Tutorial 8
Transcription
Tutorial 8
Elec308: Engineering Optics Tutorial 8 Zeng Yan [email protected] 5/10/2011 ELEC 308 Engineering Optics Outline • Interference of waves – Two sources interference – Young’s double slit experiment • Thin film interference – Equal inclination – Equal thickness – Newton’s rings • Interferometer – Michelson, Mach-Zender, Fizeau 5/10/2011 ELEC 308 Engineering Optics Example • The figure below shows a laser beam incident on a wet piece of filter paper atop a sheet of glass whose index of refraction is to be measured. The photograph shows the resulting light pattern. Explain what is happening and derive an expression for ni in terms of R and d. 5/10/2011 ELEC 308 Engineering Optics Example • Wet paper is on optical contact with the glass. The beam scatters off the wet paper and is mostly transmitted until the critical angle is attained, at which point the light is reflected back toward the source. 5/10/2011 ELEC 308 Engineering Optics Example • On the air-glass surface of the second interface TIR occurs at the critical angle. TIR condition: 1 ni sin c ni sin c nair 1 tan c • From the geometry of the experiment: 1 1 cot sin 2 2 1 1 1 sin tan 2 1 1 4d 2 ni 1 1 2 2 sin c tan c R 5/10/2011 ELEC 308 Engineering Optics R 2d Coherent waves • Waves for which phase difference is constant in time and space E1 E 01 sin(t kx) E 2 E 02 sin(t kx ) const. • Conditions for interference – Identical polarization – Identical frequency – Constant phase difference 5/10/2011 ELEC 308 Engineering Optics Interference of waves E E (r ) exp j k r t • Two waves E E (r ) exp j k r t 1P 01 1 1 2P 02 2 2 • Resultant intensity 5/10/2011 ELEC 308 Engineering Optics Phase shift kasin sin i 5/10/2011 ELEC 308 Engineering Optics wave summation 5/10/2011 ELEC 308 Engineering Optics Wave addition (superposition) • If constructive or destructive interference is to continue occurring at a point, the sources of the waves must be coherent sources. combined waveform wave 1 wave 2 Two waves in phase 5/10/2011 Two waves 180°out of phase ELEC 308 Engineering Optics • Two sources are coherent if the waves they emit maintain a constant phase difference. Young’s double slit experiment d sin Bright fringes of a double-slit 5/10/2011 Dark fringes of a double-slit sin m sin m ELEC 308 Engineering Optics d 1 2 m 0,1,2,3, d m 0,1,2,3, Example 1 Young’s Double-Slit Experiment Red light (664 nm) is used in Young’s experiment with slits separated by 0.000120 m. The screen is located a distance 2.75 m from the slits. Find the distance on the screen between the central bright fringe and the third-order bright fringe. Solution: sin m d m 0,1,2,3, 9 1 66410 m 0.951 sin m sin 3 4 d 1.2010 m 1 y L tan 2.75 m tan 0.951 0.0456 m 5/10/2011 ELEC 308 Engineering Optics Example 2 White Light and Young’s Experiment The figure shows a photograph that illustrates the kind of interference fringes that can result when white light is used in Young’s experiment. Why does Young’s experiment separate white light into its constituent colors? In any group of colored fringes, such as the two singled out, why is red farther out from the central fringe than green is? Why is the central fringe white? 5/10/2011 ELEC 308 Engineering Optics New example 1 Young’s Experiment 9.4 Will we get an interference pattern in Young’s Experiment if we replace the source slit by S by a single long-filament light bulb? What would occur if we replaced the slit S1 and S2 by these same bulb? •Bulb-extended source The source is made up of a large number of incoherent point source. •Pattern overlapping 5/10/2011 ELEC 308 Engineering Optics New example 2 Young’s Experiment 9.6 Two 1.0MHz radio antennas emitting in-phase are separated by 600m along a north-south line. A radio receiver placed 2.0km east is equidistant from both transmitting antennas and picks up a fairly strong signal. How far North should that receiver be moved if it is again to detect a signal nearly as strong? Like the Young’s Experiment, Two antennas->two slits Receiver->screen Interference for Young' s Experiment : a sin m m Geometric condition : sin y /(s 2 y 2 )1/ 2 Wavelength : c / f 5/10/2011 ELEC 308 Engineering Optics New example 2 Young’s Experiment 9.5 Figure P.9.5 shows an output pattern that was measured by a tiny microphone when two small piezo-loudspeakers separated by 15 cm were pointed toward the microphone at a distance of 1.5 m away. Given that the speed of sound at 20 degree is 343 m/s, determined the approximate frequency at which the speakers were driven. Discuss