Gaussian Beam Calculator

Tools ‣ Gaussian Beam Calculator

The tool Gauss Beam Calculator computes some free parameters of the Gaussian beam when changing one of its parameters and fixing some others. As it is not possible to make all the parameters to be free, we introduce two fixing modes called Lock waist and Lock front.

  • Lock waist - find such values of free parameters at which the beam waist w0 stays constant when one of the parameters changes.

  • Lock front - find such values of free parameters at which the wavefront ROC R and the beam radius w at some axial distance z stay constant when one of the parameters changes.

Quadratic equations have two solutions and we have to define a way to point out which solution to take. So the tool has one additional parameter - zone - which can be Near zone or Far zone.

The first two buttons on the toolbar control the locking mode, and the next two - what solution is to be displayed.

_images/wnd_gauss_calc_w.png

Beam Parameters

These parameters are assigned and computed:

  • λ - light wavelength; it is not calculated from other parameters, only assigned

  • w0 - waist radius

  • z0 - Rayleigh range (half-confocal parameter)

  • Vs - beam divergence angle in the far-field zone

  • M2 - beam quality factor (M-square parameter)

  • z - distance from the waist to a point where the wavefront is calculated or assigned

  • w - beam radius on distance z from the waist

  • R - radius of curvature of the wavefront on distance z from the waist

  • re(q), im(q) - real and imaginary parts of complex beam parameter on distance z from the waist

  • re(1/q), im(1/q) - real and imaginary parts of inverted complex beam parameter on distance z from the waist

_images/wnd_gauss_calc_r.png

Calculation Algorithm

Change waist w0

Lock waist

Calculate the beam radius w and the wavefront ROC R at given distance z:

_images/f_calc_gauss_w0_1.png

Lock front

For specified waist radius w0, find such a distance z and beam quality M2 at which the wavefront ROC keeps its previously calculated value.

How to find: express z0 from w(z) and from R(z), equate both expressions and solve against z.

_images/f_calc_gauss_w0_2.png

Change Rayleigh length z0

Lock waist

With fixed waist radius w0, find a value of the beam quality parameter M2 which yields the specified z0.

_images/f_calc_gauss_z0_1.png

Lock front

For specified z0, find such a distance z and beam quality M2 at which the wavefront ROC keeps its previously calculated value. + sign is for the far zone, - is for the near zone.

_images/f_calc_gauss_z0_2.png

Change divergence angle Vs

Lock waist

With fixed waist radius w0, find such a value of beam quality parameter M2 that yields to the specified angle.

_images/f_calc_gauss_Vs_1.png

Lock front

For specified Vs, find such a distance z and beam quality M2 at which the wavefront ROC keeps its previously calculated value.

How to find z: substitute z0 as w0/Vs into formulas for w(z) and R(z), express w0 from both, equate them and solve against z.

_images/f_calc_gauss_Vs_2.png

Change axial distance z

Lock waist

_images/f_calc_gauss_z_1.png

Lock front

_images/f_calc_gauss_z_2.png

Change beam quality M2

Lock waist

The same as when the waist is changed.

_images/f_calc_gauss_MI_1.png

Lock front

Taking a new value of M2, find such a distance z and waist radius w0 preserving constant wavefront.

How to find z: express w02 from formula for R(z), substitute the expression into formula for w(z) and solve it against z.

_images/f_calc_gauss_MI_2.png

Change beam radius w

Lock waist

Find beam quality parameter M2 giving specified beam radius at the same axial distance z and with the same waist radius w0.

_images/f_calc_gauss_w_1.png

Lock front

Find a waist radius w0 and beam quality parameter M2 giving specified beam radius at the same axial distance z.

_images/f_calc_gauss_w_2.png

Change wavefront ROC R

Lock waist

Find beam quality parameter M2 giving specified ROC at the same axial distance z and with the same waist radius w0.

_images/f_calc_gauss_R_1.png

Lock front

Find a waist radius w0 and beam quality parameter M2 giving specified ROC at the same axial distance z.

_images/f_calc_gauss_R_2.png

Change complex beam parameter q

Assigning either real or imaginary part of complex beam parameter, we define both the beam radius w and the wavefront ROC R at the same time.

_images/f_calc_gauss_q_0.png

Lock waist

Having fixed waist radius w0, find axial distance z and beam quality M2 at which specified front is achieved.

_images/f_calc_gauss_q_1.png

Lock front

_images/f_calc_gauss_q_2.png

Change inverted complex beam parameter

Changing of real part is equivalent to changing of wavefront ROC R, and changing of imaginary part is equivalent to changing of beam radius w.

Change wavelength λ

Lock waist

The same formulas as when waist w0 changes.

Lock front

The same formulas as when beam radius w or wavefront ROC R changes.