Gaussian Waves and Open Resonators

Abstract

A large number of gas and solid state lasers as well as free-electron lasers make use of an open resonator.

Problems

11.1

Gaussian wave.

• (a) Determine the energy that is contained in a sheet (perpendicular to the beam axis) of thickness $\delta z$ at the position z.

• (b) Calculate the portion of power of radiation passing an area that has the beam radius $r_\mathrm{u, 0}$ $=$ $r_0$.

• (c) Evaluate the radius $r_\mathrm{p}$ of the area passed by radiation of a portion p of the total power of the Gaussian wave.

• (d) Evaluate $r_\mathrm{p}$ if p $=$ 95%.

• (e) Evaluate $r_\mathrm{p}$ if p $=$ 99%.

• (f) Determine the power of the radiation that passes an area of radius $r_\mathrm{p} \ll r_0$.

11.2

Determine the minimum diameter of the tube of a helium–neon laser ($\lambda _L$ $=$ 633 nm) that is necessary to keep, per round trip, 99% of the radiation within a confocal resonator (L $=$ 0.5 m).

11.3

Angle of divergence. Determine the angle of divergence of a Gaussian beam generated by a helium–neon laser (resonator length 0.5 m; radius of the energy density distribution at the beam waist $r_\mathrm{u, 0}$=0.16 mm; wavelength 633 nm).

11.4

Photon density in a Gaussian wave. An argon ion laser (length 1 m; radius of the beam waist 1 cm; wavelength 480 nm; power 1 Watt) emits a Gaussian wave. By the use of a telescope, the angle of aperture diminishes by a factor of 10. Estimate the number of photons arriving each second at a detector of 2 cm diameter at different distances between laser and detector.

• (a) 100 km.

• (b) 374, 000 km (distance earth-moon).

11.5

ABCD matrix. Determine the effective focal length of an arrangement of two thin lenses (focal lengths $f_1$ and $f_2$) in contact.

11.6

Transversality of the radiation of a Gaussian wave. If a polarizer is located in a parallel beam of polarized radiation, the amplitude of the field transmitted by the polarizer is $A=A_0 \cos \theta$, where $\theta$ is the angle between the direction of polarization of the incident wave and the direction of the radiation for which the polarizer is transparent. (We assume that the transmissivity of the polarizer is 1 for $\theta =0$.) Determine the loss of power of a Gaussian wave passing a polarizer (that is assumed to be thin compared to the Rayleigh range $z_{0}$ if the polarizer is located at different positions.

• (a) In the beam waist at $z_0$.

• (b) At $z=z_0/2$.

• (c) At $z=z_0$.

• (d) At $z \gg z_0$.

• (e) Estimate the contribution of the polarizer to the V factor of a confocal laser resonator of 1 m length if the polarizer has a thickness of 1 cm and is located in the center of the resonator.

11.7

Hermite-Gaussian wave. Given is a 10l Hermite–Gaussian wave.

• (a) Determine the radius of the wave at the beam waist and the angles of divergence in the far-field.

• (b) Compare the results with corresponding values of a 00l Gaussian wave.

11.8

Calculate the Gouy phase of a Gaussian wave ($\lambda$=0.6 $\upmu \mathrm{m}$; $w_0=1\,\mathrm{mm}$) for propagation from the center of the beam waist over a distance of one wavelength; 1 mm; 1 cm; and 1 m.

11.9

Calculate the Gouy phase per round trip transit through a resonator of a Gaussian wave ($\lambda$=0.6 $\upmu \mathrm{m}$).

• (a) If the resonator is a near-planar resonator with two mirrors (radius of curvature $R_1=R_2=7\,\mathrm{m}$; resonator length $=1\,\mathrm{m}$).

• (b) If the resonator is a near-confocal resonator (radius of curvature $R_1=R_2=1.10\,\mathrm{m}$; resonator length $=1\,\mathrm{m}$).

11.10

Show that a concentric resonator is not realizable. [Hint: consider the beam waist and the angle of divergence.]

11.11

Show that (11.44) is a solution of the Helmholtz equation.

11.12

Derive ray matrices for different optical arrangements.

• (a) Reflection of radiation at a plane surface of a dielectric medium.

• (b) Propagation of radiation through a thin lens.

• (c) Focusing of radiation by a spherical mirror. [Hint: for solutions, see Sect. 11.9.]

11.13

Show that the intensity of radiation in a Gaussian beam averaged over an optical period is $I=c\upepsilon _0 A^2\pi w_0^2/2$ and that

$$\begin{aligned} I(z, r) = \frac{2P}{\pi w^2(z)}\, \mathrm{e}^{-2r^2/w^2(z)}, \end{aligned}$$

(11.160)

where $P=2\pi \int {I(z, r)r\mathrm{d}r}$ is the power of the radiation.

11.14

Estimate radiance and brilliance of radiation of a helium-neon laser (power 10 mW; angle of divergence 1 mrad; beam waist in the laser 0.5 mmm; bandwidth 1 kHz) and compare the values with those of a light bulb (electric power 10 W).

11.15

Heisenberg uncertainty principle

Show that a photon in a Gaussian beam obeys the Heisenberg uncertainty relation $\varDelta y\varDelta p_y \ge \hbar ,$ where $\varDelta y$ is the uncertainty of the position y and $\varDelta p_y$ the uncertainty of the momentum $p_\text {y}$ of the photon. [Hint: Show that the full width at half maximum of the lateral energy density in the waist is equal to $D=\sqrt{2\ln 2}\hbox { w}_\text {0}$ and that the full angle of divergence, related to the full width at half maximum of the lateral energy density in the far-field, is equal to $\vartheta =\hbox {(}\sqrt{\hbox {2 ln 2}}/\pi \hbox {)}\lambda \hbox {/}w_0 .$ It follows that the product is $D\vartheta =(\hbox {2 ln 2/}\pi )\lambda .$ Now, determine the wave vector spread, according to $\vartheta =\varDelta k_\text {y} /k_\text {x} .$ It follows, with $\varDelta y=D$ and $\varDelta p_\text {y} =\hbar \varDelta k_y$ that $\varDelta y\hbox {}\varDelta p_y =\hbox { }(4\ln 2)\hbox { }\hbar \hbox {.}$

11.16

Gaussian beam in a medium.

Show that a Gaussian beam in a medium with the refractive index n (> 1) has a smaller divergence than in free space if the beam radius in the waist is the same.

11.17

Confined Gaussian beam.

A medium with a radial dependence of the refractive index of the form $n(r)=n_0 -a\hbox { }r^{2},$ with $a> 0$, is able to guide a wave without divergence. Show that the Helmholtz equation has the solution $\psi (z, r)=\psi _0 \exp \hbox { }(-r^{2}/w_1^2 +\hbox {i }\lambda \hbox { }z/w_{_1 }^2 ),$where the beam radius $w_{1}$ is given by $w_1^2 =\lambda /(\pi \sqrt{2a})$ and where $\lambda$ is the vacuum wavelength. [Hint: make use of (11.4) and (11.13), with the relation $k=n\hbox { }\omega /c$.]