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University of Leicester Department of Physics and Astronomy Lecture Notes 1st Year Optics Professor R. Willingale April 7, 2012 Contents 1 Waves, Rays, Image Formation 112-3 2 1.1 Light Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Wavefronts - Huygen’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Rays - Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 The shape of a lens from first principles . . . . . . . . . . . . . . . . . . . . 7 1.6.1 The properties required of the lens . . . . . . . . . . . . . . . . . . 7 1.6.2 Deviation by a small angled prism . . . . . . . . . . . . . . . . . . . 8 1.6.3 An equation for the lens surfaces . . . . . . . . . . . . . . . . . . . 9 1.6.4 A spherical approximation . . . . . . . . . . . . . . . . . . . . . . . 10 1 1.7 Lens-maker’s equation using Fermat’s Principle . . . . . . . . . . . . . . . 10 2 Interference and Diffraction 112-4 12 2.1 Diffraction through a slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Young’s Slits - Two Source Interference . . . . . . . . . . . . . . . . . . . . 12 2.3 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Phasor Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Complex Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.7 Diffraction Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Interference from Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9 Fraunhofer diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.10 Diffraction from a Single Slit . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.11 Diffraction from a Double Slit . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.12 The Limit of Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . 22 1 Waves, Rays, Image Formation 112-3 The material covered is in the following sections of Tipler: • 33-1 Wave-Particle duality • 33-2 Light spectrum • 33-3 Sources of light • 33-4 The speed of light • 33-5 Propogation of light 2 • 33-6 Reflection and refraction • 34-1 Mirrors • 34-2 Lenses 1.1 Light Waves Monochromatic light (1 colour) is a travelling harmonic wave. It is actually an electro-magnetic wave but for the present analysis, amazingly, this is not important. If it is travelling along the x-axis in the +ve direction then it has the mathematical form ψ(x, t) = A sin(kx − ωt + φ) (1) where the frequency is ν = ω/(2π) and the wavelength is λ = 2π/k. φ is an arbitary constant phase angle. The speed of the wave is v = ω/k (2) This is called the phase velocity because it is the speed at which the peaks (or troughs) of the wave move. The phase velocity of a light wave depends on the medium. In a vacuum it is a constant v = c. We define the refractive index of a medium as n= c v (3) As light propogates the frequency remains constant. Therefore if n varies the wavelength varies. If n is large then k is large and λ is small or if n is small then k is small and λ is large. 3 1.2 Wavefronts - Huygen’s Principle Light always fills some volume in space. So ψ above has a value at every point within a volume. We can write ψ(r, t) = A sin(k.r − ωt + φ) (4) where r is a position vector and k is called the wavevector. |k| = 2π/λ and the wavevector points in the direction in which the wave is travelling. The argument of the harmonic sine function is called the phase angle. If this is kept constant then a 2-d surface is defined. This surface is called a wavefront. The wavevector k is perpendicular to the wavefronts. Huygen’s Principle is a geometrical construction that tells us how wavefronts will move (but not why). It states that: Every point on a wavefront acts as a source of spherical secondary wavelets such that after some time (∆t) the primary wavefront lies on the envelope defined by all the secondary wavelets. The radius of the secondary wavelets will be v∆t. If the refractive index varies with position we must take small time steps to get the correct answer. 1.3 Reflection and Refraction When light hits a plane interface between 2 media then some of the light is reflected and some is transmitted. The intensity of the reflected and transmitted beams depends on the materials and the angle of incidence. We can use Huygen’s Principle to find the reflection and refraction angles. θr = θ1 and Snell’s Law n1 sin θ1 = n2 sin θ2 (5) The refraction (change in direction from the incident to the transmitted beam) arises because of the change in wavelength of the light. Note that Huygen’s principle assumes there is no phase change at the boundary. 4 If n2 < n1 what happens if θ1 > θc where sin θc = n2 /n1 ? TIR. 1.4 Rays - Fermat’s Principle It is often useful to imagine light as consisting of rays. Such rays will always point in the direction of the wavevector k. Fermat’s Principle states that the path a light ray takes is such that the time taken to travel along the path is a minumum or stationary value compared with neighbouring paths. But the time taken to travel a distance d is given by t= nd d = v c (6) The product ∆ = nd is called the optical path length or simply the optical path. Since c is constant Fermat’s Principle also states that the optical path is a minimum or stationary value along the actual path. If n varies with position then we must integrate along the path to calculate the optical path length. ∆= Z P n(r)dl (7) S We can use this principle to prove the laws of reflection and refraction. See Tipler 33-8. 