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**Optics, Eugene Hecht, Chpt. 5**

Lenses Optics, Eugene Hecht, Chpt. 5

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**Lens designed to project and reproduce scattering centers**

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

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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 ni li + nt lt = const lt li nt ni

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**Spherical lenses Hyperbolic and elliptical lenses hard to make**

First focal length = object focal length 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 n1 l0 + n2 li = const From geometry: Paraxial approximation: Collimation Object distance Image distance Vertex Optic axis Second focal length = image focal length Focussing

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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 (L1 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 Special case -- double convex**

general case Special case -- double convex Sign conventions for radii

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**Lens aberrations Focusing or collimating Spherical lens shape**

Additional refraction when spherical wave encounters planar boundary Refraction angle too shallow Hyperbolic lens best Aberration reduction 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

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**Non-axial focusing Extended object Focus to points on sphere**

Light enters lens from several angles Focus to points on sphere Approximate by plane Focal plane Focal plane Parallel ray focus to points on sphere

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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 3 1 2 f f

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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 Lens translation Lens tilt f f f f ’

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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 = f2/f1 transverse = longitudinal f so si f f f f f1 f1 f2 f2 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 4 f imaging 2 f imaging illumination illumination f 2 f Scattering Transmission f 2 f illum. illum.

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**Beam expanders Analogous to 4 f imaging Two types**

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 Galilaen - f1 f2 d Spatial-filter arrangement

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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 Focus to set collimation Tilt to correct far-field alignment Far-field alignment Translate to center spot in output lens center spot

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**Spatial filters Laser beam intensity noise Example: beamsplitter**

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 Sources of laser aberrations beamsplitter destructive Spatial filter for laser beam cleanup Pinhole aperture Cleaned laser beam Aberrated laser beam f f

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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 Spatial filter alignment: Translate pinhole until light comes through Pinhole aperture Cleaned laser beam Aberrated laser beam f f

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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 Better spatial filter alignment technique: Translate lens instead of pinhole Pinhole aperture Cleaned laser beam Aberrated laser beam f f

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**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 Grating resolution d q Path difference d sin q = n l Path difference N d sin q1/2 = n l N d = D D f 2 x1/2 Lens resolution Like array of sources limit of zero separation

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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 Airy disk = 2-D Sinc function Lens resolution Like array of sources limit of zero separation 2 x1/2 =2.44 f l / D Sinc function D f

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**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 x1/2 D f

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**Lens example Microscope objectives Example:**

Spot size = 1.22 l / (2 NA) NA = n D / 2 f = n sin f Example: NA = 1.3, spot size: x1/2 = l / 2 z x1/2 D f

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**Review Gaussian beams Zero order mode is Gaussian Intensity profile:**

beam waist: w0 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 lens Gaussian Aperture size D w Focal spot size f l / D w0 = (4/p) f l / 2w = 1.27 f l / 2w Depth of focus l (2f / D)2 z = 1.27 l (2f /2w)2

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**Fresnel lenses Start with conventional lens**

Constrain optical thickness to be modulo l 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 GRIN fiber coupler index Radial position GRIN rod lens 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 position Position maps to angle

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**Fourier transform example**

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

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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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