448844Phone: 1-800-222-6440 • Fax: 1-949-253-1680OpticsTECHNICAL REFERENCE ANDFUNDAMENTAL APPLICATIONSLENS SELECTION GUIDESPHERICAL LENSESCYLINDRICAL LENSESKITSOPTICAL SYSTEMSMIRRORSGaussian Beam OpticsThe Gaussian is a radially symmetricaldistribution whose electric field variationis given by the following equation :Its Fourier Transform is also a Gaussiandistribution. If we were to solve theFresnel integral itself rather than theFraunhofer approximation, we would findthat a Gaussian source distributionremains Gaussian at every point along itspath of propagation through the opticalsystem. This makes it particularly easy tovisualize the distribution of the fields atany point in the optical system. Theintensity is also Gaussian:This relationship is much more than amathematical curiosity, since it is noweasy to find a light source with a Gaussianintensity distribution: the laser. Mostlasers automatically oscillate with aGaussian distribution of electrical field.The basic Gaussian may also take onsome particular polynomial multipliersand still remain its own transform. Thesefield distributions are known as higher-order transverse modes and are usuallyavoided by design in most practical lasers.The Gaussian has no obvious boundariesto give it a characteristic dimension likethe diameter of the circular aperture, sothe definition of the size of a Gaussian issomewhat arbitrary. Figure 1 shows theGaussian intensity distribution of atypical HeNe laser.Figure 1The parameter ω0, usually called theGaussian beam radius, is the radius atwhich the intensity has decreased to 1/e2or 0.135 of its axial, or peak value.Another point to note is the radius of halfmaximum, or 50% intensity, which is0.59ω0. At 2ω0, or twice the Gaussianradius, the intensity is 0.0003 of its peakvalue, usually completely negligible.The power contained within a radius r,P(r), is easily obtained by integrating theintensity distribution from 0 to r:The on-axis intensity can be very high dueto the small area of the beam.Care should be taken in cutting off thebeam with a very small aperture. Thesource distribution would no longer beGaussian, and the far-field intensitydistribution would develop zeros andother non-Gaussian features. However, ifthe aperture is at least three or four ω0indiameter, these effects are negligible.Propagation of Gaussian beams throughan optical system can be treated almostas simply as geometric optics. Because ofthe unique self-Fourier Transformcharacteristic of the Gaussian, we do notneed an integral to describe the evolutionof the intensity profile with distance. Thetransverse distribution intensity remainsGaussian at every point in the system;only the radius of the Gaussian and theFigure 2The beam size will increase, slowly atfirst, then faster, eventually increasingproportionally to x. The wavefront radiusof curvature, which was infinite at x = 0,will become finite and initially decreasewith x. At some point it will reach aminimum value, then increase with largerx, eventually becoming proportional to x.The equations describing the Gaussianbeam radius w(x) and wavefront radius ofcurvature R(x) are:radius of curvature of the wavefrontchange. Imagine that we somehow createa coherent light beam with a Gaussiandistribution and a plane wavefront at aposition x=0. The beam size andwavefront curvature will then vary with xas shown in Figure 2.where ω0is the beam radius at x = 0 andλ is the wavelength. The entire beambehavior is specified by these twoparameters, and because they occur inthe same combination in both equations,they are often merged into a singleparameter, xR, the Rayleigh range:In fact, it is at x = xRthat R has itsminimum value.Note that these equations are also validfor negative values of x. We onlyimagined that the source of the beamwas at x = 0; we could have created thesame beam by creating a larger Gaussianbeam with a negative wavefront curvatureat some x < 0. This we can easily do witha lens, as shown in Figure 3.Figure 3When normalized to the total power ofthe beam, P(∞) in watts, the curve is thesame as that for intensity, but with theordinate inverted. Nearly 100% of thepower is contained in a radius r = 2ω0.One-half the power is contained within0.59ω0, and only about 10% of the poweris contained with 0.23ω0, the radius atwhich the intensity has decreased by 10%.The total power, P(∞) in watts, is relatedto the on-axis intensity, I(0) (watts/m2),by:448855Email: [email protected] • Web: newport.comOpticsTECHNICAL REFERENCE ANDFUNDAMENTAL APPLICATIONSLENS SELECTION GUIDE SPHERICAL LENSES CYLINDRICAL LENSES KITS OPTICAL SYSTEMS MIRRORSThe input to the lens is a Gaussian withdiameter D and a wavefront radius ofcurvature which, when modified by thelens, will be R(x) given by the equationabove with the lens located at -x from thebeam waist at x = 0. That input Gaussianwill also have a beam waist position andsize associated with it. Thus we cangeneralize the law of propagation of aGaussian through even a complicatedoptical system.In the free space between lenses, mirrorsand other optical elements, the positionof the beam waist and the waist diametercompletely describe the beam. When abeam passes through a lens, mirror, ordielectric interface, the diameter isunchanged but the wavefront curvature ischanged, resulting in new values of waistposition and waist diameter on theoutput side of the interface.These equations, with input values for ω and R, allow the tracing of a Gaussianbeam through any optical system withsome restrictions: optical surfaces needto be spherical and with not-too-shortfocal lengths, so that beams do notchange diameter too fast. These areexactly the analog of the paraxialrestrictions used to simplify geometricoptical propagation.It turns out that we can put these laws in a form as convenient as the ABCDmatrices used for geometric ray tracing.But there is a difference: ω(x) and R(x) donot transform in matrix fashion as r and udo for ray tracing; rather, they transformvia a complex bi-linear transformation:where the quantity q is a complexcomposite of ω and R:We can see from the expression for q thatat a beam waist (R = ∞ and ω = ω0), q ispure imaginary and equals ixR. If we knowwhere one beam waist is and its size, wecan calculate q there and then use thebilinear ABCD relation to find q anywhereWe have invoked the approximation tanθ≈ θ since the angles are small. Since theorigin can be approximated by a pointsource, θ is given by geometrical
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