Topic 2: Scalar Diffraction

Transcription

1 V Topic 2: calar iffraction Aim: Covers calar iffraction theory to derive ayleigh-ommerfled diffraction. Take approximations to get Kirchhoff and resnel approximations. Contents:. Preliminary Theory. 2. eneral propagation between two planes. 3. Kirchhoff iffraction 4. resnel iffraction 5. ummary APPL PTC P PATMT of PYC calar iffraction -- Autumn Term

2 V calar Wave Theory Light is really a vector electomagnetic wave with and field linked by Maxwell s quations. ull solution only possible in limited cases, so we have to make assumptions and approximations. Assume: Light field can be approximated by a complex scalar potential. (amplitude) Valid for: Apertures and objects λ, (most optical systems). T Valid for: Very small apertures, ibre ptics, Planar Wave uides, gnores Polarisation. Also Assume: calar potential is a linear super-position of monochromatic components. o theory only valid for Linear ystems, (refractive index does not depend on wavelength). APPL PTC P PATMT of PYC calar iffraction -2- Autumn Term

3 V calar Potentials or light in free space the and field are linked by : = 0 : = 0 ^ = ε 0 µ 0 t ^ =, t which, for free space, results in the Wave quation given by 2, c 2 2 t 2 = 0 Assume light field represented by scalar potential Φ(r; t) which MT also obey the Wave quation, so: 2 Φ, c 2 2 Φ t 2 = 0 Write the Component of scalar potential with angular frequency ω as Φ(r; t) =u(r) exp(ıωt) then substituting for Φ we get that, [ 2 + κ 2 ]u(r) =0 where κ = 2π=λ or wave number. o that u(r) must obey elmholt quation, (starting point for calar Wave Theory). APPL PTC P PATMT of PYC calar iffraction -3- Autumn Term

4 V iffraction etween Planes y y x x 0 P 0 u(x,y;0) P u(x,y;) n P 0 the 2- scalar potential is: u(x;y;0)=u 0 (x;y) where in plane P the scalar potential is: u(x;y;) where the planes are separated by distance. Problem: iven u 0 (x;y) in plane P 0 we want to calculate in u(x;y;) is any plane P separated from P 0 by. Looking for a 2- calar olution to the full wave equation. APPL PTC P PATMT of PYC calar iffraction -4- Autumn Term

5 V ourier Transform Approach Take the 2- ourier Transform in the plane, wrt x;y. soinplanep we get (u;v;) = sincewehavethat u(x;y;) = ZZ ZZ u(x;y;)exp(,ı2π(ux + vy))dxdy (u;v;)exp(ı2π(ux + vy))dudv then if we know (u;v;) in any plane, then we can easily find u(x;y;) the required amplitude distribution. The amplitude u(x;y;) is a Linear Combination of term of (u;v;)exp(ı2π(ux + vy)) These terms are rthogonal (from ourier Theory), o each of these terms must individually obey the elmholt quation. ote: the term: (u;v;)exp(ı2π(ux + vy)) in a Plane Wave with Amplitude (u;v;) and irection (u=κ;v=κ), where κ = 2π=λ. APPL PTC P PATMT of PYC calar iffraction -5- Autumn Term

6 V Propagation of a Plane Wave rom elmholt quation we have that [ 2 + κ 2 ](u;v;)exp (ı2π(ux + vy)) = 0 oting that the exp() terms cancel, then we get that 2 (u;v;) 2 + 4π 2 λ 2, u2, v 2 (u;v;) =0 so letting then we get the relation that γ 2 = λ 2, u2, v 2 With the condition that 2 2 +(2πγ)2 = 0 8u;v we get the solution that (u;v;0)= 0 (u;v) = fu 0 (x;y)g (u;v;) = 0 (u;v)exp(ı2πγ) This tells us how each component of the ourier Transform propagates between plane P 0 and plane P,so: u(x;y;) = ZZ 0 (u;v)exp(ı2πγ)exp (ı2π(ux + vy))dudv which is a general solution to the propagation problems valid for all. APPL PTC P PATMT of PYC calar iffraction -6- Autumn Term

