Optical Design Essentials

1. Optical Design Essentials • Reflective optics (reflection from flat and curved mirrors) • Refractive optics (transmission through lens) • Aberration of light (dispersive and non-dispersive aberration) • Optical instrumentation

2. Reflection (physics) • Reflection is the abrupt change in direction of a wave front at an interface between two dissimilar media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. • Reflection of light may be specular (that is, mirror-like) or diffuse (that is, not retaining the image, only the energy) depending on the nature of the interface.

3. A mirror provides the most common model for specular light reflection and consists of a glass sheet in front of a metallic coating where the reflection actually occurs. It is also possible for reflection to occur from the surface of transparent media, such as water or glass. In the diagram, a light ray PO strikes a vertical mirror at point O, and the reflected ray is OQ. By projecting an imaginary line through point O perpendicular to the mirror, known as the normal, we can measure the angle of incidence, θi and the angle of reflection, θr. The law of reflection states that θi = θr, or in other words, the angle of incidence equals the angle of reflection. In fact, reflection of light may occur whenever light travels from a medium of a given refractive index into a medium with a different refractive index. In the most general case, a certain fraction of the light is reflected from the interface, and the remainder is refracted. Solving Maxwell's equations for a light ray striking a boundary allows the derivation of the Fresnel equations, which can be used to predict how much of the light is reflected, and how much is refracted in a given situation. Total internal reflection of light from a denser medium occurs if the angle of incidence is above the critical angle. Specular (mirror-like) reflection θi = θr. the angle of incidence equals the angle of reflection

4. Diffuse reflection • Light bounces off in all directions due to the microscopic irregularities of the interface; this is an omnipresent phenomenon, applicable for all non-shiny objects that are not black.

5. Image formation Formation of image by large plane mirrors Plane Mirror Image Object

6. Image Formation Formation of image by curved mirrors Concave mirror Reduced and upright virtual image Magnified and inverted real image Convex mirror

7. Convergent lens Inverted and real image Object Divergent lens Object Virtual and upright image Image formation Formation of image by lens

8. Light beam changes propagation direction in traversing through the interface between two media of different refractive indices Air Water

9. qi qR n1 n2 qr Refraction • Light from a point source will travel in a straight line with constant velocity in a homogeneous medium. • Light changes velocity and direction as it passes from one transparent medium into another with different refractive index, n. qi is the light incident angle. qR is the reflected beam angle. qr is the refracted beam angle. qi = qR Snell’s Law:

10. Light beam bent by prism

11. Deflection of light beam by lens

12. f f Divergent lens -f Focusing light with lens Convergent lens

13. Refractive index n • The index of refraction of a material is related to the speed of light propagating in that material. • Experiments show that the index of refraction of a medium was inversely proportional to the speed of light in that medium. Since c is defined as the speed of light in vacuum, and n = 1 is defined as the index of refraction of vacuum, we have n = c/n where n = medium’s index of refraction, v = speed of light in that medium, c = speed of light in a vacuum ~ 3 x 108 ms-1.

14. Propagation of light • When light propagates in a transparent material medium, its speed is in general less than the speed in vacuum c. An interesting consequence of this is that a light ray will change direction when passing from one medium to another. Since the light ray appears to be “broken”, the phenomenon is known as refraction.

15. Snell’s Law of refraction Huygens’ Principle of wavefront propagation in two media explains Snell’s law nicely. Alternatively one can use Fermat’s “Principle of Least Time” A Fast Slow B In the figure, it is indeed plausible that the bending of the ray serves to minimize the time required to get from a point A to point B. If the ray followed the unbent path shown with a dashed line, it would have to travel a longer distance in the medium in which its speed is slower. By bending the correct amount, it can reduce the distance it has to cover in the slower medium without going too far out of its way. It is true that Snell’s law gives exactly the set of angles that minimizes the time required for light to get from one point to another. The proof of this fact isleft as an exercise.