the nature of the pattern and explain why it has a central minimum. y ms a sin m a m y s a f v/ 5/10/2011 ELEC 308 Engineering Optics New example 3 Young’s Experiment 9.10 White light falling on two long narrow slits emerges and is observed on a distant screen. If red light (780nm) in the first order fringe overlaps violet in the second-order fringe, what is the latter’s wavelength? Interference for Young' s Experiment : a sin m m 5/10/2011 ELEC 308 Engineering Optics New example 4 Interference of Lloyd’s mirror 9.23 Using Lloyd’s mirror, X-ray fringes were observed, the spacing of which was found to be 0.0025 cm. The wavelength used was 0.833nm. If the source-screen distance was 3 m, how high above the mirror plane was the point source of X-ray placed? Interference condition : a sin m Geometric condition : y1 sin θ1s; y2 sin θ2 s Spacing : y y1 - y 2 s 5/10/2011 a ELEC 308 Engineering Optics New example 4 Interference of Lloyd’s mirror 9.24 Imaging that we have an antenna at the edge of a lake picking up a signal from a distant radio star, which is just coming up above the horizon. write expression for deta and for the angular position of the star when the antenna detects its first maximum. For Lloyd’s mirror: α α 5/10/2011 α k (r1 r2 ) k (a /(2 sin ) [sin(90 2 )]a /(2 sin )) ka(1 cos 2 ) /(2 sin ) when 2 , sin 1 ( / 2a) ELEC 308 Engineering Optics Thin film Interference • Phenomenon commonly seen as colored patches on thin layers of leaked oil, grease and soap bubbles. • To obtain a nice colored pattern, the thickness of the film has to be on the order of the wavelength of light. 5/10/2011 ELEC 308 Engineering Optics Thin film interference • In thin film interference the phase difference is due to reflection at either side of a thin film of transparent material. • The phase difference is due to two factors: – Path difference through the film (corrected for the change in speed of light in the material) – Phase shift at the interface The wavelength of light in oil though is not the same as in air: 0 n In this case, constructive interference takes place when: 5/10/2011 ELEC 308 Engineering Optics Example 3: anti-reflection coating 5/10/2011 ELEC 308 Engineering Optics New example 5 Thin film interference Q6. The refractive index of a dense flint glass is about 1.75. Design an antireflection coating to maximally reduce the reflection from glass surface at 0deg incident angle (1)The index of the coating material (2)Phase shift exist? (3)Maximun interference condition 5/10/2011 ELEC 308 Engineering Optics New example 6 Thin film interference 9.27 A thin film of ethyl alcohol (n=1.36) spread on a flat glass plate and illuminated with white light shows a color pattern in reflection. If a region of the film reflects only green light (500nm) strongly, how thick is it? (1)Phase shift in the interface (2) Reflectionenhanced : 2nd 2 / 2m 5/10/2011 ELEC 308 Engineering Optics Equal inclination interference The phase difference: 5/10/2011 ELEC 308 Engineering Optics New example 6 Thin film interference 9.29 Consider the circular pattern of Haidinger’s fringes resulting from a Film with a thickness if 2mm and an index of refraction of 1.5. For Monochromatic illumination of λ=600 nm, find the value of m for the central Fringe (θ=0). Will it be bright or dark? δ 2nktcosθ π 2nktcosθ 2 1.5 2/600 2 1000001 2π 10000 Order : m 10000,considering the phase shift, minimum 5/10/2011 ELEC 308 Engineering Optics 1)when thin film thickness increases: fringes extend outwardly i i i' i' e λ δ 2e n n sin i 2 2 2 2 1 2 k k 1.2.3. (max) 2)when thin film thickness decreases: fringes contract inwardly 5/10/2011 ELEC 308 Engineering Optics Equal inclination interference 3) if white light falls on, color fringes appear from red to violet. 2e n22 n12 sin 2 i Since e.k 2 k fixed, n22 n12 sin 2 i 5/10/2011 k 1.2.3. (max) should be larger w. r. t. longer i Therefore, it should be closer to the center. ELEC 308 Engineering Optics Thin Air Wedge 5/10/2011 ELEC 308 Engineering Optics Thin Air Wedge As the thickness of air d increases, the fringes move down; While the thickness d decreases, the fringes move up; When the angle θ increases, the fringes move down; When the angle θ decreases, the fringes move up; 5/10/2011 ELEC 308 Engineering Optics Thin Air Wedge d *Measure the distance change by the moving fringes. e m 2 d 5/10/2011 ELEC 308 Engineering Optics Application of equal thickness interference Thermal expansion instrument M:test object C:made of In, low thermal expansion coefficient M C We count the number of passing fringes m when the temperature Increase t. l m M heighten: 2 Thermal expansion coefficient: 5/10/2011 ELEC 308 Engineering Optics l l t New example 7 Equal thickness interference 9.34 Suppose a wedge-shaped air film is made between two sheets of glass, with a piece of paper 7.618*10-5m thick used as the spacer at their very ends. If light of wavelength 500nm comes down from directly above, determine the number of bright fringes that will be seen across the wedge? 