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Consider Fraunhofer (far-field) Diffraction from an arbitrary aperture whose width and height are about the same. Let A = the source strength per unit area. Then each infinitesimal area element dS emits a spherical wave that will contribute an amount dE to the field at P (X, Y, Z) on the screen The distance from dS to P is which must be very large compared to the size (a) of the aperture and greater than a 2 / in order to satisfy conditions for Fraunhofer diffraction. Therefore, as before, for OP , we can expect A /r A /R as before (i.e., the behavior is approximated as that of a plane wave far from the source).

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Fig. 10.19 A rectangular aperture. At point P (X, Y), the complex field is calculated as follows:

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For x, y << R, we can approximate r as follows: Thus, for the specific geometry of the rectangular aperture:

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Therefore, the resulting complex field at point P on the screen for a rectangular aperture having area A = ab is given by where I(0) is the intensity at the center of the screen at point P 0 (Y = 0, Z = 0). A typical far-field diffraction pattern is shown in Fig. 10.20. Note that when = 0 or = 0, we get the familiar single slit pattern. The approximate locations of the secondary maxima along the -axis (which is the Y- axis when = 0 or Z = 0) is given by m 3 /2, 5 /2, 7 /2... Since sin = 1 at these maxima, the relative irradiances along the -axis are approximated as where m 3 /2, 5 /2, 7 /2...

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Square aperture in which a = b. where m = m 3 /2, 5 /2, 7 /2... are the positions of the secondary maxima and (Z) (Y) Distribution of irradiance, I(Y,Z) Distribution of electric field, E(Y,Z)

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Fig. 10.21 Circular Aperture Geometry A very important aperture shape is the circular hole, as this involves the natural symmetry for lenses. Such a symmetry suggests the need for cylindrical coordinates: We are calculating the field E on the screen as a function of the screen’s radial coordinate q.

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The circular shape of the aperture results in complete axial symmetry. Therefore, the solution must be independent of . Therefore, we are permitted to set = 0. We are therefore led to evaluate the integral: where J 0 (u) is the Bessel function (BF) of order zero. More generally the BF of order m, J m (u), is represented by the following integral: Bessel functions are slowly decreasing oscillatory functions very common in mathematical physics. Therefore the field is expressed as Fig. 10.22

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Another important property of the Bessel Function is the recurrence relation that connects BFs of consecutive orders:

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We can express the field, using the recurrence relation, as Therefore, the irradiance at point P on the screen is It is useful to examine the series representation of the Bessel Functions:

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Consider the limit near x = 0 : Therefore, the irradiance at P 0 when q = 0 is and the Irradiance becomes (Fig. 10.21) We usually express the irradiance as a function of the angular deviation from the central maximum at point P 0. The large central maximum is called the Airy Disk, which is surrounded by the first dark ring corresponding to the first zero of J 1 (u). J 1 (u) = 0 when u = 3.83 or kaq 1 /R = kasin 1 = 3.83.

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D = 2a is the diameter of the circular hole Note that ~84% of the light is contained in the Airy Disk (i.e. 0 kasin 3.83) Airy Disk

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D = 0.5 mm D = 1.0 mm Airy Rings with different hole diameters Airy Disk Suppose that the aperture is a lens which focuses light on a screen: Entrance Pupil (Aperture) screen f Converging lens D which gives the radius of the Airy disk on the screen.

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Fig. 10.24 Overlapping Images Suppose that we image two equal irradiance point sources (e.g., stars) through the objective lens of a telescope. The angular half-width of each image point is q 1 /f = sin . If the angular separation of the stars is and >> the images of the stars will be distinct and well resolved. Rays arriving from two stars and striking a lens Analysis of overlapping images using Airy rings

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) l( min Half-angle of an airy disk: Rays from two stars If the stars are sufficiently close in angle so that the center of the Airy disk of star 1 falls on the first minimum (dark ring) of the Airy pattern of star 2, we can say that the two stars are just barely resolved. In this case, we have 1 = ( ) min = q 1 /f 1.22 / D ( l) min 1.22 f /D. This is Rayleigh’s criterion for angular or spatial resolution. Fig. 10.25 f Rayleigh’s criterion for the minimum resolvable angular separation or angular limit of resolution

