Introduction to Condensed Matter Physics

Transcription

1 Introduction to Condensed Matter Physics Diffraction I Basic Physics M.P. Vaughan Diffraction Electromagnetic waves Geometric wavefront The Principle of Linear Superposition Diffraction regimes Single slit diffraction Multiple slit diffraction Multiple slits as a 1D lattice 1

2 Electromagnetic waves Maxwell s Equations D=ρ f, B= 0, B E=, t D H= j f +, t (Gauss Law) (no magnetic monopoles) (Faraday s Law of Induction) (Ampere s Law & Continuity) where ρ f and j f are the free charge density and current respectively,

3 Wave solutions of Maxwell s Equations Using Maxwell s Equations, a wave equation may be found of the form 1 E E=, v t where, in free space, the wave speed is v= c= µ ( ε ) 1/. Similar forms may be found in materials provided that the material is Wave solutions of Maxwell s Equations linear homogenous The electrical polarisation does not introduce nonlinear terms into the wave equation The optical properties of the material are the same everywhere. isotropic If these conditions are met, then c v=, n The optical properties of the material do not depend on the direction of the optical polarisation. where n is the refractive index. 3

4 Wave solutions of Maxwell s Equations Plane wave solutions may be found of the form i( k r ωt ( t) = E e ), Er, 0 where k is the wave vector and w is the angular frequency. From the wave equation, we find that ω v=. k This speed is often called the phase velocity, for reasons we shall illuminate. Phase velocity The condition k r ω t=φ 0 where φ 0 is a constant must be satisfied for all points in space having the same phase. Differentiating with respect to time, we obtain k v = k v cosα =ω where v is the wave speed (or phase velocity) and α is the angle between k and v. 4

5 Phase velocity Hence, putting v = v and k = k, v= ω. k cosα If v and k are parallel, then ω v=. k Note that the phase velocity is not the same as the group velocity. Wave propagation in free space Taking the gradient of k r ω t=φ 0, where φ 0 is a constant, will yield a vector normal to the surface of constant optical phase. Thus φ = k r= k. Therefore, for a linear, isotropic and homogeneous material, k is parallel to the phase velocity v. 5

6 Spherical waves In this case, it is convenient to use spherical polar coordinates. The wave vector k has constant magnitude and always points away from the centre of radiation. Thus we put and k= ke r r= re r so that the equation for surfaces of constant phase becomes kr ω t=φ 0. Spherical waves Since the intensity of an EM wave is proportional to the squared modulus of the amplitude, by the conservation of energy, the amplitude must vary as 1/r. Moreover, the requirement that the amplitude be finite at r = 0 means that the spherical wave must be of the form E E r r iωt ( r, t) = e sin kr. Note that such an equation cannot exactly model an EM wave. 6

7 Spherical waves A D section of the amplitude of a spherical wave. Geometric wavefront 7

8 Geometric wavefront A geometric wavefrontis the surface in space containing all points in an optical field that have the same phase. A rayis a path through space that is everywhere perpendicular to the wavefront. Geometric wavefront - spherical Wavefronts contours of constant phase 8

9 Geometric wavefront - spherical Wavefronts contours of constant phase Rays everywhere perpendicular to wavefronts Geometric wavefront - plane Wavefronts contours of constant phase 9

10 Geometric wavefront - plane Wavefronts contours of constant phase Rays everywhere perpendicular to wavefronts Huygens Principle Each point on a wavefront acts as a source of secondary, spherical wavelets. At a later time, t, a new wavefront is constructed from the sum of these wavelets. 10

11 Huygens Principle rectilinear propagation Consider the wavefront of a plane wave at z = 0 0 z Huygens Principle rectilinear propagation All points on the wavefront act as sources of spherical wavelets constant phase over surface of sphere 0 z 11

12 Huygens Principle rectilinear propagation Every point on the line contributes a similar sphere 0 z Huygens Principle rectilinear propagation Since all points on the spheres must have the same phase, the tangent to the leading edge of all the spheres must also be at a constant phase. 0 z 1

