Chapter 4 Nanscale Reslutin in the Near and Far Field Intensity Prfile f Optical Diple Radiatin Xin Li * and Henk F. Arnldus Mississippi State University * xl@msstate.edu hfa@msstate.edu Jie Shu Rice University js35@rice.edu When an electric diple mment rtates, the flw pattern f the emitted energy exhibits a vrtex structure in the near field. The field lines f energy flw swirl arund an axis which is perpendicular t the plane f rtatin f the diple. This rtatin leads t an apparent shift f the diple when viewed frm the far field. The shift is f the rder f the spatial extend f the vrtex, which is abut a fractin f an ptical wavelength. We als shw that when an image f the radiatin is frmed n an bservatin plane in the far field, the rtatin f the field lines in the near field leads t a shift f the diple image. 4.. Intrductin When light is emitted by a lcalized surce, it appears as if the light travels alng straight lines frm the surce t the bserver, when viewed frm the far field (many wavelengths frm the surce). These light rays are the flw lines f energy, and they are usually referred t as ptical rays. The rays are the rthgnal trajectries f the wave frnts. In the gemetrical ptics limit f light prpagatin certain terms in Maxwell s equatins can be neglected under the assumptin that the wavelength f 379
380 X. Li, H. F. Arnldus & J. Shu the light is small cmpared t ther relevant distances. It can then be shwn [] that in a hmgeneus medium, like the vacuum, the rays are straight lines, running frm the surce t the far field, and they cincide with the field lines f energy flw. Hwever, when the light is detected within a fractin f a wavelength frm the surce, r with nanscale precisin at a larger distance, the gemetrical ptics limit des nt apply, and we have t cnsider the exact slutin f Maxwell s equatins at all distances. The flw lines f energy will in general be curves. Far away frm the surce, each curve will asympttically apprach a straight line, but, when measured with nanscale reslutin, these asympttic lines will nt cincide with the ptical rays. The rays run radially ut frm the surce, but an asymptte f a field line f energy flw culd be displaced with respect t this directin. When the dimensin f a lcalized surce f radiatin is small cmpared t the wavelength f the emitted light, then the surce is in first apprximatin an electric diple, lcated at the center f the surce, and when viewed frm utside the surce, the radiatin is identical t the radiatin emitted by a pint diple. Als, radiatin emitted by atms and mlecules is usually electric diple radiatin, and since atms and mlecules are much smaller than the wavelength f the light they emit, we can cnsider them as pint surces. Since electric diple radiatin is the mst elementary type f radiatin, we shall cnsider the nanscale structure f this type f radiatin. We shall cnsider the spatial distributin f the energy flw in the near field, and we shall shw that the curving f the field lines in the near field has an effect n the bservable intensity prfile in the far field, prvided that the measurement is carried ut with sub-wavelength reslutin. 4.. Electric Diple Radiatin When the current density f a lcalized surce scillates harmnically with angular frequency ω, fr instance when the surce is placed in a laser beam scillating at the same frequency ω, then the induced electric diple mment has the frm iωt d ( t) = d Re( ε e ), (4.)
Nanscale Reslutin in the Near and Far Field Intensity 38 with d > 0 and ε a unit vectr, nrmalized as ε ε* =. The radiated electric field will als have a harmnic time dependence, and can be written as iωt E ( r, t) = Re[ E( r) e ], (4.) with E (r) the cmplex amplitude. A similar expressin hlds fr the magnetic field B ( r, t). When the diple is lcated at the rigin f crdinates, the cmplex amplitudes f the electric and magnetic fields are given by [] 3 d k i = E( r) ε - ( ε r ˆ)ˆ r + [ ε - 3( ε r ˆ)ˆ] r q + 4πε iq i e q q, (4.3) 3 d k B( r) = ε r ˆ + 4πε i iq e q q, (4.4) where k = ω / c is the wave number in free space and rˆ is a unit vectr which is directed frm the lcatin f the diple t the field pint, represented by r. We have als intrduced the dimensinless variable q = kr, which represents the distance between the field pint and the diple. In this way, a distance f π in terms f q crrespnds t a distance f ne ptical wavelength in terms f r. The prperties f the electric diple mment enter the expressins fr E (r) and B (r) nly thrugh the cmplex-valued unit vectr ε (apart frm an verall d ). 