Experiment 0– Gaussian beams 1Experiment 0Properties of a GaussianBeam1 IntroductionWe will look at the intensity distribution of a laser beam. The out-put of a laser is different than that of most other light sources. Thelaser resonator determines the spatial characteristics of the laser beam.Most Helium Neon (HeNe) lasers have spherical-mirror Fabry-Perot res-onators that have Hermite-Gaussian spatial modes. Usually only thelowest order transverse resonator (TEM00) mode oscillates, resulting ina Gaussian output beam.2 Background - see Pedrotti3, Chap. 27The irradiance (proportional to the square of the electric field) of aGaussian beam is symmetric about the beam axis and varies with radialdistance r from the axis asI(r) = I0exp(−2r2/w20) (1)Here w0is the radial extent of the beam where the irradiance hasdropped to 1/e2of its value on the beam axis, I0.A Gaussian beam has a waist, where w0is smallest. It either divergesfrom or converges to this beam waist. This divergence or convergenceis measured by the angle θ which is subtended by the points on eitherside of the beam axis where the irradiance has dropped to 1/e2of itsvalue on the beam axis, this is the place where the electric field hasdropped by 1/e.Under the laws of geometrical optics a bundle of rays (a beam) con-verging at an angle of θ should collapse to a point. Because of diffrac-tion, this does not occur. However, at the intersection of the asymptotesExperiment 0– Gaussian beams 2that define θ, the beam diameter reaches a minimum value d0= 2w0,the beam waist diameter.The variation of the beam waist w as a function of propagationdistance z is:w(z) = w0s1 +zz02(2)with the Rayleigh length z0given by:z0=πw20λ(3)A TEM00mode w0depends on the beam divergence angle as w0=2λ/πθ, where λ is the wavelength of the radiation.the product w0θ is constant for a Gaussian beam of a particularwavelength. A beam with a very small beam waist w0requires thedivergence θ must be large, while for a highly collimated beam withsmall θ the beam waist w0must be large.The most important characteristic of the beam is the phase. Thephase is flat (infinite curvature) at the waist w0, then grows to a max-imum at z0and returns to flat at infinity. The curvature of the wavefront is given by the Radius of Curvature R.R(z) = zr1 +z0z2(4)3 ExperimentPlease be very careful when using a laser. Parallel light gets focusedand that can happen with a laser beam in your retina.In the following experiments, you will find the divergence of yourlaser θ, the beam waist of the laser w0. Use the appropriate limit(z >> z0) of equation 2 to define the divergence angle θ in terms of theother parameters (see figure 1).Use the diverging lens to have a large laser beam. Take the pho-todetector and place the small aperture on it. You will measure theGaussian profile of the laser using a scanning detector and the computerinterface. The data will be in the form of a tex file with two columns ofnumbers. One for time the other for voltage that will be proportionalto the irradiance. You will acquire data with the computer and thenfit the data to a Gaussian. Make sure you understand the software youuse for the fit.Experiment 0– Gaussian beams 3Figure 1: Gaussian Beam PropagationThe calibration of the scanner displacement vs time (as recordedby the computer) is obtained in the following way. The motor drivingthe scanner screw rotates at 600 rev/min and the screw pitch is 10turn/cm A check of the calculated scanner speed should be done byactually measuring the time taken to travel a known distance.4 Some web linkshttp://www.mellesgriot.com/products/optics/gb 2 1.htmVersion 1, January 29,
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