Voorblad bij tentamen Optica 3BOX1

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1 333 Voorblad bij tentamen Optica 3BOX Tentamen/vakcode: Optica/3BOX Aantal deelnemers: 9 Datum: Februari 7 Begintijd: 9: Eindtijd: : schriftelijk / notebook (*) open vragen / meerkeuzevragen (*) Aantal pagina s: / een bijlage (*) 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 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 (na ca. Februari): j.t.kohlhepp@tue.nl Let op: toiletbezoek is alleen onder begeleiding toegestaan binnen 5 minuten na aanvang en 5 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 Bijlage 3, behorende bij artikel 5 Regeling voor centrale tentamenafname

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3 EINDTOETS 3BOX Februari uur Opgaven 6 van dit tentamen dienen geheel schriftelijk ingeleverd te worden op het uitgereikte TU/e papier. Denk er aan de bijlage in te leveren, zie pagina. 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 8 en 9. Bij ieder onderdeel wordt aangegeven wat de maximale score is. Het eindcijfer is het totale aantal behaalde punten gedeeld door. De (beknopte) antwoorden worden na afloop van het tentamen op OASE geplaatst. Veel succes!

4 Exercise. Short conceptual questions ( 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!.. Light travels from medium into medium, where the index of refraction, n, of medium is greater than the index of refraction, n, of medium. In order for the light to be totally internally reflected at the boundary the angle of incidence must satisfy a certain equation. Derive this equation! (3 points).. Stars as viewed through a refracting telescope often appear to be surrounded by blurry, rainbow-colored circular fringes. Which of the colors blue or red is characterized by a larger fringe radius? Explain why! (3 points).3. Why are Polaroid sunglasses better than ordinary sunglasses? What is the direction of the transmission (polarizing) axis of the Polaroid sheets used for these sunglasses to be efficient (horizontal or vertical)? Give a short explanation! (3 points).4. You design a quarter-wave plate by using calcite which has nx.6584 (slow axis SA along horizontal x-axis) and ny.4864 (fast axis FA along vertical y-axis). Suppose you want to use this plate to produce right circularly polarized light from linearly polarized light which is travelling along the +z-direction. Under which angle with respect to the quarter-wave plate axes must the linear polarization be oriented to achieve this goal? Maybe a drawing helps! (3 points).5. Until the 9 th century two conflicting theories for the nature of light existed in parallel. () The corpuscular theory strongly supported by Newton, and () Huygens wave theory. Young s and Fresnel s experiments, and Maxwell s theory finally unambiguously demonstrated the wave character of light. However, at the end of the 9 th and at the beginning of the th century several unexpected experimental observations were made, which where only explainable by a particle character of light. Name, if possible, two of these observations! ( bonus-points)

5 Exercise. Electromagnetic waves (3 points).. A plane electromagnetic wave travelling in vacuum is described by (,,, ) ( ) i ( kx t ) Bxyzt Be e. ˆ z Find an equation for the electric field of this wave. (6 points).. Describe completely the state of polarization of the following EM wave. Explain why! (7 points) sin tkz3 E E cos tkz Exercise 3. Geometric Optics / Concave mirror under water ( points) A concave mirror with radius R is placed upright in a cubic aquarium filled with water. The center C of the spherical concave mirror coincides with the left wall of the aquarium and is positioned at z (see figure). The index of refraction of the water and of the surrounding air are given by n 4 W and n 3 air, respectively. We assume that the glass the aquarium is made of is so thin that any refraction effects due to the glass can be safely neglected. Furthermore, we will work in the paraxial approximation limit. 3.. Calculate the position of the focus point, z f, of the mirror for parallel light impinging perpendicular on the aquarium from the left side. (3 points) 3

6 The observer looks now at an object V which is positioned at z R 4 V (again see figure above). He actually observes two images of the object V: () A direct image V () via a refraction at the water/air interface of light coming directly from the object, and () a secondary image V () formed by light coming from the object which is first reflected at the mirror and then refracted at the water/air interface. 3.. We will exclusively analyse the second image V () generated via the mirror. Construct the image V () by utilizing a ray diagram (please use the attachment (bijlage) on page ). Draw at least two characteristic rays to construct your image as accurately as possible (more characteristic rays certainly can improve the consistency of your construction!). Also describe the main features of your construction in words. Both imaging steps (reflection at mirror, refraction at interface) must be clearly visible in the drawing. Of course in your drawing you will not really satisfy the paraxial approximation and it will not be completely correct but we will ignore these distortions. (8 points) 3.3. Before we actually calculate the image V (), we first analyze the image which would be formed if there would be no refraction at the water/air interface, just the reflection at the mirror. We name this virtual image V virtual(). Calculate its position z(v virtual()) on the z-axis, its size (magnification), and its orientation (upright or inverted). (4 points) 3.4. With the help of the results 3.3., and now also taking into account the refraction process at the water/air interface, determine the characteristics of the image V (), e.g. calculate its position z V'(), its size (magnification), its orientation, and determine if the observed image is real or virtual. (6 points) Hint: Compare your results with the ray diagram of 3.. (in the limits used they should be more or less consistent!) Exercise 4. Polarization ( points) A linear polarized light beam is described by an electric field amplitude E, or alternatively by an irradiation I, and propagates in the positive z-direction. The beam passes successively three optical elements: () a half-wave -plate with its slow axis (SA) aligned along an axis 4

