Diffraction Theory Diffraction Theory 0 Irradiance pattern (“Fraunhofer Diffraction Pattern” ) on an observation screen far from a square aperture illuminated by an incident plane wave. The dimensions of the aperture are on the order of the wavelength. Diffraction Theory “Interference” and “Diffraction” are arbitrarily distinguished from each other. Diffraction Theory 1 Interference irradiance due to a collection of discrete sources. Diffraction irradiance due to a continuous distribution of sources. “Fraunhofer Diffraction” the observation point P is far away from the distribution of sources or the irradiance in the focal plane of a lens is observed. Diffraction Theory We will analyze “Fraunhofer Diffraction” from apertures (illuminated by incident plane waves) whose dimensions are on the order of the wavelength. In our discussion of N slit interference, we assumed the slit width to be infinitesimally small. Diffraction Theory 2 Diffraction Theory The superposition field at the observation point is generally determined by an integral over the distribution of Huygens emitters in the aperture. Diffraction Theory 3 Diffraction Theory Fraunhofer Diffraction also describes the irradiance pattern in the focal plane of a lens. Diffraction Theory 4 Single Slit Diffraction Rather than evaluating an integral, we will determine the field from a single slit (of width b on the order of the wavelength ) by filling the slit with an array of Huygens emitters (N slits of infinitesimal width) and let N as the separation between emitters 0. Single Slit Diffraction 0 The slit: L>> and b . The N emitters we place inside this (vertical) slit are infinitesimally wide vertical slits. Single Slit Diffraction Rather than evaluating an integral, we will determine the field from a single slit (of width b on the order of the wavelength ) by filling the slit with an array of Huygens emitters (N slits of infinitesimal width) and let N as the separation between emitters 0. Single Slit Diffraction 1 S1 is at the edge of the slit and the distance between adjacent Huygens slit emitters is “a” (which we will shrink to zero as N ). Point C is the centre of the slit. To point P, far away (or in lens focal plane). Single Slit Diffraction The field at P due to the N Huygens slit emitters is: Single Slit Diffraction 2 Our earlier grating result. is the phase change undergone by a field propagating from C P. is the field amplitude at each Huygens emitter. Single Slit Diffraction The field at P due to the N Huygens slit emitters is: Single Slit Diffraction 3 As we let N and a 0 : Single Slit Diffraction So, as N and a 0 : Single Slit Diffraction 4 where the last step is true by the small angle approx. Single Slit Diffraction So, as N and a 0 , the field becomes: Single Slit Diffraction 5 with Physical realism requires that, as such that This has to be true in order for the overall field passing through the slit to remain finite as the number of Huygens emitters becomes infinite! Thus: Single Slit Diffraction The irradiance at observation point P is: Single Slit Diffraction 6 2 with We can put this in a more useful form by noting that as as Single Slit Diffraction We define the irradiance in the = 0 (forward) direction as Single Slit Diffraction 7 and we can express the irradiance in some general direction, , in terms of this quantity: This is a practical formula as we are able to calculate the irradiance in some general direction, , relative to the irradiance in the forward direction. This type of (relative) quantity can be measured easily in an experiment. Single Slit Diffraction We can express the single slit irradiance function in a more common way by introducing the “sinc” function: Single Slit Diffraction 8 with with Properties of the “sinc” function: Zeroes: Maxima: Single Slit Diffraction The sinc function. Single Slit Diffraction 9 sinc() vs sinc2() vs Single Slit Diffraction The single slit diffraction pattern. Single Slit Diffraction 10 Single Slit Diffraction The single slit diffraction pattern: Variation with slit width. Single Slit Diffraction 10a Single Slit Diffraction The single slit diffraction pattern: Limiting Cases. Narrow Slit. Single Slit Diffraction 11 sin =0 Single Slit Diffraction The single slit diffraction pattern: Limiting Cases. Wide Slit. Single Slit Diffraction 12 Except near = 0 ! sin =0 Single Slit Diffraction Geometric optics limit: (wide slit) Single Slit Diffraction 13 Lens focal plane. P P The irradiance function for a wide slit and a point source for s . This approximates the point image expected in geometrical optics. For a narrow slit, the irradiance function (diffraction pattern) is “spread out” in comparison