Test Questions & Answers An Introduction To Optics Chapter.3

Chapter 3

Problem 3.1: An argon ion laser has a power of 0.5 W at 415 nm.

(a) What is the energy of a 415 nm photon?

(b) What is the wavenumber of a 415 nm photon?

(c) For this power, how many photons strike a sample in 1 second? Recall that a W is a J/s.

(d) If the wavelength was tuned to 418 nm and we assume the same power at 0.5 W, do any results for parts (a) to (c) change?

Problem 3.2: Two waves are described by the following two equations for amplitude I as a function of time, t:

Assume that both waves occur at the same spatial location. Use a calculator to calculate the sum of I₁ and I₂ for times 0, 1, 2, and 3. Decide if the sum of these waves is fully constructive, fully destructive, or likely to result in a beat pattern.

Problem 3.3: The equations for the intensities of three waves are presented below.

Assume that all three waves occur at the same spatial location. Plot the sum of these waves and comment on the interference.

Problem 3.4: Consider visible light incident on particles in a sample. For each of these particle size dimensions, roughly compare the wavelength dependence of the scattered photons:

(a) O₂ molecule (size ~0.29 nm)

(b) Small aerosol (size ~0.1 μm)

(c) Very large aerosol (size ~2 μm)

Problem 3.5: A light source is composed of wavelengths from 550 nm to 1.2 μm. The beam is incident on a block of BK7 glass at an angle of 45 degrees from the normal. What happens to the beam as it refracts through the glass? Answer this quantitatively by describing the refraction of the 550 nm and 1.2 μm extremes in wavelength. See Figure 3.17.

Problem 3.6: In your laboratory you have a converging lens with a focal length of 6 cm that you plan to use to collect light from a sample. If your source aperture is 0.5 cm in height, what focal length lens would you need to acquire if you wished to focus the collimated beam onto a 1 mm diameter detector aperture? Discuss the feasibility of focusing the beam to such a small diameter. See Figure 3.18.

Problem 3.7: Calculate NA and the f/# for each of the lenses here. Rank the lenses in terms of their light collecting ability.

Problem 3.8: You are comparing the beam waste for two different situations with the goal of using the smallest beam waste possible. A Nd-YAG laser system emits light at 532 nm and the beam is 8 mm in diameter. You also have a Ti-sapphire laser that emits at 855 nm and has a beam diameter of 6 mm. Compare the beam waist for both laser systems using a focusing lens with a focal length of 10 mm. Assume the light fills the lenses in each case.

Problem 3.9: Suppose that you found out that the actual wavelength of the laser from Example 3.4 was 632.8 nm.

(a) Calculate the angle of the first-order diffracted light. Compare this angle to what was measured in the example.

(b) Show how you could use the known wavelength of the laser light to determine the line spacing of the grating.

Problem 3.10: Given a reflective diffraction grating with a groove distance of 3 mm, calculate the diffraction angle for a first-order (m = 1) diffraction of a 526.5 nm beam with an incident angle α of 20 degrees. At what angle would you expect to find the second order (m = 2) diffracted beam?

Problem 3.11: You need to be able to resolve a wavelength difference of 0.1 nm around 500 nm. You have a reflective diffraction grating that has 600 lines/mm and is 12.7 mm long. You plan to use the first-order diffracted light. Will this grating have the resolving power needed? Support your response with a calculation.

Problem 3.12: You have purchased a 25 mm wide reflection grating with 1,200 lines/mm. What is the resolving power and what minimum wavelength can be resolved around 750 nm?

Problem 3.13: White light contains wavelengths that range from 400 to 700 nm. Given a reflective diffraction grating with a groove distance of 5 μm,

(a) Calculate the diffraction angle for first-order diffraction of a 400 nm, a 550 nm, and a 700 nm beam with an incident angle of α = 20 degrees.

(b) If the exit slit for this monochromator is 5 cm from the grating, what will the dispersion distance (Δx) be between these three beams at the exit slit?

(c) If the exit slit is 20 μm wide, what is the bandwidth of radiation that passes through the exit slit for the 550 nm beam?

(d) Does the bandwidth change for the other two beams?

Problem 3.14: An optical signal has an intensity of 1.1 (arbitrary units) at 500 nm and 1.0 at 700 nm after diffracting off the 500 nm blazed grating (see Figure 3.24). By taking the absolute efficiency of the grating into account, what was the intensity of the 500 nm and the 700 nm signal before interacting with the grating?

Problem 3.15: A signal composed of wavelengths from 375 to 780 nm is being measured. The light is sent through a grating monochromator. Should you be concerned with diffraction order overlap? Justify your response with a calculation.

Problem 3.16: You are trying to separate wavelengths that are 505.1 nm and 505.18 nm. The grating is 25 mm long and has 1,800 grooves/mm. Use the m = 1 order diffraction to determine the dispersion angle between the two wavelengths. Repeat the calculation for m = 2. Comment on the difference.

Problem 3.17: You have access to five ND filters with optical densities of 0.1, 0.3, 0.5, 1.0, and 2.0. You have a helium-neon laser at 632.8 nm at 13 mW. What optical powers can you attain with these filters, making sure to use all possible combinations of the filters? Create a spreadsheet to answer this question.

