- Email: [email protected]
Scalar “beams of finite width”
Optics Communications 227 (2003) 57–59 www.elsevier.com/locate/optcom
Scalar ‘‘beams of finite width’’ Pierre Hillion
*
Institut Henri Poincar e, 86 Bis Route de Croissy, 78110 Le V esinet, France Received 13 March 2003; received in revised form 24 June 2003; accepted 2 September 2003
Abstract Beams with a cylindrical symmetry around their direction of propagation and with a transverse amplitude strongly decreasing from some distance w of this direction give a manageable approximation to scalar beams of finite width. Ó 2003 Elsevier B.V. All rights reserved. PACS: 41.20 Jb; 42.25 Bs; 03.40 kf Keywords: Scalar beams; Finite width beams; Wave propagation; Wave scattering
Wave scattering by obstacles of acoustic, TE, TM, electromagnetic beams of finite width is a more close approach to experimental conditions that does the assumption of plane waves [1] making possible a better comparison between numerical computations and measurements. The finite width beam used in [1] has the form Z p=2 wðx; zÞ ¼ dhvðh h0 Þ exp ½ikðx sin h þ z cos hÞ; p=2
ð1Þ 1=2
vðh h0 Þ ¼ kap
h
i exp k a ðh h0 Þ ; 2 2
2
ð1aÞ
but in fact, this expression represents an aggregate of harmonic plane waves all propagating in
directions close to h0 for ka large enough; we observe easily that vðh h0 Þ tends to the Dirac distribution dðh h0 Þ for ka ) 1 To look for scalar waves of finite width, solutions of the 2D-Helmholtz equation, we proceed as follows: we first consider in the ðx0 ; z0 Þ-coordinates a plane wave propagating in the z0 -direction and transversally truncated at jx0 j ¼ a wðx0 ; z0 ; aÞ ¼ expðikz0 ÞV ðx0 ; aÞ; V ðx0 ; aÞ ¼ U ðx0 þ aÞ U ðx0 aÞ;
where U is the unit step function. A simple calculation gives with D0 ¼ o2x0 þ o2z0 ðD0 þ k 2 Þwðx0 ; z0 ; aÞ ¼ expðikz0 Þ½d0 ðx0 þ aÞ d0 ðx0 aÞ; ð3Þ 0
*
Tel.: +33-1397-66401; fax: +33-1396-91764. E-mail address: [email protected].
ð2Þ
in which d is the derivative of the Dirac distribution. We now define a scalar beam with transverse evanescent amplitude by the following relation; where f ðaÞ is a convenient weight function:
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.039
58
P. Hillion / Optics Communications 227 (2003) 57–59
Wðx0 ; z0 Þ ¼
Z
1
daf ðaÞwðx0 ; z0 ; aÞ;
ð4Þ
1
and substituting (2) into (4) gives Z 1 da½f ðaÞ f ðaÞ; Wðx0 ; z0 Þ ¼
ð4aÞ
x0
while according to (3) and (4) with the distributions in (3) written d0 ða þ x0 Þ þ d0 ða x0 Þ ðD0 þ k 2 ÞWðx0 ; z0 Þ Z ¼ expðikz0 Þ
This result may be generalized to 3D-scalar beams with a cylindrical symmetry around the direction of propagation as follows. Using x0 -coordinates ðx0 ¼ x0 ; y 0 ; z0 Þ; we start from a plane wave propagating in the z0 -direction inside a cylinder of radius q0 ¼ ðx02 þ y 02 Þ1=2 wðx0 ; aÞ ¼ expðikz0 ÞU ða q0 Þ; 0
and with D ¼ 0
0
þ 1=q
ð10Þ
dq0 ; þo2z0 ;
0
0
0
2
0
daf ðaÞ½d ða þ x Þ þ d ða x Þ ¼ expðikz0 Þ f 0 ðx0 Þ þ f 0 ð x0 Þ ; ð5Þ
we get
1=q0 dða p0 Þ:
x0
in which f 0 is the derivative of f . Then, we look for a weight function making of Wðx0 ; z0 Þ a quasitruncated plane wave and a convenient choice is f ðaÞ ¼ w1 expða2 =w2 Þ ½sinða=wÞ a=w cosða=wÞ;
which implies 2 0 k ðD þ k 2 ÞWðx0 ; z0 Þ 6 2k 2 w2 x0 =w exp x02 =w2 0
ð7aÞ
0