4.3. Energy Flw and the Pynting Vectr In an electrmagnetic field the energy flw is determined by the Pynting vectr, defined as S ( r, t) = E( r, t) B( r, t), (4.5) µ fr prpagatin in vacuum. If da is an infinitesimal surface element at the psitin r, with unit nrmal nˆ, then the pwer flwing thrugh da is
38 X. Li, H. F. Arnldus & J. Shu equal t dp = S nˆ da. Therefre, at psitin r the energy flws int the directin f the Pynting vectr S. Fr time-harmnic fields, as in Eq. (4.), the Pynting vectr simplifies t S ( r) = Re [ E( r) B( r)*], (4.6) µ and here terms that scillate at twice the ptical frequency ω have been drpped, since these average t zer in an experiment, n a time scale f an ptical cycle. The Pynting vectr in Eq. (4.6) nly invlves the cmplex amplitudes, rather than the fields themselves, and S (r) is independent f time t. With expressins (4.3) and (4.4) the Pynting vectr fr the radiatin emitted by a diple can be evaluated immediately. We btain 3P = S( r) ζ ( θ, φ)ˆ r + Im[(ˆ r ε ) ε* ] 8π r q q, (4.7) where θ and φ are the angles f the bservatin pint r in a spherical crdinate system. The functin ζ ( θ, φ) is given by ζ ( θ, φ) = (ˆ r ε)(ˆ r ε* ), (4.8) and P 4 ck = d, (4.9) πε is the pwer emitted by the diple. The unit vectr rˆ is in spherical crdinates r ˆ = ( e csφ + e sinφ)sinθ + e csθ, (4.0) x y and this determines the dependence f ζ ( θ, φ) n θ and φ, given the diple mment vectr ε. When we take da as part f a sphere, centered arund the rigin, and with radius r, then da = r dω, and the radiated pwer per unit slid angle becmes z
Nanscale Reslutin in the Near and Far Field Intensity 383 P = dω r d S( r) rˆ. (4.) With Eq. (4.7) this becmes d P 3P = ζ ( θ, ) dω 8π φ, (4.) since ( rˆ ε) ε * rˆ is real. The emitted pwer per unit slid angle is independent f r, and this may give the impressin that the pwer flws radially utward, as in the gemetrical ptics limit f light prpagatin. We shall see belw that this is usually nt the case. When vectr ε in Eq. (4.) is real, the diple mment is d ( t) = dε cs( ωt). This crrespnds t a linear diple mment, scillating back and frth alng an axis thrugh vectr ε. Fr this case, the Pynting vectr becmes 3P S( r) = ζ ( θ, φ) rˆ, (4.3) 8π r since ( rˆ ε) ε * is real. Then S (r) is prprtinal t rˆ at any field pint, and hence the pwer flw is exactly in the radial directin. The field lines f S (r), which are the field lines f energy flw, are straight lines, running frm the lcatin f the diple t infinity. 4.4. Elliptical Diple Mment In its mst general state f scillatin, the diple mment d (t) traces ut an ellipse in a plane [3,4]. We take this plane t be the xy-plane, and we parametrize vectr ε as ε = ( β e x + ie y ) β +, (4.4) with β real. The parametrizatin is chsen such that fr β = ±, vectr ε reduces t the standard spherical unit vectrs
384 X. Li, H. F. Arnldus & J. Shu e ± = ( ± e x + ie y ), (4.5) with respect t the z-axis. With ε frm Eq. (4.4), the diple mment becmes d d ( t) = [ β e x cs( ωt) + e y sin( ωt)]. (4.6) β + As time prgresses, vectr d (t) fllws the cntur f an ellipse, as shwn in Fig. 4.. Fr β = ± the ellipse reduces t a circle. When β is psitive, the rtatin is cunterclckwise as in the figure, which is the psitive directin with the z-axis, as given by the right-hand rule. Fr β < 0 the rtatin is in the ppsite directin. Fr β = 0, vectr ε becmes ie y, and the scillatin is linear alng the y-axis. In the limit β ± we have ε me x, and the scillatin is linear alng the x-axis. Fr an elliptical diple in the xy-plane, the functin ζ ( θ, φ) frm Eq. (4.8) becmes d β + d (0) y β d β + x d (t) Fig. 4.. In its mst general state f scillatin, the diple mment d(t) rtates alng an ellipse. The parameter β (psitive in the figure) determines the lengths f the majr and minr axes, as shwn in the figure, and the sign f β determines the directin f rtatin.
