Lenses and Imaging (Part I) Why is imaging necessary: Huygen’s principle – Spherical & parallel ray bundles, points at infinity Refraction at spherical surfaces (paraxial approximation) Optical power and imaging condition Matrix formulation of geometrical optics The thin lens Surfaces of positive/negative power Real and virtual images MIT 2.71/ /15/04 wk2-b-1

MIT 2.71/ /15/04 wk2-b-2 Parabloid mirror: perfect focusing (e.g. satellite dish) f: focal length

MIT 2.71/ /15/04 wk2-b-3 Lens: main instrument for image formation air optical axis airglass Point source (object) Point image The curved surface makes the rays bend proportionally to their distance from the “optical axis”, according to Snell’s law. Therefore, the divergent wavefront becomes convergent at the right-hand (output) side.

MIT 2.71/ /15/04 wk2-b-4 Why are focusing instruments necessary? Ray bundles: spherical waves and plane waves Point sources and point images Huygens principle and why we can see around us The role of classical imaging systems

MIT 2.71/ /15/04 wk2-b-5 Ray bundles point source spherical wave (diverging) wave-front ⊥ rays wave-front ⊥ rays point source very-very far away plane wave

MIT 2.71/ /15/04 wk2-b-6 Huygens principle Each point on the wavefront acts as a secondary light source emitting a spherical wave The wavefront after a short propagation distance is the result of superimposing all these spherical wavelets optical wavefronts

MIT 2.71/ /15/04 wk2-b-7 Why are focusing instruments necessary? incident light... is scattered by the object object

MIT 2.71/ /15/04 wk2-b-8 Why are focusing instruments necessary? incident light... is scattered by the object object image ⇒ need optics to “undo” the effects of scattering, i.e. focus the light

MIT 2.71/ /15/04 wk2-b-9 Ideal lens air optical axis airglass Point source (object) Point image Each point source from the object plane focuses onto a point image at the image plane; NOTE the image inversion

MIT 2.71/ /15/04 wk2-b-10 Summary: Why are imaging systems needed? Each point in an object scatters the incident illumination into a spherical wave, according to the Huygens principle. A few microns away from the object surface, the rays emanating from all object points become entangled, delocalizing object details. To relocalize object details, a method must be found to reassign (“focus”) all the rays that emanated from a single point object into another point in space (the “image.”) The latter function is the topic of the discipline of Optical Imaging.

The ideal optical imaging system optical elements objectimage MIT 2.71/ /15/04 wk2-b-11 Ideal imaging system: each point in the object is mapped onto a single point in the image Real imaging systems introduce blur...

MIT 2.71/ /15/04 wk2-b-12 Focus, defocus and blur Perfect focusDefocus

MIT 2.71/ /15/04 wk2-b-13 Focus, defocus and blur Perfect focusImperfect focus “spherical aberration”

MIT 2.71/ /15/04 wk2-b-14 Why optical systems do not focus perfectly Diffraction Aberrations However, in the paraxial approximation to Geometrical Optics that we are about to embark upon, optical systems do focus perfectly To deal with aberrations, we need non-paraxial Geometrical Optics (higher order approximations) To deal with diffraction, we need Wave Optics

MIT 2.71/ /15/04 wk2-b-15 Ideal lens air optical axis air glass Point source (object) Point image Each point source from the object plane focuses onto a point image at the image plane

MIT 2.71/ /15/04 wk2-b-16 Refraction at single spherical surface for each ray, must calculate point of intersection with sphere angle between ray and normal to surface apply Snell’s law to find direction of propagation of refracted ray R: radius of spherical surface center of spherical surface medium 1 index n, e.g. air n=1

MIT 2.71/ /15/04 wk2-b-17 Paraxial approximation /1 In paraxial optics, we make heavy use of the following approximate (1 st order Taylor) expressions: where ε is the angle between a ray and the optical axis, and is a small number (ε «1 rad). The range of validity of this approximation typically extends up to ~10-30 degrees, depending on the desired degree of accuracy. This regime is also known as “Gaussian optics” or “paraxial optics.” Note the assumption of existence of an optical axis (i.e., perfect alignment!)