Consequently, Wðx ; z Þ satisfies the 2D-Helmholtz equation with an 0ðk 2 w2 Þ error, negligible for kw large enough and substituting (6) into (4a), we get Wðx0 ; z0 Þ ¼ 2Aðx0 Þ expðikz0 Þ; Z 1 0 Aðx Þ ¼ db expðb2 Þ½sin b b cos b:
ð8Þ
Taking zero as lower bound of the integral (8) for x0 w; this expression is the plane wave 2Að0Þ expðikz0 Þ in this region while the amplitude Aðx0 Þ is strongly decreasing for jx0 j > w. Then, beams of finite width, solutions to the 0ðk 2 w2 Þ order of the 2D-Helmholtz equation and propagating in the direction h, use in (8) the coordinates x; z obtained by a rotation h x0 ¼ x cos h z sin h:
A scalar beam with evanescent transverse amplitude is obtained from the integral relation Z 1 Z 1 Wðx0 Þ ¼ daf ðaÞwðx0 ; aÞ ¼ expðikz0 Þ daf ðaÞ; q0
1
so that according to (11) and (12) ðD0 þ k 2 Þwðx0 ; aÞ ¼ expðikz0 Þ f 0 ðq0 Þ þ 1=q0 f ðq0 Þ : ð13Þ As we now prove it, a natural generalization of (6) is f ðaÞ ¼ w1 exp a2 =2w2 ½a=wJ0 ða=wÞ J1 ða=wÞ; ð14Þ where J0 and J1 , are the Bessel functions of the first kind of order zero and one. Then, with the variable s ¼ p0 =w, taking into account the relations J00 ¼ J1 , J10 ¼ J0 J1 =s f 0 ðq0 Þ ¼ w1 os expð s2 =2Þ½sJ0 ðsÞ J1 ðsÞ ¼ w2 expðs2 =2Þ ð1 s2 ÞJ0 ðsÞ þ s1 J1 ðsÞ ; ð15aÞ
x0 =w
z0 ¼ z cos h þ x sin h;
ð11Þ
ð12Þ ð6Þ
with the parameter w characterizing the beam width. Indeed, substituting (6) into (5) gives ðD0 þ k 2 ÞWðx0 ; z0 Þ ¼ 2w2 expðikz0 Þ x0 =w sinðx0 =wÞ expð x02 =w2 Þ ; ð7Þ
6 ðe=2Þ1=2 k 2 w2 :
0
0
ðD þ k Þwðx ; aÞ ¼ expðikz Þ½d ða q0 Þ
1
0
o2q0
ð9Þ
and
1=q0 f ðq0 Þ ¼ w2 expðs2 =2Þ J0 ðsÞ s1 J1 ðsÞ ; ð15bÞ so that f 0 ðq0 Þ þ 1=q0 f ðq0 Þ ¼ s2 w2 expðs2 =2ÞJ0 ðsÞ; ð16Þ and since ½J0 ðsÞ 6 1 we get from (13) and (16) jk 2 ðD0 þ k 2 Þwðx0 ; aÞj 6 k 2 w2 s2 expðs2 =2Þ 6 2e1 k 2 w2 :
ð17Þ
P. Hillion / Optics Communications 227 (2003) 57–59
So, Wðx0 Þ satisfies the 3D-Helmholtz equation to the 0ðk 2 w2 Þ order and substituting (14) into (12) gives Wðx0 ; z0 Þ ¼ Aðq0 Þ expðikz0 Þ; ðq0 Þ Z 1 ¼ db expðb2 Þ½bJ0 ðbÞ J1 ; ðbÞ; q0 =w
ð18Þ that reduces, with zero as lower bound of the integral (18), to the plane wave Að0Þ expðikz0 Þ for q0 w and with a transverse amplitude decreasing for q0 > w. Beams propagating in the (h; /) direction are obtained by performing a rotation with the Euler angles from x0 to x giving [2] z0 ¼ x sin h cos / þ y sin h sin / þ z cos h;
ð19aÞ
and since r2 ¼ x2 þ y 2 þ z2 ¼ r02 ¼ q02 þ z02 , we get 02
2
2
q ¼ r ðx sin h cos / þ y sin h sin / þ z cos hÞ : ð19bÞ Substituting (19a) and (19b) into (18) gives beams with cylindrical symmetry around their direction of propagation and with evanescent transverse amplitude. These finite width beams which will be used to investigate acoustic, TE, TM wave scattering on
59
rough infinite dielectric films, are made of quasitruncated plane waves and they differ from the non-diffractive localized solutions of the Helmholtz equation, object of many works starting with Bateman [3] and still flourishing [4,5]. An excellent discussion is given in [6] where further references on this class of solutions can be found. But, wave scattering from infinite rough surfaces requires boundary conditions easier to get from a simple generalization of the Idemen technique [7] with quasi-plane waves than with localized waves.
