INTERFERENCE. Addition and Interference Between Sinusoidal Waves. E(t) = sin( ω t + φ ) 2(t) = sin( 2 π t + π/3) = 3 sin( 2 π t + π/6)

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1 INTERFERENCE In this and the following lab the wavelike nature of light is explored. We have learned that waves usually exhibit some periodic or oscillatory nature such as the water level at a particular point on a pond steadily rising and falling as waves pass. For light (electromagnetic radiation) it is the electric and magnetic field which oscillate in strength and direction. At visible frequencies these oscillations occur so fast that they are exceedingly difficult to observe directly ( 4 Hz.) Another important characteristic of waves is that they may interfere with each other. In simple terms interference is the result of oscillating dips and rises adding together or canceling out. A good example of interference is seen when the circular wave patterns from two stones thrown into a pond meet. High points from the separate waves add to make even higher points, dips from each wave combine making lower dips, and when a dip from one wave passes the same point as a high from the other they cancel. E(t) = sin( ω t + φ ) E (t) = sin( π t + ) E (t) = sin( π t + ) E (t) = E (t) + E (t) T = sin( π t + ) E(t) = sin( ω t + φ ) E (t) = sin( π t + ) π π E (t) = sin( t + ) E (t) = E (t) + E (t) T = E(t) = sin( ω t + φ ) E (t) = sin( π t + ) E (t) = sin( π t + π/3) E (t) = E (t) + E (t) T = 3 sin( π t + π/6) Figure π π π/3 π Addition and Interference Between Sinusoidal Waves t t t

2 Screen Plane Wave L d Slits P Interference Fringes (Periodic intensity minima and maxima) Figure Young s Two Slit Interference Experiment With Narrow Slits Figure shows how two sinusoidal waves add together to produce a new wave. In this lab we will see how interference affects the intensity pattern of light that has passed through two narrow adjacent slits, many parallel slits (diffraction grating), and a special bullseye pattern called a Fresnel zone plate. Experiment : Two Slit Interference (Young s Double Slit Experiment) In 8 Thomas Young devised a classic experiment for demonstrating the wave nature of light similar to the one diagrammed in Figure. By passing a wide plane wave through two slits he effectively created two separate sources that could interfere with each other. (A plane wave is a wave consisting of parallel wavefronts all traveling in the same directions.) For a given direction, the intensity observed at point P on the screen will be a superposition of the light originating from each slit. Whether the two waves add or cancel at a particular point on the screen depends upon the relative phase between the waves at that point. (The phase of a wave is the stage it is at along its periodic cycle.) The relative phase of the two waves at the screen varies in a oscillatory manner as a function of angle causing periodic intensity minima and maxima called fringes (see Figure ). The spacing of the fringes depends on the distance between the two slits. If the width of each individual slit is narrow compared to a wavelength, the intensity of the two-slit interference pattern is given by, Two Slit Interference E ( ) E() cosβ I ( ) I() cos β β π d sin () where E is the electric field. I is the field intensity with units of energy per area per second and is proportional to the square of the electric field E. Maximum intensity peaks occur when β = m π (m = ±, ±,...) and thus the angles of maximum intensity satisfy, sin = m, m =, ±, ±,... () d

3 Equation () is easily derived. The amplitude of the electromagnetic wave emanating from each slit may be written as E = E o sin(ωt ). After the wave emanating from one of the slits travels a distance l the wave undergoes a phase shift φ = π l/ so that the electric field is of form E = E o sin(ωt + φ). The Plane Wave d a a Figure 3 l = d sin difference in distance light from each slit travels to a point on a very distant screen is shown in Figure 3 as a function of angle. It is seen that the wave front emanating from one slit lags behind the other by l = d sin. This gives a relative phase difference of π d sin /. Since we are interested only in relative differences we can write the fields due to each slit as, E = E o sin(ωt + πd sin ) E = E o sin(ωt - πd sin ) Using the trigonometric identity, A B A + B sin A + sin B = cos sin E Phase Difference Between Waves Coming from each Slit E 3 P Plane Wave L a d a P Figure 4 Diffraction Intereference Fringes Generated by Two Finite Width Slits

4 4 gives, E T = E + E = E o sin(ωt) cos(πd sin /) (3) which when squared and averaged over time produces Equation ()..5 sinc( ) = sin( ) If the width of the slits is greater than a wavelength the.5 4 interference fringes are seen to oscillate in brightness as seen in Figure 4. This is due to diffraction. Diffraction is interference of one wave with itself. According to Huygen s Principle waves propagate such that each Figure 5 sinc Function point reached by a wavefront acts as a new point source. The sum of the secondary waves emitted from all points on the wavefront propagate the wave forward. Interference between secondary waves emitted from different parts of the wave front can cause waves to bend around corners and cause intensity fluctuations much like interference patterns from separate sources. Diffraction will be discussed in more detail in the next lab. The amplitude of the field of a diffraction pattern from one slit is given by, E ( ) E() ( sin π a = ) sin( ωt) sin (4) Just as the two waves were added above (without the diffraction term) the relative phase difference of each wave is considered, the two waves are added and squared resulting in, I ( ) I() ( sin = ) cos β (5) The diffraction term ( sin ), is of the form of a sinc function squared, where sinc = sin. A graph of the sinc function is shown in Figure 5. Note that if the slit width a is less than then is always less than one and diffraction causes little change in intensity with angle. In this case Equation () becomes a good approximation. Figure 6 shows how the separate terms of Equation (5) influence the final intensity pattern for a particular set of slits. cos² β sin Intensity I ( ) = I () sin cos²β Slit Width = 4e 5 Slit Seperation =. Wave Length = e 6 β = π d sin Figure 6 = π a sin (radians) Two Slit Interference/Diffraction

