Theory of Mie caustics

Optics Communications 103 (1993) 339-344 North-Holland

OPTICS COMMUNICATIONS

Theory of Mie caustics L.G. G u i m a r a e s 1

Departemento de Fisica, ICE, UFJF,Juiz de Fora, 36036-330,MG, Brazil Received 2 June 1993;revised manuscript received 13 August 1993

The purpose of this work is to explain the origin, magnitude and position of the caustics in resonant Mie scattering. For this, the fields are analyzedby a method based on complexangular momentum (CAM) theoryand internal reflection decomposition (Debye expansion).

1. Introduction In the last decade, quantum optic effects like stimulated Raman scattering, stimulated Brillouin scattering and lasing, were observed in micrometer size droplets [1-3]. The generation of these nonlinear optical processes occurs in regions where the intensity of the electromagnetic field is sufficiently high. In the case of spherical droplets, with complex refractive index N, for Re{N} > 1, the geometrical optics analysis of the Mie scattering theory shows that the caustics can arise in forward and backward directions (focusing effect), in regions inside and near the spherical surface [4,5 ]. Meanwhile, on resonant Mie scattering the sphere can support inside and near the surface, a spectacular magnification of the electromagnetic field [5-7], these high fields are shown in fig. la. In this figure we plotted the source function S versus p, the source function is the averaged angle intensity (S = fdg-2 UEI[2/4n), and p the relative distance to the sphere center (p=-r/a). The main features of the internal resonant field are the following: On the resonance characterized by family number (order number) n, the number of the maximum intensity peaks is just n plus one (see fig. lb). The magnitude of the maximum intensity peaks is associated with the resonance width, very narrow Permanent address: Departemento de Fisica Nuclear, Instituto de Fisica - UFRJ, Cx. Postal 68528, 21945, RJ, Brazil.

resonances give rise to the highest peaks (see fig. lb). In this letter, using the Debye expansion and CAM theory, we will give a physical explanation for the behavior of the near and internal resonant field and location of the caustics.

2. Resonances, the Debye expansion and CAM theory On Mie scattering an incident plane wave with a vector wave k is scattered by a dielectric sphere of radius a. Here we will consider a transparent sphere with refractive index N > 1. According to the Mie theory, the fields are given by infinite partial wave series, the resonances are associated with the complex poles of these series [8 ] and the lth resonant multipole satisfies the transcendental equation ln'~}l)(fl) =NEj ln' g l ( a ) ,

(1)

where ln' denotes the logarithmic derivative with respect to the arguments, ( and q/are respectively the Ricatti-Hankel and Ricatti-Bessel functions. We define the size parameters fl==-ka and a = nil, the symbol ej assuming the values el -= 1 ( j = 1 perpendicular polarization, magnetic waves - M ) , ~2~ 1/N 2 (j= 2 parallel polarization, electric waves E). The poles of the eq. ( 1 ) are complex numbers labeled in the following form, ~ ( l ) = f l f i l ) - i b j ( 1 ) , where flj represents the resonance position and bj its width. The physical interpretation for resonances can be

0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

339

Volume 103, number 5,6

/,

(a)

S(p)

OPTICS COMMUNICATIONS

I0

--Resonant ..... Non-Resonant

N = 1.33

105 S(p)

t 1

I 0.8

0.6

I 1.2

I 1.4

p

(b) N :

2). F o r the lth partial w a v e Ueff takes the f o r m o f an a t t r a c t i v e square well o f d e p t h k2(N 2 - 1 ) plus the r e p u l s i v e centrifugal p o t e n t i a l 22/r 2 w i t h 2 = l + ½ ( L a n g e r m o d i f i c a t i o n ) r e p r e s e n t i n g the angular momentum. In this context, r e s o n a n t m o d e s c o u l d be seen as s t a n d i n g w a v e s oscillating in the i n t e r n a l region rl < r < a a n d t u n n e l i n g t h r o u g h the f o r b i d d e n region a w h e r e q = ( 2 / a ) a a n d r2(2/fl)a are the classical t u r n i n g points. C o n s e q u e n t l y the r e s o n a n t m u l t i p o l e 2 is r e s t r i c t e d to a strip fl < 2 < a . U s i n g the localization principle [ 8 ] a n d the lowest W K B a p p r o x i m a t i o n for c y l i n d r i c a l f u n c t i o n s [ 10 ], a gross e s t i m a t e for the l o c a t i o n o f the r e s o n a n c e is g i v e n by

1.33

2tan-l(M)

/\'~

M"

/\

10 3

1 December 1993

/?n;j(2) ~

n = 0

7,

1

.....