Nanscale Reslutin in the Near and Far Field Intensity 385 β ζ ( θ, φ) = sin θ + cs(φ ). (4.7) β + This functin is, apart frm an verall cnstant, the emitted pwer per unit slid angle int the directin ( θ, φ), accrding t Eq. (4.). Fr the Pynting vectr, Eq. (4.7), we find 3P = β S( r) ζ ( θ, φ)ˆ r + + e θ φ 8π r q q β + sin, (4.8) with e = e x sinφ + e csφ. (4.9) φ y The term prprtinal t rˆ is respnsible fr the radial pwer utflw, and the term with e φ gives a rtatin in the flw f energy arund the z- 5 axis. Clse t the diple this term is f rder r, whereas the radial term is f rder r. Therefre, in the near field the rtatin will dminate the energy flw pattern, even thugh this rtatin des nt cntribute t the pwer utflw at any distance frm the diple. In the limit f a linear diple, β = 0 r β, the term with e φ vanishes, and the Pynting vectr nly has a radial cmpnent, crrespnding t pwer flwing radially utward frm the diple, withut any swirling arund the z-axis. 4.5. Field Lines f the Pynting Vectr In rder t investigate in detail the energy flw ut f a diple, we cnsider the field lines f the Pynting vectr S (r). Expressin (4.8) fr S (r) determines a vectr field in space, and a field line f S (r) is a curve fr which at any pint alng the curve the vectr S (r) is n its tangent line. When r is a pint n the field line, then we can parametrize such a field line as r (u), with u a dummy variable. A field line is determined nly by the directin f S (r), and nt its magnitude, and hence a field line is a slutin f dr f ( r) S( r) du =, (4.0)
386 X. Li, H. F. Arnldus & J. Shu with f (r) any psitive functin f r. Equatin (4.9) is an autnmus differential equatin, since the variable u des nt appear n the righthand side. The variable u itself has n physical significance. We nw use spherical crdinates ( q, θ, φ) t represent a pint r, s that r = ( q / k ) rˆ, with rˆ given by Eq. (4.0). Fr a pint n a field line, q, θ and φ then becme functins f u, and when we write ut Eq. (4.0) we btain dq du = k f ( r) rˆ S( r), (4.) with dθ q = k f ( r) eθ S( r), du (4.) dφ q sinθ = k f ( r) eφ S( r), du (4.3) eθ = e csφ + e sinφ) csθ e sinθ. (4.4) ( x y z Fr the functin f (r) we take 8π r /(3P k ), and with Eq. (4.8) we then find the set f equatins dq du = ζ ( θ, φ), (4.5) d θ = 0 du, (4.6) dφ β = + du q q β +, (4.7) fr the crdinates q, θ and φ as functins f u fr pints n a field line. Frm Eq. (4.6) it fllws that θ is cnstant alng a field line, and let us indicate this cnstant by θ. Any field line starts at the lcatin f the diple, and we then see that it remains n a cne which has an angle
Nanscale Reslutin in the Near and Far Field Intensity 387 θ with the z-axis. Then we can replace θ by θ n the right-hand side f Eq. (4.5), and divide Eq. (4.7) by Eq. (4.5). This yields dφ β = + dq (, ) q q ζ θ φ β +. (4.8) When we see φ as a functin f q, rather than u, this is a nnlinear firstrder equatin fr the functin φ (q). The equatin is separable, and is mst easily slved by cnsidering q as a functin f φ. The result is [5] q( φ ) =, (4.9) [( + where A is a functin f φ, defined as A = 3 A ) + A] [( + A ) ] 3 4 4 A 3 β sin θ [sin(φ ) sin(φ )] 8 β 3 β + sin θ ( φ φ ). (4.30) β We see frm Eq. (4.8) that fr q large we have dφ / dq 0, and therefre φ appraches a cnstant φ at a large distance. This is angle φ in the expressin fr A, and this φ serves as the integratin cnstant. Angle φ is nw the free parameter, and it fllws frm the derivatin in Ref. [5] that, given φ, angle φ has t be taken in the range < φ < φ, β > 0, (4.3) φ < φ <, β < 0. (4.3) Therefre, each field line is determined by a chice f θ and φ, which are the values f θ and φ at a large distance. We indicate by x = k x, y = k y and z = k z the dimensinless Cartesian crdinates f a field pint. We then have alng a field line
388 X. Li, H. F. Arnldus & J. Shu x φ) = q( φ)sinθ csφ, (4.33) ( y φ) = q( φ)sinθ sinφ, (4.34) ( z ( φ) = q( φ)csθ, (4.35) with φ the free variable, and q (φ ) given by Eq. (4.9). Figure 4. shws several field lines fr θ = π / 4 and θ = 3π / 4 and fr different values f φ. Fr the figure we tk β =, which crrespnds t a circular diple mment, rtating cunterclckwise in the xy-plane. The field lines swirl arund the z-axis with the same rientatin as the rtatin f the diple mment. The field line pattern has a vrtex structure, with the diple at the center f the vrtex. All field lines wind arund the z-axis, while remaining n a cne. The scale in the figure is such that π crrespnds t ne ptical wavelength, and we see that the spatial extend f the vrtex is well belw a wavelength. Figure 4.3 shws field lines fr β =, s fr the same diple as in Fig. 4., but nw fr different values f θ. Angle φ is the same fr all field lines, and is taken as π /. Therefre, all field lines apprach asympttically a straight line parallel t the psitive y-axis. z 0 - - - x 0 0 - - y Fig. 4.. The figure shws several field lines f the Pynting vectr fr a circular diple, with a diple mment that rtates cunterclckwise in the xy-plane. Each field line lies n a cne which makes an angle f 45º with the z-axis.
