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M squared

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. This relationship between beam width and divergence is due to diffraction. Non-Gaussian beams also exhibit this effect, but a Gaussian beam is a special case where the product of width and divergence is the smallest possible.

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 ("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)}} expleft(-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)}expleft(-i kfrac{r^2}{2 {q}(z)}right) .

Power and intensity

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 = 1.224cdot 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_{rto 0} frac {P_0 left[1 - e^{-2r^2 / w^2(z)} right]} {pi r^2}
= frac{P_0}{pi} lim_{rto 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).

Higher-order modes

Gaussian beams are just one possible solution to the paraxial wave equation. Various other sets of orthogonal solutions are used for modelling laser beams. In the general case, laser beams can be described by superpositions of these solutions. The design of the laser determines which basis set of solutions is most useful. In some cases the output of a laser may closely approximate a single higher-order mode. Hermite-Gaussian modes are particularly common, since many laser systems have Cartesian reflection symmetry in the plane perpendicular to the beam's propagation direction.

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_nleft(frac{sqrt{2}x}{w(z)}right) expleft[-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

Considering the problem in cylindrical coordinates, one can write higher-order modes using Laguerre polynomials instead of Hermite polynomials. The resulting Laguerre-Gaussian modes are described by the equation

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

where L_p^m(r) are the generalized Laguerre polynomials, the radial index pge 0, the azimuthal index is m and delta_{0m} represents the Kronecker delta: delta_{0m} = 1 if and only if m = 0.

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

uleft(varepsilon ,eta ,zright) = frac{w_{0}}{wleft( zright) }mathrm{C}_{p}^{m}left(ixi ,varepsilon right) mathrm{C} _{p}^{m}left(eta ,varepsilon right) exp left[-ikfrac{r^{2}}{ 2qleft(zright) }-left(p+1right) psi _{GS}left(zright) right] , where xi and eta are the radial and angular elliptic coordinates defined by

x = sqrt{varepsilon /2}wleft(zright) cosh xi cos eta ,
y = sqrt{varepsilon /2}wleft(zright) sinh xi sin eta , {C}_{p}^{m}left(eta ,epsilon right) are the even Ince polynomials of order p and degree m, varepsilon is the ellipticity parameter, and psi _{GS}left(zright) =arctan left(z/z_{R}right) is the Gouy phase. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for varepsilon=infty and varepsilon=0 respectively.

See also

Notes

References

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