Voorblad bij tentamen Optica 3BOX1

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1 Voorblad bij tentamen Optica 3BOX1 Tentamen/vakcode: Optica/3BOX1 Aantal deelnemers: 66 Datum: 11 April 17 Begintijd: 18: Eindtijd: 1: schriftelijk / notebook (*) open vragen / meerkeuzevragen (*) Aantal pagina s: 13 (*) doorhalen wat niet van toepassing is Instructies voor studenten en surveillanten Toegestane hulpmiddelen (mee te nemen door student): Notebook Rekenmachine (grafisch ja/nee) Dictaat/boek 1 A4-tje met aantekeningen Woordenboek(en). Zo ja, welke: Desgewenst simpel Engels woordenboek, zonder aantekeningen geen technisch woordenboek Opmerkingen (bijvoorbeeld over inzage): Inzage: voor individuelle afspraken (vanaf ca. Mai 17): j.t.kohlhepp@tue.nl Let op: toiletbezoek is alleen onder begeleiding toegestaan binnen 15 minuten na aanvang en 15 minuten voor het einde mag de tentamenruimte niet worden verlaten er dient altijd tentamenwerk (volledig ingevuld tentamenpapier: naam, studentnummer e.d.) te worden ingeleverd tijdens het tentamen dienen de huisregels in acht te worden genomen aanwijzingen van examinatoren en surveillanten dienen opgevolgd te worden etui ligt niet op tafel onderling worden geen hulpmiddelen geleend/uitgewisseld Tijdens het maken van schriftelijke tentamens wordt onder (poging tot) fraude in ieder geval verstaan: gebruik van andermans ID-bewijs/campuskaart mobiele telefoon of enige andere media dragende devices liggen op tafel of zijn opgeborgen in de kleding (poging tot) gebruik van ongeoorloofde bronnen en hulpmiddelen, zoals internet, mobiele telefoon e.d. het gebruik van een clicker die niet je eigen clicker is ander papier voor handen hebben dan door de TU/e is verstrekt toiletbezoek (of naar buiten lopen) zonder toestemming of begeleiding 1 Bijlage 3, behorende bij artikel 5 Regeling voor centrale tentamenafname

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3 EINDTOETS 3BOX1 11 April uur Opgaven 1 6 van dit tentamen dienen geheel schriftelijk ingeleverd te worden op het uitgereikte TU/e papier. Denk er aan de bijlagen in te leveren, zie pagina 11 en 13. Vragen zijn in Engelse taal gesteld, maar mogen desgewenst in het Nederlands of in het Engels worden beantwoord. Het gebruik van een eenvoudige (niet grafische) rekenmachine en een eenvoudig Engels woordenboek is toegestaan. Het formuleblad is bijgevoegd ná de opgaven, op pagina 9 en 1. Bij ieder onderdeel wordt aangegeven wat de maximale score is. Het eindcijfer is het totale aantal behaalde punten gedeeld door 1. De (beknopte) antwoorden worden na afloop van het tentamen op CANVAS geplaatst. Veel succes! 1

4 Exercise 1. Short conceptual questions (1 points) Please answer the following questions as compact as possible, thus, in a few and short sentences. Always use arguments and / or equations to support your answers. For instance, a simple yes or no will never be sufficient! 1.1. Until the 19 th century two conflicting theories for the nature of light existed in parallel. The corpuscular theory, which was strongly supported by Newton, and Huygens wave theory. Name two (experimental / theoretical) discoveries in the 19 th century which finally confirmed the wave character of light. (3 points) 1.. An observer on earth experiences light form the sun that is scattered in the atmosphere under certain conditions as linear polarized. Give a short explanation for this observation by using electric dipole radiation arguments. Don t forget here to mention the polarization direction and under which conditions the scattered light is linearly polarized! (3 points) 1.3. When a thin film of a liquid with index of refraction n l spreads out on a puddle of water with index of refraction n w, the thinnest part of the film looks dark in the resulting interference pattern. What does this tell you about the ratio of the refractive nl indices? Explain! (3 points) n w 1.4. Right-circularly polarized light passes through a half-wave retarder plate. What is the polarization state of the light emerging from the half-wave plate? Explain! (3 points) Exercise. Electromagnetic waves (15 points).1. Determine for the travelling plane wave described by (c is the speed of light) V 1 1 Eyt (, ) m cos 4 m y 3 m ct,