1.5 Image Formation A point source produces a set of spherical wavefronts moving away from the point or alternatively it produces a set of radial diverging rays. A real image of such a point source is formed by transforming these into a set of spherical wavefronts moving towards a point or alternatively a set of radial converging rays. A virtual image of a point source is formed by transforming these into a set of spherical wavefronts moving away from another point or alternatively a set of radial diverging rays from another point. 5 In image formation a set of rays from a source point S travel along different paths to an image point P. By Fermat’s Principle ALL these paths (rays) MUST have the same optical path length. Or equivalently they contain the same number of wave cycles. If we follow the passage of a single wavefront from S using Huygen’s Principle then eventually that wavefront will converge to a single image point P. Consider a simple imaging system S to P. The direct path SP is the shortest distance between the object and image. All other paths are longer. A curved mirror can compensate for this by making the peripheral paths shorter by just the correct amount to satisfy Fermat’s Principle. A convex lens can compensate for this by increasing the optical path length for rays near the axis. A virtual image is seen in a plane mirror. In this case the curvature of the wavefronts or divergence of the rays is unaltered. A virtual image is seen when looking through a refracting interface, for example the water surface of a swimming pool. The images seen are at a different position to the objects. The image of the bottom of the pool appears to be nearer than the actual bottom of the pool. In this case the curvature of the wavefronts from the object has been changed by refraction. Real and virtual images are formed by spherical mirrors and lenses. In these cases the curvature of the wavefronts is changed or the divergence of the rays is changed. Paraxial rays are such that they hit the spherical surfaces close to the normal or alternatively they are nearly parallel to the optical axis which is a normal to the surfaces. If we consider just paraxial rays then for a mirror of radius of curvature R using the law of reflection we can show that 1 2 1 1 + 0 = = s s R f (8) where s is the distance of the object from the mirror and s0 is the distance of the image from the mirror. f is called the focal length. Similarly for refraction across a single spherical interface of radius R between two media of refractive indices n1 and n2 using Snell’s Law we can show 1 n2 − n1 1 1 + 0 = = s s R f 6 (9) A thin lens consists of 2 spherical surfaces of radius of curvature Ra and Rb close together. In this case we can show 1 1 n1 = (n2 − n1 )( − ) f Ra Rb (10) This is the lens-makers’ equation. Lenses and spherical mirrors can convert plane wavefronts into spherical wavefronts and vice versa. In such a case the object or the image appears at the primary focus a distance f from the lens or spherical mirror. Note that plane wavefronts correspond to a source or image at infinity. 1.6 1.6.1 The shape of a lens from first principles The properties required of the lens 1) Parallel rays near the optical axis are brought to a common focus at a distance f from the lens (f is the focal length). For rays incident at distance x from the optical axis the angle of deviation is given by: tan(α) = x ≈α f providing x small (paraxial rays). 2) Diverging rays from a point source on the optical axis a distance s from the lens are converted into converging rays which produce an image of the point at a distance s0 the other side of the lens. If a diverging ray makes angle γ with the axis and a converging ray angle β with the axis then the deviation angle is given by: α=γ+β If the rays meet at the lens at a distance x from the axis and the angles are small we can substitute for the angles: 7 α= x x + s s0 We can substitute for α using the focal length f giving: x x x = + 0 f s s which reduces to the Gaussian imaging equation: 1 1 1 = + 0 f s s The incoming and outgoing rays intersect at a single plane called the Principal Plane of the lens. This plane is perpendicular to the optical axis. In order for this equation to hold the deviation angle at a distance x from the axis is given by (from above): α= 1.6.2 x f Deviation by a small angled prism If δ is the prism angle then the deviation of a light ray is given by: α = (n − 1)δ where n is the refractive index. Providing δ is small this is independent of the incidence angle. This is easy to prove using Snell’s Law for refraction. So if we have two nearly parallel surfaces on a dielectric with an angle δ between them the deviation of a ray passing through the surfaces is given by the above equation. 