Nanscale Reslutin in the Near and Far Field Intensity 389 z x -3 0 0 - - 0 3 3 x Fig. 4.3. The figure shws field lines fr the same diple as in Fig. 4., but nw fr different values f θ, and with φ = π / fr all field lines. The result fr the field lines depends parametrically n the value f β. Fr β = the field lines shw a vrtex structure near the lcatin f the diple, as illustrated in Figs. 4. and 4.3, and the dimensin f this vrtex is a fractin f a wavelength. Fr β 0 and β the ellipse reduces t a line, and the scillatin f the diple mment becmes linear. As shwn abve, fr a linear diple the field lines are straight lines emanating frm the diple. In rder t see the transitin frm the vrtex pattern fr a circular diple t the straight-line pattern fr a linear diple we have graphed field lines fr three values f β in Fig. 4.4. It is seen frm the figure that when β decreases frm unity t zer, the size f the vrtex diminishes, until it reaches a pint fr β = 0. Fr β =, the field lines bend arund the z-axis ver an extend f abut a wavelength frm the diple. Fr smaller values f β, the field line becmes straight already in the very neighbrhd f the diple. 4.6. Field Lines in the Far Field At a distance f many wavelengths frm the diple, each field line appraches a straight line, as can be seen frm the figures abve. This line, hwever, des nt appear t cme exactly frm the lcatin f the diple, but it is slightly displaced with respect t the radial directin, as
390 X. Li, H. F. Arnldus & J. Shu β =.0 z 0.5 0.0-0.5 0.0 x 0.5 y.0 -.0-0.5 0.0 0.5.0 β = 0..0 z 0.5 0.0-0.5 0.0 x 0.5 y.0 -.0-0.5 0.0 0.5.0 β = 0.00.0 z 0.5 0.0-0.5 0.0 x 0.5 y.0 -.0-0.5 0.0 0.5.0 Fig. 4.4. Shwn are field lines fr diples with different values f β. Fr each, θ = π / 4 and φ = π /.
Nanscale Reslutin in the Near and Far Field Intensity 39 4 z ˆr 3 0 - x 0 q d 0 - - y Fig. 4.5. Due t the rtatin f a field line near the surce, it appears as if the surce is displaced when viewed frm the far field. illustrated in Fig. 4.5. Fr an bserver, indicated by the eye in the figure, the field line f energy flw seems t cme frm a pint in the xyplane that des nt cincide with the psitin f the surce, and hence it appears as if the surce is displaced with respect t its actual psitin. This apparent displacement, indicated by the displacement vectr q d, is a result f the rtatin f the field lines near the surce, and as such this near field effect shuld be bservable in the far field. In rder t cmpute this displacement, we cnsider q as the independent variable alng a field line, rather than φ, even thugh in the explicit slutin (4.9) it is the ther way arund. Fr q large, φ appraches the value φ, and the value f θ alng a field line is θ at any pint. Therefre, fr q large we can expand φ in an asympttic series as φ ( q ) = φ + c / q + c / q..., with the cefficients c, c,... t be determined. In expressin (7) fr ζ ( θ, φ) we set θ = θ and fr φ we substitute the asympttic expansin. This gives ζ ( θ, φ) = ζ ( θ, φ ) + (/q). We use this t expand the right-hand side f Eq. (4.8), which yields
39 X. Li, H. F. Arnldus & J. Shu dφ = dq q ζ ( θ, φ ) β + β + 3 q. (4.36) The right-hand side f Eq. (4.36) is an asympttic series in q, and termby-term integratin gives where we have set φ( q ) = φ Y ( θ, φ ; β ) +..., (4.37) q β Y ( θ, φ; β ) =, (4.38) ζ ( θ, ) φ β + which is a cnstant alng a field line. Equatins (4.33) (4.35) give the Cartesian crdinates f a pint n a field line. We nw view q as the independent variable, and we expand cs φ and sin φ in Eqs. (4.33) and (4.34) with the help f Eq. (4.37). We then btain csφ( q ) = csφ + Y ( θ, φ; β )sinφ +..., (4.39) q sinφ( q ) = sinφ Y ( θ, φ; β )csφ +.... (4.40) q Fr a pint n a field line, q is the dimensinless distance between this pint and the rigin f crdinates. When the terms represented by ellipses in Eq. (4.37) are mitted, parameter q lses this significance. Therefre, we shall write t instead f q. Equatins (4.33) (4.35) then becme x = sinθ[ t csφ + Y ( θ, φ; β )sinφ ], (4.4) y = sinθ[ t sinφ Y ( θ, φ ; β )csφ ], (4.4)
