Lenses Optics, Eugene Hecht, Chpt. 5 Lenses for imaging • Object produces many spherical waves – scattering centers • Want to project to different location Lens designed to project and reproduce scattering centers Object is collection of scattering centers Diverging spherical waves Converging spherical waves Plane wave approximation • • • • Distant object Radius of curvature large Approximate by plane wave Image approximately at focal plane Distant object gives plane waves 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 path A • Use Snell’s law and geometry – Result is equation of hyperbola – ni li + nt lt = const lt li ni nt Spherical lenses • Hyperbolic and elliptical lenses hard to make • Spherical lenses easy to make – Good enough approximation in many cases Collimation First focal length = object focal length fo n1 R n2 n1 • Example: condition for imaging – When path lengths from object to image are equal – n1 l0 + n2 li = const • From geometry: n1 o n2 1 n2 si n1so i R i o • Paraxial approximation (s ~ l): n1 n2 n2 n1 so si R Focussing Second focal length = image distance fi Optic axis Vertex Object distance Image distance n2 R n2 n1 Real lenses • High index material finite • Two radii of curvature • Lensmakers formula 1 1 1 1 n2 1 so si R1 R2 • Focal length 1 1 1 n2 1 f R1 R2 • Thin lens equation 1 1 1 so si f Variable focal length • Positive and negative lens combos – Effective focal length (L1 first) f 2 d f1 d ( f1 f 2 ) • 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 d Types of lenses • Focal length – general case 1 1 1 n2 1 f R1 R2 • Special case -- double convex 1 1 1 n2 1 f R1 R2 Sign conventions for radii Lens aberrations Aberration reduction • Focusing or collimating – hyperbolic lens shape is ideal • Spherical lens shape – gives insufficient refraction near edges – use plano-convex Hyperbolic lens best • Face flat toward spherical wavefront – extra refraction – spherical wave on flat interface • Why not double convex ? • Computer solution Refraction angle too shallow – plano convex better – only for collimation/focusing • 4 f imaging – double convex better – symmetry argument Additional refraction when spherical wave encounters planar boundary 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 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 3 1 2 f f Lens alignment • Position important • Angle less important – slightly changes focal length in one dimension – aberration • Use translation mount instead of tilt plate Lens translation f f Lens tilt f f ’ Lenses for imaging • Single lens -- image • Two lenses -- depends on seperation • Interesting case -- telescope so si MT – equal focal lengths • 4 f imaging – unequal focal lengths f f f f f1 4 f imaging M L M T2 1 1 1 so si f • magnification = f2/f1 • transverse = longitudinal f si so f1 f2 Imaging telescope f2 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 illumination 4 f imaging 2 f imaging illumination Scattering f f f f 2f 2f Transmission illum. illum. f f f f 2f 2f 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 f2 Galilaen - f1 d Spatial-filter arrangement 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 Focus to set collimation Far-field alignment Tilt to correct far-field alignment center spot Translate to center spot in output lens Spatial filters • Laser beam intensity noise – can view as interference of intersecting beamlets • Example: beamsplitter Sources of laser aberrations 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 destructive – use pinhole to filter out one position • Result is spatial filter Spatial filter for laser beam cleanup Pinhole aperture Cleaned laser beam Aberrated laser beam f f 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 Spatial filter alignment: Translate pinhole until light comes through Pinhole aperture Cleaned laser beam Aberrated laser beam f f 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 Better spatial filter alignment technique: Translate lens instead of pinhole Pinhole aperture Cleaned laser beam Aberrated laser beam f f Resolution of lenses • First find angular resolution of aperture – Like multiple interference – Diffraction angles: d sin q = n l – Diffraction halfwidth (resolution of grating): N d sin q1/2 = l • Take limit as d --> 0, but N d = a (constant) – Diffraction angle: sin q = n l / d • only works for n = 0, q = 0 -- (forward direction) – Angular resolution: sin q1/2 = l / N d = l / D • Lens converts angle resolution to position resolution – x1/2 = f l / D (n = 1) – circular lens: x1/2 = 1.22 f l / D Lens resolution Like array of sources limit of zero separation Grating resolution Path difference N d sin q1/2 = n l q 2 x1/2 Path difference d sin q = n l d D f Nd=D More on lens/aperture resolution • Lens exchanges angle for position – Fourier transform • Lens is rectangular aperture Airy disk = 2-D Sinc function – F.T. of rectangle is sinc(x) = sin(x)/x Lens resolution Like array of sources limit of zero separation 2 x1/2 =2.44 f l / D D f Sinc function Lens formulae • F-number: F/# = (M+1) f / D, (M is magnification) • Numerical aperture: NA = n sin f , (n is refractive index) – for small angles NA = D/2f = 1/(2 F#) • Focal spot size x1/2 = 1.22 f l / D = 1.22 l F# = 1.22 l /(2 NA) • Depth of focus z = 1.22 x 4l (f/D)2 cos f – small angles z = 1.22 l /NA2 z f D f x1/2 Lens example • Microscope objectives – Spot size = 1.22 l / (2 NA) = 0.61 l /NA – NA = n D / 2 f = n sin f • Example: – NA = 1.3, spot size: x1/2 = l / 2 z D f f x1/2 Microscope objectives Review Gaussian beams • Zero order mode is Gaussian • Intensity profile: I I e 2 r 2 / w2 0 • beam waist: w0 lz w w0 1 2 w0 2 • confocal parameter: z w02 zR l • far from waist w lz w0 • divergence angle 2l l 0.637 w0 w0 Gaussian propagation Lens resolution with laser light (Gaussian beams) • Laser beam diameter is effective lens diameter: D = 2w – Fourier transform of Gaussian is Gaussian Standard lens Gaussian Aperture size D 2w Focal spot size 1.22 f l / D w0 = (4/) f l / 2w = 1.27 f l / 2w Depth of focus 1.22 l (2f / D)2 z = 1.27 l (2f /2w)2 Divergence 1.22 l / D q 1.27 l / 2w Fresnel lenses • Start with conventional lens • Constrain optical thickness to be modulo l • Advantage -- thinner and lighter Fresnel vs conventional lens 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 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 Radial position GRIN rod lens GRIN fiber coupler epoxy index GRIN periodic focusing 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 position Position maps to angle Fourier transform example • 4 f configuration -- transform plane in center Fourier transform of mesh Fourier transform of letter “E” 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 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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