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Another criterion for resolving two objects has been proposed by C. Sparrow. At the Rayleigh limit there is a central minimum between adjacent peaks. Further decrease in the distance between two point sources will cause this minimum (dip) to disappear such that Sp The resultant maximum will therefore have a broad flat top when the distance between the peaks is r = Sp, and serves as the Sparrow criterion for resolving two point objects.

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Diffraction Gratings Transmission Grating is made by scratching rulings or notches onto a clear flat plate of glass. Each notch serves as a source of scattering that affects radiating secondary sources, in much the same way as for a multiple-slit diffraction array. When the phase conditions are met through OPD = = m, constructive interference is observed. Oblique incidence ( i > 0) m = 0 (zeroth order), m = 1 (first order), m = 2 (second order), m = 3 (third order) from the geometry.

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The diffraction grating can also be constructed as a reflection grating. The principals and conditions for constructive interference are the same as that for a transmission grating. Most commercial gratings for spectroscopy are constructed with a Blaze angle to control the efficiency of diffraction for a particular and order m.

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Most of the incident light undergoes specular reflection, similar to a plane mirror, and this occurs when i = m and m = 0 for the zeroth order beam. The problem is that most of the irradiance is wasted for the purpose of spectroscopy. It is possible to shift the reflected energy distribution into a higher order (m = 1) in which m depends on. It is possible to change the distribution of the specular reflection by changing the blaze angle so that the first order diffraction is optimized for a particular range of wavelengths. Consider the situation such that: i = 0 so m = 0, 0 = 0. For specular reflection i - r = 2 and so most of the diffracted irradiance is concentrated near r = -2 . This will correspond to a particular non- zero order in which m = -2 and asin(-2 ) = m. Controlling the irradiance distribution of diffracted orders using a Blazed grating.

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Schematic from the Fluorescence Group, University of California, Santa Barbara, USA

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Exit slit determines spectral resolution of the instrument Resolution is determined by the product of the monochromator linear dispersion (nm/mm) and the slit width Monochromator resolution depends on the grating pitch and the (focal) length of the monochromator For PTI monochromator with 1200 groove/mm grating the reciprocal linear dispersion is 4 nm/mm. 1 turn of the slit micrometer = 0.5 mm slit opening = 2 nm spectral resolution. Note that since E = h = hc/ E = (-hc/ 2 ) Monochromators with micrometer adjustable entrance and exit slit widths Example of luminescence spectra measured with a grating monochromator on the left for GaN/InGaN multiple quantum well (QW) samples. The maximum spectral resolution is obtained for the narrowest slit widths. Sit widths narrow (top) and wide (bottom)

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Cathodoluminescence (CL) - Light emitted by the injection of high-energy electrons. Photomultiplier Tube (PMT) or Ge p-i-n detector From Prof. Rich’s Laboratory for Optical Studies of Quantum Nanostructures

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Fresnel (Near-Field) Diffraction The basic idea is to start again with the Huygen’s-Fresnel principle for secondary spherical wave propagation. At any instant, every point on the primary wavefront is envisioned as a continuous emitter of spherical secondary wavelets. However, no reverse wave traveling back toward the source is detected experimentally. Therefore, in order to introduce a realistic radiation pattern of secondary emitters, we introduce the inclination factor, K( ) = (1+cos )/2 which describes the directionality of secondary emissions. K(0) = 1 and K( ) = 0.

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A rectangular aperture in the near-field (Fresnel Diffraction) The monochromatic point source S and the point P on a screen are placed sufficiently close to the aperture where far-field conditions are no longer applicable. Consider a point A in the aperture whose coordinates are (y,z). The location of the origin O is determined by a perpendicular line from the source S to the aperture . The field contributions at P from the secondary sources on dS (area element at point A) is given by where 0 is the source strength at S, A is the secondary wavelet source strength per area, and A = 0 is obtained from the Huygen’s-Fresnel formalism.

