Presentation on theme: " Light can take the form of beams that comes as close"— Presentation transcript:
1 Light can take the form of beams that comes as close Beam Optics: Light can take the form of beams that comes as closeas possible to spatially localized and nondivergingwaves. Two extremes: (a) Plane wave: no angular spread;(b) spherical wave: diverge in all directions;Paraxial waves satisfy the paraxial Helmholtz equation!An important solution of this equation that exhibits thecharacteristics of an optical beam is the a wave called theGaussian beam. The Gaussian beam:The complex amplitude of a paraxial waves isA(r) is a slow varying function of position => the envelope isassumed to be approximately constant locally (within )
2 U(r): satisfy the Helmholtz equation => A(r): satisfy the paraxial Helmholtz equationOne simple solution to the Paraxial Helmholtz EquationParaboloidal waveAnother solution of the Paraxial Helmholtz EquationGaussianbeamA paraboloidal wave centered about the point z = . could be complex value; dramatically differentproperties acquired when is real or complex.
3 When is purely imaginary, i.e. =-iz0; z0: real => the complex envelope of the Gaussian beamz0: Rayleigh rangeSeparate amplitude and phase of this complex envelope(z2+z02)1/2z(z)z0
4 Put all this equation into the complex envelope of the Gaussian beamGaussian-Beam Complex Amplitudewavefronts radius of curvatureBeam widthA0 and z0 are determinedfrom the boundaryconditionsDefine beam parameters
5 and radial ()distance * Properties:Intensitya function of axial (z)and radial ()distanceIntensity is a Gaussian function of the .yxI/I0I/I0I/I0z = 0z = z0z = 2z0I/I01Power (at a given z; transverse)0.5-z0z0zFor = 0Independent of z!
6 Express I0 interms of P.The ratio of the power carried within a circle of radius0 in the transverse plane at position z to the totalpowerFor 0 = W(z); the ratio is ~ 86%; for 0 = 1.5W(z)the ratio is ~ 99%.Beam Radius (width): 86% of the power is carried within a circle of W(z);W(z); is regarded as the beam radius. The rms widthof the intensity distribution is = W(z)/2.The beam width is governed by
7 W(z) has the minimum value of W0 at z = 0, called the beam waist. W0 is the waist radius; 2W0 is called thespot size. The beam radius increases gradually with z.At z = z0, the beam radius = 21/2W0.At z >> z0, the beam radiusW00-z0z0Beam Divergence :If the waist is squeezed => the beam diverges;A short wavelength and fat beam waist => ahighly directional beam!
8 A small spot size (2W0)and a long depth of focus Defined as twice theRayleigh range 2z0.W021/2W0-z0z02z0A small spot size (2W0)and a long depth of focus(2z0)can not be obtained simultaneously unless isshort.Phase : of the Gaussian beam= -i(, z)On the beam axis = 0 =>kz: phase of a plane wave; (z): retardation
9 The total accumulated excess retardation as the wave travels from z = - to z = is . Guoy effect(z)-/2-/4z0z-z0-/4-/2Wavefronts: the third component in the aboveequation is responsible for wavefront bending.The surface of constant phase satisfy(z) and R(z) are relatively slowly varying =>~ constant within the beam radius on each wavefrontequation of paraboloidal waveR(z)z = z02z0zz0
10 Properties of the Gaussian Beam at Special Points At the plane z = z0, the wave has the followingproperties(i) the beam radius is 21/2 times greater than theradius at the beam waist, and the area is largerby a factor of 2.(ii) The intensity on the beam axis is 1/2 thepeak intensity(iii) The phase on the beam axis is retarded by anangle /4 relative to the phase of a plane wave(iv) The radius of the curvature of the wavefront isthe smallest, so that the wavefront has thegreatest curvature (R = 2z0)Near the beam center. At points for which |z|<<z0and << W0,
11 so that the beam intensity is approximately constant. Also, R(z) z02/z and (z) 0, so that thephaseAs a result, the wavefronts are approximatelyplanar. => Gaussian beam ~ a plane wave near itscenter.Far from the beam waist. At points within thebeam-waist radius ( << W0), but far from thebeam waist (z >> z0) the wave ~ like a sphericalwave. Since W(z) W0z/z0 >> W0 and < W0 =>so that the beam intensity is~ uniform. Since R(z) ~ z the wavefronts areapproximately spherical.spherical
12 Parameters of a Gaussian Laser Beam: * A 1-mW He-Ne laser produces a Gaussian beam ofwavelength = 633 nm and a spot size 2W0 = 0.1mm.ºAngular divergence: 0=W0/z0= /W0 = 63310-9/3.1416/0.0510-3 4.0310-3 rad.º Depth of focus: 2z0 =2W02/ = 23.1416(0.0510-3 )2 /63310-9=2.4810-2 (m)=2.48 (cm)º At z = 3.5105 km (~ distance to moon), thediameter of the beam 2W(z) ~ 20 z = 24.0310-33.5105 km = 2.821103 km = 2.821106 mº The radius of curvature(z0=1.24 cm); at z = 0 is R(z) = 0;at z = z0 is R(z) = 2z0 = 2.48 cm;at z = 2z0 is R(z) = 2.5z0 = 3.1 cm;
13 º Optical intensity at the beam center: z = 0, = 0 I( 0, 0) = 2P/W02 = 21//(0.005)2 mW/cm2= mW/cm2I(0, z0) = 2P/W(z0)2 = P/W02 = 1//(0.005)2mW/cm2 = mW/cm2Point source of 100W at z = 0. At z = z0, 100W isdistributed over 4z02.=> I(z0)=100x1000/4//(1.24)2 = 5175 mW/cm2.Parameters required to characterize a Gaussian Beam:* Peak amplitude, direction (beam axis), location of itswaist, and the waist radius (W0) or the Rayleighrange (z0).