8 1.6.3 An equation for the lens surfaces We can combine the results in the last two sections to calculate the equation of the lens surfaces which will give the desired result. Assume the lens consists of two identical surfaces back-to-back. If we take the case of a convex lens the semi-thickness on-axis with be a maximum h0 . The semi-thickness h will decrease with radius x until we reach a point where h = 0, the largest possible radius for the lens. What is the equation of h as a function of x which will produce a focal length f? If the angle between the front and back surfaces of the lens is δ(x) the gradient of h is given by: δ(x) = −2 dh dx The deviation angle is therefore: α = −2(n − 1) dh dx But this is required to be the ratio x/f for imaging: x dh = −2(n − 1) f dx Rearranging we have: dh 1 = −x dx 2f (n − 1) Integrating this equation and imposing the boundary condition that h = h0 for x = 0 gives: h = h0 − x2 4(n − 1)f 9 1.6.4 A spherical approximation We can approximate the parabola above using a spherical surface of radius R. If ∆ = h0 − h(x) using Pythagoras we have: (R − ∆)2 + x2 = R2 Neglecting the second order term in ∆ gives: ∆= x2 2R But from the previous section: ∆= x2 4(n − 1)f So we have that the radius of the spherical surface must be: R = 2(n − 1)f or 1 1 1 = (n − 1)( + ) f R R which is the lens-maker’s equation for the case where both surfaces are the same radius and convex. QED. 1.7 Lens-maker’s equation using Fermat’s Principle We can derive the lens-maker’s equation directly using Fermat’s Principle. Consider paths from object to image intersecting at principal plane. Since all these are equivalent (possible) then All paths have the same optical path length. i.e. path is stationary w.r.t. intersection point on Principal plane. 10 Let maximum thickness of lens be 2h0 as before. Distance from object to lens surface s and lens surface to image s0 . We implicitly assume 2h0 << s and 2h0 << s0 . The optical path from object to sphere touching lens and from image to sphere touching lens is same for all paths. So we need only consider region close to Principal plane. The optical path has three components; from sphere about object to plane touching centre of lens, length t; from plane touching otherside of centre of lens to sphere about image, length t0 ; region of thickness h0 about Principal plane which contains glass of thickness 2h and refractive index n. All angles w.r.t. the axis are assumed small such that cos θ ≈ 1. So optical path is: ∆ = ((2h0 + t + t0 ) − 2h) + 2nh But for spheres: x2 t≈ 2s t0 ≈ x2 2s0 where x is the distance of the ray from the optical axis at the lens. Therefore we can substitute for t and t0 : ∆ = 2h0 + x2 1 1 ( + 0 ) + 2(n − 1)h 2 s s When x = 0 then h = h0 so ∆ = 2nh0 . We can replace the bracket containing s and s0 using the focal length f as before: x2 = 2(n − 1)(h0 − h) 2f This is the same parabola derived using Snell’s law above. 11 2 Interference and Diffraction 112-4 The material covered is in the following sections of Tipler: • 35-1 Phase difference and coherence • 35-2 Interference in thin films • 35-3 Two slit interference pattern • 35-4 Diffraction from a single slit • 35-6 Fraunhofer and Fresnel diffraction • 35-7 Diffraction and resolution • 35-8 Diffraction gratings 2.1 Diffraction through a slit When light passes through a narrow slit it spreads out or diffracts reaching regions on the far side of the slit which are inside the classical shadow. This is a consequence of Huygen’s Principle. A more detailed discussion of this will be given later. 2.2 Young’s Slits - Two Source Interference If monochromatic light from a distant point source passes through 2 narrow slits separated by a small distance d then there is a region on the far side where there are 2 sets of overlaping wavefronts. Within this region the wave amplitude is the sum of 2 components, E1 = E0 sin(ωt + φ1 ) and E2 = E0 sin(ωt + φ2 ) where φ1 and φ2 depend on the position. The resultant is E12 = E1 + E2 = 2E0 cos(δ/2) sin(ωt + φ/2) (11) where δ = φ1 − φ2 and φ = φ1 + φ2 . This is still a travelling wave of the same frequency but now the amplitude depends on δ which in turn depends on position. The intensity of light (the power level detected or the number of photons detected) is proportional to the square of the amplitude. If there is one slit I0 ∝ E02 . Therefore 12 I12 = 4I0 cos2 (δ/2) (12) If we consider a point far from the slits at angle θ wrt the axis then the path difference between the slits is ∆ = d sin θ and hence the phase difference is given by δ = d sin θ 2π = kd sin θ λ (13) If δ = 2mπ where m is an integer then we see a bright fringe. This is called constructive interference. If δ = (2m0 +1)π where m0 is an integer then we see a dark fringe. This is called destructive interference. 