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Lens Equation

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  1. Lens Equation ( < 0 ) • True for thin lens and paraxial rays. • magnification m = h’/h = - q/p

  2. Signs in the Lens Equation for Thin Lenses • pis positive for real object • q is positive for real image • q is negative for virtual image • m is positive if image is upright • m is negative if image is inverted • f is positive if converging lens • f is negative if diverging lens • p is negative for virtual object

  3. Aberrations Chromatic aberration Cameras, … correct nblue > nred Spherical aberration Parabolic mirrror Large telescopes, …

  4. Geometric Optics vs Wave Optics • Geometric optics is a limit of the general optics where wave effects such as interference and diffraction are negligible. • Geometric optics applies when objects and apertures involved are much larger than the wavelength of light. • In geometric optics, the propagation of light can be analyzed using rays alone. • Wave optics (sometimes also called physical optics) - wave effects play important roles. • Wave optics applies when objects and apertures are comparable to or smaller than the wavelength of light. • In wave optics, we must use the concepts relevant to waves such as phases, coherence, and interference.

  5. - laser light and light transmitted through a small aperture are coherent. - light from a light bulb and sun light over some area are incoherent. Coherence • When the difference in phase between two (or more) waves remains constant (i.e., time-independent), the waves are said to be perfectly coherent. • A single light wave is said to be coherent if any two points along the propagation path maintains a constant phase difference. Only coherent waves can produce interference fringes! Coherence length: the spatial extent over which light waves remain coherent.

  6. Interference of Two Coherent Waves Snapshot of wave fronts at a given instant Constructive interference (in phase) Destructive interference (completelyout of phase) A B,C

  7. Intensity is proportional to E2 Intensity of Interference Fringes Let the electric field components of the two coherent electromagnetic waves be The resulting electric field component point P is then I=0 when f = (2m+1)p , i.e. half cycle + any number of cycle.

  8. Destructive interference eliminates (or minimizes) the reflected light! e.g., non-reflecting lens coating Phase change by  Thin Film • Thinhere meansthat thethickness is comparable to the wavelength of the light. • The reflected light waves from the two sides of a thin film interfere. • Phase difference could come from: • reflection, path length difference, • different indices of refraction • If the incident light propagates from a medium of lower index of refraction toward one of higher index of refraction, the phase of the reflected wave shifts by . (neither highlow index nor transmitted light)

  9. (Assume near-normal incidence.) Path length difference: destructive constructive where Thin-Film Interference-Cont’d • ray-one got a phase change of 180o due to reflection from air to glass. • the phase difference due to path length is: • then total phase difference: f = f’+180.

  10. If the film has an intermediate index of refraction Conditions for maxima/minima will reverse! The equations fail for some of the following situations. Which one(s)? Thin-Film Continued The previous discussion was for the situation in which n2 > n1 and n2 > n3 , i.e., the index of refraction of the film is larger than those of the surrounding media, but they are also valid if the index of refraction of the film is smaller than those of the surrounding media (n2 < n1 and n2 < n3 ).

  11. 1 2 oil (n1>1.33) water (n=1.33) Warm-up quiz Monochrome light of wavelength  in air is incident normal to a thin layer of oil film floating on water as shown. If the film thickness is 5/(4n1). Which of the following statement is true? • The reflection is dark by destructive interference of rays 1 and 2 • b) The reflection is bright by constructive interference of rays 1 and 2 • c) The reflection is colorful by interference of rays 1 and 2.

  12. Newton’s Ring The air between the glass plates acts like a thin film. • Since the thickness of the film changes over the radius of the plates, alternating bright and dark fringes form, when the plates are illuminated. Because of the curvature of the upper piece, the film thickness varies more rapidly at larger radius. Thus the fringe separation is smaller toward the outside.

  13. Young’s double-slit experiment • According to Huygens’s principle, • each slit acts like a wavelet. The • the secondary wave fronts are • cylindrical surfaces. • Upon reaching the screen C, the • two wave interact to produce an • interference pattern consisting of • alternating bright and dark bands • (or fringes), depending on their • phase difference. Constructive vs. destructive interference Two (narrow) slit Interference

  14. Interference Fringes For D >> d, the difference in path lengths between the two waves is • A bright fringe is produced if the path • lengths differ by an integer number of • wavelengths, • A dark fringe is produced if the path • lengths differ by an odd multiple of • half a wavelength, y ~ D*tan(θ)

  15. Intensity is proportional to E2 Intensity of Interference Fringes Let the electric field components of the two coherent electromagnetic waves be The resulting electric field component point P is then I=0 when f = (2m+1)p , i.e. half cycle + any number of cycle.

  16. For Young’s double-slit experiment, the phase difference is Intensity of Interference Fringes-Cont’d

  17. Physics 241 –Quiz A Light of wavelength  in air is incident normal to a thin layer of glass held in air as shown. If the reflection is suppressed (dark) by interference of rays 1 and 2, what is a possible thickness d of the glass layer? 1 a) /(4n1) b) /4 c) /(2n1) d) /2 e) 3/4 2 glass (n1>1)

  18. 1 2 oil (n1>1.33) water (n=1.33) Physics 241 –Quiz B Light of wavelength  in air is incident normal to a thin layer of oil film floating on water as shown. If the reflection is suppressed (dark) by interference of rays 1 and 2, what is a possible thickness d of the oil film? a) /(4n1) b) /4 c) /(2n1) d) /2 e) 3/4

  19. 1 2 oil (n1>1.33) water (n=1.33) Physics 241 –Quiz C Light of wavelength  in air is incident normal to a thin layer of oil film floating on water as shown. If the reflection is bright (constructive interference) by interference of rays 1 and 2, what is a possible thickness d of the oil film? a) /(4n1) b) /4 c) /(2n1) d) /2 e) 3/4