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In the case where the dimensions of the aperture are small compared to and r, we can assume primarily forward propagation in the secondary spherical waves so that K( ) 1 and 1/ r 1/ 0 r 0. Also, from the figure the geometry yields: Expand both terms in a binomial series for small y and z: Note that this approximation contains quadratic terms that appear in the phase whereas the Fraunhofer approximation contains only linear terms. Thus, we can expect a greater sensitivity in the phase of the cosine for this near-field treatment. The complex field at point P on the screen is therefore:

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This is the unobstructed disturbance at P. where A(w) and C(w) are called Fresnel Integrals; note that both are odd functions of w.

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Very often, we work in the limit of incoming plane-waves striking the aperture. For example, a laser beam could strike the aperture. In this limit we let the radius from the source to the aperture 0 . This results in an immediate simplification for the change of variables: Cornu Spiral Elegant geometrical description of the Fresnel Integrals (Fig. 10.50).

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Fig. 10.50 The Cornu Spiral for a graphical representation of the Fresnel integrals. A(w)A(w) C(w)C(w)

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A(w) C(w) w A(w) C(w)

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Therefore, values of w in B(w) correspond to arc length on the Cornu spiral. Consider a 2-mm square aperture hole: y z (1 mm, 1 mm) (-1 mm, 1 mm) (-1 mm, -1 mm) (1 mm, -1 mm) We are given that = 500 nm, r 0 = 4 m, plane wave approx. is valid. Find the irradiance at a point P on the screen along the axis x, directly behind the center of the aperture. P r0r0 O (y 1, z 1 ) (y 2, z 2 ) (y 1, z 2 ) (y 2, z 1 )

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Notice that there is an increase of the irradiance at the center point P on the screen by 256% relative to the unobstructed intensity due to a redistribution of the energy. In order to find the irradiance 0.1 mm to the left of center, move the aperture to the right relative to the OP line. While y 1 and y 2 are shifted, z 1 and z 2 remain unchanged. Then we have u 2 = 1.1, u 1 = -0.9, v 2 = 1.0, v 1 = -1.0. y z (1.1 mm, 1 mm) (-0.9 mm, 1 mm) (-0.9 mm, -1 mm) (1.1 mm, -1 mm) r0r0 (y 2, z 1 ) (y 1, z 1 ) (y 1, z 2 ) (y 2, z 2 )

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The decrease in the irradiance (2.485I 0 < 2.56I 0 ) for a small 0.1 mm shift to the left (or right) of center on the screen shows that the center position is a relative maximum (see Cornu spiral on the next slide). Note that if the aperture is completely opened: which must equal to the unobstructed intensity as a check.

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C(w)C(w) A(w)A(w) The decrease in the complex vector length from the position of the central peak shows that the central position is a maximum. 1.1 -0.9

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We can apply this formalism for Fresnel diffraction by a long narrow slit in which b = z 1 – z 2 = slit width and let v = v 2 – v 1 which is a string of length v lying along the Cornu spiral (next slide).

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Suppose that v = 2. At point P, opposite point O in the aperture, the aperture and the screen are centered symmetrically and the string is centered at point O s. If the aperture is moved up or down, the arc length of the string remains constant, but the length of the vector B 12 (v) changes, as before. It should be apparent that the length of B 12 (and the intensity at point P on the screen) will oscillate as the string slides around one of the spirals, which is equivalent to the slit moving up or down with respect to a reference point on the screen, as shown in the previous slide.

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It is also possible to visualize a clear minimum at the center of the near field diffraction pattern on the screen by considering the an arc-length of w = 3.5. Any change in the slit position will give and increase in B 12 and therefore an increase in irradiance. It is apparent that the slit width has a marked effect on whether the central position is a maximum or local minimum. Also note the oscillation in irradiance for positions beyond the width of the slit in both cases.

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