Laser Diffraction PDF

Diffraction of Light Aim: To observe the intensity patterns generated by the diffraction/interference of a laser beam when passed through a double slit and to measure the value of slit separation up to micro-meter (micron) scale accuracy to find out the value of slit s...

Diffraction of Light Aim: To observe the intensity patterns generated by the diffraction/interference of a laser beam when passed through a double slit and to measure the value of slit separation up to micro-meter (micron) scale accuracy to find out the value of slit separation. Apparatus: X y (wave length 650nm) Figure 1. Experimental Set-up with part list [Ref.1,2]  Diffraction of Light The device consists of a one meter long optical rail on which the light source (laser), the diffracting element (slit, hole or wire) and the photo detector which measures the intensity of the diffracted light, are mounted. Their locations along the rail (x) and perpendicular to the rail (y), in case of the detector, can be controlled with a precision of mm and micron, respectively. Theoretical Background: For many of the phenomena involving light, more generally electromagnetic waves, geometric optics is sufficient which relies on the observation that light travels in straight line. But when light rays are partially obstructed, for example, by an opaque screen with a circular aperture or a knife edge held on the path of light, the light rays tend to show some amount of bending around the obstacle which results in complex pattern of illuminated and dark spots on a screen located on further downstream. This is known as diffraction. This bending of light can be explained by Huygen’s principle (1678) which states that that every point on the wave front that arrives at the aperture acts as a source of spherical waves and these secondary waves interfere on the screen to generate the diffraction pattern (see Figure.2). This also requires that the width (d) of the aperture be only few times larger than the wave-length (λ) of the monochromatic light. If the distance between the diffracting obstacleand the screen (D) is large compared to the slit width (d) then the waves arriveat the screen as plane waves and creates Fraunhofer or far field diffraction pattern. When the distance is less one get the Fresnel or near field diffraction pattern. In this experiment we will be concerned with Fraunhofer patterns only. 1. Diffraction of light by single slit y Figure 2. Each point on the slit opening generates spherical waves (www.cronodon.com)  Diffraction of Light Figure 3. Intensity pattern on the screen y Figure 4. Location of the 1st minima corresponds to path difference δ = λ/2. Adapted from hyperphysics.phy-astr.gsu.edu Using superposition principle one can derive that the intensity pattern on the screen 𝑠𝑖𝑛𝛽 2 𝜋𝑑 should follow 𝐼(𝜃) = 𝐼0 ( ), where 𝛽 = ( ) 𝑠𝑖𝑛𝜃 𝛽 𝜆 From this expression it follows that the minima occur when βm = ± m ,m = 1, 2,.. and correspondingly at angles θ = ± sin−1(mλ / d). The central maxima occurs at θ=0 and the other maxima when βm = ± ( 2m+1)π / 2, m= 1,2,.. occurs at θ= ±sin−1[(2m +1)λ / 2d] Note that m = 0 is not a local maxima. Minimas and maximas are caused by destructive (for path difference of odd multiples of λ / 2 ) and constructive interference (for path difference m λ ) of the diffracted waves,respectively. Thus from the positions of the maxima, we can find out the slit width using, d = mλ/sin θ m , Experimentally, we measure θm by recording ym , the y- positions of the minima, and then using the relation tan θm = ym/D.  Diffraction of Light 2. Diffraction of light by double slit In the two slit interference, light from the laser hits two very narrow slits, which then act like in-phase point sources of light. When light waves are travelling from the slit to screen, both the waves are travelling different distance from two slits. As shown in Figure 5, light hitting point P from the bottom slit travels longer distance than the light from the top slit. This extra path introduces a phase shift between the two waves and leads to a position dependent interference pattern on the screen. Here the extra path is δ =d sin  = λ , whichmakes a phase shift φ =2π. 𝜋𝑑𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝛽 2 Intensity pattern now 𝐼(𝜃) = 𝐼0 𝑐𝑜𝑠 2 ( 𝜆 )( 𝛽 ), where β = (πa / λ)sinθ and a is the width of each slit while d is their separation. The phase shift that are even multiples π, gives constructive interference and odd multiples of π gives destructive interference. Therefore the conditions for maxima are d sin  = mλ , and conditions for minima are d sin = (m+(1/2))/λ; where m = 0, ±1, ±2, ±3,… Figure 5. Location of the 1st maxima for shift δ = λ Figure 6. Diffraction pattern for double slit [Ref.3].  