1. Lenses and imaging MIT 2.71/2.710 09/10/01 wk2-a-1 Huygens principle and why we need imaging instruments A simple imaging instrument: the pinhole camera Principle of image formation using lenses Quantifying lenses: paraxial approximation & matrix approach “Focusing” a lens: Imaging condition Magnification Analyzing more complicated (multi-element) optical systems: – Principal points/surfaces – Generalized imaging conditions from matrix formulae

2. The minimum path principle MIT 2.71/2.710 09/10/01 wk2-a-2 (aka Fermat’s principle) Consequences: law of reflection, law of refraction ∫Γ n(x, y, z) dl Γ is chosen to minimize this “path” integral, compared to alternative paths

3. The law of refraction MIT 2.71/2.710 09/10/01 wk2-a-3 nsinθ = n’sinθ’ Snell’s Law of Refraction

4. Ray bundles MIT 2.71/2.710 09/10/01 wk2-a-4 point source plane wave point source very-very far away spherical wave (diverging)

5. Huygens principle MIT 2.71/2.710 09/10/01 wk2-a-5 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

6. Why imaging systems are needed MIT 2.71/2.710 09/10/01 wk2-a-6 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.

7. The pinhole camera MIT 2.71/2.710 09/10/01 wk2-a-7 The pinhole camera blocks all but one ray per object point from reaching the image space ⇒ an image is formed (i.e., each point in image space corresponds to a single point from the object space). Unfortunately, most of the light is wasted in this instrument. Besides, light diffracts if it has to go through small pinholes as we will see later; diffraction introduces artifacts that we do not yet have the tools to quantify.

8. Lens: main instrument for image formation MIT 2.71/2.710 09/10/01 wk2-a-8 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. Point source (object) Point image

9. Analyzing lenses: paraxial ray-tracing MIT 2.71/2.710 09/10/01 wk2-a-9 Free-space propagation Refraction at air-glass interface Free-space propagation Free-space propagation Refraction at glass-air interface

10. Paraxial approximation /1 MIT 2.71/2.710 09/10/01 wk2-a-10 In paraxial optics, we make heavy use of the following approximate (1 st order Taylor) expressions: sinε ≒ ε ≒ tanε cosε ≒ 1 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.” Note the assumption of existence of an optical axis (i.e., perfect alignment!)

11. Paraxial approximation /2 MIT 2.71/2.710 09/10/01 wk2-a-11 Valid for small curvatures & thin optical elements Ignore the distance between the location of the axial ray intersection and the actuall off-axis ray intersection off-axis ray axial ray Apply Snell’s law as if ray bending occurred at the intersection of the axial ray with the lens

12. Example: one spherical surface, translation+refraction+translation MIT 2.71/2.710 09/10/01 wk2-a-12

13. Translation+refraction+translation /1 MIT 2.71/2.710 09/10/01 wk2-a-13 Starting ray: location x 0 direction α 0 Translation through distance D 01 (+ direction): -------------------------------------------------------- Refraction at positive spherical surface:

14. Translation+refraction+translation /2 MIT 2.71/2.710 09/10/01 wk2-a-14 Translation through distance D 12 (+ direction): --------------------------------------------------------- Put together:

15. Translation+refraction+translation /3 MIT 2.71/2.710 09/10/01 wk2-a-15

16. Sign conventions for refraction MIT 2.71/2.710 09/10/01 wk2-a-16 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

17. On-axis image formation MIT 2.71/2.710 09/10/01 wk2-a-17 All rays emanating at x 0 arrive at x 2 irrespective of departure angle α 0 ∂x 2 / ∂α 0 = 0 “Power” of the spherical surface [units: diopters, 1D=1m -1 ]

18. Magnification: lateral (off-axis), angle MIT 2.71/2.710 09/10/01 wk2-a-18 Lateral Angle

19. Object-image transformation MIT 2.71/2.710 09/10/01 wk2-a-19 Ray-tracing transformation (paraxial) between object and image points

20. Image of point object at infinity MIT 2.71/2.710 09/10/01 wk2-a-20

21. Point object imaged at infinity MIT 2.71/2.710 09/10/01 wk2-a-21

22. Matrix formulation /1 MIT 2.71/2.710 09/10/01 wk2-a-22 translation by distance D 10 refraction by surface with radius of curvature R form common to all ray-tracing object-image transformation

23. Matrix formulation /2 MIT 2.71/2.710 09/10/01 wk2-a-23 Power Refraction by spherical surface Translation through uniform medium

24. Translation+refraction+translation MIT 2.71/2.710 09/10/01 wk2-a-24

25. Thin lens MIT 2.71/2.710 09/10/01 wk2-a-25

26. The power of surfaces MIT 2.71/2.710 09/10/01 wk2-a-26 Positive power bends rays “inwards” Negative power bends rays “outwards” Simple spherical refractor (positive) Plano-convex lens Bi-convex lens Simple spherical refractor (negative) Plano-concave lens Bi-concave lens

27. The power in matrix formulation MIT 2.71/2.710 09/10/01 wk2-a-27 (Ray bending)= (Power)×(Lateral coordinate) ⇒ (Power) = −M 12

28. Power and focal length MIT 2.71/2.710 09/10/01 wk2-a-28

29. Thick/compound elements: focal & principal points (surfaces) MIT 2.71/2.710 09/10/01 wk2-a-29 Note: in the paraxial approximation, the focal & principal surfaces are flat (i.e., planar). In reality, they are curved (but not spherical!!).The exact calculation is very complicated.

30. Focal Lengths for thick/compound elements MIT 2.71/2.710 09/10/01 wk2-a-30 generalized optical system EFL: Effective Focal Length (or simply “focal length”) FFL: Front Focal Length BFL: Back Focal Length

31. PSs and FLs for thin lenses MIT 2.71/2.710 09/10/01 wk2-a-31 The principal planes coincide with the (collocated) glass surfaces The rays bend precisely at the thin lens plane (=collocated glass surfaces & PP) 1/(EFL) ≡ P = P 1 + P 2 (BFL) = (EFL) = (FFL)

32. The significance of principal planes /1 MIT 2.71/2.710 09/10/01 wk2-a-32

33. The significance of principal planes /2 MIT 2.71/2.710 09/10/01 wk2-a-33

34. Imaging condition: ray-tracing MIT 2.71/2.710 09/10/01 wk2-a-34 Image point is located at the common intersection of all rays which emanate from the corresponding object point The two rays passing through the two focal points and the chief ray can be ray-traced directly

35. Imaging condition: matrix form /1 MIT 2.71/2.710 09/10/01 wk2-a-35

36. Imaging condition: matrix form /2 MIT 2.71/2.710 09/10/01 wk2-a-36 Imaging condition: 1 Output coordinate x´ must not depend on entrance angle γ

37. Imaging condition: matrix form /3 MIT 2.71/2.710 09/10/01 wk2-a-37 Imaging condition: S’/n’ + S/n – PSS’/nn’ = 0 system immersed in air, n=n’=1; power P=1/f n/S + n’/S’ = P 1/S + 1/S’ = 1/f

38. Lateral magnification MIT 2.71/2.710 09/10/01 wk2-a-38

39. Angular magnification MIT 2.71/2.710 09/10/01 wk2-a-39

40. Generalized imaging conditions MIT 2.71/2.710 09/10/01 wk2-a-40 Power: Imaging condition: Lateral magnification: Angular magnification: image system matrix object P = -M 12 ≠ 0 M 21 = 0 m X = M 22 M a = n/n’M 11