The angle between this straight line and the central axis of the beam is called the divergence of the beam. It is given by


 * $$\theta \simeq \frac{\lambda}{\pi w_0} \qquad (\theta \mathrm{\ in\ radians.}) $$

The total angular spread of the beam far from the waist is then given by
 * $$\Theta = 2 \theta\ .$$

Because of this property, a Gaussian laser beam that is focused to a small spot spreads out rapidly as it propagates away from that spot. To keep a laser beam very well collimated, it must have a large diameter.

Since the gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the direction of propagation. From the above expression for divergence, this means the Gaussian beam model is valid only 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 $$w_0$$. 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.

Gouy phase
The longitudinal phase delay or Gouy phase of the beam is


 * $$\zeta(z) = \arctan \left( \frac{z}{z_R} \right) \ .$$

Complex beam parameter
The complex beam parameter is


 * $$ q(z) = z + q_0  = z + iz_R \ .$$

It is often convenient to calculate this quantity in terms of its reciprocal:


 * $$ { 1 \over q(z) }   =   { 1 \over z + iz_R } =   { z \over z^2 + z_R^2  }  -  i  { z_R \over z^2 + z_R^2  } = {1 \over R(z) } - i { \lambda \over \pi w^2(z)  }$$

The complex beam parameter plays a key role in the analysis of gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.

In terms of the complex beam parameter $${q}$$, a gaussian field with one transverse dimension is proportional to



{u}(x,z) = \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right) $$.

In two dimensions one can write the potentially elliptical or astigmatic beam as the product



{u}(x,y,z) = {u}(x,z)\, {u}(y,z) $$,

which for the common case of circular symmetry where $${q}_x = {q}_y = {q}$$ and $$x^2 + y^2 = r^2$$ yields



{u}(r,z) = \frac{1}{{q}(z)}\exp\left( -i k\frac{r^2}{2 {q}(z)}\right) $$.

Power through an aperture
The power P passing through a circle of radius r in the transverse plane at position z is


 * $$ P(r,z) =  P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]\ ,$$

where


 * $$ P_0 = { 1 \over 2 } \pi I_0 w_0^2 $$

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


 * $${ P(z) \over P_0 } = 1 - e^{-2} \approx 0.865\ .$$

Similarly, about 95 percent of the beam's power will flow through a circle of radius $$r = 1.224\cdot w(z) \, $$.

Peak and average intensity
The peak intensity at an axial distance $$z$$ from the beam waist is calculated using L'Hôpital's rule as the limit of the enclosed power within a circle of radius $$r$$, divided by the area of the circle $$\pi r^2$$:


 * $$I(0,z) =\lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2}

= \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)}. $$

The peak intensity is thus exactly twice the average intensity, obtained by dividing the total power by the area within the radius $$w(z)$$.

Hermite-Gaussian modes
In one transverse dimension higher order Hermite-gaussian modes exist. These are natural extensions of the fundamental lowest-order Gaussian solution. In terms of the previously defined complex $$q$$ parameter these modes have intensity distributions proportional to

{u}_n(x,z) = \left(\frac{2}{\pi}\right)^{1/4} \left(\frac{1}{2^n n! w_0}\right)^{1/2} \left( \frac{{q}_0}{{q}(z)}\right)^{1/2} \left[\frac{{q}_0}{{q}_0^\ast} \frac{{q}^\ast(z)}{{q}(z)}\right]^{(2n+1)/4} H_n\left(\frac{\sqrt{2}x}{w(z)}\right) \exp\left[-i \frac{k x^2}{2 {q}(z)}\right] $$ where the function $$H_n(x)$$ is the Hermite polynomial of order $$n$$ (physicists' form, i.e. $$H_1(x)=2x\,$$), and the asterisk indicates complex conjugation. For the case $$n=0$$ the equation yields a Gaussian transverse distribution.

For two dimensional rectangular coordinates one constructs a function $${u}_{m,n}(x,y,z)=u_m(x,z) u_n(y,z)$$ from a product of two one dimensional functions as given above. Mathematically this property is due to the separation of variables applied to the paraxial Helmholtz equation for Cartesian coordinates.

Hermite-Gaussian modes are typically designated "TEMm,n", where m and n are the polynomial indices in the x and y directions. A Gaussian beam is thus TEM0,0.

Laguerre-Gaussian modes
If we consider the problem in cylindrical coordinates we can write higher order modes using Laguerre- instead of Hermite-polynomials. In this manner two dimensional Laguerre-Gaussian modes may be written as

{u}(r,\theta,z)=\sqrt{\frac{2 p!}{(1+\delta_{0m})\pi(m+p)!}} \frac{\exp\left(i (2 p + m + 1)(\psi(z) - \psi_0)\right)}{w(z)} \times \left(\frac{\sqrt{2}r}{w(z)}\right)^m L_p^m\left(\frac{2 r^2}{w(z)^2}\right) \exp\left[ -i k \frac{r^2}{2{q}(z)}+i m \theta\right] $$

where $$L_p^m(r)$$ are the generalised Laguerre polynomials, the radial index $$p\ge 0$$, the azimuthal index is $$m$$ and $$\delta_{0m}$$ represents the Kronecker delta, $$\delta_{0m} = 0$$ if $$m\neq0$$ but 1 otherwise.