Diffraction of Light Here the diffraction envelope is set by the width of the slits a, while the individual wiggles are due to the interference between the light coming from the two different slits at separation d. Since here d>a , the locations of the minima are more closely packed here compared to the diffraction pattern of theearlier case, generated by a single slit of width a. Thus from the positions of the maxima we can now find the slit separation, d =mλ / sin θm. Figure.6. Left: Diffraction of water waves at the opening of the port of Alexandria in Egypt (adapted from google earth). Right:Diffraction of radio waves from a tower ( www.atis.org) Some interesting facts about diffraction 1. Babinet’s principle: the diffraction pattern for a single slit of width d is same as that of a thin wire of width d. This can be explained by Babinet’s principle: a slit and wire, when superposed, produce a completely opaque sheet which will produce a dark screen. This implies that the net electric field E = Eslit + Ewire= 0 at the screen, which means Eslit = - Ewire , but since intensity I  E 2 at the centre. both slit and wire produce the same pattern, except at the center 2. In the near field Fresnel diffraction pattern, for a circular aperture, a central minima replaces the bright central Airy disc, characteristic of thefar field pattern. 3. Diffraction limits the resolution of optical microscopes to 0.25 micron. The airy discs of two points closer than this will overlap and hence cannotbe resolved.  Diffraction of Light Some applications 1. To determine the track width on a CD by reflecting laser light andmeasuring the diffraction pattern. 2. To measure the wavelength λ of the laser light using double slit width. 3. To measure thickness of fine fibers, eg, human hair. 4. To measure size of red blood cells. 1. Procedures for double slit/single slit: 1. The laser, the double slit and the detector all are fixed on a rail and made co- linear, so that the laser beam in incident on the detector after passing through the slit. 2. Observe the diffraction/interference pattern and adjust the distance between slit and detector (D) to get a sharp pattern. The detector shows maximum intensity (adjust for > 50 micro-Amp) when laser beam falls on the detector. This should correspond to detector position in the range 10-12mm on the circular scale. Also measure D. 3. 1st scan the diffraction pattern (of varying intensity) by turning thecircular scale. When you actually take readings turn the circular scale in one direction only and go from one end of the pattern to the other across the central maxima. Do not move back and forth which will generate backlash error. Experimental set up and the measured quantities are same for both single and double slit cases, except that while we need to locate the maxima in the double slit case we need the minima in the single slit case. Data and analysis for double slit : Record current (microA) versus detector position (millimeter). While recording positions show, a) the main scale, b) the Vernier scale and c) the net readings. Plot current versus position, showing current on the y-axis. From the graph identify total five maxima : the central one and two each, on both sides of the central maxima. Try to use not more than 40-50 points to cover all the 5 maxima. Compute the slit separation d using the distance between the central maxima and the side maxima. Compute average d and its error (standard deviation) and present it as dd , keeping only significant digits. Since the angles m are small gap between successive maxima are almost same. So you can, alternatively, use these gaps in the formula for m=1 to get corresponding 1 and then use average 1 and its error to compute d  d.  Diffraction of Light Safety instructions 1. Laser radiation predominantly causes injury via thermal effects; avoid looking directly into the laser beam. The beam of laser source is so intense that it can cause damage to retina. Do not shine them on others or on yourself. 2. Care should be taken while handling diffraction slits and the other components. 3. Do not touch the inner surface of diffraction cell. 4. Avoid backlash error while moving the micrometer scale on the detector. Appendix: Least count = (Smallest div on Main Scale) / (No. of div on circular scale) = 0.05 cm /50 = 0.001 cm Micrometer reading = Main scale reading + (number of div. on circular scale xL.C.) References: 1. HOLMARC opto –Mechanics Pvt. Ltd Instruction manual for Detection based Diffraction Experiments Model: HO-EQ-D-02. 2. website for HOLMARC opto –Mechanics Pvt. Ltd http://www.holmarc.com/differ_detector.php , accessed on June 2015 3. Department of Physics, Massachusetts Institute of Technology,” Expt. No. 9. Interference and diffraction”. https://ocw.mit.edu/courses/8-02-physics-ii- electricity-and-magnetism-spring- 2007/80de81c8897a6835c7771736d6385fa3_experiment9.pdf 4. Abbei Tippie, Tammy Lee “experiments with Diffraction”, RochesterEducation www.optics.rochester.edu/workgroups/.../EDay2008_Diffraction.pdf ,accessed on June 2015 5. Double slit simulation https://www.geogebra.org/m/ynwgtfk8