M

+ (n+~)n

~

0.459

+ ejM221/3

+ O ( 2 -2/3) ,

(2)

2

w h e r e M - Nx/NT2- 1 a n d the o r d e r o f r e s o n a n c e s n can a s s u m e the f o l l o w i n g v a l u e s 101

/

/

~I I /

....

\\

.4 ~

n = 0 , 1, 2, ..., nn~ax

,./N

(3)

.'.nmax ~< ( 2 / n ) [ M - t a n - l ( M ) ] - ¼ 1 0 -1

[

0.6

0.8

N

[

[

I

I

1.0

1.2

1.4

1.6

P

Fig. 1. (a) For refractive index N= 1.33 we compare the behavior of the source function S in two distinct situations as p, the relative distance to the sphere center, varies. In the first case the resonance is on (full line) and in the other the resonance is off (dotted line). The nonresonant size parameter is distant from the resonance approximately seventeen times the resonance width. We note that during a resonance the enhancements of the source function (the caustics) are close to the spherical surface. (b) Plot of the source function S for three different resonances (n=0, 1, 2 ). For a given multipole l the resonances are associated with the poles ofeq. ( 1 ), we label the resonance by symbol j?, where j is the polarization and n is the order of the resonance, the width of the resonance decreases exponentially with n (see table 1). Observe that for a resonance j7 the source function has n+ 1 peaks and very narrow resonance generates the highest peaks. The caustics are limited to the region between the spherical surfaces

r=a/N and r=aN. u n d e r s t o o d w i t h the h e l p o f t h e w e l l - k n o w n analogy b e t w e e n scattering in optics a n d q u a n t u m m e c h a n ical scattering [9] (in units such t h a t h = 2 m = l ) t h r o u g h the e f f e c t i v e p o t e n t i a l b a r r i e r Ueff (see fig. 340

.

O n r e s o n a n t M i e scattering, all types o f cross sections show ripple fluctuations, that a p p e a r in the f o r m the q u a s i - p e r i o d i c p a t t e r n a s s o c i a t e d w i t h resonances [ 11 ], so eq. (2) p e r m i t us to catalogue, in a g i v e n cross section, the characteristics distances bet w e e n resonances. F o r e x a m p l e , eq. ( 2 ) shows that the d i s t a n c e s b e t w e e n r e s o n a n c e s w i t h c o n s e c u t i v e

Ueff

T

k:

7. $ L

L

1

r I

a

r2

/ r

Fig. 2. The effective potential barrier U~ for a transparent sphere with N>~ 1 and radii a. The positions q and rE are the classical turning points.

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OPTICS COMMUNICATIONS

angular momentum and order are respectively, approximately tan- ~( M ) / M and n/M [ 11,12 ], while that the distance associated with the resonances with different polarization is O ().- ~/3). The width b,j of these resonances is proportional to the barrier penetration factor [ 10,13,18 ], \~/2

b,,.,yocexp[-2i(~2-k2 )

7 drJ

permits the external E (ext) and internal E (int) electric fields be written in terms of the its Debey expansion [9,10,15,16], namely p 2 E ( e x t ) = E E __p;jF(ext)+Af(ext) , p=0 j= 1

(5)

p E(int) = ~

(6)

p=l

a 2

= e x p l - 2 ~ j (~2 - l)l/2 d x l .

(4)

Consequently, in the relevant limit 2 >> 1 the resonances become extremely sharp (see table 1 ). This brings some computational difficulties, because any algorithm to solve eq. ( 1 ) needs an accuracy greater than b~j [ 1 l, 12 ]. By this reason, the accuracy of the eq. (2) and eq. (4) are not sufficient to locate sharp resonances, but in the case of large resonances, we can use these results as an initial guess to some numerical procedure to solve eq. ( 1 ). The powerful algorithm to supply this deficiency is based on uniform asymptotic expansion to Bessel functions [ 14]. During a sharp resonance, associated with the/th multipole, it is intuitive to consider that this multipole gives the main contributions for the intensity [7]. It can be better understood if we observe that, in the Mie theory, the coefficients of the partial wave series expansion can be rewritten in terms of the interactions with the spherical surface by using their expansions in terms of the Fresnel spherical reflections and transmission coefficients. This procedure Table 1 For refractive index N = 1.33 some magnetic (M) resonances are shown. The symbol j7 for angular m o m e n t u m l describes the resonance which polarization and order are j and n, respectively. The resonance location and width are fl and b, respectively.