2.3 Michelson Interferometer A light beam can be split using a semi-reflecting interface between two media. These two beams can be recombined using mirrors. Now the path difference between the 2 components depends on twice the difference in the arm lengths 2(l2 − l1 ). The factor of 2 arises because the light travels to and from the returning mirrors. Hence the phase difference for light travelling along the axes is given by δ= 2π 2(l2 − l1 ) = k2(l2 − l1 ) λ (14) Hence if we move one of the mirrors we see interference fringes of the same intensity form as seen in Young’s slits. 2.4 Coherence It is implicit in the discussion of interference above that the 2 sources are travelling harmonic waves of the same frequency. Furthermore the waves are in phase at the slits in Young’s arrangement and in phase at the beam splitter in the Michelson Interferometer. 13 Real light waves can only approximate this ideal. If the light is a single colour then the range of frequencies is very small so the first condition is reasonably well satisfied. However light effectively comes in bursts or wave trains which only last a finite time and the phase difference between successive bursts is random. Therefore if the path difference is greater than the length of a typical wavetrain the phase between the 2 sources is random and the 2 sources are said to be incoherent. Incoherent sources will not interfere and will not produce fringes. Thus the visibility of interference fringes is a measure of how coherent the light is. Note that in all interference experiments the sources which are made to interfere ultimately always originate from the same source. Therefore the visibility of interference fringes in some experiment is a measure of the coherence of that source. It tells us how monochromatic the source is and how long the wavetrains from the source are. If there is more than one source or if the source has significant spatial extent then this can decrease the coherence. For example in Young’s slits experiment if the angular size of the source is too large when viewed from the slits the interferece fringes will disappear. In such a case the slits are exposed to overlapping wavefronts from different parts of the original source and these produce overlapping interference patterns with different lateral positions on the screen. 2.5 Phasor Diagrams Phasor diagrams are often used to illustrate how harmonic waves of a single frequency add together. See Tipler 35-5. A phasor is a vector with one end at the origin with a length equal to the amplitude of a harmonic wave. This vector rotates around the origin at angular velocity ω so after time t it will have rotated ωt radians. If the harmonic wave is cos(ωt + φ) then at time t = 0 the phasor makes an angle φ radians with the x-axis. Thus the amplitude of the wave at any time is represented by the projection of the vector onto the x-axis. i.e. the x-component of the phasor. It is conventional to draw phasors at t = 0 or alternatively to rotate the axes so that the phasor always makes an angle φ with the x-axis. We can find the sum of waves of the SAME frequency by adding their phasors by vector addition. The amplitude of the resultant phasor is the amplitude of the resultant harmonic wave. The intensity of the resultant wave is given by the square of the amplitude of the resultant phasor. This is proportional to the power in the wave (Watts m−2 say) or the number of photons s−1 m−2 . The amplitude is actually proportional to the transverse electric field vector and the energy density is proportional to the square of the transvers electric field 14 vector. 2.6 Complex Amplitudes It is very often convenient to represent a wave by a complex amplitude. ψ = A(cos(ωt + φ) + i sin(ωt + φ)) This can then be plotted as a point on an Argand Diagram. This will look exactly the same as a phasor diagram. The amplitude of the actual harmonic wave is given by the real part of the complex amplitude in just the same way as it is represented by the x-component of the phasor. So addition of harmonic waves can be done by complex addition of their complex amplitudes. Such a complex addition is equivalent to phasor addition. Finally the intensity of the harmonic wave is given by the square of the modulus of the complex amplitude. We can calculate this by taking the complex amplitude and multiplying by the complex conjugate. I = ψψ ∗ This is one of the aspects of complex numbers which is central to their application in physics. In the maths course you will find that A(cos(ωt + φ) + i sin(ωt + φ)) = A exp i(ωt + φ) This so-called complex exponential representation of harmonic waves is very powerful and exploited a great deal in physics. 2.7 Diffraction Gratings What happens if we replace 2 slits (Young’s Slits) with a series of N narrow, parallel, equally spaced slits? 15 At a position on a distant screen which subtends angle θ with the normal through the centre of the slits the path difference between adjacent slits is ∆ = d sin θ where d is the slit spacing so the phase difference is: δ = kd sin θ If the amplitude from each slit is S then ignoring any obliquity factor the total complex amplitude at angle θ is: A(θ) = S(1 + exp(iδ) + exp(i2δ) + · · · + exp(i(N − 1)δ)) Summing the geometric series gives: A(θ) = S 1 − exp(iN δ) 1 − exp(iδ) So the intensity (amplitude squared) at angle θ is: I = AA∗ = S 2 sin2 (N δ/2) sin2 (δ/2) Principle maxima occur at δ/2 = 0, ±π, ±2π, . . . Minima occur at δ/2 = ±π/N, ±2π/N, . . . Secondary maxima occur at δ/2 = ±3π/2N, ±5π/2N, . . . If N is large only the principal maxima are visible. These correspond to the orders of diffraction. The mth order is given by mλ = d sin θ For values of θ that satisfy this equation a very large peak is seen. Different wavelengths will appear at different angles so the grating is said to produce a dispersed spectrum. It splits up the light into the constituent colours. How wide are these peaks? We have to change the phase difference between adjacent slits by 16 ∆δ = 2π N to reach the first minimum (zero) either side of the peak. We also know by differentiating the expression for δ above that dδ = dk cos θ dθ Therefore the change in θ required to reach the minimum for the mth order is ∆θ = λ N d cos θm If N is large this angle is very small and hence the peaks are very narrow. If the peaks are narrow then the spectral resolution of the grating will be high. In practice gratings are constructed by scoring or etching grooves in a glass substrate. In modern gratings the grooves can be optimized in shape or blazed to maximize the intensity that is diffracted into certain orders for a chosen range of wavelengths. 2.8 Interference from Thin Films Consider a transparent parallel sided film of dielectric thickness d, illuminated by a monochromatic source at some distance. Plane wavefronts will hit the film at incidence angle θi . Some of the light will be reflected and some will be transmitted into the film. The refracted wavefronts meet the other side of the film and again a fraction is reflected while the remainder is transmitted. Finally the beam inside the film will meet the front face again and a fraction is transmitted to form a beam parallel to the original reflected beam. We want to calculate the condition for interference in the general case when θi is not zero. So we have 2 beams reflected from the film, 1 from the front surface and 1 from the rear. The optical path difference between these 2 components is: ∆ = nf (AB + BC) − n1 AD 17 AB = BC = d/ cos θt , AD = AC sin θi = AC(nf /ni ) sin θt using Snell’s law and AC = 2d tan θt so ∆= 2nf d (1 − sin2 θt ) = 2nf d cos θt cos θt If n1 = n2 then 1 beam suffers an internal reflection and the other beam an external reflection at a n1 : nf interface which introduces a phase difference of π if there is no absorption. So the phase difference between the beams is: δ = (2π/λ)2nf d cos θt ± π Hence for a maximum d cos θt = (2m + 1)λf /4 where m is an integer and λf = λ/nf . Note that this formula is very similar to that derived for the interference condition in the Michelson Interferometer. Actually the light suffers multiple reflections in such a film. We should sum a large number of reflection terms. If the surfaces of the film are highly reflecting the sum is similar to the case of the diffraction grating. Two parallel mirrors form a resonant cavity. Such cavities are used in lasers to increase the gain and/or to tune the light to a particular frequency. 2.9 Fraunhofer diffraction Fraunhofer diffraction is the diffraction of plane wavefronts through small diffracton angles. It is a limiting case which is mathematically easy to handle and is very important in the analysis of optical instrumentation. We can use lenses to produce an experimental arrangement that approximates the conditions for Fraunhofer diffraction quite accurately. This requires a collimator lens and a telescope lens. Without these lenses the source and screen must both be a large distance from the diffracting object so that the wavefronts are approximately planar. Thus using lenses increases the thoughput of light for the system. Note that diffraction is actually the propogation of light through or around objects. The so called diffraction pattern is created by interference at a screen or in some light detector. 18 2.10 Diffraction from a Single Slit How do we calculate the diffraction/interference pattern? We do it in the same way as for Young’s slits or a diffraction grating, by summing component amplitudes. We divide up the slit into a large number of small slits and sum the contributions from all these slits. Because the wavefront is continuous across the slit in the limit we take the sum of an infinite number of infinitesimal slits. i.e. we perform an integration. If the the wavefronts incident on the slit are plane and parallel to the slit plane all the components are in phase at the slit. Suppose the slit is width a centred on the origin. Then the wavefronts in region −a/2 < y < +a/2 are transmitted. At an angle θ wrt the axis the path difference wrt the origin is ∆ = y sin θ Therefore