Nanscale Reslutin in the Near and Far Field Intensity 393 z = t csθ. (4.43) Fr a given bservatin directin ( θ, φ ), these Cartesian crdinates are linear functins f t, and hence Eqs. (4.4) (4.43) are the parameter equatins f a straight line. This is line l in Fig. 4.5, which is the asymptte f the field line in the directin ( θ, φ ). Vectr ˆr in Fig. 4.5 is the unit vectr in the ( θ, φ ) directin, which fllws frm Eq. (4.0) by replacing ( θ, φ) by ( θ, φ ). When we intrduce the vectr qd = Y ( θ, φ; β )sinθ ( e x sinφ e y csφ ), (4.44) then Eqs. (4.4) (4.43) can be written in vectr frm as q = q d + t ˆr, (4.45) where q = kr is a pint n the line l. Accrding t Eq. (4.43), t = 0 gives z = 0, s this crrespnds t the intersectin pint f l and the xyplane. The psitin vectr f this pint is q d, as fllws frm Eq. (4.45), and therefre vectr q d is the displacement vectr f the surce as shwn in Fig. 4.5. It represents the apparent psitin f the surce in the xy-plane, when viewed frm the far field. Frm Eqs. (4.0) and (4.44) we see that q d rˆ = 0, s q d is perpendicular t ˆr. 4.7. The Displacement The magnitude f the displacement vectr q d is given by β sinθ q d =, (4.46) β + ζ ( θ, φ ) with ζ ( θ, φ ) given by Eq. (4.7). This surce displacement depends n the bservatin angles θ and φ, and n the parameter β f the ellipse. Fr bservatin alng the z-axis we have sinθ = 0 and the
394 X. Li, H. F. Arnldus & J. Shu displacement is zer. As a functin f θ, the displacement is maximum fr θ = π /, s fr bservatin alng the xy-plane. In the xy-plane and fr a given β, the displacement depends n the lcatin in the xy-plane, e.g., n the angle φ. We find that the displacement is maximum fr cs( φ ) = when β >, and fr cs( φ ) = when β <. Frm Fig. 4. we then see that this crrespnds t an bservatin alng the majr axis f the ellipse in bth cases. In this directin, the displacement is given by β, β > q d =, (4.47), β < β fr a given β. Fr a circular diple we have β =, and the maximum displacement is q d =. Fr β 0 r β the eccentricity f the ellipse increases and the diple becmes mre linear. We then have q d > and the displacement can grw withut bunds when the diple appraches a linear diple. 4.8. Intensity in the Image Plane The bservatin directin ( θ, φ ) can be represented by a unit vectr ˆr, as in Fig. 4.5. We nw cnsider an bservatin plane, shwn in Fig. 4.6, which is a plane perpendicular t ˆr, and a distance r away frm the diple. The rigin f crdinates in this plane is represented by the vectr r. We then define λ and µ axes in this plane, such that the axes run int the directin f the the spherical unit vectrs e θ and e φ, as shwn in the figure. Therefre, λ and µ are the Cartesian crdinates f a pint in this plane, and the psitin vectr r f a pint in this plane can be written as r + = r + λ eθ µ eφ. (4.48)
Nanscale Reslutin in the Near and Far Field Intensity 395 µ eφ rˆ λ eθ r r d γ rˆ Fig. 4.6. The figure shws the crdinate system fr the image plane and several field lines f the Pynting vectr. Pint r in the plane, represented by, has Cartesian crdinates ( λ, µ ). The dimensinless displacement vectr (4.44) is a vectr in this plane, and can be expressed as q = e φ Y ( θ, φ ; β ) sinθ. (4.49) d The crrespnding vectr r d = q d / k is shwn in the figure. Therefre, the displacement vectr is alng the µ -axis. The asymptte l f the field line fr this directin ( θ, φ ) ges thrugh the pint r d, and when the distance r is sufficiently large, the field line thrugh this pint is perpendicular t the bservatin plane. When an image is frmed n the bservatin plane, r image plane, it may be expected that the majr cntributin cmes frm the field lines in the neighbrhd f r d, since these crss the plane almst perpendicularly. In that case, the image in this plane wuld be