7 having an angle relative to the x-axis (horizontal), () a quarter-wave -plate with its 4 slow axis (SA) along the x-axis (horizontal), and (3) a linear polarizer with its transmission axis (TA) along the y-axis (vertical). 4.. Show, that the Jones matrix M cos sin sin cos describes the first optical element (), namely the horizontal. -plate with its SA rotated by an angle relative to the To do so, apply M first to light which is linear polarized along the x-axis (horizontal), thus, calculate J MJ x. Then, apply M to light which is linear polarized along the y- axis (vertical), thus, calculate J MJ vertical linear polarized light ( J x and y. Here, the Jones vectors for horizontal and J y, respectively) you must know/derive yourself. Then compare your results with what you would expect from the working principle of a half-wave plate. Hint: Make a drawing of the electrical field vectors (Jones-vectors) before and after the half-wave plate for the two light polarizations; here explicitly the sizes of the x- and y- components of the Jones-vectors should be shown and give you information. (7 points) 4.. Now we use a light beam with a linear polarization along the x-direction (horizontal) and we align the -plate at 3. Calculate the Jones-vector of the light after passing the first () and the second () optical element. The Jones-matrix of the second optical element, the -plate, is given by 4 M QWP i. What is the polarization state of the light after those two elements (i.e. linear with certain orientation, left- or right circular, or left- or right elliptical)? (7 points) 5

8 4.3. Finally, we use again a light beam with a linear polarization along the x-direction (horizontal), but the angle is now arbitrary. The Jones-matrix of the third element is given by M LP, but actually you could know this or be able to derive yourself. Calculate the Jones-vector and the irradiation after the beam passed all three optical elements. Calculate and/or argue for which angles min the irradiation has minima, and for which angles max the irradiation has maxima. What is the irradiation after the three optical elements for these angles? (7 points) Exercise 5. Interference ( points) A ray of white light impinges at an angle of incidence of 45 i on a thick glass slab ( nglass.5 ) which is covered with a thin polymer layer of homogenous thickness l.8 m (index of refraction nl.35 ). 5.. Determine a possible extra phase shift ref accumulated due to the reflection of the two light rays at the two polymer interfaces. (4 points) 5.. We want to calculate now for how many and which vacuum wavelengths (colors) between 4 nm and 7 nm we see a maximum in the reflected light irradiation. (8 points) Exercise 6. Spectroscopy ( points) We perform in vacuum an experiment using a diffraction grating with N slits of equal width b and separation d. We illuminate the whole grating under perpendicular incidence with a parallel light beam emerging from a monochromatic laser having a vacuum wavelength i 6

9 of 5 nm. The far-field (Fraunhofer) diffraction pattern is observed on a screen via a positive lens with focus length f 5 cm. The diffracted light is measured on the screen with a detector (see figure). Obviously, certain principal maxima are missing at certain positions in the diffraction pattern as indicated in the figure! 6.. Determine the number of slits N of the diffraction grating. (3 points) 6.. Calculate the width b and the separation d of the slits. (9 points) 6.3. We use this grating now for a spectroscopic experiment, again in vacuum. If we utilize the 7 th -order diffraction peak for our experiment, we know that the grating has a certain chromatic resolving power e grating. Imagine now, we would perform the same experiment with a Fabry-Perot interferometer built from two bad mirrors with a distance t m. What is the reflectance (irradiation reflectivity) R of the mirrors, if the Fabry-Perot has exactly the same chromatic resolving power efp e grating as the grating? (9 points) 6.4. With the described grating we actually could unambiguously resolve the smallest wave length difference which is determined by the in 6.3. calculated chromatic resolving power. However, with our Fabry-Perot interferometer this is not possible; with this FP we run into serious problems! Calculate / explain why! (5 bonus-points) 7

10 Wave equation: Maxwell equations: Poynting vector: Reflection: i r OPTICS equation chart ( x, t) ( x, t) x v t E B B E t E B E t S EB Phase velocity: vp Irradiance: in vacuum: Group velocity: k I S E T E E t B B t c c Refraction (Snell): n sin n sin v i i t t g d dk dk d 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 E cos t E n cos n ni i cos i i i t t nicosi 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: nl n m s si nm R R s si f 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 T k r k r R i Young s double-slit: dy I Icos Icos L 8

11 4 n l l l Thin film interference: cos ref Fabry-Perot: Transmission: I I F t sin F 4 R( R) 4Y Diffraction: Chromatic resolving power: sin Single slit: I I e my Free spectral range: m fsr, kb sin N slits: sin sin N 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 Free spectral range: m fsr min i Math: e i cos isin cos sin cos e i e i sin e i e i sin 45 cos 45 ; sin 3 cos6 ; sin 6 cos3 3 sin sin sin ( )cos ( ) sin sin cos ( )sin ( ) cos cos cos ( )cos ( ) cos cos sin ( )sin ( ) sin( ) sincos cossin sin( ) sincos cossin cos( ) coscos sinsin cos( ) coscos sinsin tan tan tan( ) tan tan cosinusregel: a b c bccos a b c sinusregel: sin sin sin i 9

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13 Attachment: Exercise 3.3. (turn drawing by 9 ; landscape!) Naam: Identiteitsnummer: