CLOSED BOOK. Equation Sheet is provided. YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT. ALL NUMERICAL ANSERS MUST HAVE UNITS INDICATED. (Except dimensionless units like index of refraction). Problem 1: A prism of a transparent material has an apex angle φ o of 40 degrees and an index of refraction of 1.5. As illustrated in the figure below, a beam of light is incident on the first surface, refracts through the prism and then exits the prism from the second interface. (a) For an incident angle of 30 degrees, what is the value of refracted angle θ r in the prism of the first interface? (b) Knowing that the sum of the internal angles of the triangle φ + φ1+ φ = 180 o, what is θ i at the second surface? o (c) At what angle θ r does the light exit the prism? (d) Now assume that the refractive index is NOT constant but instead exhibits dispersion such that the refractive index is INCREASING as the wavelength of visible light increases. If white light is incident on the prism, do all the colors of light emerge at the same angle from the prism? EXPLAIN in a few sentences. φ ο θ i φ 1 φ θ r θ i θ r Problem : LAB PRACTICAL For this problem, there are three parts. Do the first two parts (just calculations) BEFORE you attempt the last part (measurements on optical table). Page 1 of 1
(a) If you insert a 0.3 ND neutral density filter in the path of a laser beam, what percentage of the light is transmitted through this filter? (b) If instead, you insert a 1.0 ND neutral density filter in the path of a laser beam, what percentage of the light is transmitted through this filter? (c) There is an optics setup in the lab which you will use to determine if the detector is linear for the power level of the laser source. Using the provided neutral density filters, determine if the detector is responding LINEARLY to the power level of the laser source. You may use the neutral density filters on the table as well as your answer(s) to (a) and/or (b) above to answer this question. EXPLAIN in a few sentences WHY your answer is correct based on your observations. THE DETECTOR IS (Circle your answer) LINEAR NOT-LINEAR Problem 3: (a) Determine the optical path difference for the two waves A and B both having vacuum wavelengths of 500nm depicted in the figure below. In the diagram, the distances are L1=1mm, L=3mm, and L5=3mm. A B n=.0 n=.0 L1 L L3 (b) If the waves start out in-phase, find their relative phase difference (in units of radians) when they emerge from the stack of materials. (c) After the beams emerge from the stack of materials, they are overlapped so that they interfere with each other. Assuming that the BOTH waves emerge from the stack of materials with a maximum amplitude of 1 in the appropriate units, what is the amplitude of the resulting wave when the two beams overlap and interfere? Page of 1
Problem 4: This question concerns Lab 5. Remember for this experiment it was emphasized that one should NOT move the box of colored liquid during the measurements. This question will explore that statement. As illustrated in the figure below, the path of the light through the plastic of the box and the liquid is illustrated. The power incident on the box is Po while the power incident on the detector is P. We can represent the relationship between Po and detected power P as a function of dye concentration using the following equation P( C ) = P(1 R )exp( α L )exp( σ C L )(1 R )exp( α L ). (1) dye 0 p p p dye dye dye p p p The term (1 R p ) represents the reflective losses through one layer of plastic. The term exp( α L ) represents the absorption losses through the plastic. The absorption coefficient of p p dye, concentration of the dye, and path length through the dye are given by σ dye, C dye, and L dye, respectively. (a) Show that the detected power with a concentration of dye given by Co DIVIDED by the detected power BEFORE the dye is added to clean water DOES NOT DEPEND on the absorption and reflection losses of the plastic. (b) Why is your answer to (a) above dependent on your NOT moving the box during the experiment? EXPLAIN. L p L dye L p P o P Problem 5: An object is 0cm from a f = -10cm negative lens shown in the figure below, (a) On the diagram DRAW a ray diagram locating the image. (b) Calculate the location of the image relative to the lens. Page 3 of 1
(c) (d) Is the image to the left or to the right or the lens? What is the magnification of the image? Problem 6: An object is 0cm from a f = -10cm negative lens as in the previous problem (you can use the results of Problem 5 here). In addition a positive lens f=30cm lens is added to the optical system as shown in the figure below. The distance between the two lenses is 10cm. f=10cm f=30cm (a) Calculate the location of the FINAL image relative to the f=30cm lens. (b) Is the image to the left or to the right or the f=30cm lens? (c) What is the TOTAL magnification of the image after passing through both lenses? Problem 7: For the past several years, the concept of wireless power transfer has been explored. One application is drones for which a high power laser beam is incident on a photocell on the drone. The incident power from the laser beam is converted to electrical power by the photocell to recharge the batteries on the drone while it is inflight. The goal of this problem is to estimate how large an optical beam would be on the drone after it expands due to diffraction. (a) Assume that the laser beam as it is emitted from the laser has a circular cross-section with a diameter of 5cm. Its wavelength is 1500nm. If the power in the laser beam is 10W, what is the intensity of the laser beam (in W/cm ) as it leaves the laser? POWER TOO LOW CHARGING TAKES TOO LONG (b) Due to diffraction, what would be the expected beam diameter a distance of 1km away from the laser? Page 4 of 1
(c) Based on your answer to (a), what should be the minimum size of the photocell on the drone? Explain your answer in a few sentences. Ie. Why can the size of the photocell not be SMALLER than your minimum size? NEED TO ADJUST, BEAM DOESN T EXPAND MUCH Problem 8: You can use the results of Problem 7 in this problem. (a) What is the intensity (in units of W/cm ) of the laser at the location of the drone which is 1km away from the laser? (b) Assuming that all of the laser power incident on the drone hits the drone, and that the photocell converts 40% of the optical power in to usable electrical power, how much power can the laser provide to the drone as electrical power? (c) Assuming that the laser power/ photocell combination is used to recharge a D-cell rechargeable battery, approximately how long will it take for the D-cell battery to be recharged is it can store a total of 75 kjoules of energy? Problem 9: In this problem, you will be using complex notation for waves to solve for the interference of two overlapping plane waves which overlap each other at an angle as illustrated in the figure. The wave functions for the two waves are E1 = E exp( ikz iωt) and E = Eo exp( ik( z cosθ + y sin θ) iωt). Please note that if you can not do one part of the problem, you can use the Equations given below to finish the following parts. (a) Use the principle of superposition to show that the total wave at z=0 from the overlapping wave E1 and E can be written as E ( z = 0) = E (1 + exp( iky sin θ))exp( iωt) () total o o (b) Based on Equation (), show that the intensity of the wave at the z=0 plane is given by I( y) ~ E(1 + cos( kysin θ )) (3) o Page 5 of 1
(c) Based on Equation (3), show that the locations of the y position where the intensity equals zero occurs at mλ ym = (4) sinθ where m is an ODD integer. (d) Based on Equation (4), what is the spacing on the z=0 plane between fringes of zero intensity? y θ z Problem 10: In class, we discussed refractive index many times. When it was first introduced in class, it was defined as n = c/ v where v is the velocity of the wave. (a) Explain in a few sentences and/or equations if the refractive index can be less than 1 meaning that the velocity v is LARGER than the speed of light in a vacuum. (b) Explain in a few sentences and/or equations under what conditions that the refractive index can vary as the frequency (or wavelength) of light changes. (c) Explain in a few sentences and/or equations the concept of a COMPLEX refractive index. What does the REAL part represent? What does the IMAGINARY part represent? Page 6 of 1
(d) Can the real part of the refractive index be negative? EXPLAIN in a few sentences your answer. Problem 11: A thin soap film (n=1.3) is surrounded on both sides by air. The film is illuminated with white light at near normal incidence and shows a color pattern in reflection. If a region of the film reflects only red light (600nm) strongly, what is the minimum (non-zero) thickness of the film at that location? Problem 1: The equation for single slit diffraction through a CIRCULAR aperture is J1( ka sin θ ) I( θ ) = Io ka sinθ, where J1 is a Bessel function of the first kind. For this problem, you do not need to know the mathematical details of the Bessel function EXCEPT that the first zero of the Bessel function J1(z)=0 occurs when z=3.83. Using this information, your goal is to show that the ANGULAR width of the central maximum of the circular diffraction pattern is λ given by θ = 1.. D (a) Using the illustration below as a guide, show that the location in angle θ 1 at which the intensity of the diffraction pattern corresponds to zero occurs at 3.83 λ sinθ1 = π D (b) Using the result of (a), show that in the small angle limit, the angular width of the central maximum of the diffraction pattern is λ θ = 1. D Page 7 of 1
D θ θ 1 I( θ) Screen Problem 13: Michelson Interferometer Lab Practical. On the optical table is the basic setup of a the Michelson interferometer. Please finish the alignment so that you can observe localized fringes which are oriented PERPENDICULAR to the floor. Call over your instructor when you are ready to be graded. Problem 14: For non-magnetic materials, we described in class that the complex refractive index n = n + in is related to the complex relative permittivity (or dielectric function) ε through r i ( ) ε = = +. (5) n nr ini (a) By assuming that the permittivity has a real and imaginary part, ε = εr + iεi, show that the real part of the permittivity can be written as ε = n n (6) r r i (b) Similarly, derive an expression for the imaginary part of the permittivity ε i as a function of the real and imaginary refractive indices. (c) Using your results from (a) and (b) above, show that one can write the real part of the refractive index in terms of the real and imaginary parts of the permittivity as n r ε + ε + ε = r r i HINT: you will need to solve a quadratic equation. If you don t remember the quadratic formula, it is ax bx c + + = 0 has a solution of b ± b 4ac x =. a Page 8 of 1
Problem 15: Light from a Blue laser (λ=450nm) impinges on a TWO parallel slits in an opaque screen. The separation between the slits is 0.mm wide. Kindly calculate the expected separation between successive interference fringes in the interference pattern on a screen which is 1.5m from the slits. Problem 16: Typically in class, we have been using laser beams with power levels of roughly 3mW. The typical beam diameter is mm. (a) What is the average intensity of the laser beam with this power and beam diameter? (b) What is the corresponding electric field of a light wave with this power and beam diameter? (c) If one were to focus down a laser beam to a small enough spot, it is possible to increase the local electric field in the laser beam to a large enough value to strip air molecules of electrons to produce an ionized gas of positively charged molecules and electrons. Assuming that the required electric field to ionize air is 3 10 6 V/m, estimate the intensity of the laser beam required. (d) From your answer to part (c) and assuming a laser power of 3mW, estimate the required beam diameter to which the laser beam must be focused to achieve ionization of the air. (e) Compare your answer in part (d) to the wavelength of light. Would you be able to use a lens to focus the laser beam down to the required diameter? EXPLAIN YOUR ANSWER. Page 9 of 1
Problem 17: One method to monitor the health of plants is to monitor the so called red edge spectra of green plants. The large change of reflectance of plants in the red hued colors is due to the fact that chlorophyll (which is green) no longer strongly absorbs red and infrared colors of light. Plants that are stressed (eg. not enough water) exhibit different reflectance spectra in the red edge compared to healthly plants. Consider the spectra of green leaves in the figure below. (a) Let s consider the a few specific wavelengths which might be used to monitor the stress in plants. Explain in a few sentences why the choice of wavelength 500nm would not be helpful in distinguishing healthy from stressed leaves. (b) Explain in a few sentences why the choice of wavelength 875nm would allow some ability to tell the stressed from normal leaves. (c) Similar to Lab 5, you could measure the REFLECTION of leaves at two different wavelengths. For this problem assume that you measure the reflection at 550nm and 875nm. Using the data from the figure above, which of the two wavelengths would give you the LARGEST change in reflectance between normal and stressed leaves? WHY? (d) The measured power which is reflected from the leaf may possibly change because the power of the illumination source may vary (for example if you use sunlight to illuminate the leaves). One method to remove this variation is to calculate reflection power ratios. For example, one can measure the reflected power at 875nm and divide by the measured reflected power at 550nm. Briefly explain how this ratio will remove the problem of an unknown illumination power or an illumination power that may vary in time. Page 10 of 1
Problem 18: A beam of unpolarized, incoherent red (600nm wavelength) light incident in air on a glass (n=1.5) interface at 50 degrees is partially reflected and partially transmitted. (a) Calculate the REFLECTION coefficient of POWER of the interface for light polarized PARALLEL to the plane of incidence. (b) Calculate the transmission coefficient of POWER of the interface for light polarized PARALLEL to the plane of incidence. (c) Calculate the REFLECTION coefficient of POWER of the interface for light polarized PERPENDICULAR to the plane of incidence. (d) Assuming that the total power of the unpolarized incident light is 10mW, how much total power (in mw) is REFLECTED from the glass interface? Problem 19: It is well-known that optical fibers are birefringent. The birefringence can be due to stress applied to the fiber from bending/ twisting or residual stress from the manufacturing 6 process. Assume that difference in the refractive index for the birefringent fiber is n = 4 10. (a) Calculate the length of fiber required to use the fiber as a fiber-optic equivalent of a ½ waveplate. (b) Now assume that two orthogonally linearly polarized light wave pulses enter the fiber at the same time. One pulse is polarized parallel to the optical axis (and therefore experiences one index of refraction). The other pulse is polarized perpendicular to the optical axis. Both pulses exit the fiber after traveling a distance of 1km in the fiber. Use a wavelength of 1500nm (He-Ne laser wavelength). Calculate the time difference between the exit of the faster moving pulse compared to the slower moving pulse. Page 11 of 1
Problem 0: The polarization rotation of a chiral protein in solution is measured. It is observed that the linear polarization is rotated 3 degrees. The concentration of the protein is 15 g/l and the pathlength through the solution is 0cm. Please calculate the specific rotation of the protein. You MUST specify the units of your final answer. Problem 1: Unpolarized light is incident upon a linear polarizing filter. A second polarizer is oriented so that its transmission axis is polarized 5 degrees with respect to the transmission axis of the first polarizer. If the unpolarized light has a total power of 5mW, how much power is transmitted through the second polarizer? Page 1 of 1