Resonance

fl

b

M°5 M~5 M°I M~I M~ M°2 M~2 M~2

53.8076592 58.2808441 58.4883093 63.0903566 66.9546011 73.2517041 78.2083369 82.3783624

3.35 × 5.26× 9.49× 2.07× 4.43 × 1.55× 8.01 × 4.57 ×

10 -5 10 -3 10 -6 10 -3 10 -2 10 -7 I0 -s 10 -3

1 December1993

2 Z __p;jF'(int)-~-,~E(int) . j=l

For the external field and a given polarization j and p~> 1, the fields F (e~') represent the effect of the ~p;J transmission through spherical surface following ( p - 1 ) internal reflections, while the fields E ~ xt) are associated with the contribution of the direct reflection from surface effects. In the case of the internal field, the fields ]bT(int) represent the effect of the ~p;J , ( p - 1 ) internal reflections only. For large P, the remainder terms AEp in eqs. (5), (6) do not have significant contributions, except in the case of resonance, in which they are divergent [10], so that we rewrite eqs. (5), (6) as E(eX') =E~ ~xt) + E ~ ~xt) ,

(7)

E (int) = g ~ int) + g E nt) ,

(8)

where EB denotes the backgroundcontributions, associated with nonresonant effects like diffraction and geometrical optics interference [ 15,16 ], while ER gives the resonant contributions. Figures 3a and 3b show the Debye expansion analyses for the fields in two different situations. In the first the resonance is on and in the second the resonance is off, in both figures the Debye expansion for the fields has about eighteen terms, P = 17 in eqs. (5) and (6), while we build the background contributions using only the three and four first terms in the Debye expansion for internal and near fields, respectively. This procedure permits us to interpret the backgroundcontributions as the effect of these optical rays which undergo some internal reflections before emerging from the spherical surface. The resonantcontributionsfields ER are described by partial wave series, the CAM theory is introduced by means of the Watson-Sommerfeld transform performed on ER, in such a way that the computation of ER is reduced to the calculation of the contour integral on the CAM plane. The main contribution for this integral arise from the residue series evaluated 341

Volume 103, number 5,6

OPTICS COMMUNICATIONS

(a)

S(p)

1 December 1993

0.20 (~

S(p)b

M29o

/?l

0.15

10

/

I:".

1 /

0.10 •

--n

M"

= 0

7z

. . . . . n=l

~

--

-n=2

N

= 1.33

0.05 i

L

0.6

0.8

I

L

1.0

1.2

"l p

1.4

,"-"

0.8

"

,' "" - .". . .... . . 1.0

r .....

1.2

I

I

1.4

1.6

P

Fig. 4. The product of the source for three different resonances function by the respective resonance width. Note that the height of the peaks, the maximum intensity, is inverselyproportional to the resonance width.

(b)

3.0

0.00 ~ 0.6

s(p) 2.4

[ iil ]

1.8

1.3

N = 1.33 = 82.3

-~

J

0.7 0.6

0.8

1.0

1.2

p

1.4

Fig. 3. The present theory for the internal and near field. (a) During the resonance MgZo,the exact Mie result (full line, curve [i] ) is compared with the following Debye expansion cases. In all cases the backgroundcontributions are representedby the four first terms in the Debye expansion, since the full Debye expansion has about eighteen terms. In curve [ii] we consider both contributions background and resonance. In curve [iii] we consider only the background contributions. We observe that the resonance contributions are responding to the behavior of the fields on caustics regions. (b) Curves [i] and [iii] are the same as in (a), but in this case the resonance is off, so that the background is the dominant contribution. on Regge poles, the poles of eq. (1) in the CAM plane. T h e n on the neighborhood of the resonance ft,;j(2) and width b,;j(2), if we use the connection between the CAM plane and energy plane (fl-plane) [ 10 ], we can write the resonant c o n t r i b u t i o n for the internal field as E~int) ( an;jp ) ~ E o a

(fl.;j, 2) ~ -

i/2 (ce";JP)

,

(9)