16. qr = 74.60 qr = 900 qr = 48.60 Reflection n1 = 1.0 n2 = 1.5 qi= qc = 41.80 qi> 41.80 qi= 400 qi= 300 Total internal reflection • One important consequence of Snell’s law of refraction is the phenomenon of total internal reflection. If light is propagating from a more dense to a less dense medium (in the optical sense), i.e. n1 > n2, then sin qr > sin qi. The angle of incidence for which refraction is still possible is given by sin qi < n2/n1 • For qi = qc = n2/n1 (critical angle), the refracted ray propagate along the interface, i.e. no transmission! • For larger angles of incidence, the incident ray does not cross the interface, but is reflected back instead. This is what makes optical fibres possible. Light propagates inside the fibre, which is made of glass which has a higher refractive index than the air outside. Since the fibre is very thin, the light beam inside strikes the interface at a large angle of incidence, large enough that it is reflected back into the glass and is not lost outside. Thus fibres can guide light beams in any desired direction with relatively low losses of radiant energy.

17. Attenuation (dB/km) Rayleigh scattering Attenuation of silica fibre Wavelength nm Attenuation dB/km 1310 0.38 1380 0.6 1550 0.2 3 Silica Fibre Infrared Absorption of Silica 2 1 900 1100 1300 1500 1700 Wavelength (nm) Optical Fibers

18. Protective sheath q Refractive Index n Core Cladding Step index Graded index 9 mm 200mm 125mm 240mm Single-mode Fibre Types of optical fibres

19. Dispersion of light • What is dispersion? And how it comes about? • Optical Properties of diamonds • Brilliance, fire, and flashes • Rainbow • how does it happen? • Why everyone sees a different rainbow?

20. Dispersion of light i light rays incident on a sheet of glass: i' r r' • normally incident ray passes through without refraction • obliquely incident ray must refract - how? • angles i and r = angles r' and i' respectively • all light passing through a sheet of glass obliquely undergoes this offset • only noticeable at the edge of the sheet.

21. r' Dispersion of light what happens if the two surfaces are not parallel? e.g., in a prism: i r' i i' r r i' nprism > nmedium nprism = nmedium nprism < nmedium these examples assume a single ray, of a single wavelength what happens if different wavelengths are used?

22. Light travel at different velocity in different materials. Since the refractive index is defined as n = c/v, it implies that the value of n is different for different materials. Light in transparent bulk materials Refractive indices (n) for the mean sodium D line (689.3 nm)

23. Maxwell equation suggested that different wavelength of light will propagate at different velocity v. Since the refractive index is defined as n = c/v, it implies that the value of n is wavelength dependent. White Light in transparent bulk materials Dispersion of Crown Glass. The wavelengths are those of a He discharge tube. • The index of refraction is not a constant! • It depends on the frequency of the light!This is because the light with different frequencies wiggles the charges with a different speed. • Examples: • water glass diamond • Red =656 nm 1.331 1.571 2.410 • Yellow =589 nm 1.333 1.575 2.418 • Blue =434 nm 1.340 1.594 2.450

24. The Physics of the Rainbow The picture shows a primary rainbow, a fainter secondary rainbow above it, and several pastel-shaded rainbows inside the primary rainbow.

25. Newton’s Discovery • Newton (at the age of 23) reached the revolutionary conclusion that white light is not a simple, homogenous entity, as natural philosophers since Aristotle had believed. • When he passed a thin beam of sunlight through a glass prism, he noted the oblong spectrum of colors: red, yellow, green, blue,violet– that formed on the wall opposite.

26. Dispersion of white light How different colors are diffracted Different colors are diffracted by different amounts producing a spectrum Air Incident white light with all the colors Glass Prism

27. Dispersion of white light by prism

28. THE SECRETDifferent frequencies of the light bend different amount as it passes through a dispersive media.

29. Optical properties of diamondsWhy are the diamonds expensive? • Diamonds has a large index of refraction (n=2.4) and has a small critical angle 24.5°. The diamond surfaces are cut so that most of light entering a diamond is eventually reflected back out the front  brilliance! • if you look from the back, the diamond is black!

