Optics Vac Work MT 2008 1. Explain what is meant by the Fraunhofer condition for diffraction. [4] An aperture lies in the plane z = 0 and has amplitude transmission function T(y) independent of x. It is illuminated by coherent monochromatic light of wavelength λ at normal incidence. Show that, in the Fraunhofer case, the diffracted intensity is proportional to A(θ) 2, where A(θ) = T(y) exp[ i(2π/λ)y sin θ] An aperture consists of two long slits parallel to the x axis. The centres of the slits are separated by a distance d, and each slit has a width a. Write down T(y) for this aperture and use it to calculate A(θ). Hence sketch the diffracted intensity as a function of sinθ. A two-slit aperture of this type has d = 0.1 mm and a d. It is illuminated at normal incidence with light of wavelength λ = 650 nm. A lens of focal length 90 mm and diameter 10 mm is used to form an image of the aperture on a screen situated 1 m behind the aperture. Calculate the two possible positions of the lens, and explain which one of these positions gives the sharper image of the aperture. [7] 2. Describe with the aid of a labelled diagram the important features of a transmission-grating spectrograph. A diffraction grating has a line spacing of d and a total of N lines. The grating is illuminated at normal incidence by light of wavelength λ. Show that the minima adjacent to a principal maximum in the diffracted intensity are separated in angle by θ = 2λ Nd cosθ m where θ m is the angle of diffraction for the mth order. A transmission-grating spectrograph has a grating with 500 lines mm 1 and total width 50 mm. The spectrograph is used at normal incidence to examine a spectral line with a wavelength of 540 nm consisting of two components separated by 0.02 nm. Determine whether the two components can be resolved with the grating provided. The resolution is also affected by the size of the entrance slit of the spectrograph. Assuming that the entrance slit has width 0.01 mm and the collimator lens has focal length 0.5 m, estimate the contribution of the entrance slit to the resolution and hence discuss whether the two components of the spectral line can be resolved in practice. [7] 1
3. Explain what is meant by polarized light and describe the different types of polarization that are possible. Describe briefly the principle of operation of a polarizing device made from a birefringent material. A beam of light is elliptically polarized with the major axis vertical. The ratio of the major and minor axes of the ellipse is a : b. Explain how you would use a quarter-wave plate to obtain linearly polarized light, and determine the angle of the plane of polarization to the vertical. A beam of light consisting of a mixture of elliptically polarized and unpolarized light is passed through a linearly polarizing filter. The maximum of the transmitted light intensity is observed when the transmission axis of the filter is vertical, and is twice the minimum intensity. In a second experiment, the beam is passed through a quarter-wave plate with the fast axis vertical followed by the polarizing filter. The maximum is now observed when the transmission axis is at 33.21 to the vertical. Calculate the ratio of the intensities of the polarized to unpolarized components of the light. How could the handedness of the elliptically polarized component be determined? [3] 4. Describe the principles of operation of a Michelson interferometer. Include in your account a labelled diagram of the apparatus. [9] What are localized and non-localized fringes? Discuss, for both point and extended sources, the localization of (a) circular fringes and (b) straight-line fringes in a Michelson interferometer. Discuss the application of the Michelson interferometer to both (i) tests of Special Relativity, and (ii) analysis of spectral lines. 5. A transmission diffraction grating has N slits, each of width a and spacing d between centres. Using the Convolution Theorem, or otherwise, show that for light of wavelength λ at normal incidence the intensity I(θ) diffracted at angle θ is given by where u = 2π sin θ/λ. I(θ) = I(0) sin2 (ua/2) sin 2 (Nud/2) (ua/2) 2 sin 2 (ud/2) A grating with 500 lines per mm is used at normal incidence in a spectrometer where the focal lengths of the collimating and imaging lenses are both 1 m. Two spectral lines of wavelengths 670.96 nm and 670.99 nm are examined in the second order of diffraction. For this arrangement calculate: (a) the angular dispersion of the grating, (b) the minimum width of the grating necessary to resolve these lines, (c) the maximum useful entrance slit width. [12] Sketch the optical layout of a spectrometer, or spectrograph, and describe the procedure for correct illumination. 2