Eo being the amplitude of the incident electric field, 342

the module jiG(fin;j, 2)11 a smooth function in this d o m a i n and f~_ t/2 the analytic c o n t i n u a t i o n in CAM plane of the Ricatti-Bessel function or its first derivative depending on the polarization j. Equation (9) explains the principal properties of the internal field on resonant conditions, for example, it shows that in this case the internal field has m a x i m u m intensity at the same neighborhood where the resonant multipole f~_ 1/2 has m a x i m u m module, besides the height of the peaks the m a x i m u m intensity is inversely proportional to resonance width

( Ioc llEIIZocb-l ). Figure 4 confirms this last result. In this figure is plotted, for a given resonance, the product of the source function by the respective resonance width, for the case of three different Regge trajectories (n = 0, 1 and 2 ). We note that the vertical axis is now in linear scale (compare with fig. l b ) .

3. C a u s t i c s

locations

Equation (9) permits us to associate the resonant multipole critical points with the points where the intensity has a m a x i m u m . O n the other hand, we can interpret the optical rays associated with caustics inside the sphere as being the rays for which the optical difference path generates a constructive interference. This criterion is satisfied if

x/'-dTp 2- 2 2 1 2 2 - tan-1 ( %/-~p 2__),2/)2)

=(m+¼)rc/,~, m=0,1,....

(10)

Observing eq. (10), we can see that in the limit 2>> 1, one of its solution occurs on the neighborhood of the point p=2/ot [7], but it is not a critical point, it is only the classical turning point r~ divided by sphere radii a. During a resonance we have 2 ~/3+ O (fl~/3) so that we can rewrite eq. (10) as x/~

2 - 1 - tan-1 ( x / ~

1 December 1993

OPTICS COMMUNICATIONS

Volume 103, number 5,6

order of n/2M, which is the asymptotic limit of the WKB result rip,,,_ l /.v/-( N fl, p,,_ ~) 2 _ 2 2 . Equation ( 12 ) shows that the peaks closer to the spherical surface are associated with greater values of m. Equation (10) can be obtained using the WKB expansion to Bessel functions [ 10,13 ], but a more accurate result is obtained if we use the Sch~Sbe asymptotic expansion to Bessel functions [ 9,12 ]. Using this approach an estimate to the points where the intensity has a m a x i m u m is given by P~.)n ~ 1 + ½{(3/;t) [ m + -~(2i + 1)In}

2 - 1) ~ ( m + ¼)re/2.

(11) The existence of real solutions to above equations imply that the caustics regions inside the sphere are limited by the sphere r = J'= a/N. Using an analog argument to caustics outside the sphere, we can show that an upper limit for these regions is the sphere r=?=Na (see fig. l b ) , the spherical surfaces f and being known as aplanatic spheres [4], each one being the perfect stigmatic image of the other. The shell t'~
p,,.~ t a n - t ( M ) / M + (m+ ¼)re/2M.

(12)

Consequently, the distance between two consecutive peaks, centered respectively at p,, and Pro- ~, is of the

2/3_

1 + ½[ (3/,~) (n+ ~)~12/3 +O(1/N2),

m = 0 , 1, ..., ram,x,

(13)

where the subscript i can assume the values, i = 0 in the critical points of the Ricatti-Bessel and i = 1 in the critical points of its derivative, besides as p}d)n ~< 1 we can see from eq. (13) that mm,x-~n, consequently, for a given resonance of order n, the number of the m a x i m u m intensity peaks is just n + 1. This fact is in complete accordance with the analogy between the behavior of the resonant field and the quantum mechanical case of the wave function ¢, (p) on a state characterized by radial quantum n u m b e r n [ 18 ], s o that we can interpret the caustic positions as the points where the radial probability density IP~,(P) 12 has a m a x i m u m . Besides, it is interesting to observe that in eq. ( 13 ) the refractive index is independent until order

Table 2 For N= 1.33 and for magnetic resonances of order n and angular momentum l, a comparison is made between the point where the source function has a maximum (/~) with the point of the Ricatti-Bessel function (if). The fourth column shows the estimate given by eq. ( 13 ) for these points/~. The symbol [ 1]~ labels the location of the ruth critical point. [/1~,

#.'.O~,l(p)lOplp=~,=O

A.'.OS(p)IOpIo=o=O

Eq. (13)