5 5 Experiment Procedure Two-Slit Interference You are provided with a slide that has Slit Number of Width of each Slit Space between Slits Slits ( mm ) several sets of slits on it with separations and widths as shown in Figure 7. You will. 4 x - mm x - mm be using a helium-neon (HeNe) laser as x - mm.3 mm your plane wave light source. The x - mm.8 mm experimental set up is as shown in x - mm.35 mm Figure 8. If the laser beam is pointed onto x - mm.7 mm a pair of slits, an interference pattern similar to Figure 9 may be observed on a screen placed nearby. The provided screen has a Figure 7 Dimensions of Slits on Slide set of magnets attached to it which can be used to fasten a sheet of paper to it. This allows the distance between fringes be easily marked and measured. a) Observe the interference pattern on the screen for each of the four double-slits on the slide. Draw what you see for each. How does the slit separation effect the interference pattern? Can you detect any intensity variations in the different fringes? Why does this occur? For each set of double slits identify the intensity distribution caused by the width a of each slit as well as the interference fringes whose spacing is related to the distance d between the two slits. The diffraction from the sets of double slits labeled 4, 5, and 6 in Figure 7 are particularly interesting to observe because they all have the same slit width but different slit separations d. Consequently the minimum of the diffraction pattern must occur at the same value but the spacing of the interference fringe pattern varies. Laser Slide Screen Alignment Screw Figure 8 Setup for Interence Experiments

6 6 Figure 9 Two Slit Interference Pattern as Seen on Screen b) You will now make measurements using double-slits #3 and #4. Measure the spacing of the interference fringes (the finely spaced minima/maxima.) Also measure the distance from the double slits to the screen. Using = 63.8nm as the wavelength of the laser-light and Equation (), calculate the slit separation. c) Calculate an average value for each double-slit separation and compare with those listed in Figure 7. (Note that if there is a blue dot somewhere on the end of the slide, its slits separations do not match the Figure 7 and it is thus the values will not compare well.) Experiment : Diffraction Gratings A system consisting of a large number equally spaced of parallel slits is called a grating. The diffraction pattern expected from a grating can be calculated in a way similar to the two slit system. A simple way to do this calculation is to represent the amplitude and phase of each wave originating from one of the grating slits and impinging on a point on the screen by a vector called a phasor (see textbook). Summing the phasors corresponding to each slit gives the correct amplitude and phase of the total wave. The intensity distribution of the light scattered off a grating with N slits is, ( ) ( ) Diffraction Off Grating I = I sin sin Nβ sin β (6) This relation is very similar to the one for the diffraction of two slits ( and β are defined in Equations sin Nβ () and (3)). The interference maxima are described by the function. With increasing number of β slits N this function has a rapidly varying numerator and a slowly varying denominator. The effect of each term on the final pattern is shown in Figure for a particular grating. As the number of illuminated slits N increases, the intensity peaks will narrow and brighten. The position of the fringe maxima generated by a grating is given by,

7 7 Fringe Maxima (7) sin = m, m = ±, ±,... d where d is the grating constant (the distance between the grating slits). Gratings are commonly used to measure the wavelength of light. This may determined using Equation (7) and measurements of the angle between the maxima. Procedure -- Experiment : Diffraction Gratings sin² (N β ) sin² (N β ) sin² ( β ) sin² ( β ) sin² ( β ) sin² (N β ) sin² ( ) Slit Width = e 5 Slit Seperation =. Wave Length = e 6 Number of Slits = (radians) sin² ( ) Figure Interference Fringes from a a) Place the diffraction grating in Diffraction Grating the path of the laser. The number of slits per inch should be written on the side of the slide. Because the spacing between slits is so small the angles of the maxima are quite large. You will need to keep the screen close so that the st and nd order maxima can be seen. Measure the distance from each of them to the central maximum. Also measure the distance of the screen from the slide. How much does the intensity vary for the central st and nd order maxima? b) For both measurements made in step a) calculate a values of the wavelength of the light. Compare these values to the given wavelength of the laser, 63.8nm. c) Calculate the maximum number of fringes that you are expected to observe with the grating. (Hint sin ). Do you observe all the expected fringes? Experiment 3: The Fresnel Zone Plate In the Fraunhofer formalism light is represented by plane waves, and the distance between light-source and scattering object or scattering object and observer is assumed to be very large compared to the dimensions of the obstacle in the light path. If you drop these conditions, the light phenomena observed are described by the Fresnel formalism and the light waves are represented by spherical waves rather than plane waves.

