Chapters 34, 35, 36. Geometry of Optical Instruments, Interference, and Diffraction + Desert Island addons.

Chapter 34. Geometric Optics and Optical Instruments.

Constructing the image from a plane mirror I Following the light rays to form an image of an object. Consider Figure 34.4. Consider Figure 34.5.

Constructing the image from a plane mirror II Consider Figure 34.6, below left. Consider Figure 34.7, below right. Images from a plane mirror show left/right reversal.

Mirrors form images by reflection.   Plane mirror: using law of reflection to make a ray diagram, we find that the lateral magnification, m = y’/y = + or -1. + sign is erect image, - sign is inverted. Spherical mirror: (for paraxial ray approximation—rays nearly parallel to axis or R large) 1/s + 1/s’ = 1/f, where s = object distance, s’ = image distance, f = focal length (rays from infinity will focus at that distance from the vertex of the mirror.) If R is the radius of curvature, f = R/2. Magnification, m = y’/y = -s’/s.

Reflections from a spherical mirror Images from a concave mirror change depending on the object position. Consider Figure 34.10. Concave and convex mirror sign convention. Consider Figure 34.11.

Petersen’s Rules for signs of optical distances for s, s’, f, and r: real images produce +ive, virtual images –ive. For m, erect is +ive, inverted is -ive. Refraction at a spherical surface: n1/s + n2/s’ = (n2 - n1)/R. Note that if R∞, we have a flat surface and n1/s + n2/s’ = 0.

The Hubble Space Telescope The HST has a spherical mirror. Unfortunately, it was ground to the wrong dimensions by 1/50 the width of a human hair over a mirror as large as a person. Replacing the mirror was unthinkable. The final solution was an electronic adjustment to the photodiode array which converts the optical image to digital data. Consider Figure 34.12 at right to see the dramatic “before” and “after” images.

The focal point and focal length of a spherical mirror The focal point is at half of the mirror’s radius of curvature. All incoming rays will converge at the focal point. Consider Figure 34.13.

A system of rays may be constructed to reveal the image Rays are drawn with regard to the object, the optical axis, the focal point, and the center of curvature to locate the image. Consider Figure 34.14 to view one such construct.

The convex spherical mirror I If you imagine standing inside a shiny metal ball to visualize the concave spherical mirror, imagine standing on the outside to visualize the concave spherical mirror. Consider Figure 34.16 to introduce the convex spherical mirror.

Refraction within a sphere Perhaps as light traveling through a raindrop. Consider Figure 34.21 and Figure 34.22.

Refraction at a spherical surface: n1/s + n2/s’ = (n2 - n1)/R Refraction at a spherical surface: n1/s + n2/s’ = (n2 - n1)/R. Note that if R∞, we have a flat surface and n1/s + n2/s’ = 0. You can use the above equation applied twice to derive: The thin lens equation (the same as for paraxial spherical mirrors): 1/s + 1/s’ = 1/f. Petersen’s rules for signs also apply. The lens-makers equation: for grinding two sides with radii R1 and R2, 1/f = (n-1)( 1/R1 - 1/ R2).

Thin lenses I The converging lens is shown in Figure 34.28. Note symmetrical focal points on either side of the lens. Figure 34.29 uses the same ray formalism as we used with mirrors to find the image.

Thin lenses II Figure 34.31 at bottom left illustrates a diverging lens scattering light rays and the position of its second (virtual) focal point. Figure 34.32 illustrates some assorted common arrangements of lens surfaces. Follow Example 34.8.

Lenses have chromatic aberration (different n and thus refraction for different colors) and spherical aberration (a perfect lens would be ground with parabolic surfaces). Spherical mirrors have only spherical aberration. More expensive cameras and optical devices correct for these with added lenses or parabolic grinding.

The microscope Optical elements are arranged to magnify tiny images for visual inspection. Figure 34.52 presents the elements of an optical microscope.

The astronomical telescope Optical elements are arranged to magnify distant objects for visual inspection. Figure 34.53 presents the elements of an astronomical telescope.

The reflecting telescope Optical elements are arranged to reflect collected light back to an eyepiece or detector. This design eliminates aberrations more likely when using lenses. It also allows for greater magnification. The reflective telescope is shown in Figure 34.54.

Chapter 35 Interference

Monochromatic (one λ) and coherent (in phase, same φ) light, like from a laser, can be used to gauge the effects of interference.   Amplitudes add in constructive interference: Path difference: δ = r2 – r1 = mλ, m = 0, +-1, +-2, +-3, etc. Amplitudes cancel in destructive interference: Path difference: δ = r2 – r1 = (m+1/2)λ, m = 0, +-1, +-2, +-3, etc.

Two-source interference of light Figure 35.4 shows two waves interfering constructively and destructively. Young did a similar experiment with light. See below.