[71]8 [711~ [7111 [71]~ [7117 [71]~ [92]~ [92]~ [92]~ [92]~ [92]7 [92]~

0.9635 0.8932 0.9855 0.8416 0.9286 0.9931 0.9671 0.9058 0.9857 0.8599 0.9358 0.9918

0.9615 0.8913 0.9851 0.8376 0.9276 0.9948 0.9656 0.9044 0.9853 0.8584 0.9355 0.9918

0.9496 0.8845 0.9681 0.8377 0.9168 0.9745 0.9570 0.9003 0.9724 0.8589 0.9277 0.9779 343

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OPTICS COMMUNICATIONS

0(~-2/3), in

other words, the caustics p o s i t i o n s are n o t to m u c h sensitive to the b a r r i e r height, the validity o f this result b e i n g restricted to the r e s o n a n c e s closer to a . We can use eq. ( 1 3 ) as a n i n i t i a l a p p r o x i m a t i o n to start s o m e n u m e r i c a l m e t h o d to f i n d the critical p o i n t s o f the i n t e r n a l field d e s c r i b e d b y p a r t i a l wave series or the p o i n t s o f the c o r r e s p o n d e n t r e s o n a n t m u l t i p o l e . I n b o t h cases we have a r a p i d c o n v e r gence, for e x a m p l e in the case o f table 2 a n accuracy of four d e c i m a l places was required, so that the convergence occurs in less t h a n ten steps.

4. Conclusion U s i n g the analogy b e t w e e n Mie scattering a n d the scattering t h r o u g h effective b a r r i e r p o t e n t i a l Uerr, we can interpret the resonance as light quasi-bound states so that we c a n c o m p a r e the b e h a v i o u r o f the reson a n t field with a wave f u n c t i o n associated to these q u a s i - b o u n d states. Based o n this a n d with help o f the C A M t h e o r y a n d the D e b y e e x p a n s i o n to the fields, we have the m a t h e m a t i c a l tools to o b t a i n the r e s o n a n c e c o n t r i b u t i o n s for caustics, their m a g n i t u d e a n d location.

Acknowledgements T h i s work was s u p p o r t e d b y the B r a z i l i a n agency C o n s e l h o N a c i o n a l de D e s e n v o l v i m e n t o Cientifico

344

1 December 1993

e Tecnol6gico - C N P q a n d I t h a n k the Prof. H e r c h Moys6s N u s s e n z v e i g a n d Prof. Joao Torres de Melo for suggestions a n d m a n y helpful discussions.

References [ 1 ] P.G. Pinnick et al., Optics Lett. 13 ( 1988 ) 494. [ 2 ] I. Zhang and R.K, Chang, J. Opt. Soc. Am. B 6 ( 1989 ) 151; S.M. Chitanvis and C.D. Cantrell, J. Opt. Soc. Am. B 6 (1989) 1326. [ 3 ] S.X. Qian et al., Science 231 ( 1986 ) 486. [4 ] M. Born and E. Wolf, Principles of optics (Pergamon, New York, 1970). [ 5 ] M.A. Jarzembski and V. Srivastava, Appl. Optics 28 ( 1989 ) 4962; V. Srivastava and M.A. Jarzembski, Optics Lett. 16 ( 1991 ) 126. [6] P. Chylek, J.D. Pendleton and R.G. Pinnick, Appl. Optics 24 (1985) 3940. [7] R. Bhandari, Appl. Optics 25 (1986) 2464. [ 8 ] H.C. van de Hulst, Light scattering by small particles (Wiley, 1957). [9] H.M. Nussenzveig, J. Math. Phys. 10 (1969) 82. [10]L.G. Guimar~es, Ph.D. thesis (Centro Brasileiro de Pesquisas Fisicas - CBPF, 1991 ). [ 11 ] P Chylek, J. Opt. Soc. Am. A 7 (1990) 1609. [ 12] C.C. Lan, P.T. Leung and K. Young, J. Opt. Soc. Am. B 9 (1992) 1585. [ 13] L.G. Guimaraes and H.M. Nussenzveig, Optics Comm. 89 (1992) 363. [ 14] L.G. Guimar~es and H.M. Nussenzveig, to he published. [ 15 ] J.A. Lock, J. Opt. Soc. Am. A 5 (1988) 2032. [ 16 ] J.A. Lock and J.R. Woodruff, Appl. Optics 28 ( 1989 ) 523. [ 17 ] M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1965 ). [18] B.R, Johnson, J. Opt. Soc. Am. A 10 (1993) 343.