8 8 Figure shows the spherical surface corresponding to the primary wave front at some arbitrary time t after it has been emitted from S at t =. As illustrated the wave front is divided into a number of annular regions. The boundaries of these regions correspond to the intersections of the wave front with a series ρ r o r o of spheres centered at P of radius r + /, r +, r + 3/ etc. These are Fresnel or half-period zones. The sum of the optical disturbances from all m zones of P is E = E + E + E 3 + E E m (8) Because of the phase of the light passing through each consecutive zone increases by / ( change in phase by π) the amplitude E p of the zone l alternates between positive and negative values depending on whether m is odd or even. As a result contributions from adjacent zones are out of phase and tend to cancel. This suggests that we would Figure Fresnel Zone Plates observe a tremendous increase in irradiance (intensity) of P if we remove all of either the even or odd zones. A screen which alters the light, either in amplitude or phase, coming from every other half-period is called a zone plate. Examples are shown in Figure. Suppose that we construct a zone plate which passes only the first odd zones and obstructs the even ones, E = E + E 3 + E E 39 (9) and that each of these terms is approximately equal. For a wavefront passing through a circular aperture the size of the fortieth zone, the disturbance at P (as was demonstrated by Fresnel) would be E / with corresponding intensity I = (E /). However, with the zone plate in place E E at P and I (E ). r o + 3/ r o Figure Fresnel Zone + + / P

9 9 The intensity I of the light at P has been increased by a factor of 6. The zone plate acts as a lens with the focusing being done by interference rather than by refraction! S ρ m ρ o A m R O m r m r o P To calculate the radii of the zones shown in Figure, refer to Figure 3. Figure 3 Fresnel Zone Radii for Fresnel Zone Plates The outer edge of the m th zone is marked with the point A m. By definition, a wave which travels the path S - A m -P must arrive out of phase by m/ with a wave which travels the path S-O-P, that is, phase difference = (ρ m + r m ) - (ρ + r ) = m/. () According to the Pythagorean theorem ρ m = (R m + ρ ) / and r m = (R m + r ) /. Expand both these expressions using the binomial series and retain only the first two terms (ρ m ρ + R m /r and r m r + R m /r ). Substituting into Equation () gives the criterion the zone radii must satisfy to maintain the alternating / phase shift between zones. The width of zone m is proportional to m Rm = + ro ro m. Rewriting Equation () as, () m + = ρ r R = f ()) m puts it in a form identical to the thin lens equation (/o + /i = /f) with primary focal length f, f Rm =, m =,, 3,... m (3) Procedure -- Experiment 3: The Fresnel Zone Plate 3a) Put the Fresnel Zone Plate in the laser path and observe the diffraction pattern. Move the plate back and forth so that you observe the results when light hits the center and the outside of the plate. Is the slit separation smaller on the outside or center of the plate? What have you observe that supports this answer?

10 3b) Measure the primary focal length of your setup s zone plate. Do this by centering the laser beam on the double concave lens which will diffuse the beam and generate on the screen a uniformly illuminated circle. Now place the zone plate between the lens and the screen, and by changing the position of the screen along the direction of the beam, observe the refocusing of the laser beam. Measure the two quantities ρ and r (Figure 3) for at least different initial values of ρ and then use Equation () to calculate f. Often zone plates are made of plastic material or metal with a self-supporting spoked structure so that the transparent regions are devoid of any material. These will function as lenses in the range from ultraviolet to soft (that is low energy) x-rays ( to Å) where ordinary glass is opaque. Zone plates have also been used to focus low energy neutron beams. QUESTIONS: Following is a list of questions intended to help you prepare for this laboratory session. If you have read and understood this write up, you should be able to answer most of these questions. The TA may decide to check your degree of preparedness by asking you some of these questions: What is a plane wave? The relation of sin / for = is an undefined expression of /. What is the limit of this relation for? Sketch two sinusoidal waves out of phase by 45. Are the eyes sensitive to the amplitude or to the intensity of light? sin a The distribution of light from two slits is represented by the product cos b a ( ). Which one of these two terms is called the diffraction term and which one is the interference term? Which term is responsible for the interference fringes? Two waves in phase and each one with an amplitude A meet at the same point. What is the intensity of the signal observed? What are gratings useful for? What is the difference between Fraunhofer and Fresnel scattering? What is a Fresnel half-period zone? Sketch a zone plate. What are the zone plates useful for? Sections of this write up were taken from: Physics Part, D. Halliday & R. Resnick, John Wiley & Sons. Physics Volume, Electricity, Magnetism, and Light, R. Blum & D. E. Roller, Holden-Day. Optics, E. Hecht/A. Zajac, Addison-Wesley Publishing