phase difference is δ= 2π y sin θ λ The amplitude from an infinitesimal slit at y of width dy is dE = Ec dy a where Ec is the amplitude at the centre of the viewing screen (on-axis). This implicitly assumes that the wavelet amplitude from a narrow slit is proportional to the width, doubling the slit width doubles the amplitude. So the sum of all the infinitesimal slits is given by the integral E= +a/2 Z −a/2 Ec sin(ωt + δ)dy a We can change the integration variable from y to δ E= Z +β −β Ec sin(ωt + δ)dδ 2β 19 where β= πa sin θ λ We can evaluate the definite integral by recalling that sin(A + B) = sin A cos B + cos A sin B so that Z +β Ec Z +β E= ( sin ωt cos δdδ + cos ωt sin δdδ) 2β −β −β Because the limits are symmetrical about the origin and sine is an odd function the second integral is zero. Therefore E= Ec sin ωt sin δ]+β −β 2β E = Ec sin ωt sin β β The intensity of the diffraction pattern is given by the time average of the square of the amplitude. I = Ic sin2 β β2 The function sin β/β is called the sinc function. Note that at β = 0 we must use L’Hospital’s rule to evaluate the value sinc(0) = cos(1) =1 1 The zeros of the sinc function occur when sin β = 0 excluding the point β = 0 which is the peak refered to above. Therefore for the minima 20 β = π, 2π, 3π, ..., nπ where n is an integer. These zeros correspond to sin θ = n 2.11 λ a Diffraction from a Double Slit We are now in a position to calculate the diffraction pattern from a Young’s slits arrangement in which the slits have a finite width a < d where d is the separation of the centres of the slits. We split the slits up into an infinite number of infinitesimal slits as above. The integral becomes E= Z +α+β Ec Z −α+β sin(ωt + δ)dδ) sin(ωt + δ)dδ + ( 2β −α−β +α−β where β is defined as above and α= πd sin θ λ We can evaluate the integrals using the same trigonometric substitution as before. E = 2Ec sin ωt sin β cos α β The intensity is therefore given by sin2 β I = 4Ic 2 cos2 α β This is the product of the single slit intensity pattern and the interference fringes expected from Young’s slits. The factor of 4 arises because now there are 2 slits which doubles the amplitude on axis and hence quadruples the intensity on axis. 21 Note if the width of the slits (a) is very small the first zero of the sinc function sin θ = nλ/a occurs at a very large angle and we recover the unmodulated cosinusoidal interference fringes as expected. 2.12 The Limit of Angular Resolution Consider a Fraunhofer diffraction setup consisting of a collimating lens (to produce plane wavefronts) and a telescope lens (to focus the diffracted plane wavefronts). We can consider any aperture between the collimator and telescope as defining the size of the aperture of the telescope. If the aperture is a single slit of width a then the first zero of the focused spot (in 1-D) occurs at an angle sin θ = λ/a. As the width of the slit increases so the width of the diffraction pattern (the focused spot) decreases. In practice circular apertures are much more common than slits or rectangular apertures. The diffraction pattern of a circular hole consists of a bright central spot surrounded by much fainter rings. In between the bright rings are dark rings. The pattern is calculated in the same way as for the slit above but the aperture is broken into infinitesimal areas rather than slits and the integral has to be performed in polar coordinates. The resulting pattern is called the Airy disk. The angular distance from the centre of the pattern to the first dark ring is given by ∆θ = 1.22 λ d where now d is the diameter of the circular aperture (rather than the width of the slit). So the change from a 1-D slit to a 2-D circular hole just introduces a numerical factor of 1.22. Again, as the diameter of the aperture (d) increases so the angular size of the central spot decreases. Now suppose a second point source is introduced in the focal plane of the collimator. This will produce a second image in the focal plane of the telescope. The Rayleigh criterion states that the two images will be resolved if the centre of the Airy disk of one lies at an angular radius larger than the first zero dark ring of the Airy disk of the other. 22 ∆θR ≈ 1.22 λ d Take a look back at the angular width of the diffraction orders of a grating. We found that ∆θ = λ N d cos θm The factor N d = a is just the total width of the grating so we have ∆θ = λ 1 a cos θm This formula has exactly the same form but now we find a numerical factor 1/ cos θm which usually lies in the range 1 to 1.5 for small orders m. The angular resolution of an optical system is always given by an equation of the form ∆θ ≈ λ d where d is some aperture size. This is true across the whole electromagnetic spectrum, from radio waves through to gamma rays. For example, consider the angular resolution of a satellite dish used to receive radio transmissions from a TV satellite. If the carrier frequency is 12GHz and the diameter of the dish is 0.75m what is the angular width of the beam? 23