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Lenses Optics, Eugene Hecht, Chpt. 5

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Lenses for imaging Object produces many spherical waves –scattering centers Want to project to different location Object is collection of scattering centers Lens designed to project and reproduce scattering centers Diverging spherical waves Converging spherical waves

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Plane wave approximation Distant object Radius of curvature large Approximate by plane wave Image approximately at focal plane Distant object gives plane waves

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Lenses for collimation Convert diverging spherical wave to plane wave –Plane wave like spherical wave with infinite radius of curvature First step toward imaging –plane wave like intermediate To flatten wavefront –distance from S to D must be constant –independent of A Use Snell’s law and geometry –Result is equation of hyperbola –n i l i + n t l t = const ntnt nini lili ltlt

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Spherical lenses Object distance Image distance Vertex Optic axis Collimation Focussing Hyperbolic and elliptical lenses hard to make Spherical lenses easy to make –Good enough approximation in many cases Example: condition for imaging –path lengths from object to image are equal –n 1 l 0 + n 2 l i = const From geometry: Paraxial approximation: First focal length = object focal length Second focal length = image focal length

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Real lenses High index material finite Two radii of curvature Lensmakers formula Focal length Thin lens equation

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Variable focal length Positive and negative lens combos –Effective focal length (L 1 first) Long focal-length lenses –Curvature of incoming light becomes important –Result: Lens does not behave as expected –Solution: Variable focal length Achromats –Different wavelength dispersions –Dispersion ratio = 1/ (focal length ratio) –All colors focus at same point

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Types of lenses Focal length –general case Special case -- double convex Sign conventions for radii

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Lens aberrations Focusing or collimating –hyperbolic lens shape is ideal Spherical lens shape –gives insufficient refraction near edges –use plano-convex Face flat toward spherical wavefront –extra refraction –spherical wave on flat interface Why not double convex ? Computer solution –plano convex better –only for collimation/focusing 4 f imaging –double convex better –symmetry argument Additional refraction when spherical wave encounters planar boundary Refraction angle too shallow Hyperbolic lens best Aberration reduction

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Non-axial focusing Extended object –Light enters lens from several angles Focus to points on sphere Approximate by plane Focal plane Parallel ray focus to points on sphere Focal plane

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Basic lens ray tracing tricks 1. Rays through lens center –undeflected 2. Rays parallel to optic axis –go through focal point 3. Parallel rays –go to point on focal plane f f 1 2 3

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Lens alignment Position important Angle less important –slightly changes focal length in one dimension –aberration Use translation mount instead of tilt plate f f f ’ f Lens translationLens tilt

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Lenses for imaging Single lens -- image Two lenses -- depends on seperation Interesting case -- telescope –equal focal lengths 4 f imaging –unequal focal lengths magnification = f 2 /f 1 transverse = longitudinal f soso sisi f fff f1f1 f1f1 f2f2 f2f2 4 f imaging Imaging telescope

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Imaging: transparent vs. scattering objects Scattering object acts as array of sources –image is replica -- one or two lenses –4 f configuration puts image at a distance w/o magnification -- “relay” lenses Transmission object -- curvature important –4 f configuration better Scattering Transmission ff f f 2 f f fff illumination 4 f imaging2 f imaging 2 f illumination illum.

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Beam expanders Analogous to 4 f imaging –wavefront curvature preserved –magnification is focal length ratio independent of lens spacing Two types –Galilaen and spatial-filter arrangements –Galilaen easier to to set and maintain alignment Spatial-filter arrangement Galilaen - f 1 f2f2 d

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Alignment of telescope Need both tilt and translation (2 lenses) –first tilt to correct far field spot position –second translate to center spot in output lens –interate –focus to adjust collimation Tilt to correct far-field alignment Far-field alignment Translate to center spot in output lens center spot Focus to set collimation

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Spatial filters Laser beam intensity noise –can view as interference of intersecting beamlets Example: beamsplitter –front surface 4% reflection 4% intensity = 20% field reflected field modulated between 0.8 and 1.2 intensity modulation between 0.64 and 1.4 –large effect Lens converts angle to position –use pinhole to filter out one position Result is spatial filter beamsplitter destructive ff Pinhole aperture Aberrated laser beam Cleaned laser beam Sources of laser aberrations Spatial filter for laser beam cleanup