5/10/2011 ELEC 308 Engineering Optics Newton’s rings ---A special case of equal thickness interference 5/10/2011 ELEC 308 Engineering Optics Newton’s rings 5/10/2011 ELEC 308 Engineering Optics Newton’s rings Newton’s rings effect on scanned picture 5/10/2011 • Newton’s ring is often used to measure the curvature of radius of a convex lens. ELEC 308 Engineering Optics New example 8 Newton ring 9.31 Figure illustrates a setup used for testing lenses. Show that d x 2 ( R2 R1 ) / 2 R1 R2 When d1 and d2 are negligible in comparison with 2R1 and 2R2, Respectively. (Recall the theorem from plane geometry that relates the Products of the segments of intersecting chords.) Porve that the radius of The mth dark fringe is then xm [ R1 R2 m f /( R2 R1 )]1/ 2 Geometry condition : R 12 x 2 (R1 d1 ) 2 R 12 x 2 (R 12 2R 1d1 d12 ) x 2 2R 1d1 d12 d1 x 2 /2R1 The same for R 2 R 22 x 2 (R 2 d 2 ) 2 R 22 x 2 (R 22 2R 2 d 2 d 22 ) x 2 2R 2 d 2 d 22 d 2 x 2 /2R 2 5/10/2011 ELEC 308 Engineering Optics New example 8 Newton ring 9.31 Figure illustrates a setup used for testing lenses. Show that d x 2 ( R2 R1 ) / 2 R1 R2 When d1 and d2 are negligible in comparison with 2R1 and 2R2, Respectively. (Recall the theorem from plane geometry that relates the Products of the segments of intersecting chords.) Porve that the radius of The mth dark fringe is then xm [ R1 R2 m f /( R2 R1 )]1/ 2 Differenceof optical path : d d1 d 2 ( x 2 /2)(1 / R 1 1 / R 2 ) x 2 (R 2 - R 1 )/2R1R 2 For the dark fringe : 2d mλ m/2 x 2m (R 2 - R 1 )/2R1R 2 x m [R 1R 2 mλ f /(R 2 R 1 )]1/ 2 5/10/2011 ELEC 308 Engineering Optics Problem 4: Newton’s ring 5/10/2011 ELEC 308 Engineering Optics New example 8 Thin film interference 9.32 Newton rings are observed on a film with quasimonochromatic light that has a wavelength of 500 nm. If the 20th bright ring has a radius of 1 cm, what is the radius of curvature of the lens forming one part of the interfering system? Newton ring approximat ion : r 2 2 yR Interference condition : y (m 1/2)0 / 2n 5/10/2011 ELEC 308 Engineering Optics The Michelson Interferometer • Precise distance measurements can be made with the Michelson interferometer by moving the mirror and counting the interference fringes which move by a reference point. • The distance d associated with m fringes is 5/10/2011 ELEC 308 Engineering Optics The Michelson Interferometer Haidinger fringes 5/10/2011 Fizeau fringes ELEC 308 Engineering Optics The Michelson Interferometer White light fringes • Applications – The core of autocorrelator/crosscorrelator, Fourier transform spectroscopy. – Precise measurement of the wavelength. – Tested the dependence of speed of light on the motion of the Earth An interesting Demo: http://www.youtube.com/watch?v=V yePASErr5Q&feature=related 5/10/2011 ELEC 308 Engineering Optics Mach-Zender Interferometer • A device used to determine the phase shift caused by a small sample which is placed in the path of one of two collimated beams. • In contrast to the Michelson interferometer, there are two output ports. 5/10/2011 ELEC 308 Engineering Optics Fizeau Interferometer Fizeau interferometers are commonly used for measuring the shape of an optical surface. 5/10/2011 ELEC 308 Engineering Optics Fizeau interferometer Surface test: make use of the principle of thin air wedge. 5/10/2011 ELEC 308 Engineering Optics New example 9 Michelson Interferometer 9.35 A Michelson Interferometer is illuminated with monochromatic light. One of its mirrors is then moved 2.53*10-5m, and it is observed that 92 fringe-pairs, bright and dark, pass by in the process. Determine the wavelength of the incident beam. 9.36 One of the mirrors of a Michelson Interferometer is moved, and 1000 finge-pairs shift past the hairline in a viewing telescope during the process. If the device is illuminated with 500-nm light, how far was the mirror moved? For 9.35, a motion of /2 causes a single fringe pair to shift past, hence 92 / 2 2.53*10 5 and 550nm For 9.36 d N(0 /2) (1000)(5.00 *10 - 7m)/2 2.50 *10 - 4m 5/10/2011 ELEC 308 Engineering Optics 5/10/2011 ELEC 308 Engineering Optics