13 Huygens Principle rectilinear propagation This leading edge will be a new wavefront. Since it is parallel to the original wavefront, the light must be propagating in a straight line. 0 z Huygens Principle spherical waves Application of Huygens Principle to spherical waves. 13

14 Huygens Principle rectilinear propagation We have used Huygens Principle to prove the Law of Rectilinear Propagation for a plane wave. Note: we did not invoke the Principle of Superposition to prove this result. When the Principle of Superposition is explicitly added to Huygens Principle, it is becomes the Huygens-Fresnel Principle. Linear Superposition 14

15 Linear superposition Consider the differential equation E x 1 v E t = 0. Suppose E 1 and E are both solutions of this. Then we may construct the linear superposition E = ae 1 + be. Linear superposition We then have E 1 x v E t E = a x 1 E + b x = 0. 1 v 1 v E t 1 E t In other words, the linear superposition of E 1 and E is also a solution of the differential equation. 15

16 Linear superposition The differential equation E x 1 v E t = 0. is an example of a linear differential equation. On the other hand, consider the equation E t = 0 and suppose E 1 and E are both solutions of this. Linear superposition If we substitute in the linear superposition as before, we find E t = t = ab ( a E + abe E + b E ) 1 ( E E ) t Thus, in general, we cannot find new solutions by linearly adding known solutions. This type of differential equation is called nonlinear. 16

17 Linear superposition In free space or a linear medium, the electric field is linearly additive. Hence, we may apply the Principle of Linear Superposition to the electric field vector. Interference effects Linear superposition of the electric field vector will lead to interference effects. Diffraction is an interference effect. Physically interference and diffraction are one and the same phenomenon. 17

18 The Huygens-Fresnel Principle For light of a given frequency, every point on a wavefront acts as a secondary source of spherical wavelets with the same frequency and the same initial phase. The wavefront at a later time and position is then the linear superposition of all of these wavelets. Diffraction regimes: near field and far field 18

19 Light passing through a narrow aperture Maximum possible path difference max = AP BP = AB = D. Limiting cases: λ >> D max always less than λ wavelets add constructively in all directions. Emergent field looks like point source. 19

20 Limiting cases: λ << D Wavelets add constructively in this region Both constructive and destructive interference outside shaded region Fresnel and Fraunhofer diffraction Near field (Fresnel diffraction) Far field (Fraunhofer diffraction) 0

21 Fresnel and Fraunhofer diffraction Fresnel diffraction Diffraction pattern varies with increasing distance from aperture Fraunhofer diffraction Diffraction pattern settles down to a constant profile Applies for when the radial distance from the aperture satisfies the Fraunhofer condition (see later). Single slit diffraction 1

22 Single slit diffraction E L is the field strength per unit length E P is the total field a the point P Single slit diffraction Field at x de = E L dx. Contribution to field E P due to de de P Total field E P E EL = sinω r ( x) / = D P D / E r L ( x) [ t kr( x) ] dx. ( x) sin[ ωt kr ] dx.

23 Single slit diffraction r(x) is given by the cosine rule r r ( x) R + x Rx cos( π θ) = or x R x R ( x) = R 1+ sinθ. x 1/ To find a closed form solution, we must approximate this expression. Taylor series expansion of r(x) The Taylor series expansion for a function (1 + ξ) 1/ is Hence, and r ξ 1 ξ / ( 1+ ξ) = 1+ + K x x ( x) = R 1 sinθ+ cos θ+ K kr R 8 R ( x) = kr kxsinθ+ cos θ+k. kx R 3

24 The Fraunhofer condition The third term in the expression for kr(x) takes its maximum when x ± D/ and θ = 0. That is kx R cos kd θ = 8R πd 4λR. The condition that this term makes a negligible contribution to the phase is πd 4λR << π. The Fraunhofer condition Neglecting the factor of 4 in the denominator of the condition just found, it may be re-written as D R << λ. D This is the Fraunhofer condition for far field diffraction. 4

