DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS

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1 DOING PHYIC WITH MTLB COMPUTTIONL OPTIC FOUNDTION OF CLR DIFFRCTION THEORY Ian Coope chool of Physics, Univesity of ydney DOWNLOD DIRECTORY FOR MTLB CRIPT View document: Numeical evaluation of the Rayleigh-ommefeld diffaction integal of the fist kind CLR DIFFRCTION INTEGRL The integal theoem of Helmholtz and Kichhoff Kichhoff bounday conditions on apetue E( ) E ( ) and E( ) E ( ) 1 e E(P) E( ) jk 1 ˆ E( ) n d 4 jk Rayleigh-ommefeld diffaction integal of the second kind 1 e E(P) E( ) n d jk Rayleigh-ommefeld diffaction integal of the fist kind 1 e E(P) E( ) z ( 1) d 3 p jk jk Doing Physics with Matlab op_diffaction_integals_theoy.docx 1

2 The Rayleigh-ommefeld egion includes the entie space to the ight of the apetue. It is assumed that the Rayleigh-ommefeld diffaction integal of the fist kind is valid thoughout this space, ight down to the apetue. Thee ae no limitations on the maximum size of eithe the apetue o obsevation egion, elative to the obsevation distance, because no appoximations have been made. Doing Physics with Matlab op_diffaction_integals_theoy.docx

3 THE INTEGRL THEOREM OF HELMHOLTZ ND KIRCHHOFF The basic idea of the Huygens-Fesnel theoy is that the light distubance at a point P aises fom the supeposition of seconday waves that poceed fom a suface situated between this point P and the light souce. Kichhoff fomulated a sound mathematical basis fo this pinciple based upon solutions of the homogeneous wave equation, whee the value at an abitay point P in the field can be expessed in tems of the values of the solution and its fist deivatives at all points on an abitay closed suface suounding the point P. We conside stictly monochomatic scala waves that ae solutions of the wave equation 1 E(, t) E t (, ) 0 c t (1) whee E(, t ) is electic field (o monochomatic optical distubance) and can be witten as jt j k ct E(, t) E( ) e E( ) e () olutions of the wave equation fo plane waves taveling the diection + and - ae + diection: - diection: ( ) E(, t) E e j k t o ( ) (, ) e j E t E k t o (3) olutions of the wave equation fo spheical waves taveling the diection + and - ae + diection: - diection: E o ( ) E(, t) e j k t E o ( ) E(, t) e j k t In vacuum, the space-dependent pat then satisfies the Helmholtz equation E( ) k E( ) 0 (4) Doing Physics with Matlab op_diffaction_integals_theoy.docx 3

4 In poblems involving boundaies it is often convenient to study the popeties of the diffeence between two solutions of an equation athe than one of the solution alone, since the bounday conditions become simple to handle. Take the oigin at the obsevation point P, at which the electic field has the value E(0). The two wave fields, E( ) the unknown field to be calculated and E t ( ) a tial solution of the Helmholtz equation satisfy the equation E E E E E k E E k E (5) ( ) t( ) t( ) ( ) ( ) t( ) t( ) ( ) 0 at all points except at the oigin = 0. Now, integate (5) thoughout a volume V bounded by a suface and by Geen s theoem, the volume integal can be changed to a suface integal: t t t t V E( ) E ( ) E ( ) E( ) d V E( ) E ( ) E ( ) E( ) n d 0 E ( ) E ( t ) E t( ) E ( ) n d 0 (6) n being the inwad pointing nomal to the suface. We assume that both E( ) and E t ( ) possess continuous fist- and second-ode patial deivatives within and on this suface. Because the integand is zeo in (5), the integals in (6) ae also zeo. Futhemoe, the volume containing P must be fee of any souces of optical adiation (including seconday souces of eadiated o scatteed light). Note: tems such as of integation. n give the nomal component of the gadient at the suface Conside a tial solution to be E ( ) ot j k Et e (7) which is a spheical wave adiating fom the oigin at P. This spheical wave Et ( ) plays the ole of a mathematical auxiliay function a pobe which we use fo investigating the optical field E ( ). This is a vitual spheical wave and has nothing to do with any eal spheical waves that act as the souce functions. This tial function has a singulaity at the point = 0 and since E t ( ) was assumed to be continuous and diffeentiable, the point P must be excluded fom the domain of integation in equation (6). This is done by suounding P by a small sphee of adius and taking the suface to be divided into two pats, an abitay oute suface o and the small inne spheical suface i of adius. Doing Physics with Matlab op_diffaction_integals_theoy.docx 4

5 d o n V i P n Theefoe, the gadient of the tial function is jk jk Eote ˆ Eote 1 jk E ( ) 1 ˆ t jk Eote (8) Howeve, afte we make the substitutions into of equations (7) and (8) into (6) we can take these waves to descibe spheical waves which ae adiated by elements of the suface and which aive at the point P at a distance fom d. Theefoe, (6) become o E( ) E ( ) E ( ) E( ) n d E( ) E ( ) E ( ) E( ) n d 0 t t t t i (9) The contibution fom i can be evaluated diectly, since ove the small sphee of adius we conside E() to be constant and equal to E(0). The unit vecto n is paallel to, so we have n = 1 and we can substitute d fo d whee d is an element of the solid angle. Theefoe, the integal ove the inne suface can be witten as Doing Physics with Matlab op_diffaction_integals_theoy.docx 5