396 X. Li, H. F. Arnldus & J. Shu - - 0 3 4 6 5 4 3 µ y 0 x - - Fig. 4.7. Field lines f the Pynting vectr in the xy-plane fr a diple mment which rtates in the xy-plane are shwn. They swirl arund the rigin numerus times, and are then incident upn the image plane. The intensity distributin n the image plane shws a maximum in the neighbrhd f the central field line fr this directin, which ges thrugh the pint x = n the image plane. shifted ver rd with respect t the rigin, and this shift wuld then be the same as the virtual displacement f the diple in the xy-plane. Hwever, since a bundle f field lines passing thrugh this plane determines the image, rather than the single field line frm Fig. 4.5, the image f the diple is nt necessarily exactly lcated at the psitin rd. Figure 4.7 shws a bundle f field lines and the crrespnding intensity distributin ver a plane (defined belw), and we bserve that indeed the maximum f the image des nt cincide with the displacement f the crrespnding field line fr this case ( q d = e x ). It shuld als be nted that the shift in the xy-plane is defined thrugh the extraplatin f the
Nanscale Reslutin in the Near and Far Field Intensity 397 asymptte l f the field line in the bservatin directin, as in Fig. 4.5, and, frm an experimental pint f view, this may nt be a directly bservable shift. In rder t investigate this issue in detail, we nw cnsider the intensity distributin ver the image plane. Since ˆr is the unit nrmal vectr at all pints in the image plane, the intensity (pwer per unit area) at pint r is ˆ I ( r ; λ, µ ) = S( r) r. (4.50) This intensity depends n r, the distance between the surce and the image plane, the bservatin directin ( θ, φ ), and the crdinates λ and µ f pint r in the plane. In additin, it depends parametrically n the parameter β f the ellipse. We intrduce dimensinless crdinates λ = kλ and µ = kµ in the bservatin plane, and the dimensinless distance between the rigin f the plane and the diple is q = kr. Fr pint r in the plane we then have q = k r, and the relatin t its Cartesian ( λ, µ ) is Putting everything tgether yields [6] + λ + µ q = q. (4.5) 3 q = I ( r ; λ, µ ) I q q + β [ β ( ρ csφ µ sinφ ) + ( ρ sinφ + µ csφ ) ] q q β + µ sinθ q β +, (4.5) fr the intensity distributin in an bservatin plane fr the case f an elliptical diple mment, rtating in the xy-plane. In Eq. (4.5) we use the abbreviatin
398 X. Li, H. F. Arnldus & J. Shu and the verall factr I is given by ρ = q sinθ + λ csθ, (4.53) I 3P =. (4.54) 8π r The first tw lines in Eq. (4.5) cme frm the angular dependence f the emitted pwer, which is accunted fr by the functin ζ ( θ, φ) in Eq. (4.) fr dp / dω. The appearance in Eq. (4.5) f this angular dependence is nt thrugh the functin ζ ( θ, φ), since here we cnsider the pwer flw thrugh a plane, whereas dp / dω refers t the pwer flw thrugh a sphere. The third line in Eq. (4.5) riginates in the pssible rtatin f the field lines near the surce. This is mst easily seen frm the fact that this term changes sign with the sign f β. The verall factr f /( q q ) indicates that this term vanishes rapidly in the far field, as cmpared t the remaining terms in braces. We shall see belw, hwever, that a finite and bservable effect f this term survives in the far field. 4.9. Linear Diple Let us first cnsider β = 0, crrespnding t a linear diple, scillating alng the y-axis. The intensity distributin (5) n an image plane then becmes 3 q I( r ; λ, µ ) = I ( ρ sinφ + µ csφ ). (4.55) q q Fr a linear diple, all field lines are straight, and any dependence f the intensity n r, λ and µ cmes frm the angular dependence f dp / dω and frm the fact that we cut thrugh dp / dω with a plane. As an illustratin, let us cnsider an bservatin plane perpendicular t the y-axis at the psitive side. We then have θ = π /, s ρ = q, and φ = π /. We intrduce angle γ as the bservatin directin fr a pint