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Spatial filter alignment Standard alignment procedure –Translate pinhole aperture until light comes through Difficult procedure –usually no light until position almost perfect –random walk in 2D not efficient Solution: Defocus input lens –larger spot at aperture –easy to align Refocus input lens –spot at aperture shrinks –fine tune alignment Iterate ff Pinhole aperture Aberrated laser beam Cleaned laser beam Spatial filter alignment: Translate pinhole until light comes through

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Problem with spatial filter design Pinhole and output lens define alignment for rest of system Translating pinhole destroys alignment Better option: Translate input lens Leave output fixed -- alignment reference for rest of system –independent of changes in laser input ff Pinhole aperture Aberrated laser beam Cleaned laser beam Better spatial filter alignment technique: Translate lens instead of pinhole

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Resolution of lenses First find angular resolution of aperture –Like multiple interference –Diffraction angles: d sin = n –Diffraction halfwidth (resolution of grating): N d sin = Take limit as d --> 0, but N d = a (constant) –Diffraction angle: sin = n / d only works for n = 0, = 0 -- (forward direction) –Angular resolution: sin = / N d = / D Lens converts angle resolution to position resolution –x 1/2 = f / D (n = 1) –circular lens: x 1/2 = 1.22 f / D d Path difference d sin = n Path difference N d sin = n N d = D D f 2 x 1/2 Lens resolution Like array of sources limit of zero separation Grating resolution

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More on lens/aperture resolution Lens exchanges angle for position –Fourier transform Lens is rectangular aperture –F.T. of rectangle is sinc(x) = sin(x)/x D f 2 x 1/2 =2.44 f / D Lens resolution Like array of sources limit of zero separation Sinc function Airy disk = 2-D Sinc function

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Lens formulae F-number: F/# = (M+1) f / D, (M is magnification) Numerical aperture: NA = n sin , (n is refractive index) –for small angles NA = D/2f = 1/(2 F#) Focal spot size x 1/2 = 1.22 f / D = 1.22 F# = /NA Depth of focus z = 1.22 x 4 (f/D) 2 cos –small angles z = 1.22 /NA 2 z x 1/2 D f

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Lens example Microscope objectives –Spot size = 1.22 / (2 NA) –NA = n D / 2 f = n sin Example: –NA = 1.3, spot size: x 1/2 = / 2 Microscope objectives z x 1/2 D f

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Review Gaussian beams Zero order mode is Gaussian Intensity profile: beam waist: w 0 confocal parameter: z far from waist divergence angle Gaussian propagation

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Lens resolution with laser light (Gaussian beams) Laser beam diameter is effective lens diameter: D = 2w –Fourier transform of Gaussian is Gaussian Standard lensGaussian Aperture sizeD 2w Focal spot size 1.22 f / D w 0 = ( 4/ ) f / 2w = 1.27 f / 2w Depth of focus 1.22 (2f / D) 2 z = 1.27 (2f /2w) 2

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Fresnel lenses Start with conventional lens Constrain optical thickness to be modulo Advantage -- thinner and lighter Fresnel vs conventional lens

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Other fresnel lenses Spherical waves intersect plane Phase depends on distance from optic axis Block out negative phase regions Fresnel lens construction Block out one phase

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Graded index (GRIN) lens Glass rod with radial index gradient Quadratic gradient -- high index in center –like lens –optical path length varies quadratically from center Periodic focusing –laser spot size varies sinusoidally with distance index Radial position GRIN rod lens GRIN fiber coupler epoxy GRIN periodic focusing

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Lenses as Fourier transformers Angle at front focal plane --> position at back focal plane Position at front focal plane --> angle at back focal plane Angle maps to positionPosition maps to angle

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Fourier transform example 4 f configuration -- transform plane in center Fourier transform of letter “E” Fourier transform of mesh

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Lenses as retro-reflectors Angle of input –defines position in focal plane Mirror in focal plane –converts position back to angle at output Output angle = input angle –translations still possible

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Other retro-reflectors Right angle reflectors, 90 ° –reflection angles complementary, add 90 ° Net result is 180 ° reflection –translation can still occur -- off axis Corner cube

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