5 a) the propagation speed and propagation direction; b) the index of refraction of the medium in which the wave is travelling; what medium could this be? c) The plane wave impinges now on an opaque screen with a small hole of diameter (8 points) d 1mm. What is the irradiation power (in Watt) which leaks through the hole?.. Describe completely the state of polarization of the following EM wave. Explain why! (7 points) E sin tkz3 E E cos tkz Exercise 3. Geometric Optics / Water droplet hanging on plain mirror ( points) An observer looks along the z-direction through a water droplet which hangs on a plain mirror (see figure). The vertex of the spherical shaped water surface is located at position z and has a radius of curvature R. The distance of the plain mirror to the vertex of the water surface is 3R. The index of refraction of water and of the surrounding air are given by n 4 W and n 1 3 air, respectively. 3

6 3.1. For an optical analysis of the water droplet / plain mirror system we must know the positions of the two focus points of the spherical air / water surface. Thus, calculate a) the focus point position for the propagation of light impinging from the airside parallel to the z-axis on the air / water surface (right focus point): z f, air/ water b) and the focus point position for the propagation of light impinging from the waterside parallel to the z-axis on the water / air surface (left focus point): z f water/ air. (4 points) The observer looks now at a tiny object V which is positioned in the water droplet at z 1 V R (see figure). He actually observes two images of the object V: (1) A direct image V (1) via a refraction at the water/air interface of light coming directly from the object V, and () a secondary image V () via first a reflection of light (coming from the object) at the plain mirror, with then a subsequent refraction at the water/air interface. 3.. Construct the image V (1) generated by only the refraction of the direct light at the water/air surface (thus, not via the reflection at the mirror) by utilizing a ray diagram (please use the attachment (bijlage) on page 11). Construct your image by using two characteristic rays (Hint: see problem 3.1.) and shortly describe the main features of your construction in words. (5 points) 3.3. Calculate the position of the image V (1) on the z-axis, its size (magnification), its orientation, and determine if the observed image is real or virtual. (6 points) We shift the tiny object V now to a new position z V (see figure). Again we can observe two images of the object V Construct now the image V () which is generated via the reflection at the mirror by utilizing a ray diagram R 4

7 (please use the attachment (bijlage) on page 13). Construct your image again by using two characteristic rays and describe the main features of your construction in words. Where do you expect to find your image? (A calculation is not necessary!) (5 points) Exercise 4. Polarization ( points) The linear polarized light beam emerging from a laser is described by an electric field amplitude E, or alternatively by an irradiation I, and propagates along the positive z- direction. The laser light has a wavelength and the electric field oscillates parallel to the x- axis (horizontally polarized). The beam passes sequentially through two polarizers: (1) Polarizer P1 which has a transmission axis (TA) along an angle relative to the x-axis, () Polarizer P which has a transmission axis (TA) along the y-axis (vertical). Finally the laser beam hits a detector which measures the irradiance of the impinging light (see figure below for the experimental setup) a) Determine the Jones vector J 1 of the electric field after the light beam has passed polarizer P1, expressed in terms of the angle. Here, you do not need to know the 5

8 Jones-matrix of the polarizer. You can determine J 1 by geometric/trigonometric considerations. b) Determine the Jones vector J of the electric field after the light beam has also passed polarizer P, again expressed in terms of the angle. c) Determine the light irradiation I which is measured with the detector. Calculate and/or argue for which angles the irradiance has maxima and minima. What is the size of the irradiance for these angles? (1 points) Now we position a retarder-plate (wave-plate) between the two polarizers. The wave-plate is made from a material with an index of refraction n F for light polarized along the fast axis (FA) and an index of refraction n S for light polarized along the slow axis (SA), thus, n S n. F The slow axis of the retarder-plate is aligned along the x-axis (horizontal). 4.. Calculate the minimal thickness for which the wave-plate works as a three-quarterwave-plate ( 3 -plate) for a given wavelength. (4 points) Calculate the Jones-matrix of the 3 -plate. (3 points) 4 1 Hint: Jones-matrix -plate with SA horizontal: M QWP 4 i 1 Jones-matrix -plate with SA horizontal: M HWP Calculate the light irradiance I which is now measured with the detector for the situation that the 3 -plate is positioned between the two polarizers as described above. 4 (5 points) 6

