Study of the Optical Parameters of the 40m LIGO Prototype
Lisa Maria Goggin
Mentor: Professor Alan Weinstein
September 2000
Abstract:
The objective of my project was to model the interferometer optics of the 40m LIGO prototype with regard to cavity lengths, mirror radii of curvature and beam spot sizes and beam radii of curvature in both the cases of flat and curved input test mass. This study also includes the design of a 12m modecleaner and evaluation of its performance on suppression of higher order modes. We present the optical design of modematching telescopes, which are necessary to match the beams from the resonant cavities of the prestablized laser, mode cleaner and interferometer.
LIGO, the Laser Interferometer GravitationalWave Observatory is a project dedicated to the detection of gravitational waves and the harnessing of these waves for scientific research. Gravitational waves, which are emitted by accelerated masses, were first predicted by Einstein in 1916 in his general theory of relativity. They have not yet been observed although their presence has been indirectly verified.
LIGO consists of two widely separated sites, one at Hanford, Washington, the other in Livingston, Louisiana. These sites house power recycled Michelson interferometers with FabryPerot arms 4 kilometers in length. These sites have been constructed and the detectors are currently being installed. The LIGO I data run is to commence in 2003.
Even before LIGO1 comes online, plans for modifications and improvements to the current setup are already well under way. The advanced LIGO II configuration will be installed in 2005. Before these changes can be implemented they have to be tested, and this is the purpose of the 40m lab on campus. This lab contains a 40m LIGO prototype, that is a power recycled Michelson interferometer with FabryPerot arms 40m in length. This is currently being upgraded to become as LIGO 1 like as possible. The features of the upgrade that motivated my project are the replacement of the green laser by an infrared laser and the adoption of the LIGO 1 mode cleaner design.
For my SURF I worked at the 40m lab, my project being concerned with modeling the input optics and interferometer and measuring the properties of the beam throughout using Matlab. Section 2 of this report discusses relevant properties of Gaussian beam optics and resonators. The components of the input optics and interferometer that my project was concerned with are introduced in section 3. Section 4 presents a more indepth discussion of the methods that led to our results.
2.Theory:
2.1 Gaussian Beams:
The scalar wave equation for electromagnetic fields in free space is given by
(2.1.1)
where k = 2, where is the wavelength of light in the medium.
For light traveling in the z direction
(2.1.2)
where the function represents a spatial modulation of the plane wave.
Substitution of (2.1.2) into (2.1.1), taking account of the paraxial approximation:
(2.1.3)
i.e. that the longitudinal variation in the modulation function changes very slowly, gives
(2.1.4)
This is the paraxial approximation of the wave equation.
A trial solution of (2.1.4) is
(2.1.5)
where g is a function of x and z and h is a function of y and z. (z) is a measure of the decrease of the field amplitude with distance from the axis, and p(z) is a complex phase shift. q(z) is a complex beam parameter, which describes the Gaussian variation in the beam intensity with distance from the optic axis as well as the curvature of the phase front.
Insertion of (2.1.5) into (2.1.4) yields a differential equation of a Hermite polynomial
(2.1.6)
which is satisfied if
(2.1.7)
Thus the intensity pattern in the cross section of such a beam is a product of a Hermite and a Gaussian function. The primary E and H field components in these beams are polarized transverse to the direction of propagation and hence these waves are referred to as TEM_{mn} optical waves, where m and n are integers known as transverse mode numbers. The most important solution for the paraxial equation however is that with a purely Gaussian intensity profile, the TEM_{00} mode. This is the only mode that is spatially coherent.
Fig. 2.1.1. Amplitude distribution of a Gaussian beam

Physical Properties of Gaussian Beams
The parameter is referred to as the beam radius or spot size. It is the distance normal to the direction of propagation at which the amplitude is 1/e times that on the axis. A Gaussian beam propagating through a homogeneous medium will have one unique minimum value of beam radius, _{ 0 }at a particular position, the beam waist.
Fig. 2.2.1. Beam Radius versus Distance from Waist
The distance that the beam travels on the optic axis at either side of the waist before the beam radius increases by , or equivalently before the area doubles, is called the Rayleigh range, z_{R}. This marks the approximate dividing line between the ‘near field’ or Fresnel and the ‘far field’ or Fraunhofer regions for a beam propagating out from a Gaussian waist. The radius of curvature, R, of the wavefront is planar at the waist. As the beam propagates outward the wavefront gradually becomes curved and the radius of curvature rapidly drops to finite values. For distances well beyond the Rayleigh range the radius of curvature increases again as R(z)~z. The radius of curvature is taken to be positive if the wavefront is convex as viewed from z = .
Fig. 2.2.2 Radius of Curvature of Wavefront versus Distance from Waist
The farfield beam angle, that is, the angle that the gaussian beam spreads at z>>z_{R}, is defined by the width corresponding to the 1/e point of the amplitude;
(2.2.1)
The beam divergence is the half angular spread
(2.2.2)
Both and R can be expressed in terms of z_{R }and z
(2.2.3)
(2.2.4)
The complex beam parameter q is defined in terms of R and
(2.2.5)
hence, at the waist, q is purely imaginary
(2.2.6)
and a distance z away from the waist
(2.2.7)
The real part of the complex phase shift p, is known as the Guoy phase shift ,
(2.2.8)
This has the effect of giving the lowest order mode a phase shift of 180^{0} on passing through the waist, with most of this occurring within one or two Rayleigh ranges on either side of the waist. In physical terms this means that the phase velocity and the spacing between wavefronts are slightly larger than for an ideal plane wave. Higher order modes have larger Guoy phase shifts in passing through the waist region.
Fig. 2.2.3. Guoy phase shift through the waist region of a Gaussian beam
2.3 Optical Resonators:
It two curved mirrors, of radius of curvature R_{1} and R_{2}, are placed anywhere in the path of a Gaussian beam, and if the radius of curvature of the wavefront exactly matches that of the mirrors, then an optical resonator is formed. The mirrors, a distance L apart, produce a standing wave, reflecting the beam back on itself with exactly reversed radius of curvature and direction. This is depicted in Fig. 2.3.1
Fig. 2.3.1. Gaussian beam resonant cavity
The optical resonator can support both the lowest order Gaussian mode and the higher order HermiteGaussian modes as resonant modes of the cavity. These mirrors define a unique Rayleigh range for the Gaussian beam
(2.3.1)
The distances of the mirrors R_{1} and R_{2} from the waist respectively are;
(2.3.2)
We can define a pair of ‘resonator gparameters’ for each mirror.
(2.3.3)
The product of the gfactors is a measure of the stability of the cavity. The stability range is 0 < g_{1}g_{2} < 1, as otherwise real and finite solutions for the gaussian beam parameters and spot sizes cannot exist. As the gfactor decreases below 1 the Guoy phase difference of higher order modes gets larger and only one mode resonates in the cavity. The total Guoy phase shift along the resonator length is given in terms of the gparameters;
(2.3.4)