25 Far field approximations Assuming that the Fraunhofer condition is valid, the third term in the expression for kr(x) may be neglected and we have kr ( x) kr kx sinθ. The 1/r(x) factor appearing in the integral for E P is less sensitive to changes in r(x) than the phase and we may simply put 1 r ( x) 1. R Integrating over x Using these approximations, the expression for the total field E P becomes E To perform this integral, we note that sin P = EL R D / [ ωt kr kxsinθ] dx. sin + D / Im{ }. i( ωt kr+ kxsinθ) [ ωt kr+ kxsinθ] = e 5

26 The total field EP Integrating over the x-dependent part where D / D / e ikxsinθ ikxsinθ e dx= ik sinθ β = Hence, the total field E P is E P D / kd sinθ. D / ELD sinβ = sin( ωt kr). R β sinβ = D, β Intensity profile for a single slit Averaging E P over time gives E P = ELD sinβ. R β The squared modulus of this will be proportional to the intensity, i.e. sinβ β ( θ) I( 0). I = 6

27 Intensity profile for a single slit Intensity profile for a single slit The zeros of the peaks occur at values of β = kd sinθ = mπ, where m is an integer. Hence, the first zeros around the central peak are given by sinθ = λ. D Note that this result is only valid for λ < D. In other cases, there are no zeros from π to π. 7

28 Multiple slit diffraction Multiple slit diffraction 8

29 Multiple slit diffraction It will be useful to define α = ka sinθ, in analogy to the previously defined β = kd sinθ. Multiple slit diffraction Again, we make use of the earlier approximations for far field diffraction. For N slits, the total contribution of the field E P is E P = N 1 n= 0 na+ / na D / E R L sin [ ωt kr+ kxsinθ] dx. 9

30 Multiple slit diffraction Focussing on the x-dependent part of the integral and factorising as before, we find + / N 1 ikxsinθ na N 1 n= 0 e ik sinθ na D / = n= 0 e inα sinβ D. β The new factor is a geometric progression with common factor e iα N 1 n= 0 S N = e inα. Multiple slit diffraction Multiplying S N by e iα so which gives S N S S N e iα = N n= 1 e inα iα inα ( 1 e ) = 1 e, N = e i ( N 1), α sin Nα. sinα 30

31 Intensity profile for multiple slits The phase factor may be dropped from this expression. Note also that since sin N α lim = N, α 0 sinα it is useful to incorporate a normalising factor 1/N into this ratio. Hence, the intensity takes the form I sin Nα N sinα sinβ β ( θ) = I( 0). Intensity profile for multiple slits 31

32 Diffraction condition Note that the condition for constructive interference is a sinθ = nλ. Diffraction condition We can re-write this as But this is just ka sinθ =πn. α =πn, which gives the condition for the local maxima of the intensity I sin Nα N sinα sinβ β ( θ) = I( 0). 3

33 Multiple slits as a 1D lattice We can begin to see the connection between the diffraction pattern and crystal structure by imagining the array of slits to be a 1D lattice with lattice vector a= aˆx. The associated reciprocal lattice vector would then be π b = x ˆ. a Diffraction condition Incident and diffracted wavevectors: k= kẑ, k ' = k( xˆ sinθ+ zˆ cosθ). 33

34 Multiple slits as a 1D lattice We take the wavevector of the incident wave to be k= kẑ, and that of the diffracted wave to be k ' = k( xˆ sinθ+ zˆ cosθ). We then have k = k' k = k [ xˆ sinθ+ zˆ ( cosθ 1) ]. Multiple slits as a 1D lattice Considering just the x component, k x = k sinθ. Asserting the diffraction condition this becomes ka sinθ =πn, πn k x =. a 34

35 Multiple slits as a 1D lattice But this is just an integral multiple of the x component of the reciprocal lattice vector πn k = = a x nb x Writing this as a G vector, assertion of the diffraction condition then gives k x = G x.. 35