7 V ree pace Propagation unction ach ourier Component (or patial requency) propagates as: (u;v;) = 0 (u;v)exp(ı2πγ) efine: ree pace Propagation unction so we can write: (u;v;) =exp(ıπγ) Look at the form of (u;v;). (u;v;)= 0 (u;v) (u;v;) u 2 + v 2 + γ 2 = λ 2 Case : f u 2 + v 2 =λ 2 then γ is AL (u; v) Phase hift so all spatial frequency components passed with Phase hift. Case 2: f u 2 + v 2 > =λ 2 them γ is MAAY (u; v) =exp(,2πjγj) o ourier Components of 0 (u;v) with u 2 + v 2 > =λ 2 decay with egative xponential. (evanescent wave) APPL PTC P PATMT of PYC calar iffraction -7- Autumn Term

8 V requency Limit for Propagation n plane P where λ then the negative exponential decay will remove high frequency components. (u;v;) =0 u 2 + v 2 > =λ 2 so ourier Transform of u(x;y;) is of limited extent. Maximum spatial frequency when u = =λ, this corresponds to a grating with period λ. d θ d sinθ = nλ for d > λ d f o diffraction when d < λ. nformation not transferred to plane P. APPL PTC P PATMT of PYC calar iffraction -8- Autumn Term

9 V We have that Convolution elation (u;v;) = (u;v;) 0 (u;v;) so by the Convolution Theorem, we have that u(x;y;) =h(x;y;) u 0 (x;y;) With u(x;y;) the distribution in P due to u 0 (x;y;) in P 0. We then have that ZZ h(x;y;)= exp(ı2πγ)exp (ı2π(ux + vy)) dudv u 2 +v 2 <=λ 2 where r γ = λ 2, u2 + v 2 t-can-e-hown (with considerable difficulty), that where be have that h(x;y;) =, 2π λ 2 exp(ıκr) κr r 2 = x 2 + y and κ = 2π λ we therefore get that h(x;y;) = λ κr, ı exp(ıκr) r r which is known as The mpulse esponse unction for ree pace Propagation APPL PTC P PATMT of PYC calar iffraction -9- Autumn Term

10 V Point bject a δ( x,y ) y x r y x 0 P 0 n P 0 we have a elta unction, so: P u(x,y;) u 0 (x;y) =aδ(x;y) o in plane P we have u(x;y;) = ah(x;y;) a = λ κr, ı exp(ıκr) r r where r is the distance from: (0;0;0) ) (x;y;) o the intensity in plane P is given by: i(x;y;) = b2 r 2 κ 2 r 2 + where the λ 2 has been incorperated into the constant b. r 2 APPL PTC P PATMT of PYC calar iffraction -0- Autumn Term

11 V hape of unction The shape of i(x;0;) is shown below:.2 i(x,) i(x,2) i(x,3) i(x,4) for = λ;2λ;3λ;4λ, x-scale in λs. Compare with a isolated 3- point source, spherical expanding waves, so intensity in plane at distance of: (x;y;)= b2 r (x,) (x,2) (x,3) (x,4) ote We have a 2- elta unction, (whole in a screen), and T an 3- point source. APPL PTC P PATMT of PYC calar iffraction -- Autumn Term

12 V The full convolution expression is u(x;y;) = ull xpression ZZ P 0 u 0 (s;t)h(x, s;y,t;)dsdt where s;t are variables in plane P 0 The h() term will contain terms of the form (x, s) 2 +(y,t) = l 2 so l is istance from (s;t;0) ) (x;y;) t s y x 0 l u(s,t;0) P 0 P u(x,y;) ull xpression is u(x;y;) = λ ZZ P 0 u 0 (s;t) κl, ı exp(ıκl) dsdt l l ayleigh-ommerfeld iffraction quation APPL PTC P PATMT of PYC calar iffraction -2- Autumn Term

13 V Kirchoff iffraction Look at mpulse esponse unction: h(x;y;) = λ κr, ı exp(ıκr) r r Most Practical cases, P 0 and P separated by MAY wavelength, Approximate the term: so that ) λ ) r λ κr, ı,ı h(x;y;) exp(ıκr) ıλ r r Look at Terms exp(ıκr) r pherically expanding wave from point (0; 0; 0) r bliquity actor which forces propagation in direction. (n oodman s notation the obliquity is written as a cos() term). The other terms are just constants. APPL PTC P PATMT of PYC calar iffraction -3- Autumn Term