9 Exercise 5. Interference (1 points) White light impinges at an angle of incidence of 45 on a thick glass slab ( n 1.5 ) which is covered with a thin polymer layer of thickness l.8 m (index of refraction l n 1.35 ). Under these conditions we see maxima in the reflection for two vacuumwavelengths, namely 46. nm and nm. Now we move the glass slab a bit until the light beam hits a broad scratch in the polymer layer. We measure the reflection of the light under the same conditions as described above and we find now reflection maxima at vacuumwavelengths of 45. nm and 6. nm, respectively. Calculate the depth of the scratch in the polymer layer! You can ignore in your calculations any extra phase shifts reflections at the interfaces (they are indeed zero). (1 points) i glass ref due to the Exercise 6. Diffraction grating (1 points) We perform an underwater diffraction experiment using a diffraction grating comprised of 5 slits/cm (index of refraction nw 1.33 ). The grating is homogenously illuminated with parallel light form a laser beam of vacuum wavelength 63. nm under perpendicular incidence. The far-field (Fraunhofer) diffraction pattern is observed at a distance i l 1. m to the grating on a screen. In the diffraction pattern the principal maxima corresponding to the 5 th, 1 th, 15 th, orders are missing Calculate the separation d of the slits. (3 points) 6.. Calculate the total number of intensity maxima you would expect to observe on a screen of width s 3.5 m. (6 points) 6.3. Calculate the height of the irradiance maximum of the 4 th order diffraction peak and compare this with the irradiance you observe at the position of the th order diffraction peak maximum I (). (6 points) 7

10 6.4. Suppose, we illuminate under perpendicular incidence the grating simultaneously with two overlapping homogenous laser beams both having a diameter of 1mm, but having slightly different wavelengths 63. nm and nm, respectively. Would it be possible to resolve the wavelengths of the two lasers by using the third order diffraction peaks of the grating? Calculate why yes or no! (6 points) i 8

11 Wave equation: Maxwell equations: Poynting vector: Reflection: i r OPTICS equation chart ( x, t) 1 ( x, t) x v t E B B E t E B E t 1 S EB Phase velocity: vp Group velocity: k E E t B in vacuum: B t 1 c Irradiance: c I S E T v g d dk dk d 8 c 3.1 m/s F/m H/m Refraction (Snell): n sin n sin i i t t 1 Fresnel: r E n cos r E n cos n n cos cos i i t t i i i t t n cos cos E n cos r r E n cos n i t t i i i t t i t nicos i cos E t E n cos n i i i t t nicos i cos E t t E n cos n i i t t i Refraction at a spherical surface: n n n n s s R m l l m i Thin lens: lensmakers formula: Thin-lens formula: Mirror: 1 1 nl n m s si nm R1 R s si f 1 1 s s R i Numerical aperture: NA nsin Malus s law: f max f-number: f /# D Brewster s angle: tan I I cos Interference (vacuum): nt B n I c E ce ce ce E cos 1 1 T k r k r R i Young s double-slit: dy I Icos Icos L 9

12 4 n l l l Thin film interference: cos ref 1 Fabry-Perot: Transmission: I I F t 1 sin F 4 R(1 R) 4Y Diffraction: Chromatic resolving power: sin Single slit: I I 1 e my Free spectral range: m fsr 1, kb sin N slits: sin sin N 1 I I, N sin kd sin Fraunhofer condition: h r Airy disk angular diameter:.44 f D d sin sin m Grating equation: Chromatic resolving power: e Nm m 1 Free spectral range: m fsr min i i Math: e cos isin e cos sin sin cos 1 e i i e i e i sin 45 cos 45 ; sin 3 cos6 ; sin 6 cos sin sin sin ( )cos ( ) 1 1 sin sin cos ( )sin ( ) 1 1 cos cos cos ( )cos ( ) 1 1 cos cos sin ( )sin ( ) 1 1 sin( ) sincos cossin sin( ) sincos cossin cos( ) coscos sinsin cos( ) coscos sinsin tan tan tan( ) 1 tan tan cosinusregel: a b c bccos a b c sinusregel: sin sin sin i 1

13 Attachment: Exercise 3.. (turn drawing by 9 ; landscape!) Naam: Identiteitsnummer: 11

14 1 EMPTY

15 Attachment: Exercise 3.4. (turn drawing by 9 ; landscape!) Naam: Identiteitsnummer: 13