An Analysis of Interference as a Source for Diffraction

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1 J. Electromagnetic Analysis & Applications, 00,, doi:0.436/jemaa Published Online October 00 ( 60 An Analysis of Interference as a Source for Diffraction Armand Waksberg Consultant, Lasers and Electrooptics Montreal, Canada. chamonix@total.net Received September st, 00; revised September 9 th, 00; accepted October 6 th, 00. ABSTRACT An analysis is made of interference of to plane aves hich in turn give rise to subsequent diffraction phenomena. This is done using to different approaches. One of them is straight forard but is difficult to interpret. The second is less conventional but more intuitive thus giving more insight into the process. The geometry is kept very simple to focus on the process itself. Both approaches give rise to the same results thus offering a choice in tackling such problems. Keyords: Electromagnetic Wave Analysis, Diffraction Phenomena, Interference. Introduction Interference and diffraction theory have been developed as a result of the realization of the avelike nature of light. This has led to the thousands of successful applications, ranging from the many forms of interferometers to the development of optical components such as Fresnel lenses and other Fourier transform components, making possible the ide field of present day Photonics. The basics of interference and diffraction theory can easily be obtained from textbooks [,]. To get a better insight into these phenomena, hoever, ould require a closer examination, such as, by looking at them from different perspectives. The purpose of this paper is to provide such an insight by looking at a simple situation comprising an interference situation, hich in turns gives rise to a diffraction phenomenon. The problem is then solved using to different approaches. Although the first is more straightforard, the results are difficult to interpret. The second, is less conventional but more intuitive. The geometry is kept very simple to focus on the process itself. It consists of a all on hich to plane aves intersect to form interference fringes, a slit in the all through hich the interference ave passes and a screen at a distance behind the all hich exhibits the resulting diffracted ave that as produced by the slit. Waveforms are calculated at the all and at the screen for each of the to approaches and compared. The second approach is then used to offer an insight for the solution obtained by the first approach. The poer transfer is calculated for each approach to ascertain that the process obeys conservation of energy. As an example, e shall discuss ho to beams intersecting and thus interfering at a point behave as they move aay from it. As both approaches give rise to the same results, a side benefit of this analysis is that it provides a choice in tackling such problems depending on the type of initial information that is supplied.. Configuration Selection Referring to Figure, e assume to infinite plane aves ith amplitude E and E, polarized in the Y direction (hich is perpendicular to the paper), ith their normals in the XZ plane approaching an infinitely large all in the XY plane at Z = 0. Their normals make an angle i and i ith the Z-axis respectively. It is further assumed that the all has an infinitely long slit in the Y direction of idth a along the X-axis centered at X = 0. On the right of the slit, a large screen, parallel to the all, is placed at a distance d from it. For discussion purpose, e have chosen the distance d so that the geometrical overlap of the to beans entering the slit and reaching the screen has a idth a/ along the X s ordinate. It is to be expected that there ill be interference effect at the all due to the superposition of the to aves hich ill result in fringes on the all. In particular, this interference field ill penetrate the slit and end up on the Copyright 00 SciRes.