Nanscale Reslutin in the Near and Far Field Intensity 399 in the image plane, as seen frm the site f the diple (see Fig. 4.6). We then have cs γ = q / q, and the intensity becmes 3 I ( r ; λ, µ ) = I cs γ sin γ. (4.56) Since the right-hand side f Eq. (4.56) nly depends n angle γ, the intensity distributin in the image plane is circularly symmetric arund the rigin. At the rigin f the image plane we have γ = 0 and therefre I = 0, which crrespnds t the well-knwn fact that n radiatin is emitted alng the diple axis fr a linear diple. The intensity has a maximum fr cs γ = 3/ 5, crrespnding t an angle f γ = 39, and this defines a ring in the bservatin plane. The radius f this ring is q / 3. The intensity prfile is shwn in Fig. 4.8. 4.0. Rtating Diple and the Far Field The intensity distributin in Fig. 4.8 scales with the distance q between the diple and the bservatin plane, which is a reflectin f the fact that the field lines f the Pynting vectr are straight fr a linear diple. When the image plane mves further away, the picture remains the same, apart frm a scale factr. We nw cnsider the effect f the rtatin f the field lines near the surce fr a rtating diple mment n the intensity prfile in the far field. We first lk at a circular diple fr which β = ±. The intensity n the image plane, given by Eq. (4.5), simplifies fr β = ± t 3 q I( r ; λ, µ ) = I ( ρ + µ ) q q β + µ sinθ qq q. (4.57) There is n dependence n the bservatin angle expected fr a circular diple mment in the xy-plane. φ, as culd be
400 X. Li, H. F. Arnldus & J. Shu I.0 0.5 4 0.0-0 4-4 0 λ µ Fig. 4.8. The figure shws the intensity distributin n an image plane fr a linear diple. The plane is perpendicular t the diple axis. We are lking fr effects in the far field that are due t the rtatin f the field lines near the surce. The displacement frm Sec. 4.7 was maximum fr bservatin in the xy-plane, e.g. θ = π /, s here we cnsider the same situatin in rder t find the maximum effect. Frm Eq. (4.53) we then have ρ = q, and frm Eq. (4.57) we see that the dependence n the crdinate λ in the image plane nly enters thrugh the λ dependence f q in Eq. (4.5). Therefre, the intensity nly depends n λ as λ, and cnsequently there is an extremum at λ = 0. With θ = π / and λ = 0, Eq. (4.57) becmes 3 q β I( r ;0, µ ) = I + µ, (4.58) q q q q which is the intensity alng the µ axis in the image plane. T find the extrema, we set I / µ = 0. This yields
Nanscale Reslutin in the Near and Far Field Intensity 40 3 β = + µ q + 3µ. (4.59) q q q In the far field we have q >>, and Eq. (4.59) becmes 3 β µ = ( q 3µ ). (4.60) q q We are lking fr a pssible shift in the intensity prfile due t the rtatin f the field lines. Such a shift has t remain finite fr q, as can be seen frm Fig. 4.7, s at such an extremum we must have µ / q 0 fr q large. If we indicate by µ p the slutin f Eq. (4.60), we find fr the lcatin f the peak in the intensity prfile µ p = β. (4.6) 3 The magnitude f this shift is /3 in dimensinless crdinates, and since π crrespnds t ne wavelength, the shift is equal t a wavelength divided by 3 π. Apparently, this is a nanscale shift fr ptical radiatin, but it is a shift that persists in the far field. Figure 4.9 shws the intensity distributin fr this case. Fr Fig. 4.9 we tk q = 4 (and I = ), which is / π times a wavelength. This is just enugh t justify the far-field apprximatin q >>. Fr larger values f q, the backgrund becmes very large since it scales with q, and it may nt be pssible experimentally t reslve the shift f the peak. In an experiment, the diple is set in scillatin with a laser beam. A circular diple mment in the xy-plane is induced by a circularly plarized laser beam, prpagating alng the z- axis, and the sign f β is determined by the helicity f the laser. In Eq. (4.57), the first line is the brad backgrund and the secnd line is due t the rtatin f the field lines. The backgrund des nt depend n the sign f β, s if we wuld measure the intensity fr left- and righthanded helicity then the difference f the prfiles wuld be twice the secnd line f Eq. (4.57), and any resulting difference prfile wuld be due entirely t the rtatin f the field lines. This experiment was perfrmed recently [7] and such an asymmetric prfile was fund indeed.
