Midterm Examination 2 Introduction to Optics & Photonics EECS 334 Instructor: Professor S.C. Rand Date: November 24, 2009 Time: 10:30-12:30 p.m. Duration: 2.0 hours PLEASE read over the entire examination before you start. DO ALL QUESTIONS and show all your work in submitted material to be eligible for partial credit. No textbook or lecture materials or notes of any kind may be consulted. Additionally, no homework solutions or electronic aids of any kind, with the exception of simple calculators, are permitted. Having a ruler available for measurements and making clear diagrams is recommended. Honor Pledge: I have neither given nor received aid on this exam. __ Name (Print) ____________________________________ Student ID ___________________ Signature Total 1 Useful Formulae: Imaging, refraction & propagation: ni sin i nt sin t ; ni sin c nt 1 1 1 s s' f 1 1 n2 n1 1 1 ' s1 s 2 n1 R1 R2 h s' Magnification: m i i ho o s 2 ( z, t ) 1 2 ( z, t ) 2 z 2 v t 2 WE 1 2 E 2 ; WM 1 2 H 2 ; WTOT 2WE S Irradiance Power / Area E H ; S EB / EH n v v v g v p (1 ) ; 0 1 ; 0 1 ; n c c ; t 1 Coherence: R I max I min I max I min Coherent waves with random phase fluctuations emitted from two points S1, S2 of an extended source and observed at a single point P: I TOT I 1 I 2 0 c Re E1 (t ) E 2* (t ) I 1 I 2 2 I 1 I 2 Re( ) l c ; Visibility: V ( ) exp( i )( 0 ) / 0 and 0 is the source coherence time. l S1 s S2 P1 Screen r P2 2 Incoherent waves emitted from two points S1, S2 of an extended source and observed effectively at TWO points P1 and P2 by forming their fringe pattern: I TOT I 1 I 2 0 c Re E1,tot (t ) E 2*,tot (t ) I 1 I 2 2 I 1 I 2 Re( ) , where ( ) sin( sl / r ) /( sl / r ) lt r / s / 1 / 2 Interferometry: m 2nd cos Interference in thin films: t / 4n f and n f n0 ns T02 IT , where T0 (tt ' t * t '*)1 / 2 , I 0 1 R02 2 R0 cos 2 R0 (rr ' r * r '*)1 / 2 and 2n f t cos SFR R d c SFR ; Q ; mF ; F 2nd FW HM 1 R d FW HM Multiple-beam interference: T ( ) sin N N slits: I P I 0 sin Lasers: I ( z ) I (0) exp( z ) ( N 2 N1 ) 2 2 sin where 12 ka sin and 12 kb sin Bh n 2 g ( ) ( N 2 N1 ) c 8n 2 Diffraction: Diffraction by a circular aperture of diameter D: 1 / 2 1.22 / D Grating Eqn: m a(sin i sin m ) General: Stefan-Boltzman Law: I T 4 Wien displacement Law: max T constant Miscellaneous Mathematical Relations: sin 31! 3 51! 5 ... cos 1 21! 2 41! 4 ... 3 1. (10 marks total) Light from a star of unknown diameter located at a distance 5 1011 km is observed through a filter at a wavelength of =400 nm. Starlight is collected by two small, stable mirrors of variable spacing l in a plane perpendicular to the line of sight and redirected to form a fringe pattern on a screen. When the mirrors are spaced by 50 cm, fringes with a visibility of V=0.9 are observed. Determine the diameter of the star. (Note: If you don't have a calculator, you may want to use the first two terms of the small angle expansions for trigonometric functions to make the estimate. See formulas) (10 marks) 4 2. (10 marks total) (a) A 100 micron thick piece of glass (n=1.5) is inserted into an empty Fabry-Perot cavity of length d=1 cm. At a wavelength of =500 nm, what is the change in mode number m due to the insertion? Draw a picture and write an equation for m to get started. (5 marks) (b) In a second Fabry-Perot, the refractive index n=1.5 of material between the mirrors slowly drifts by n 10 2 n . If the separation of the mirrors is 10 cm and the wavelength of incident light is again =500 nm, how many transmission maxima are observed during this change? (5 marks) 5 3. (10 marks) The oscillating mode spectrum of a gas laser is given in the figure below. (1014s-1) 4.5 5 5.5 (a) Is this laser an ultraviolet, visible, near infrared or far infrared laser? Justify. (1mark) (b) Presuming it is a simple, linear 2-mirror laser, what is its length? (4 marks) 6 (c) Assuming that the envelope of the emission spectrum results entirely from Dopplerbroadening of a narrowband emission, estimate the average velocity of atoms in the gas. (3 marks) (d) To achieve single-mode operation with an etalon inserted into the cavity, what should the spectral free range and thickness of the etalon be? (2 marks) 7 4. Short problems (worth 2-3 marks each - total of 20 marks) 4.1. (a) In an optical waveguide, such as a fiber, should the central core have a refractive index higher, lower or equal to that of the cladding? (1 mark) 4.1. (b) In a desert mirage, state whether the index of the air nearest ground level is higher, lower, or equal to that of the cooler air above and on this basis explain the phenomenon. (An imaging diagram is adequate for the second part). (2 marks) 4.2. By using the uncertainty relation xk 1 (or by considering properties of the Fourier transform pair x and k 2 / ) justify whether precision optical metrology of distance intervals should make use of a large or a small wavelength range ? (3 marks) 4.3. The light intensity inside a Fabry-Perot (FP) cavity is I c . If light of intensity I 0 impinges on a cavity with identical mirrors of 99% reflectivity and the FP is tuned to a transmission peak (T=100%) so that the output is also I 0 , what is I c ? Justify. (2 marks) 4.4. Sketch the orientations of two prisms whose relative positioning can be used to compensate normal dispersion inside an ultrashort pulse laser. (2 marks) 8 4.5. In order to make the most efficient laser with continuous output, is it preferable to choose a gain medium with 4 levels or to eliminate unnecessary levels by choosing a 2level system? Why? (3 marks) 4.6. Name three different laser gain media. (3 marks) 4.7. The angular spread of light due to diffraction is unavoidable and is completely determined by the diameter D of the limiting aperture according to 1 / 2 / D . Use the law of reversibility of linear optics to decide whether the diameter of a focal spot is proportional to the diameter of the light beam impinging on the focusing lens. (2 marks) 4.8. The spectrum of a laser that initially is a delta function versus frequency becomes a 2 sin c function if it passes through an electro-optic shutter that opens and closes for a brief time. True or false? Why? (2 marks) 9