MIT 2.71/ /15/04 wk2-b-18 Paraxial approximation /2 Ignore the distance between the location of the axial ray intersection and the actual off-axis ray intersection Apply Snell’s law as if ray bending occurred at the intersection of the axial ray with the lens Valid for small curvatures & thin optical elements axial ray off-axis ray

MIT 2.71/ /15/04 wk2-b-19 Refraction at spherical surface Refraction at positive spherical surface: Power x: ray height α: ray direction R: radius of curvature optical axis off-axis ray

MIT 2.71/ /15/04 wk2-b-20 Propagation in uniform space Propagation through distance D: x: ray height α: ray direction optical axis off-axis ray

MIT 2.71/ /15/04 wk2-b-21 Paraxial ray-tracing air glass Free-space propagation Refraction at air-glass interface Free-space propagation Free-space propagation Refraction at glass-air interface

MIT 2.71/ /15/04 wk2-b-22 Example: one spherical surface, translation + refraction + translation Non-paraxial ray (approximation gives large error) medium 1 index n, e.g. air n=1 center of spherical surface Paraxial rays (approximation valid) R: radius of spherical surface

MIT 2.71/ /15/04 wk2-b-23 Translation + refraction + translation /1 Starting ray: location x 0 directionα 0 Translation through distance D 01 (+ direction): Refraction at positive spherical surface:

MIT 2.71/ /15/04 wk2-b-24 Translation + refraction + translation /2 Translation through distance D 12 (+ direction): Put together:

MIT 2.71/ /15/04 wk2-b-25 Translation + refraction + translation /3

MIT 2.71/ /15/04 wk2-b-26 Sign conventions for refraction Light travels from left to right A radius of curvature is positive if the surface is convex towards the left Longitudinal distances are positive if pointing to the right Lateral distances are positive if pointing up Ray angles are positive if the ray direction is obtained by rotating the +z axis counterclockwise through an acute angle optical axis +z

MIT 2.71/ /15/04 wk2-b-27 On-axis image formation Point object Point image

MIT 2.71/ /15/04 wk2-b-28 On-axis image formation All rays emanating at arrive at x 2 irrespective of departure angleα 0

MIT 2.71/ /15/04 wk2-b-29 On-axis image formation All rays emanating at arrive at x 2 irrespective of departure angleα 0 “Power” of the spherical surface [units: diopters, 1D=1 m - 1 ]

MIT 2.71/ /15/04 wk2-b-30 Off-axis image formation Point object (off-axis) Point image optical axis

MIT 2.71/ /15/04 wk2-b-31 Magnification: lateral (off-axis), angle Lateral Angle

MIT 2.71/ /15/04 wk2-b-32 Object-image transformation Ray-tracing transformation (paraxial) between object and image points

MIT 2.71/ /15/04 wk2-b-33 Image of point object at infinity Point image

MIT 2.71/ /15/04 wk2-b-34 Image of point object at infinity object at ∞ image : image focal length Note: ambient refractive index at space of point image 1/Power

MIT 2.71/ /15/04 wk2-b-35 Point object imaged at infinity Point object

MIT 2.71/ /15/04 wk2-b-36 Point object imaged at infinity object Image at ∞ : object focal length Note: ambient refractive index at space of point image 1/Power

MIT 2.71/ /15/04 wk2-b-37 Image / object focal lengths Point object Image at ∞ object at ∞ Point image Power

MIT 2.71/ /15/04 wk2-b-38 Matrix formulation /1 translation by Distance D 01 refraction by surface with radius of curvature R ray-tracing object-image transformation form common to all

MIT 2.71/ /15/04 wk2-b-39 Matrix formulation /2 Refraction by spherical surface Power Translation through uniform medium

MIT 2.71/ /15/04 wk2-b-40 Translation + refraction + translation translation by D 12 Refraction by r.curv. R translation by D 01 result…

MIT 2.71/ /15/04 wk2-b-41 Thin lens in air Objective: specify input-output relationship

MIT 2.71/ /15/04 wk2-b-42 Thin lens in air Model: refraction from first (positive) surface + refraction from second (negative) surface Ignore space in-between (thin lens approx.)

MIT 2.71/ /15/04 wk2-b-43 Thin lens in air

MIT 2.71/ /15/04 wk2-b-44

Thin lens in air Power of the first surface Power of the second surface MIT 2.71/ /15/04 wk2-b-45

Thin lens in air MIT 2.71/ /15/04 wk2-b-46 Lens-maker’s formula thin lens

MIT 2.71/ /15/04 wk2-b-47 Thin lens in air thin lens Ray bending is proportional to the distance from the axis

MIT 2.71/ /15/04 wk2-b-48 The power of surfaces Positive power bends rays “inwards” Simple spherical refractor (positive) Plano-convex lens Bi-convex lens Negative power bends rays “outwards” Simple spherical refractor (negative) Plano-convex lens Bi-convex lens