7.3 Properties of Gaussian beams

The basic or ‘lowest-order’ Hermite–Gauss mode is equivalent to what is usually called a Gaussian beam and is given by
∘ -- ( ) 2- --1-- x2-+-y2- x2-+-y2- u(x, y,z) = π w (z ) exp (iΨ (z)) exp − ik2RC (z) − w2 (z) . (119 )
The parameters of this equation are explained in detail below. The shape of a Gaussian beam is quite simple: the beam has a circular cross section, and the radial intensity profile of a beam with total power P is given by
--2P--- ( 2 2) I(r) = πw2 (z) exp − 2r ∕w , (120 )
with w the spot size, defined as the radius at which the intensity is 2 1 ∕e times the maximum intensity I(0). This is a Gaussian distribution, see Figure 40View Image, hence the name Gaussian beam.
View Image

Figure 40: One dimensional cross-section of a Gaussian beam. The width of the beam is given by the radius w at which the intensity is 2 1∕e of the maximum intensity.

Figure 41View Image shows a different cross section through a Gaussian beam: it plots the beam size as a function of the position on the optical axis.

View Image

Figure 41: Gaussian beam profile along z: this cross section along the x-z-plane illustrates how the beam size w(z) of the Gaussian beam changes along the optical axis. The position of minimum beam size w0 is called beam waist. See text for a description of the parameters Θ, zR and Rc.

Such a beam profile (for a beam with a given wavelength λ) can be completely determined by two parameters: the size of the minimum spot size w0 (called beam waist) and the position z0 of the beam waist along the z-axis.

To characterise a Gaussian beam, some useful parameters can be derived from w0 and z0. A Gaussian beam can be divided into two different sections along the z-axis: a near field – a region around the beam waist, and a far field – far away from the waist. The length of the near-field region is approximately given by the Rayleigh range zR. The Rayleigh range and the spot size are related by

2 πw-0 zR = λ . (121 )
With the Rayleigh range and the location of the beam waist, we can usefully write
∘ --------------- (z − z0 )2 w (z) = w0 1 + ------ . (122 ) zR
This equation gives the size of the beam along the z-axis. In the far-field regime (z ≫ zR,z0), it can be approximated by a linear equation, when
z z λ w(z) ≈ w0 ---= ----. (123 ) zR πw0

The angle Θ between the z-axis and w(z) in the far field is called the diffraction angle6 and is defined by

( ) ( ) w0 λ w0 Θ = arctan --- = arctan ---- ≈ --. (124 ) zR πw0 zR

Another useful parameter is the radius of curvature of the wavefront at a given point z. The radius of curvature describes the curvature of the ‘phase front’ of the electromagnetic wave – a surface across the beam with equal phase – intersecting the optical axis at the position z. We obtain the radius of curvature as a function of z:

z2R RC (z) = z − z0 + ------. (125 ) z − z0
We also find:
RC ≈ ∞, z − z0 ≪ zR (beam waist) RC ≈ z, z ≫ zR, z0 (far field ) (126 ) RC = 2zR, z − z0 = zR (maximum curvature ).

  Go to previous page Go up Go to next page