2 60 An Analysis of Interference as a Source for Diffraction X X s -i E a/ a/4 slit 0 Z i E -a/ d all sreen Figure. To plane aves E and E intersect on a all at Z = 0, enter through a slit of idth a and are projected on a screen at Z = d. screen. We shall calculate, using to different approaches, the resulting field at the screen and compare the results... Approach Starting ith hat e consider the most straightforard approach, e shall calculate first the interference field at the slit. Since, on entering the slit, it is being sliced by its edges, hich e assume to be very thin, e expect diffraction to take place. We shall then use diffraction theory to calculate the resulting field at the screen originating from this interference field... Approach We follo each plane ave separately, obtain their field strength at the slit, noting that each ave has been sliced by the slit, and calculate the resulting field at the screen as caused by diffraction. We end up ith to separate diffracted fields at the screen originating from the to original plane aves. We then calculate the interference field resulting from their superposition. We compare the results ith those of Approach to ensure that they are the same. 3. Relevant Theory Folloing Approach, let the to plane aves approaching the all have a functional representation given by [3], j( tkz) j( kxsinikycosi) Axyzt (,,, ) Ae e () Bxyzt (,,, ) Be e j( tkz) j( kxsinikycosi) () here A and B are amplitude constants proportional to the fields E and E respectively. They ill be evaluated later. Since e ant to calculate the interference field at the all (including the slit), e kno that z 0. Also, since the field amplitude is invariable along the Y direction, e let y = 0. Adding Equations () and () e get the amplitude of the interference field at the all to be, j( t kxsin i) jkxsini C( x, t) Ae ( Re ) (3) here R = B /A. To relate A and B to the actual field, e shall resort to the conservation of energy principle. To do this e note that the intensity at the all of this interference field is given by the time average of the square of Equation (3) hich is ell kno to require a factor of ½ to the result. The intensity is therefore given by, C ( x) A I R Rcos (4) here kxsini. The total poer entering the slit per unit length in the Y direction is therefore, A a P I dx R ka i k a/ sin( sin )/ s a/ ini (5) To simplify our interpretation, e ant to examine only hole fringes. This ill occur hen the last term, hich is associated ith partial fringes, is made equal to zero. This occurs hen ka sin i n here n 0,,,3... or an n /(sin i) (6) With n representing the number of hole fringes. The total poer per unit length at the slit is, Aa n P ( n) R (7) We are no prepared to relate A and B to the fields. Looking back at field E in Figure, its related intensity is defined as the energy transmitted per sec per unit area normal to the direction of propagation. It is proportional to E /. But the all is not normal to the direction of the Copyright 00 SciRes.

3 An Analysis of Interference as a Source for Diffraction 603 plane ave. The intensity along the all is spread out over a larger area by a factor of /cos i. The poer associated ith plane ave E per unit of Y is therefore proportional to E cos i/. Similarly, the poer associated E ith E is cosi. The total poer entering the slit from both aves per units of Y is thus E a n ( R ) cos i (8) Comparing Equations (7) and (8) hich should be equal for conservation of energy, e must have, A cos E i and B cos E i Before e can calculate the field at the screen using diffraction theory, e need the amplitude and phase of the field at the slit. We obtain it from Equation (3) by reriting it as, j C( x) A D( x) e (0) here Dx ( ) Re j and ( x) ( x) kxsini, and ( x) is the argument of D(x), or the angle the phasor D(x) makes ith the X-axis. The term e it has been omitted since it is common to all the expressions and is of no importance to calculate the amplitude at the screen. Since the field at the slit is being clipped by the edge of the slit, e shall use the Fresnel-Kirckhoff formulation of diffraction theory []. We assume that the to plane aves have their field polarized in the Y direction so that E x = E z =0. Further, the slit at the all is assumed to be infinite in length in the Y direction. This simplifies somehat the mathematics since the field should be invariable in that direction. Consequently, e shall use the simplified one-dimensional expression [4], Cs( xs) an / j jk, () s C( x){ e / s} cos d / s dx an / Where C s is the field disturbance resulting from that at the slit, d is the distance beteen the all and the screen, s [ d ( x xs) ] /, is the angle beteen the avefront at x and the all normal. It can be obtained from the relation sin ( x) / ( x). k Equation () is a good approximation provided that d > a, a > λ, and the angle i is small. Other approximations ould provide similar results [5-7]. 