Understanding Lenses: Ray Diagrams, Real and Virtual Images

1. Design Realization lecture 26 John Canny 11/25/03

2. Last time • Reflection, Scattering • Refraction, TIR • Retro-reflection • Lenses

3. This time • Lenses reviewed: convex spherical lenses. • Ray diagrams. Real and virtual images. • More on lenses. Concave and aspheric lenses. • Fresnel optics: • Lenses: spherical and aspheric • Lenticular arrays • Prisms

4. Refraction – ray representation • In terms of rays, light bends toward the normal in the slower material.

5. Refractive indices • Water is approximately 1.33 • Normal glass and acrylic plastic is about 1.5 • Polycarbonate is about 1.56 • Highest optical plastic index is 1.66 • Bismuth glass is over 2 • Diamond is 2.42

6. Lenses • If light comes from a point source that is further away than the focal length, it will focus to another point on the other side.

7. Lenses • When there are two focal points f1 , f2 (sometimes called conjugates), then they satisfy:

8. Ray diagrams – real & virtual images • Tracing a pair of rays from the top and bottom of the object allows us to find the orientation and size of an image. • The pair of rays from a point converge at some distance from the lens, defining the image distance. • One pair of rays are usually straight ray through the axis of the lens.

9. Real images • An object further than the focal length away from the lens forms a convergent real image.

10. Virtual images • An object closer than the focal length forms a virtual image on the same side of the lens.

11. Virtual images • Virtual images can be created with concave lenses, which are smaller than the object.

12. Spherical Lenses • If a thin lens consists of spherical surfaces with radii r1 and r2, then the focal length satisfies 1/f = ( - 1) (1/r1 - 1/r2) this is known as the “lens-maker’s formula”.

13. Thick Lenses • The above approximations apply to “thin” lenses. Thick lenses use different approximations (based on paraxial rays). • Principal planes and Gullstrands equation are used to compute focal length etc. See:http://hyperphysics.phy-astr.gsu.edu

14. Thick Lenses • The above approximations apply to “thin” lenses. Thick lenses use different approximations (based on paraxial rays). • Principal planes and Gullstrand’s equation are used to compute focal length etc. See:http://hyperphysics.phy-astr.gsu.edu • The matrix method can also be used:

15. Matrix method • Lens effects can be approximated with 2D matrices. r1 = incoming ray, r2 = outgoing. • Let r = (, y) be a ray, where  is its angle from horizontal, and y is its vertical coordinate. • A lens can be represented as a matrix M:

16. Matrix method: thin lens example • Rays through the origin do not change direction, so a = 1. • Rays through the origin do not change y-value, so c = 0. • Assume the lens is at the origin, so intercept does not change, d = 1. • If incoming angle = 0, outgoing rays converge at the focal length, so b = -1/f.

17. Matrix method: thin lens example • Thin lens matrix is:

18. Matrix method: half-lens example • For the transition from air to glass on the entry side of the lens, the incoming ray angle is weakened by the refractive index ratio, so:

19. Matrix method: translation • Within a thick lens, direction does not change but the intercept changes

20. Thick lens matrix • We derive the thick-lens matrix by multiplying two half-lenses with a translation in between. The result is (d is lens thickness):

21. Spherical aberration • Cylindrical lenses do not converge to a point – outer rays converge closer:

22. Multi-element lenses • Are used to reduce aberration.

23. Aspheric lenses • Lens shape generated to provide better convergence between two conjugates (focal points) at specified distances. • Used to replace multi-element lenses. Increasingly popular.

24. Parabolic and elliptical mirrors • Curved mirrors provide very similar performance to lenses. • A parabolic mirror perfectly focuses parallel light to a point.

25. Parabolic and elliptical mirrors • Elliptical mirrors have two focal points, and focus light from one to the other. • A pair of parabolic mirrors also does this.

26. Fresnel lenses • Thin lenses are accurate but provide weak magnification. Thick lenses provide power but increase aberration. • Much of the aberration in thick lenses comes from the thick glass (not from the surfaces). • Fresnel lenses provide magnification without thickness.

27. Fresnel lenses • Remove the thick-ness, but preservepower. • Some artifacts areintroduced, but are invisible for large viewing areas(e.g. diplays).

28. Fresnel lenses • Fresnel lenses have no “thickness”, and simplify analysis for spherical and aspheric lenses. • In particular, aspheric lens equations can be written in closed form. • Two conjugates are needed because the lens equation is exact.

29. Fresnel lenses • Fresnel lenses can be made with high precision and low cost from optical plastics by pressure molding. • They are available in arbitrarily large sizes from custom manufacturers – and off the shelf up to about 5’ x 3’. • Fresnel grooves/inch may be 100 or more. Better for display than for imaging.

30. Lenticular arrays • Many lenses printed on one sheet. • Simplest version: array of cylindrical lenses. • Used to budget 3D vision:

31. Lenticular arrays • Simplest version: array of cylindrical lenses.

32. Lenticular arrays • Lenticular screens are rated in LPI for lines per inch. Typical range is 40-60 LPI, at about $10 per square foot. • Budget color printers can achieve 4800 dpi. • At 40 LPI that gives 120 images in approx 60 viewing range, or 0.5 per image.

33. Lenticular stereograms • By interleaving images from views of a scene spaced by 0.5, you can achieve a good 3D image. • At 1m viewing distance, 0.5 translates to 1cm spacing between images. • Eye spacing is about 6 cm.

34. Diffusers • Diffusers spread collimated (parallel) light over a specified range of angles. • Can control viewing angle for a display. • Controls sense of “presence” in partitioned spaces.

35. Geometric diffusers • Arrays of tiny lenses (lenticular arrays). • Can be cylindrical (diffusion in one direction only), used in rear-projection screens. • Surface etching. Using in shower glass, anti-glare plastic coatings. • Holographic surface etching: provides tightly-controlled diffusion envelope. • Low-quality surface finish(!) on plastics gives diffusion effect.

36. Geometric diffusers • Arrays of tiny lenses (lenticular arrays). • Can be cylindrical (diffusion in one direction only), used in rear-projection screens. • Surface etching. Using in shower glass, anti-glare plastic coatings. • Holographic surface etching: provides tightly-controlled diffusion envelope. • use a material with diffusing properties: • E.g. small spheres in refractive material

37. Fresnel prisms • Similar idea to lenses. Remove the thickness of the prism and stagger the surface facets. • Useful for bending light over a large area, e.g. for deflecting daylight. • Also used for vision correction.

38. Summary • Ray diagrams. Real and virtual images. • More on lenses. Concave and aspheric lenses. • Parabolic and elliptical mirrors. • Fresnel optics: • Lenses: spherical and aspheric • Lenticular arrays • Prisms