to the point image: “diffraction limited optics”. N Slit Diffraction Expand our previous discussion of diffraction from a single slit to an array of N slits with each slit having width “b” (b ) and the separation between slits (ie centre to centre distance) is “a”. N Slit Diffraction 1 We consider N identical slits, illuminated by an incident plane wave and determine the superposition field and irradiance at an observation point P located far from the slits (or lying in a lens focal plane). To observation point P. N Slit Diffraction Expand our previous discussion of diffraction from a single slit to an array of N slits with each slit having width “b” (b ) and the separation between slits (ie centre to centre distance) is “a”. N Slit Diffraction 2 Each slit gives rise to a field component at P: where 0 is the phase change undergone by the field in travelling from the slit centre to P. The phase change 0 is different for each slit! To observation point P. N Slit Diffraction N Slit Diffraction 3 The overall superposition field at P is the sum of field components arising from each of the N slits. Identifying 0i as the phase change undergone by the field in travelling from the centre of slit “i” to P we can write this superposition field as: N Slit Diffraction N Slit Diffraction 4 Having identified 0i as the phase change undergone by the field in travelling from the centre of slit “i” to P we can define the quantity as the difference in this phase change arising from neighbouring slits. N Slit Diffraction With defined, the superposition field can be written as: N Slit Diffraction 5 We can sum the series (as we did for the grating) using the formula: with N Slit Diffraction The result: N Slit Diffraction 6 Where: (b = slit width) (a = slit separation) is the phase change undergone by a field travelling from the centre of the array to P. N Slit Diffraction The important quantity: Find the irradiance at P ! N Slit Diffraction 7 As before, for the single slit, we can write this in a more useful form by seeing what happens as 0. As 0 : N Slit Diffraction Defining the irradiance in the = 0 (forward) direction as: N Slit Diffraction 8 we can express the irradiance in some arbitrary direction, , in terms of the “forward” irradiance: N Slit Diffraction N Slit Diffraction 9 So we have the irradiance function: N Slit Diffraction Example: N=4 N Slit Diffraction 10 15 The black curve is the irradiance pattern. N Slit Diffraction Example: N=4 N Slit Diffraction 11 Interference Fringes Diffraction “Envelope” 15 Interference fringes correspond to the principal maxima of the N slit irradiance pattern. The irradiance in a given fringe depends on the value of the single slit irradiance pattern: the “diffraction envelope”. N Slit Diffraction N Slit Diffraction 11a Interference Fringes (Principal Maxima) Diffraction “Envelope” Interference fringes correspond to the principal maxima of the N slit irradiance pattern. The irradiance in a given fringe depends on the value of the single slit irradiance pattern: the “diffraction envelope”. N Slit Diffraction Properties of the irradiance function. N Slit Diffraction 12 Interference fringes (principal maxima): Diffraction envelope: Zeroes: Central maximum: Subsidiary maxima: (usually not concerned with these.) N Slit Diffraction N Slit Diffraction 12a N Slit Diffraction 12a N Slit Diffraction Example: N=2 (Young’s Double Slit with finite width slits) N Slit Diffraction 15 1. Double slit pattern for slit width b 2. Double slit pattern for slit width b << 3. Single slit pattern for slit width b in 1. N Slit Diffraction Example: N=2 (Young’s Double Slit with finite width slits) N Slit Diffraction 15a 15 N Slit Diffraction Example: N=2 (smaller slit width) As b 0 , we approach our earlier double slit result. N Slit Diffraction 16 15 N Slit Diffraction Example: N=2 (smaller slit width) As b 0 , we approach our earlier double slit result. N Slit Diffraction 17 15 N Slit Diffraction “Missing Order” in the irradiance pattern. N Slit Diffraction 13 If the 1st order diffraction zero overlaps a principal maximum location, then we have a missing order. Example: 15 The 3rd order maximum is missing here. N Slit Diffraction Calculating the missing order: N Slit Diffraction 14 Overlap of the 1st diffraction zero with a principal maximum they occur at the same angle . The order, m, of the principal maximum for which this occurs is given by: 1st diff zero: princ. max: N Slit Diffraction “Missing Order” in the irradiance pattern. N Slit Diffraction 14a For this example: a/b = 3 the 3rd order fringe is missing. 15 The 3rd order maximum is missing here.

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