6 ˆ 1 I E e E(0)( j k 1) E(0) n d jk i ot i jk Ii Eot e E(0)( j k )d E(0)d E(0) n d i i i limit 0 I 4 E E(0) i ot (10) The value of the electic field at the obsevation point P is o ˆ j k j k Eo e E( ) j k 1 Eot e E( ) n d 4 Eot E(0) 0 1 e E(0) E( ) j k 1 ˆ E( ) n d 4 j k (10) o Thus E(0) can be found if the values fo E( ) and E ( ) ae known at all points on the oute suface suounding the point P. This is one fom of the integal theoem of Helmholtz and Kichhoff. This theoem expesses E(0) in tems of both E( ) and E( ). It may, howeve be shown that fom the theoy of Geen s functions, that eithe E( ) o E( ) on o ae sufficient to specify E(0) at evey point P with o. In applying the integal theoem of Helmholtz and Kichhoff to diffaction poblems, the suface o is divided into thee: 1) the diffaction apetue (opening), ) the sceen blocking the adiation B, and 3) since the suface must be closed, we must include a mathematical suface of adius R cente about the point P, so that when R the integal ove the integal in (10) will appoach zeo o R is chosen so lage thee is insufficient time fo a contibution fom this suface to have eached the point P. Doing Physics with Matlab op_diffaction_integals_theoy.docx 6

7 p B p Q s n R P n souces o = + B + It emains to integate (10) ove + B. Howeve, the actual field and its gadient ove the suface ae almost impossible to detemine except in the simplest cases because fo eal physical apetues will patially absob, eflect and scatte the incident adiation. The fist appoximation to make in integating (10) is to ignoe the eal bounday values and apply the so-called Kichhoff bounday conditions and ae the basis of Kichhoff s diffaction theoy. These eplace the actual field in the pesence of the apetue by the incident field fo the open apetue E ( ) and by zeo elsewhee: Kichhoff bounday conditions: on B E ( ) 0 and E ( ) 0 (11a) on E( ) E ( ) and E( ) E ( ) (11b) Then (10) becomes 1 e E(0) E ( ) jk 1 ˆ E ( ) n d 4 jk (1) The Kichhoff bounday conditions ae an appoximation, since at the edge of the opening, the amplitude of the wave does not suddenly jump fom a finite value to zeo. The appoximation is bette when the apetue is lage compaed to the wavelength. Doing Physics with Matlab op_diffaction_integals_theoy.docx 7

8 tictly speaking the bounday conditions (11) ae mathematically inadmissible. theoem in Riemann s theoy of functions implies that if the value of a function and its fist deivative vanish along a segment then the function vanishes eveywhee, in fact the assumed bounday conditions even contadict each othe and if we would calculate the value at P on the suface, the bounday conditions would not be epoduced. In pinciple, a scala appoach can be pefomed fo each component of the vecto wave, but in pactice this is aely necessay. We can assume that the adiation fom the opening is much lage than fom the blocking sceen. lso, well into the opening, all possible diections of polaization ae possible fo the adiation. Often, edge effects ae only pedominant in a egion within less than a wavelength fom the edges. Theefoe, in most pactical applications the use of a scala appoach is justified. Howeve, in the scala appoach, all infomation about the polaization of the wave is lost. Doing Physics with Matlab op_diffaction_integals_theoy.docx 8

9 GREEN FUNCTION IMPLIFIED FORMULTION OF HUYGEN PRINCIPLE Rayleigh-ommefeld diffaction integal of the fist kind Rayleigh-ommefeld diffaction integal of the second kind The mathematical contadiction posed by the Kichhoff bounday conditions can be avoided by substituting fo the tial spheical wave function Et ( ) given by (7) by the Geen s function G belonging to the suface suounding the point P. This function is defined by the following conditions: within the volume V G k G 0 (13a) on B G = 0 (13b) as 0 G E t ( ) (13c) as ( G n j k G) 0 (13d) as befoe, is the distance fom the point P and (13d) is called the adiation condition. G has a singulaity only at the point P ( = 0) and is continuous eveywhee else within the volume V. G diffes fom Et ( ) because of the additional condition (13b). s a esult of this condition the tem containing the tem E ( ) in equation (9) vanishes and theefoe (9) simplifies to 1 E(P) E( ) G( ) n d 4 (14) o Now, we need to specify only the bounday values of E ( ) and not its gadient. This appoach has the pactical advantage of leading to the simple fom of the integal as given in (14) as compaed to (1). This appoach again is only an appoximation and is valid fo sufficiently small wavelengths. The eal field does not vanish completely behind the blocking sceen, no is the field entiely unaffected by the pesence of the sceen, at least not within distances of the ode of magnitude of a wavelength fom the edge of the sceen. Howeve, the applicability of the Geen s function method is esticted to the special case of a plane sceen, fo this is the only case fo which the Geen s function can be conveniently expessed. Doing Physics with Matlab op_diffaction_integals_theoy.docx 9

