pyoptics.beams

Analytical beam definitions.

Includes

• Gaussian beams

• plane waves

Convention: fields vary as exp(i(k.r - omega*t)); the time dependence exp(-i*omega*t) is implicit. A beam travelling in +z thus carries the phase exp(+i*k*z), a diverging wavefront (z > 0 behind a waist) has the phase +k*r**2/(2*R(z)) with R(z) > 0, and the Gouy phase enters as exp(-i*psi(z)).

References

1. 5. Siegman, “Lasers”, University Science Books (1986).

H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5, 1550-1567 (1966).

B. E. A. Saleh and M. C. Teich, “Fundamentals of Photonics”, 2nd ed., Wiley (2007).

Functions

gauss_hermite	Hermite-Gaussian beam HG_lm (paraxial, waist at z = 0).
gauss_laguerre	Laguerre-Gaussian beam LG_pl (paraxial, waist at z = 0).
plane_wave	Return plane wave defined by direction vector.
spherical_wave	Return spherical wave
z_r_from_divergence_angle	Rayleigh length from the waist and the far field divergence.

pyoptics.beams.plane_wave(A, dir_vec, x, y, z, wl, n=1.0)

Return plane wave defined by direction vector.

Parameters:

Anumber

Amplitude (scalar or vector-valued).

dir_vecarray

Dimensionless direction vector of unit length. Relates to wave vector by k_vector = n*2*pi/lambda dircetion_vector.

x, yarrays

x, y coordinate arrays.

znumber

z position.

wlnumber

Vacuum wavelength.

nnumber, optional

Refractive index.

Returns:

Earray

Sampled electric field E(x, y, z; k) = A*exp(+i*k_vector.r).

pyoptics.beams.spherical_wave(x, y, z, wavelength, x0, y0, z0, U0=1.0, n=1.0)

Return spherical wave

Parameters:

x, yarrays

znumber

wlnumber

x0, y0, z0numbers

U0: number, optional

Complex amplitude factor.

nnumber, optional

Refractive index of medium.

Returns:

Uarray

U0*exp(+-ikR)/R, diverging (exp(+ikR)) for z >= z0 and converging towards (x0, y0, z0) (exp(-ikR)) for z < z0, both travelling in +z for the exp(-i*omega*t) time dependence.

pyoptics.beams.z_r_from_divergence_angle(w0, theta, n=1.0, wl=None)

Rayleigh length from the waist and the far field divergence.

z_r = w0/theta (paraxial).

Parameters:

w0number

Waist radius.

thetanumber

Far field divergence half angle in the medium (radians).

n, wloptional

Not needed, only kept for backwards compatibility.

Returns:

z_rnumber

pyoptics.beams.gauss_laguerre(p, l, x, y, z, w_0, z_r, wl)

Laguerre-Gaussian beam LG_pl (paraxial, waist at z = 0).

Parameters:

p, lint

Radial and azimuthal mode index.

x, yarrays

znumber

Distance from the waist.

w_0number

Waist radius.

z_rnumber

Rayleigh length (pi*w_0**2/wl).

wlnumber

Wavelength.

Returns:

fieldarray

Field proportional to exp(i*(k*z + k*r**2/(2*R(z)) + l*phi - (2p + |l| + 1)*arctan(z/z_r))). For l > 0, the helical phase exp(+i*l*phi) carries the orbital angular momentum +l*hbar per photon along +z (exp(-i*omega*t) convention).

pyoptics.beams.gauss_hermite(l, m, x, y, z, w_0, z_r, wl)

Hermite-Gaussian beam HG_lm (paraxial, waist at z = 0).

Parameters:

l, mint

Mode indices in x and y.

x, yarrays

znumber

Distance from the waist.

w_0number

Waist radius.

z_rnumber

Rayleigh length (pi*w_0**2/wl).

wlnumber

Wavelength.

Returns:

fieldarray

Field proportional to exp(i*(k*z + k*r**2/(2*R(z)) - (l + m + 1)*arctan(z/z_r))).