Young’s 2 slit experiment:. dsinΘ = mλ,. for light maxima fringes Young’s 2 slit experiment: dsinΘ = mλ, for light maxima fringes. dsinΘ = (m+ 1/2)λ, for light minima. (d = slit spacing) Also, y = Rmλ/d where R is distance to screen, constructive fringes.

Intensity distribution Figure 35.10, below, displays the intensity distribution from two identical slits interfering. Follow Example 35.3. Intensity I = I0 cos2(πyd/λR).

Thin films will interfere The reflections of the two surfaces in close proximity will interfere as they move from the film. Figure 35.11 at right displays an explanation and a photograph of thin-film interference.

Thin film interference —the path difference for bouncing off two successive surfaces: 2t = mλ for constructive interference, m = 0, 1,2,… 2t = (m+1/2)λ for destructive interference. This is for the case of 0 or 2 phase shifts (of (1/2)λ) related to going into a higher index of refraction, n. For one phase shift at only one surface, constructive and destructive interference formulas are switched.

Newton’s rings Figure 35.17 illustrates the interference rings resulting from an air film under a glass item.

Using fringes to test quality control An optical flat will only display even, concentric rings if the optic is perfectly ground. Follow Example 35.7.

Michelson and Morley’s interferometer In this amazing experiment at Case Western Reserve, Michelson and Morley suspended their interferometer on a huge slab of sandstone on a pool of mercury (very stable, easily moved). As they rotated the slab, movement of the earth could have added in one direction and subtracted in another, changing interference fringes each time the device was turned a different direction. They did not change. This was an early proof of the invariance of the speed of light.

Chapter 36 Diffraction

Diffraction from a single slit The result is not what you might expect. Refer to Figure 36.3.

Diffraction   The same formula holds for the minima (dark fringes) for one slit, Fraunhofer distant screen, so rays are parallel) diffraction as for the maxima (bright fringes) for two slit and multi-slit (diffraction grating) interference. The formula is: sin(Θ) = mλ /d, Where theta is the angle off the center line going from the slit midpoint to the screen, m = ±1, ±2, ±3, etc., lambda is the wavelength of monochromatic light used, and d is the slit width for the one slit and the slit spacing for the two slit (or multi-slit) experiment. Intensity: I = I0 [sin(β/2)/β/2]2, where β = (2πdsinΘ)/λ. Width of the single slit maximum: Θ = λ/d .

Fraunhofer diffraction and an example of analysis Figure 36.6 (at bottom left) is a photograph of a Fraunhofer pattern from a single slit. Follow Example 36.1, illustrated by Figure 36.7 (at bottom right).

Multiple slit interference The analysis of intensity to find the maximum is done in similar fashion as it was for a single slit. Consider Figure 36.12 at right. Consider Figure 36.13 below.

For a circular aperture (like telescope):. for small angle, For a circular aperture (like telescope): for small angle, sin Θ = Θ = 1.22λ/d Diffraction grating: same formula as two slit, except maxima are narrower and more well defined, making this device excellent for providing spectra, or spreads of wavelengths.

Circular apertures and resolving power In order to have an undistorted Airy disk (for whatever purpose), wavelength of the radiation cannot approach the diameter of the aperture through which it passes. Figures 36.26 and 36.27 illustrate this point.

The grating spectrograph A grating can be used like a prism, to disperse the wavelengths of a light source. If the source is white light, this process is unremarkable, but if the source is built of discrete wavelengths, our adventure is now called spectroscopy. Chemical systems and astronomical entities have discrete absorption or emission spectra that contain clues to their identity and reactivity. See Figure 36.19 for a spectral example from a distant star.

The grating spectrograph II—instrumental detail Spectroscopy (the study of light with a device such as the spectrograph shown below) pervades the physical sciences.

X-ray diffraction X-rays have a wavelength commensurate with atomic structure. Rontgen had only discovered this high-energy EM wave a few decades earlier when Friederich, Knipping, and von Laue used it to elucidate crystal structures between adjacent ions in salt crystals. The experiment is shown below in Figure 36.21.

Bragg X ray diffraction —light bounces off two successive layers of a crystal of spacing d, and constructive interference occurs for path difference 2 d sin(Θ) = mλ, m = 0, 1, 2,... This is a way of determining the crystal spacing, d. A hologram is a wave pattern formed by the interference of monochromatic, coherent light scattered from a 3-d object, and light coming directly from the source. Information about the whole is contained in every part of the pattern.

Using multiple modes to observe the same event Multiple views of the same event can “nail down” the truth in the observation. Follow Example 36.6, illustrated by Figure 36.29.

Holography—experimental By using a beam splitter, coherent laser radiation can illuminate an object from different perspective. Interference effects provide the depth that makes a three-dimensional image from two-dimensional views. Figure 36.30 illustrates this process.