Exercise 3.1: You are using an Nd:YAG laser as an excitation source for a time-resolved fluorescence experiment. The Nd:YAG laser emits at 1064 nm. You are using a non­linear optical material (KTP) to generate the second harmonic of this beam at 532 nm. The 532 nm output still contains a percentage of the laser fundamental at 1064 nm and you want to only use the 532 nm photons. Search online or in an optics catalog for a filter that will block 1064 nm and pass 532 nm. Characterize the loss at both 532 nm and 1064 nm for the filter.

Exercise 3.2: When light of wavelength λ moves from air (n = 1) to a material with index of refraction n(λ), the fraction of light reflected, R, is:

This relationship holds when the light is incident at 90 degrees. You are to use BK7 glass in your experimental setup. What percentage of light is reflected off the BK7 if you use λ = 355 nm (recall that the index of refraction versus wavelength was provided in this chapter)? Would the percentage of reflected light increase or decrease if you use λ = 785 nm?

Exercise 3.3: A light source contains wavelengths at 501 nm and 502 nm. A grating with a groove spacing of 2.5 μm is used and the light is incident on the grating at 45 degrees. For the m = 1 order, what is the angle of reflection for the 501 nm and 502 nm components of the beam?

Exercise 3.4: The intensity, I, of a light source is:

where P is the power of the source and A is the area of the beam. A diode laser with wavelength of 488 nm and power of 10 mW is to be used in a Raman spectroscopy experiment, but you are not sure you will get a large enough intensity to study your sample (you have been told that you need at least 3 W/cm²). In order to decide the minimum radius of beam to attain this intensity, produce a plot of the intensity of the beam versus the beam radius, from r = 0.001 to 0.5 cm.

Exercise 3.5: One of your laboratory partners is doing experiments measuring the index of refraction of a new polymer film. In the experiment, two laser beams, wavelength 632.8 nm and 594 nm, enter a polymer surface from air at an angle of 35 degrees from the normal and refract. He reports that the 594 nm beam refracts at an angle of 27 degrees from the normal and the 632.8 nm beam refracts at 25 degrees. Calculate the index of refraction for the polymer film at these two wavelengths. There is something strange about the data he has proposed. What is it?

Exercise 3.6: Your boss has asked you to buy a new grating for your grating monochromator. She has told you that you need to measure peak intensities at wavelengths of 501.1 nm and 501.3 nm. Assume slit widths of 0.3 nm and a grating that must be 5 cm long. What groove spacing should you choose? Does this grating spacing seem physically possible? Search commercial gratings online to see if such a grating is commercially available.

Exercise 3.7: Load the PhET simulation “Bending Light” from http://phet.colorado.edu. Set the top materials to Mystery A and the bottom material to water. Measure the angle of refractions with the protractor in the simulation and then use Snell’s law to calculate the index of refraction of Mystery A material. Change the top material to Mystery Material B and the bottom material to air and determine the index of refraction for Mystery Material B.

Exercise 3.8: It turns out it is possible to construct an object, a metamaterial, that has a negative index of refraction. A 855 nm beam is incident at an angle of 30 degrees from the normal onto a negative index object. Assume an index of refraction of –1 for the object. Calculate the angle of refraction and comment on what this means.

Exercise 3.9: Load the PhET simulations “Wave Interference” from http://phet.colorado.edu. Go to the light tab. Set up the two-slit experiment. Qualitatively answer the following question: What is the relationship between the slit width and wavelength and the extent to which interference occurs?

Exercise 3.10: Load the PhET simulation “Bending Light” from http://phet.colorado.edu. Set the top material to glass and the bottom material to air. Is it possible for the laser light to completely reflect off the interface? If so, what is the maximum angle that allows for this? Is this angle dependent on the index of refraction of the top material?

Exercise 3.11: The third and fourth harmonic of a Nd:YAG laser produces radiation at 355 and 266 nm, respectively. Calculate the beam waist size (2w₀) for each harmonic of the Nd:YAG laser assuming a beam that is 8 mm in diameter for a lens with a focal length of 6 cm.

Exercise 3.12: Choose two focal length lenses to collect, collimate, and focus light from a xenon lamp that is 8 mm in size. Sketch your optical system and determine the size of the focused spot. Calculate the NA of your collection lens.

Exercise 3.13: In the photograph in the figure, three light rays are incident on a diverging lens. In the image is a sketch of a diverging lens and two incident light rays. Use Snell’s law to sketch the light rays as they refract from air to glass and then from glass to air.

Exercise 3.14: Two waves are described by the following two equations for their intensities:

Assume that both intensities occur at the same spatial location. Plot each of these two waves in spreadsheet software. Construct your spreadsheet so that you can easily change any of the parameters (amplitude, 100 for both; frequency, 5 for both; and phase constant, π/9 and π/3) and see the corresponding plot of I₁, I₂, and their sum. Change the phase constant for the first wave to 0 and the second to π/2. What happens to the sum of I₁ and I₂?

Exercise 3.15: When focusing a beam of light, what advantages are gained by using a focusing mirror over a focusing lens?

Exercise 3.16: Why are grating monochromators more commonly found in modern spectrophotometers than prism monochromators?