10 Note: The wavelength is eally only defined fo plane monochomatic waves and can lose its significance in wave types encounteed in diffaction. Howeve, we can always intepet as the length c / which is defined fo all monochomatic adiation pocesses and which fo plane waves is identical with the actual wavelength. The method of images is used to detemine the appopiate Geen s function. (x d, y s, z s ) sq sceen + Z z = 0 O(0, 0, 0) Q(x q, y q, z q ) n P(x p, y p, z p ) cos = z p / Constuction of the Geen s function fo a plane sceen We constuct the mio image of the point P with espect to the plane of the sceen z = 0. Fo an abitay point Q(x q, y q, z q ) whee z q > 0, we fom the Geen s function whee j k j k sq e e G (15) sq ( p q) ( p q) ( p q) x x y y z z sq ( s q) ( s q) ( s q) x x y y z z xp xs yp ys zp zs (16) We have to detemine G to evaluate (14). Fistly, j k j k sq G d e d e z d z d z q q sq sq q sq (17) Doing Physics with Matlab op_diffaction_integals_theoy.docx 10

11 If we now place Q on the sceen, we have sp z z cos z z sq p p q q j k (18) G G e G cos (19) n z q jk e 1 ( 1) j k j k e (0) ubstituting (19) and (0) into (14) we obtain and we can integate ove the aea. jk e 3 p (1) 1 E(P) E( ) z ( jk 1) d (1) is known as the Rayleigh-ommefeld diffaction integal of the fist kind. If we make the appoximation that k >> 1 then 1 ( j k 1) j then (1) can be witten as j e E(P) E( ) cos d () j k This expession () is equivalent to Huygens s pinciple a light wave falling on an apetue popagates as if evey element d emitted a spheical wave, the amplitude and phase of which ae given by that of the incident wave. The facto cos elates to Lambet s law of suface bightness. Instead of the Geen s function (15) which satisfies the bounday conditions (13) in which G = 0 at z = 0, we now fom j k j k sq e e G (3) sq Doing Physics with Matlab op_diffaction_integals_theoy.docx 11

12 This is a function which satisfies the bounday condition G/z = 0 at z = 0. s a esult of this condition the tem containing the tem E ( ) in equation (9) vanishes and theefoe (9) simplifies to 1 E(P) G( ) E( ) n d 4 (4) o Now, we need to specify only the bounday values of E ( ) and not the value of E ( ). If we substitute the values of the Geen s function on the plane z = 0 we obtain 1 e E(P) E( ) n d (5) j k (5) is known as the Rayleigh-ommefeld diffaction integal of the second kind. The Rayleigh-ommefeld egion the entie space to the ight of the apetue. It is assumed that the Rayleigh-ommefeld diffaction integals ae valid thoughout this space, ight down to the apetue. Thee ae no limitations on the maximum size of eithe the apetue o obsevation egion, elative to the obsevation distance, because no appoximations have been made. The Fesnel egion (nea field) that potion of the Rayleigh-ommefeld egion within which the Fesnel conditions ae satisfied. Both the input and output signals ae esticted to egions lying nea the z axis, i.e., the lateal dimensions of which ae much smalle than the sepaation between the input and output planes. z L L (6) 1 whee L 1 and L ae the maximum lateal extents of the apetue and obsevation planes espective. lso it is assumed that z L L 4 (7) Conditions (6) and (7) ae sometimes known as the Fesnel conditions. Doing Physics with Matlab op_diffaction_integals_theoy.docx 1

13 The Faunhofe egion (fa field) that potion of the Fesnel egion within which the Faunhofe condition is satisfied z 1 L (30) In the Faunhofe egion the size of the diffaction patten inceases with inceasing distance but its shape is invaiant. The Rayleigh distance drl The Rayleigh distance in optics is the axial distance fom a adiating apetue to a point an obsevation point P at which the path diffeence between the axial ay and an edge ay is λ / 4. good appoximation of the Rayleigh Distance d RL is 4a drl whee a is the adius of the apetue. Rayleigh distance is also a distance beyond which the distibution of the diffacted light enegy no longe changes accoding to the distance z P fom the apetue. z P < d RL z P > d RL Fesnel diffaction Faunhofe diffaction. If we conside a cicula apetue of adius a, then much of the enegy passing though the apetue is diffacted though an angle of the ode / a fom its oiginal popagation diection. When we have tavelled a distance drl fom the apetue, about half of the enegy passing though the opening will have left the cylinde made by the geometic shadow if a/ drl. Putting these fomulae togethe, we see that the majoity of the popagating enegy in the "fa field egion" at a distance geate than the Rayleigh distance drl 4 a / will be diffacted enegy. In this egion then, the pola adiation patten consists of diffacted enegy only, and the angula distibution of popagating enegy will then no longe depend on the distance fom the apetue. Refeences Bon 417 Hect 501,648 Klein 360 Lipson 156 late 168 ommefeld 197 Gaskill p385 Doing Physics with Matlab op_diffaction_integals_theoy.docx 13