40 X. Li, H. F. Arnldus & J. Shu I 0.6 0.4 0. 0.0-4 - 0 4 0 0-0 µ λ Fig. 4.9. The figure illustrates the nanscale shift f the intensity prfile fr a circular diple. The image plane is the same as in Fig. 4.8. Fr an elliptical diple, similar calculatins shw [6] that the prfile is mre cmplicated. It can have maxima, minima and saddle pints. In particular, the maximum shwn in Fig. 4.9 can becme a minimum, and the effect f the rtatin f the field lines can be a mving hle rather than a mving peak. 4.. Intensity in the Near Field With very precise near field techniques, invlving fiber tip prbes, intensities can be measured in the near field with nanscpic precisin [8,9]. In this apprach the existence f the vrtex culd be verified experimentally by measuring the intensity directly at the lcatin f the vrtex, rather than thrugh the bservatin f the displacement f the maximum f the intensity prfile in the far field. Figure 4.0 shws the intensity distributin n an image plane very clse t a circular diple, fr the case where the image plane is perpendicular t the y-axis. Near
Nanscale Reslutin in the Near and Far Field Intensity 403 0 - I -0.0-0.05 - -0.0-0.05 0.00 λ 0.05 µ 0.00 0.05 0.0 0.0 Fig. 4.0. The figure shws an intensity prfile in the near field f a circular diple. the psitive peak, the field lines pass thrugh the image plane in the utward directin. The negative peak indicates that in this regin f the bservatin plane the field lines are inward. This image is a direct cnsequence f the spiraling f the field lines near the surce. A field line first passes thrugh the image plane in the utward directin, and then spirals back inwards near the peak with negative intensity. Althugh it may nt be pssible t measure a negative intensity directly in an experiment, the figure nevertheless illustrates the winding f the field lines in an appealing manner. 4.. Cnclusins When an electric diple mment rtates in the xy-plane, the field lines f energy flw swirl numerus times arund the z-axis, while remaining n the surface f a cne. Far away frm the diple, the field lines apprach asympttically a straight line, reminiscent f an ptical ray. This line is displaced with respect t the radial directin, and this leads t an apparent shift f the diple lcatin, when viewed frm far away. This
404 X. Li, H. F. Arnldus & J. Shu displacement depends n the bservatin directin and n the eccentricity f the ellipse. In rder t define an image f the diple, we have cnsidered the intensity distributin f the emitted radiatin ver an image plane. Fr a circular diple, the image is a single peak when viewed in the plane f rtatin f the diple. It was shwn that in the far field the peak f this intensity prfile is shifted with respect t the central directin, and that this shift is due t the rtatin f the field lines near the surce. In this fashin, the near field curving f the field lines affects the image in the far field, and this makes the vrtex near the diple bservable at a macrscpic distance. The shift is f the rder f a fractin f a wavelength, as is the dimensin f the vrtex. References. M. Brn and E. Wlf, Principles f Optics, 6th edn. (Pergamn, 980), Chapter 3.. J. D. Jacksn, Classical Electrdynamics, 3rd edn. (Wiley, 99), p. 4. 3. I. V. Lindell, Methds fr Electrmagnetic Field Analysis (Oxfrd U. Press, 99), Sectin.4. 4. L. Mandel and E. Wlf, Optical Cherence and Quantum Optics (Cambridge U. Press, 995), p. 469. 5. J. Shu, X. Li and H. F. Arnldus, Energy flw lines fr the radiatin emitted by a diple, J. Md. Opt. 55, 457 47 (008). 6. J. Shu, X. Li and H. F. Arnldus, Nanscale shift f the intensity distributin f diple radiatin, J. Opt. Sc. Am. A 6, 395 40 (009). 7. D. Haefner, S. Sukhv and A. Dgariu, Spin Hall effect f light in spherical gemetry, Phys. Rev. Lett. 0, 3903 4 (009). 8. K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Chi, H. Kim, C. Rpers, D. J. Park, Y. C. Yn, S. B. Chi, D. H. W, J. Kim, B. Lee, Q. H. Park, C. Lienau and D. S. Kim, Vectr field micrscpic imaging f light, Nat. Phtnics, 53 56 (007). 9. Y. Ohdaira, T. Inue, H. Hri and K. Kitahara, Lcal circular plarizatin bserved in surface vrtices f ptical near-fields, Opt. Exp. 6, 95 9 (008).