4. Numerical Calculations For definiteness e let i = 0 o, E =, a = 7 and d = 0 both in avelength units. This ill ensure the geometry given in Figure. 4.. Approach In this approach, e calculate first the amplitude of the interference field present at the slit. It is the resultant of the superposition the to plane aves. It can be obtained from Equation (3) by taking its absolute value, C( x) A R Rcos This is the interference field, hich is responsible for the visible fringes, given by Equation (4). Plotting the interference field at the slit as a function of x, hich is the normalized distance along the slit, e obtain the aveform shon as (9) curve (a) in Figure. e / () note first that there are 5 peaks present at the slit as should be expected, the slit extends in the range -36 < x < 36. for a total idth of 7 avelengths. The amplitude varies from 0 to A. We can no use the interference field Equation (3) at the slit to calculate ho it is transformed hen it reaches the screen. For this e use the appropriate diffraction expression Equation () since the field is sliced by the edges of the slit. It turns out that for R =, φ = 0 for all values of x. Using computer numerical calculations, e obtain the field at the screen hich is plotted on the same graph in Figure as curve (b) for comparison purpose. We observe that around the central portion of the screen, the aveform pattern is very similar in shape and amplitude to that at the slit. As e move aay from this region, the field amplitude falls don to a level somehat half-ay don until around X s = 50 hen it decreases very quickly in amplitude toards zero. Also, the distance beteen peaks increased compared to that at the slit. To explain quantitatively the behaviour of the aveform from this approach is not simple. On the other hand, hen Approach is used, a much more intuitive derivation ill permit a better understanding of the process. 4.. Approach In this approach, e follo each plane ave separately as they reach the slit. Since they are sliced by the slit, e can use diffraction theory to calculate their resulting fields at the screen. We then add these fields (hich are phasors) to obtain the combined field. The results are then compared ith that of Approach. Starting ith plane ave E, e notice that hen it reaches the all, its amplitude is obviously a constant, although its phase varies ith X. C simplifies no to, j x sini C ( x) A e. (3) This is shon in Figure 3-curve (a). Since in this case φ = i = 0 o, using Equation () e obtain the diffracted field amplitude at the screen shon as curve (b). As should be expected, this is a typical ave- Copyright 00 SciRes.

4 604 An Analysis of Interference as a Source for Diffraction.5 Field amplitude (b) X or Xs X or Xs Figure. Field amplitude in arbritrary units of to plane aves E and E meeting at a slit as a function of position in avelength units. (a) interference field at all slit along X; (b) diffration field at screen along Xs. (a).4. Field amplitude (b) (a) X or Xs or Xs Figure 3. Field amplitude in arbitrary units of plane ave E meeting at a slit as a function of position in avelength units. (a) field at all slit along X from plane ave E ; (b) resulting diffration field at screen along Xs. form of Fresnel diffraction by a slit. It is, hoever, displaced by 0 o to the left of center. Obviously, a similar aveform is produced by plane ave, except that it is displaced by the same amount to the right. Its function at the all is, C ( x) A e j x sini. (4) If the to diffracted fields are added, the end result is shon in Figure (4) as curve (d). A comparison ith Figure curve (b) shos that they are identical. To get an intuitive understanding of that curve, e no plot on the same graph the aveforms of plane ave E shon in Figure 3 as curves (a) and (b). For reference purpose, curve (c) is added and is the interference field the to plane aves shon in Figure curve (a). Finally, e recall that in Section II e have selected the distance d so that the geometric overlap of the to aves at the screen covers only half the idth of the slit. This is shon as region (e). We thus observe that, for the main diffraction curve (d): ) At the edge of the aveform, on the left of the Figure, aay from region (e), here there is no overlap, the shape follos almost exactly the aveform of a single plane ave - curve (b). ) In region (e), the center portion here there is overlap, the shape is essentially that due to the sum of the to incoming aves. 3) In beteen those to regions, the ave progressively changes in shape and amplitude to blend from the overlap to the single ave region. 4) We should be gratified that our results sho that in the region, aay from (e), here there is no overlap, the aveform reverts to that of single plane aves. This is because e kno from intuition, that if to beams cross Copyright 00 SciRes.