6. Derive an expression for the intensity transmitted through a Fabry-Perot etalon with mirrors of reflectivity R, as a function of the mirror separation d, the wavelength λ and the angle of incidence θ. Define the finesse and free-spectral range. [10] A short etalon with mirrors of reflectivity R = 0.75 is used as an interference filter transmitting infrared radiation of wavelength 4.3 µm, at normal incidence. The full-width at half maximum of the transmitted intensity is about λ = 0.2 µm. Any phase change of the light on reflection may be neglected. (a) Calculate the spacing of the mirrors. (b) Calculate the change in mirror spacing required to shift the centre wavelength to 4.5 µm. (c) The mirror spacing is fixed at the value which gives maximum transmission at 4.3 µm for normal incidence and the filter is tilted so that the angle of incidence becomes 17.5. Calculate the wavelength of the light transmitted. (d) Find a wavelength longer than 4.3 µm for which the intensity transmitted at normal incidence is a maximum, and a wavelength shorter than 4.3 µm which also gives a maximum. Comment on how the transmission of other wavelengths affects the usefulness of the filter. [15] 7. Explain how a quarter-wave plate and a half-wave plate may be constructed from a piece of transparent birefringent material. Two beams of light are provided: one is elliptically polarized and the other consists of a mixture of circular and plane polarized light. Explain how a quarter-wave plate and a sheet of Polaroid can be used to (a) identify which is the elliptically polarized beam; (b) determine the orientation of the major axis of the elliptic polarization; (c) find the degree of ellipticity i.e. the ratio of the major to minor axis. [9] A source emits light over the range 550 to 630 nm. This light is passed through a filtering system consisting of a pair of crossed polarizers between which is placed a half-wave plate constructed for light of wavelength 580 nm. Explain how the crystal axis of the half-wave plate must be oriented in order to obtain maximum transmission by the filter for light of wavelength 580 nm. It is found that the system also has a maximum of transmission at 620 nm. Use this information to find the thickness of the plate given that the ordinary and extraordinary refractive indices have a mean value of 1.48 and differ by 0.7%. Any variation in refractive index with wavelength may be ignored. Suggest how a new plate could be made to ensure transmission is achieved at only one of these wavelengths. [11] 8. Describe how a Michelson interferometer may be used to measure the wavelength of monochromatic light. A Michelson interferometer is used to examine light from a vapour of atoms. The intensity of light at the centre of the circular interference pattern is monitored as a function of the mirror displacement x, and found to have the form: I(x) = 3I 0 + [2I 0 cosαx + I 0 cosβx] exp( γ 2 x 2 ) where α = 1.6405 10 7 m 1, β = 1.6319 10 7 m 1 and γ α, β. Show that the spectrum consists of two components and find 3
(a) their relative intensity and their mean wavelength; (b) the minimum distance x required to resolve the two spectral lines. [10] Make a rough sketch of the interferogram and suggest a physical origin for the term exp( γ 2 x 2 ). How could light from a helium-neon laser which provides a standard wavelength known to one part in 10 8 be used to improve the accuracy of the wavelength measurement? 9. Define the finesse and free spectral range of a Fabry-Perot etalon. For an etalon with two flat mirrors of reflectivity R a distance l apart in vacuum show that the free spectral range, in wavenumbers, is given by 1/2l. Derive an expression for the finesse, F, and discuss the accuracy of the approximation F π/(1 R) for a typical etalon. [13] To measure the spacing of the two mirrors in an etalon, light from a laser is sent normally through both mirrors and the transmission is recorded. The wavenumber ν L of the laser is scanned over a small range about two accurately known calibration lines with wavenumbers ν 1 = 1 257 890 m 1, and ν 2 = 1 259 005 m 1 Data of the form shown in the diagram are recorded where the wavenumber of the laser is measured relative to that of the calibration line in each scan. (a) Use the data from the scan about ν 1 to estimate the reflectivity of the mirrors and the free spectral range. (b) Calculate the number of peaks which would be observed if the laser was scanned continuously between ν 1 and ν 2. Hence refine your value of the free spectral range using the data from both scans. (c) Give a value for the separation between the mirrors to the full precision justified by the data. (d) How could the final uncertainty of the measurement be reduced? [12] 4
10. Light diffracted by a grating is detected in a fixed direction making an angle of 2γ with the incident direction. The angles of incidence θ i and reflection θ r are shown in diagram (a). Show that the relation between the rotation angle φ of the grating and the wavelength λ for which the diffraction peak occurs is 2d sin φcosγ = pλ, where φ is measured from the position where the normal to the grating bisects the angle between the directions of incidence and detection. Define the quantities represented by p and d. Diagram (b) shows the optical layout of a spectrometer used to record absorption spectra by sending white light through the sample. The focal length of the curved mirrors (M1 and M2) is 1 m and γ = 25. On the triangular rotatable mount there are three diffraction gratings with the following rulings. Grating 1 Grating 2 Grating 3 : 500 lines per mm : 1 000 lines per mm : 2 400 lines per mm The detector is an array of 1000 individual detecting elements evenly spaced over a 25 mm width and is sensitive to wavelengths in the range 500 950 nm. For each grating give all possible values of φ for which light at 800 nm might fall on the centre of the detector. For two of the gratings 800 nm light is detected close to normal incidence (θ i 0). For both cases, calculate the angular dispersion for a fixed grating position and the range of wavelengths which fall on the detector. Give the limit to the resolution set by the size of the individual detecting elements and the entrance slit width necessary to achieve it. Comment on any additional factors which affect the choice of grating for absorption measurements around 800 nm. 5