14 V ote: or 3- point source, get expanding spherical wave obliquity factor results from 2- source in a plane. mpulse esponse unction: pherical Wave with directional weighting term. Model: ach point in P 0 acts a source of impulse response functions, that sum is P. h h(x; y; ) =pherical Wave ri This is ygen s econdary Wavelet (weighted by obliquity factor). ygen s econdary Wavelets Model: ach point on the wavefront gives rise to pherical Waves, h(x;y;) = exp(ıκr) r Add postulate that wave propagate in positive direction. h Kirchhoff, ygen s ri APPL PTC P PATMT of PYC calar iffraction -4- Autumn Term

15 V Kirchhoff iffraction ntegral n ayleigh-ommerfeld integral, λ ) l λ make the approximation that κl, ı,ı so we get that u(x;y;) = u 0 (s;t) ıλ ZZP0 exp(ıκl) l which is valid (%), > 20λ. Typical starting point for optical calculations. Look at =l factor: l dsdt x s θ P 0 l P so that l = cosθ ame expression as in books [eg. oodman page 52, equation (3-5)] APPL PTC P PATMT of PYC calar iffraction -5- Autumn Term

16 V resnel Approximation Assume extend of P 0 and P x r x P o we have that 0 jxj&jyj We have Kirchhoff impulse response exp(ıκhr) h(x;y;) = ıλ r r Which we can write as: h(x;y;) =A(x;y;)exp(ıκr) where the amplitude A(x;y;) = ıλ ince x&y <<, we can expand r as r 2 r = + x2 + y 2 2 x 2 + y P APPL PTC P PATMT of PYC calar iffraction -6- Autumn Term

17 V Amplitude Term: irst rder approx: r ) A(x;y;) ıλ Phase Term: econd rder approx: r + x2 + y 2 2 exp(ıκr) exp(ıκ) exp o the resnel Approximation is that h(x;y;) exp(ıκ) ıλ exp ı κ 2 (x2 + y 2 ) ı κ 2 (x2 + y 2 ) resnel Approximations ) eplace pherical waves by Parabolic Waves 2) gnore bliquity factor. T: only valid jxj&jyj ) mall bjects This is also known as Paraxial Approximation. seful in many practical systems. se in the rest of the course APPL PTC P PATMT of PYC calar iffraction -7- Autumn Term

18 V We have that = exp(ıκ) ıλ resnel iffraction ZZ u(x;y;) =h(x;y;) u 0 (x;y) u 0 (s;t)exp ı κ 2 This we can expand to get the } { ıλ exp(ıκ) ZZ 2 } { exp ı κ 2 (x2 + y 2 ) 4, (x, s) 2 +(y,t) 2 dsdt } { u 0 (s;t)exp ı κ 2 (s2 +t 2 ) exp,ı κ (sx +ty) dsdt { } 3 Look the terms. Absolute amplitude and phase, depends only on (constant which is not normally important). 2. Parabolic phase term, no effect on intensity. 3. ourier Transform scaled by κ= 4. calar distribution in P 0 weighted by parabolic phase term. resnel iffraction ) ourier Transform weighted by parabolic phase term [+ xtra phase and constants] APPL PTC P PATMT of PYC calar iffraction -8- Autumn Term

19 V n ourier pace we have so we have that n ourier pace (u;v;) = (u;v;) 0 (u;v) (u;v;) = ıλ ZZ exp(ıκ) exp ı κ 2 (x2 + y 2 ) exp (,ı2π(ux + vy))dxdy This can be separated into 2 integrals in x and y, and with identity Z r π exp(,bx 2 )exp(ıax)dx =, b exp a2 4b, t-can-be-hown that (u;v;) =exp(ıκ)exp, ıπλ(u 2 + v 2 ) which is again a parabolic term. resnel diffraction ) Multiplication inourierplane byparabolic phase term. APPL PTC P PATMT of PYC calar iffraction -9- Autumn Term

20 V ummary calar Wave solution for prapagation between two plane: ull olution ) ayleigh-ommerfeld f distance between planes > a few wavelengths then caler Kirchoff iffraction This is ygen s econdary pherical wavelets plus obliquity factor due to 2- whole in plane. f distance between planes Large and planes are mall make small angle approximation to get resnel iffraction which replaces pherical waves with Parabolic and ignores obliquity factor. resnel iffraction will be used for the rest of this course. APPL PTC P PATMT of PYC calar iffraction -20- Autumn Term