Diffraction of light 9.1. Introduction The Dutch scientist Christian Huygens proposed a wave theory of light that provides a useful technique he developed for predicting the future position of a wave front when an earlier position is known. This is known as Huygens’ principle and can be stated as follows: Every point on a wave front can be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself. The new wave front is the envelope of all the wavelets – that is, the tangent to all of them. A large number of equally spaced parallel slits is called a diffraction grating, although the term ”interference grating” might be as well appropriate. Gratings can be made by precision machining of a very fine parallel lines on a glass plate. The untouched spaces between the lines serve as the slits. Photographic transparencies of an original grating serve as inexpensive gratings. Gratings containing 10 000 lines per centimetre are common today, and are very useful for precise measurements of wavelengths. A diffraction grating containing slits is called a transmission grating. Reflection gratings are also used, which are made by ruling fine lines on a metallic or glass surface from which light is reflected and analyzed. The analysis is basically the same as for a transmission grating, which we now discuss. The analysis of a diffraction grating is much like that of Young’s double-slit experiment. We assume parallel rays of light are incident on the grating as shown in Fig. 9.1. We also assume that the slits are narrow enough so that diffraction by each of them spreads light over a very wide angle on a distant screen behind the grating, and interference can occur with light from all the other slits. Light rays that pass through each slit without deviation (θ=0) interfere constructively to produce a bright line at the centre of the screen. Constructive interference also occurs at an angle such that the rays from adjacent slits travel an extra distance of ∆l=mλ, where m is an integer. Thus, if d is the distance between slits, then we see from Fig. 9.1 that ∆l=dsinθ, and 68 DIFFRACTION OF LIGHT sin m , m 0,1,2, d principal maxima (9.1) is the criterion to have a brightness maximum. This is the same equation as for the double-slit situation, and again m is called the order of the pattern. l =d sin d l Fig. 9.1 Diffraction grating. 9.2. Measurements 9.2.1. Calculation of wavelength Switch on the laser and set the diffraction grating ( d = 10 μm) perpendicularly to the direction of the ray of laser light. The effect of interference of rays bent on the diffraction grating can be seen as a pattern of light dots on the screen. The pattern is symmetrical with respect to the zero-order dot. 69 DIFFRACTION OF LIGHT Screen Laser Diffraction grating Fig. 9.2 Experimental setup. Measure the distance of the diffraction grating from the screen and the distances of the series of interferential maxima from the zero-order dot (“zeroth maximum” on the figure). That allows to measure angles of diffraction for these maxima (on both sides of the incident beam). If the angles differ by less than 5 degrees, then the position of the diffraction grating should be appropriately corrected. Compose your setup and repeat the measurements 6 times. The diffraction grating has a known constant, hence wavelength λ of a light source can be easily determined, from the equation (9.1). The data should be collected in the Table 9.1. 1st maximum 1st minimum 1st maximum P’ x A Zeroth maximum 1st minimum Zeroth maximum C B P Diffraction scale 1st minimum L Diffraction grating Retina of your eye 1st maximum Fig. 9.3. Diffraction of a light. 70 DIFFRACTION OF LIGHT 9.2.2. Diffraction grating constant Switch on the laser and set next diffraction grating perpendicularly to the direction of the ray of laser light. Measure the distance L of the diffraction grating from the screen and the distances of the series of interferential maxima from the zero-order dot (“zeroth maximum” on the figure) l1, l2, l3, l4, and so on. That allows to measure angles of diffraction for these maxima (on both sides of the incident beam). Measurements should be repeated 6 times.The light source has a known wavelength λ from the first part, hence diffraction grating constant d can be easily determined, from the equation (9.1). The data should be collected in the Table 9.1. 9.2.3. Hair width This experiment is similar to the previous one, because the diffraction can be observed on any thin obstacle like human hair. The result is also obtained in the similar procedure by measuring the angles of diffraction for several orders of the pattern. The data should be collected in the Table 9.1. Table 9.1. Order of pattern L l [nm] d [m] a[m] m 1 2 3 4 9.3. Results, calculation and uncertainty From the data collected in the table calculate the average values of the wavelength λ of a light source, diffraction grating constant d and hair width a. Estimate the uncertainty of the measured values using method of standard deviation. 71 DIFFRACTION OF LIGHT The final results reads: S (9.2) d d Sd (9.3) a a Sa (9.4) 9.4. Questions 1. Formulate the Huygens’ principle. What can you predict using this principle? 2. Discuss Young’s double – slit experiment. 3. What kind of physical phenomena can you observe on diffraction grating? 4. What are the conditions for diffraction and interference? 5. Discuss Bragg’s equation. 6. What is geometric optics? What is physical optics? 7. Explain rainbow phenomenon. 8. Can you give examples of diffraction gratings implementation ? 9. What is a laser? How is laser light different from ordinary light? What are lasers used for? 10. What are electromagnetic waves? Discuss polarization phenomena . 9.5. References 1. Dryński T., Ćwiczenia laboratoryjne z fizyki, PWN, Warszawa, 1959 2. Resnick R., Halliday D., Fizyka, Tom 2, PWN, Warszawa, 1989. 3. Szydłowski H., Pracownia fizyczna, PWN, Warszawa, 1994. 4. Young H.D., Freedman R.A., University Physics with Modern Physics, Addison-Wesley Publishing Company, 2000 72