References [1] A.A. Maradudin, T. Michel, A.R. McGurn, E.R. Mendez, Ann. Phys. 203 (1990) 255. [2] G.A. Korn, T.A. Korn, Mathematical Handbook for Scientists and Engineers, MacGraw-Hill, New York, 1968. [3] H. Bateman, Electrical and Optical Wave Motion, University Press, Cambridge, 1915. [4] A.P. Kiselev, J. Phys. A 36 (2003) L145. [5] P. Hillion, Eur. Phys. J. B 30 (2002) 527. [6] J. Salo, Propagation-Invariant Waves in Acoustic, Optical and Radiowave Fields, University Press, Helsinki, 2003. [7] H. Idemen, Essays on the Formal Aspects of Electromagnetic Theory, World Scientific, Singapore, 1993.
Scalar ‘‘beams of finite width’’ Pierre Hillion
*
Institut Henri Poincar e, 86 Bis Route de Croissy, 78110 Le V esinet, France Received 13 March 2003; received in revised form 24 June 2003; accepted 2 September 2003
Abstract Beams with a cylindrical symmetry around their direction of propagation and with a transverse amplitude strongly decreasing from some distance w of this direction give a manageable approximation to scalar beams of finite width. Ó 2003 Elsevier B.V. All rights reserved. PACS: 41.20 Jb; 42.25 Bs; 03.40 kf Keywords: Scalar beams; Finite width beams; Wave propagation; Wave scattering
Wave scattering by obstacles of acoustic, TE, TM, electromagnetic beams of finite width is a more close approach to experimental conditions that does the assumption of plane waves [1] making possible a better comparison between numerical computations and measurements. The finite width beam used in [1] has the form Z p=2 wðx; zÞ ¼ dhvðh h0 Þ exp ½ikðx sin h þ z cos hÞ; p=2
ð1Þ 1=2
vðh h0 Þ ¼ kap
h
i exp k a ðh h0 Þ ; 2 2
2
ð1aÞ
but in fact, this expression represents an aggregate of harmonic plane waves all propagating in
directions close to h0 for ka large enough; we observe easily that vðh h0 Þ tends to the Dirac distribution dðh h0 Þ for ka ) 1 To look for scalar waves of finite width, solutions of the 2D-Helmholtz equation, we proceed as follows: we first consider in the ðx0 ; z0 Þ-coordinates a plane wave propagating in the z0 -direction and transversally truncated at jx0 j ¼ a wðx0 ; z0 ; aÞ ¼ expðikz0 ÞV ðx0 ; aÞ; V ðx0 ; aÞ ¼ U ðx0 þ aÞ U ðx0 aÞ;
where U is the unit step function. A simple calculation gives with D0 ¼ o2x0 þ o2z0 ðD0 þ k 2 Þwðx0 ; z0 ; aÞ ¼ expðikz0 Þ½d0 ðx0 þ aÞ d0 ðx0 aÞ; ð3Þ 0
*
Tel.: +33-1397-66401; fax: +33-1396-91764. E-mail address: [email protected].
ð2Þ
in which d is the derivative of the Dirac distribution. We now define a scalar beam with transverse evanescent amplitude by the following relation; where f ðaÞ is a convenient weight function:
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.039
58
P. Hillion / Optics Communications 227 (2003) 57–59
Wðx0 ; z0 Þ ¼
Z
1
daf ðaÞwðx0 ; z0 ; aÞ;
ð4Þ
1
and substituting (2) into (4) gives Z 1 da½f ðaÞ f ðaÞ; Wðx0 ; z0 Þ ¼
ð4aÞ
x0
while according to (3) and (4) with the distributions in (3) written d0 ða þ x0 Þ þ d0 ða x0 Þ ðD0 þ k 2 ÞWðx0 ; z0 Þ Z ¼ expðikz0 Þ
This result may be generalized to 3D-scalar beams with a cylindrical symmetry around the direction of propagation as follows. Using x0 -coordinates ðx0 ¼ x0 ; y 0 ; z0 Þ; we start from a plane wave propagating in the z0 -direction inside a cylinder of radius q0 ¼ ðx02 þ y 02 Þ1=2 wðx0 ; aÞ ¼ expðikz0 ÞU ða q0 Þ; 0