14 * q-parameter: q(z) = z + iz0. If q(z) = 3 + i4 cm at some points on the beam axis => beam waist lies ata distance z = 3 cm to the point and that the depth offocus is 2z0 = 8 cm. q(z) is linear on z, q(z)=q1 andq(z+d) = q2 => q2 = q1+ d. The q-parameter issufficient for characterizing a Gaussian beam.* Determination of q-parameter: measure the beamwidth, W(z), and the radius of curvature, R(z), at anarbitrary point on the axis => using equationor solve z, z0, and W0 using the following equations;
15 Transmission through optical components: Transmission through a thin lens: the complextransmittance of a thin lens of focal length f exp(ik2/2f); when a Gaussian beam crosses the lensits complex amplitude is multiplied by this phasefactor => wavefront is bent, but the beam radius isnot alteredzz’0WRW'R'0'zW0', z0'W0, z0
16 * A Gaussian beam centered at z = 0 with waist radius W0 is transmitted through a thin lens located at z;The phase at the plane of the lens isThe phase of the transmitted wave is altered toR: + beam diverging;R': - beam convergingThe transmitted wave is itself a Gaussian beam withwidth W' (=W) and radius of curvature R'.The waist radius of the new beam W0' centered at z';- representing waist liesto the right of the lensSubstituting R(z) and W(z) into above equation!
17 Consider the limiting case (z-f) >> z0; Waist radiusWaist locationDepth of focusParameterTransformedby a LensDivergenceMagnificationConsider the limiting case (z-f) >> z0; r << 1 M Mr equations of ray optics* Beam shapingBeam focusing: a lens isplaced at the waist of aGaussian beamz0f2W0'2W0
18 The transmitted beam is then focused to a waist radius W0' at a distance z' given byIf the 2z0 (depth of focus) >> f (focal length)The incident Gaussian beam is well approximatedby a plane wave at its waist => focused at the focalplane! Smallest possible spot size is desired in manyapplications (laser scanning, laser printing, andlaser fusion).Smallest and f;thickest incident beam
19 The lens should intercept the incident beam, its diameter D must be at least 2W0; assume D =2W0.A microscope objective with small F-number isoften used!Beam collimation: a Gaussian beam is transmittedthrough a thin lens of focal length f ;z0/f =0Fromz'/f -1z0/f =0.5z/f -1For beam collimation, z' as distanceas possible from the lens; achievedby smallest z0/f (short depth of focus andlong focal length)
20 For a given ratio of z0/f => the optimal value of z is maximum of z'/f ; assume z/f-1 = a and z0/f = b.* Reflection from a spherical mirrorIncident Gaussian beam: width W1, roc R1;Reflected Gaussian beam: width W2, roc R2;The phase of the incident beam is modified by aphase factorThe relations between W1, R1, W2, and R2 are· If the mirror is planar => R = · If R1 = , i.e. the beam waist lies on the mirror
21 · If R1 = -R, i.e. the incident beam has the same curvature as the mirror => R2=R.* Transmission through an arbitrary optical systemModification of a Gaussian beam by an arbitraryparaxial opticalsystem characterizedby a matrix M.W1, R1, q1W2, R2, q2The ABCD Law:Transmission through free space: distance d of freespace; => A=1, B=d, C=0, D=1
22 Transmission through a thin optical component; ray position is the same, angle is alteredD = n1/n2;12In terms of beam parametersThe optical component is thin => beam width doesnot changed => W1 = W2;Paraxial approximationThe matrix used in chapter could be used inGaussian beam!
23 Hermite-Gaussian Beams: Beam of paraboloidal wavefronts are of importance;The curvature of the wavefronts could match thecurvature of spherical mirrors that form a resonatorReflection inside the resonator won’t change thecurvature of the wavefront.Consider a Gaussian beam of complex envelopeConsider a second wave whose complex envelope· Except for an excess phase of , the phase ofthe of the wave have the same phase as that of theunderlying Gaussian wave!
24 · The magnitudeFunction of x/W(z) and y/W(z) whose width in the xand y directions vary with z in accordance with thesame scaling factor W(z). As z increase, theintensity distribution in the transverse planeremains fixed (except for a magnification factorW(z). => Gaussian function modulated in the x andy directions.The existence of this wave is assured if three realfunction could be found such thatA(x, y, z) satisfies the paraxial Helmholtz equation.
25 Three independent variables => assume the first term = -1, the second term = -2, => the thirdterm = 1 + 2,(a)(b)(c)Equation (a) represents an eigenvalue problemwhose eigenvalues are 1= l = 0, 1, 2 … and whoseeigenfunctions are the Hermite polynomials (Hl(u))Hermite polynomials is defined by the recurrencerelation
26 Similarly, solution for equation (b) is (let 2=m) Solution for equation (c) is (with 1+2= l+m)Substitute all the solution into A(x, y, z) andmultiplying by the phase factor exp(-ikz)=> Ul,m(x,y,z)Hermite-Gaussian function