30. Diamond is highly dispersive. So the white light is spread out into a broad spectrum the fire of diamonds: beautifulcolors.

32. Flashes • Diamonds are cut withmany surfaces.When viewed from one angle, some light gets reflected from a particular surface and its color spreads out, reaching your eyes. • As you turn the diamonds or yourself, some other rays reach your eyes with other paths. And this causes theflash (sparkle)of the diamonds.

33. Rainbows • Because of the dispersion of water, droplets of water can break up the sun light into a spectrum rainbow • How does this happen, precisely? • A light beam is dispersed twice through raindrops and reflected once….

34. 42°

35. Since the red rays bend less than the blue, they have a steeper opening angle relative to the horizontal. • A math calculation shows that you see the red rays from all those raindrops that lie at 42° relative to the horizontal sun rays. • The blue rays bend more, and have smaller opening angle relative to the horizontal. • You see the blue rays from all those raindrops the lie at 40°.

37. Thus all rays from a rainbow come from different raindrops. And everybody sees a different rainbow!

43. Light ray Radius of curvature of the interface R Lens • The sign convention for refraction: Light is propagated from left to right. Object distances are counted as positive when the object is in front of the interface, but image distances are positive when the image is formed behind the interface. The radius of curvature follows the same convention as the image distances.

44. Types of Lens • Thin lens- lens with no thickness • Thick lens- thickness of lens considered • Spherical lens - lens with spherical surfaces • Aspherical lens- lens with aspheric surfaces (hyperbola) • Single lens - optical system with one lens • Compond lens - system with more than one lens

45. R2 < 0 R1 > 0 Spherical single thin lens • Lensmaker’s Formula (Thin Lens Equation) In this formula we have assumed that the interface is between glass and air. Glass has refractive index of nland that of air is 1.

46. CONVEX LENS CONCAVE LENS Type of Spherical Single Thin Lens Bi-concave R1 < 0 R2 > 0 Bi-convex R1 > 0 R2 < 0 Planar convex R1 > 0 R2 = infinity Planar-concave R1 = infinity R2 > 0 Meniscus concave R1 > 0 R2 > 0 Meniscus convex R1 > 0 R2 > 0

47. Radius of curvature of the interface R Light ray l0 l1 h R f p s v u n1 n0 Refraction at Spherical Surfaces Optical Path : sp = n0 lo + n1 l1 lo = [R2 +(u + R)2 -2R(u + R) cos f]1/2 l1 = [R2 +(v - R)2 -2R(v - R) cos f]1/2

48. Therefore sp = n0 [R2 +(u + R)2 -2R(u + R) cos f]1/2 + n1 [R2 +(v - R)2 -2R(v - R) cos f]1/2 From Fermat’s principle of least distance, i.e. d(sp)/df = 0 Therefore It follows that If we assume small value of f (this is approximation of geometric optics), i.e. cos f ~ 1, the expressions for l0 and l1 yeild l0 ~ u and l1 ~ v. Then we have Refraction at a Spherical SurfaceParaxial ray approximation i.e. beam size is small compared with the cross-section of the optics

49. Principal plane The thickness of lens is L n1 n0 R1 R2 Refraction from two spherical surfaces (lens) • In a lens, there are two consecutive refractions, one from air to glass, and then from the glass back into the air. • In the first refraction: Use (1) and (2) to eliminate in first image at v, and we get: For parallel input beam, i.e. u = infinity and image form at the focal plane, v’ = f. The expression becomes This is the lensmaker’s formula Comparing (3) and (4) one obtains This is the thin lens equation In the second refraction: (denoted by u’, v’) The image formed by the first refraction will be the object of the second refraction. The distance from the second surface will be L – v. In the thin lens approximation, L approach to 0. Therefore

50. f f f c c c f f f c c c Image formation and amplification by single spherical thin lens f = 1m and c = 2m v = -2.33m u = 0.7 m m = v/u = 2.33 v = 3m m = v/u = -2 u = 1.5m u = 2.5m v = 1.67m m = v/u = -0.668