Homework Set #1 Due: 1-15-14 This homework set does not cover lecture material (class just got started after all) but, instead, reviews basic optics. If some of this material is new (not uncommon for graduate students these days) and you need help, see me. The techniques required can be learned quickly and I need to be able to depend on your knowing these concepts. With the exception of (2), these problems also have a direct connection to ultrashort pulse laser systems. 1) An equilateral prism is made of glass with an index of refraction of 1.7. (a) Find the input angle such that the light travels parallel to the prism base inside the prism. (b) What is the output angle of the light when it exits the prism back into air? A prism configured this way is said to be at “minimum deviation” and has properties that are useful for some applications, including inside laser cavities. Review: When light travels from one medium (#1) to another (#2) across a sharp interface, it refracts according to Snell’s Law: n1 sin1 = n2 sin2. The angles are measured with respect to the normal to the interface. The index of refraction of air is about 1.0003 and can usually be taken to be unity. 2) An arrow is followed by a diverging lens and a converging lens. The two lenses together form an image. (a) Where is the final image? (Specify distance from the converging lens and whether it is to the left or right of it.) (b) Is the final image real or virtual? Which way is the arrow’s image pointing: up or down? (c) What is the magnification of the system? lens #1 f1 = -100 mm 50 mm lens #2 f2 = 100 mm 150 mm Review: If f is the focal length, o the distance to the object and i the image distance for a given lens: 1/f = 1/o + 1/I, with all distances measured from the center of the lens to good approximation for thin lenses. Sign convention: the object distance is positive if the object lies before the lens and the image distance is positive if it comes after. A positive image distance means the image is “real” – you can see it on a card or screen. A negative image distance means the image is “virtual” – the output light appears to be coming from the image, but a screen placed at the image will not show one. The magnification m = -i/o. A negative magnification means the image is inverted. For multi-lens problems you handle the first lens first, and then use the image of the first lens as the object for the second lens. Note this means the second lens’s object can come after the second lens itself. 3) The beam expander. Two lenses are separated by a distance equal to the sum of their focal lengths. A collimated beam (one consisting of, to good approximation, parallel rays) enters from the left with beam diameter Din. The figure shows the situation for the cases where both lenses are positive and f2 > f1. (a) Show by ray tracing that the beam that exits the system is still collimated, but has a larger diameter. (b) Show that Dout = (f2/f1) Din. This is a commonly used configuration of lenses, sometimes called a telescope. If you work in a laser lab, you’ll likely have to use one. These results hold even if one of the lenses is negative, so long as f1 + f2 > 0. Din Dout f1 f2 f1 + f2 4) Gratin ng tutorial. A grating is an a optical com mponent that modulates ligght spatiaally so that th he outgoing diffracted d ligh ht comes out at an angle thhat depen nds on its wavelength. w Gratings can n be designeed so that tthe diffraacted light is transmitted t th hrough the grrating or, as sshown in Fig.. 1, refleccted. Reflectio on gratings are a the most common. c Graatings are maade by im mposing a peeriodic, spatial modulatio on of some pproperty of tthe substrrate. The mo odulated quan ntity can be the absorptioon, reflectiviity, transm mission, thick kness, or indeex of refractio on. Althoughh somewhat leess comm mon, a substraate with spheerical curvatu ure is sometim mes used so tthe gratin ng can form m images. Th he first grating that stuudents usuallly encou unter in class is an array of slits on a sccreen. This is a transmissioon gratin ng and the mo odulated quan ntity is the trransmission (eeither 100% or 0%), as a shown in Fig. F 2. Suppo ose you havee monochrom matic light in ncident on a flat substratte, reflecction grating. There will, in n general, be many diffractted beams each at a different angle, a called d orders, coming c from m the gratinng nly one is sho own in the fig gures). The am mount of pow wer simultaneously (on y the choice of o modulationn. A sinusoiddal in eacch order is deetermined by modu ulation tends to t put most of the power into the first oorder diffracted beam (m=1), wherreas the mod dulation show wn in Fig. 3 ccan be used to place more power is a high ordeer mode. ng monochrom matic light, tthe Referrring back to Fig. 1, and still assumin input and output raays are relateed via the gratting equationn, one variantt of which h is: d(sini + sind) = m . Here, d is the “wavvelength” of tthe spatiaal modulation n, called the groove g spacin ng, i and d are the incideent and diffracted anglles, is the liight waveleng gth and m is aan integer. Noote that th he angle of th he incident an nd diffracted rays r is measurred with respeect to thee normal, as is usually th he case in op ptics. Differeent values off m selectt the differentt possible outp put angles or diffraction oorders. Note thhat for th he m = 0 ord der, the gratting acts lik ke a mirror independent of wavellength. d i d Fig. 1 Incoming wave. A diffracted orderr. Fig. 2. Grating formed ussing multiple slits.. (Picturre: http://blog.cenccophysics.com/ 2009/007/diffraction-gratiing/) d Fig. 3. G Gratings with a groove shape like this are said to be blazzed. (a) Explain E the sig gn convention n for the anglles in the grat ating equationn. According to this conveention, what w is the sign n of i and d for the case shown in Figg. 1? [Hint: coonsider the m= =0 case.] (b) A grating used d in “Littrow” configuration operates sso that a speccified diffracttion order is retroreeflected back onto the inciident beam. Short S pulse lasser systems ffrequently em mploy gratingss at or neear Littrow, as do some narrow n line tunable t laserss, including ssome diode llasers. What is the in ncident angle for a Littrow w grating with h 500 groovess/mm operatinng in first ordder (m=1) witth 800 nm m light? (c) For short pulsee work, it is often o best to not permit orrders higher than m=1. Ennergy appeariing in i usually wassted. Keeping g the angle annd wavelengthh in (b), find a value for d tthat is hiigher orders is ju ust sufficiently y small to elim minate orderss with m>1.