5 An Analysis of Interference as a Source for Diffraction (e) Field amplitude.5 (d) 0.5 (a) (b) X or Xs X or Xs (c) Figure 4. Field amplitude of plane ave(s) as a function of position for situations: (a) field at all slit from plane ave E along X; (b) diffraction field as a result of (a) at sceen along Xs; (c) interference field at all slit along X resulting from both E and E ; (d) diffraction field at screen along Xs resulting from adding curve (b) from E and its counter part from E ; (e) geometric overlap of E and E at the screen. one another, they ill eventually carry on as if the interference region as not there. It should be emphasized, of course, that this is only true if the interference takes place in a vacuum here no other interaction takes place. Approach II thus allos us to see hy the shape of the diffracted ave derived in Approach I can be evolved from the diffracted aves produced by the to individual plane aves. 5) Poer Transfer Considerations We have shon, so far, that Approaches and, although quite different, give out the same results as far as final aveforms at the screen are concerned. As a final check, e ould like to ascertain that the poer entering the slit is equal to the poer reaching the screen for both Approaches for conservation of energy. The initial poer from the to incoming aves entering the slit per unit length in the Y direction is given by Equation (7). The intensity at the screen is given by Cs( xs) Is cos s( xs) (5) here C s is the total field obtained from Equation () for each of the approaches, hile φ s is the angle beteen the resulting field at x s and the Z-direction. The total poer per unit length in the Y direction in each case is then obtained by integrating, xs s ( ) x s s s s P I x dx (6) Where x s is taken far enough from x s = 0 that the field is negligible. When the calculations are performed, it is found that the difference in poer beteen the incoming poer at the slit and that at the screen for both Approaches is of the order of %. This is ell ithin the approximations expected from Equation (). 6. Conclusion We have considered interference theory resulting from to plane aves meeting at a slit, and on entering it, being subjected to the diffraction phenomenon. This as done using a simple geometry to focus on the process itself. To approaches ere selected, one more intuitive than the other. We have seen that they provide essentially the same results. Approach hich is more intuitive helps to understand the former hich is more conventional but more difficult to visualize. It also demonstrate that interference effect in a vacuum, although manifesting itself mainly in a region of beam overlap, does not carry a permanent effect on the beams hich revert back to their original behavior once they move far enough aay from the overlapped region. As a corollary to the calculations for both approaches, e have demonstrated, by an example, that the interference field hich is made up of the superposition of to or more fields can be used as the starting source to calculate further diffraction effects caused by obstacles. On the other hand, if the components of the interference field reaching the obstacle are knon, each of the components can be used to calculate its diffraction effect. The sum of the diffracted fields ill then give rise to the same results as in the first case. Consequently, this suggests that a choice can be made in solving such problems based on the type of initial information that is supplied. As a final remark, although e have considered in this paper only cases here R = and 0, it can be shon that similar conclusions can be obtained for other values of R. Copyright 00 SciRes.

6 606 An Analysis of Interference as a Source for Diffraction REFERENCES [] M. Born and E. Wolf, Principle of Optics, Pergamon Press, Ne York, 970. [] S. Silver, Microave Antenna Theory and Design, Dover, Ne York, 965. [3] C. Slater and N. Frank, Electromagnetism, McGra Hill, Ne York, 947. [4] A. Fox and T. Li, Resonant Modes in a Maser Interferometer, Bell System Technical Journal, Vol. 40, 96, pp [5] K. Mielenz, Algorithms for Fresnel Diffraction at Rectangular and Circular Apertures, Journal of Research of the National Institute Standards and Technology, Vol. 03, No. 5, 998, pp [6] K. Abedin, Computer Simulation of Fresnel Diffraction from Rectangular Apertures and Obstacles Using the Fresnel Integrals Approach, Optics and Laser Technology, Vol. 39, No., 007, pp [7] A. Waksberg, FIR Laser Efficiency Enhancement by Double Resonance Tuning, International Journal of Infrared and Millimetre Waves, Vol. 6, No. 3, 003, pp Copyright 00 SciRes.