and with D ¼ 0
0
þ 1=q
ð10Þ
dq0 ; þo2z0 ;
0
0
0
2
0
daf ðaÞ½d ða þ x Þ þ d ða x Þ ¼ expðikz0 Þ f 0 ðx0 Þ þ f 0 ð x0 Þ ; ð5Þ
we get
1=q0 dða p0 Þ:
x0
in which f 0 is the derivative of f . Then, we look for a weight function making of Wðx0 ; z0 Þ a quasitruncated plane wave and a convenient choice is f ðaÞ ¼ w1 expða2 =w2 Þ ½sinða=wÞ a=w cosða=wÞ;
which implies 2 0 k ðD þ k 2 ÞWðx0 ; z0 Þ 6 2k 2 w2 x0 =w exp x02 =w2 0
ð7aÞ
0
Consequently, Wðx ; z Þ satisfies the 2D-Helmholtz equation with an 0ðk 2 w2 Þ error, negligible for kw large enough and substituting (6) into (4a), we get Wðx0 ; z0 Þ ¼ 2Aðx0 Þ expðikz0 Þ; Z 1 0 Aðx Þ ¼ db expðb2 Þ½sin b b cos b:
ð8Þ
Taking zero as lower bound of the integral (8) for x0 w; this expression is the plane wave 2Að0Þ expðikz0 Þ in this region while the amplitude Aðx0 Þ is strongly decreasing for jx0 j > w. Then, beams of finite width, solutions to the 0ðk 2 w2 Þ order of the 2D-Helmholtz equation and propagating in the direction h, use in (8) the coordinates x; z obtained by a rotation h x0 ¼ x cos h z sin h:
A scalar beam with evanescent transverse amplitude is obtained from the integral relation Z 1 Z 1 Wðx0 Þ ¼ daf ðaÞwðx0 ; aÞ ¼ expðikz0 Þ daf ðaÞ; q0
1
so that according to (11) and (12) ðD0 þ k 2 Þwðx0 ; aÞ ¼ expðikz0 Þ f 0 ðq0 Þ þ 1=q0 f ðq0 Þ : ð13Þ As we now prove it, a natural generalization of (6) is f ðaÞ ¼ w1 exp a2 =2w2 ½a=wJ0 ða=wÞ J1 ða=wÞ; ð14Þ where J0 and J1 , are the Bessel functions of the first kind of order zero and one. Then, with the variable s ¼ p0 =w, taking into account the relations J00 ¼ J1 , J10 ¼ J0 J1 =s f 0 ðq0 Þ ¼ w1 os expð s2 =2Þ½sJ0 ðsÞ J1 ðsÞ ¼ w2 expðs2 =2Þ ð1 s2 ÞJ0 ðsÞ þ s1 J1 ðsÞ ; ð15aÞ
x0 =w
z0 ¼ z cos h þ x sin h;
ð11Þ
ð12Þ ð6Þ
with the parameter w characterizing the beam width. Indeed, substituting (6) into (5) gives ðD0 þ k 2 ÞWðx0 ; z0 Þ ¼ 2w2 expðikz0 Þ x0 =w sinðx0 =wÞ expð x02 =w2 Þ ; ð7Þ
6 ðe=2Þ1=2 k 2 w2 :
0
0
ðD þ k Þwðx ; aÞ ¼ expðikz Þ½d ða q0 Þ
1
0
o2q0
ð9Þ
and
1=q0 f ðq0 Þ ¼ w2 expðs2 =2Þ J0 ðsÞ s1 J1 ðsÞ ; ð15bÞ so that f 0 ðq0 Þ þ 1=q0 f ðq0 Þ ¼ s2 w2 expðs2 =2ÞJ0 ðsÞ; ð16Þ and since ½J0 ðsÞ 6 1 we get from (13) and (16) jk 2 ðD0 þ k 2 Þwðx0 ; aÞj 6 k 2 w2 s2 expðs2 =2Þ 6 2e1 k 2 w2 :
ð17Þ
P. Hillion / Optics Communications 227 (2003) 57–59
So, Wðx0 Þ satisfies the 3D-Helmholtz equation to the 0ðk 2 w2 Þ order and substituting (14) into (12) gives Wðx0 ; z0 Þ ¼ Aðq0 Þ expðikz0 Þ; ðq0 Þ Z 1 ¼ db expðb2 Þ½bJ0 ðbÞ J1 ; ðbÞ; q0 =w
ð18Þ that reduces, with zero as lower bound of the integral (18), to the plane wave Að0Þ expðikz0 Þ for q0 w and with a transverse amplitude decreasing for q0 > w. Beams propagating in the (h; /) direction are obtained by performing a rotation with the Euler angles from x0 to x giving [2] z0 ¼ x sin h cos / þ y sin h sin / þ z cos h;
ð19aÞ
and since r2 ¼ x2 þ y 2 þ z2 ¼ r02 ¼ q02 þ z02 , we get 02
2
2
q ¼ r ðx sin h cos / þ y sin h sin / þ z cos hÞ : ð19bÞ Substituting (19a) and (19b) into (18) gives beams with cylindrical symmetry around their direction of propagation and with evanescent transverse amplitude. These finite width beams which will be used to investigate acoustic, TE, TM wave scattering on
59
rough infinite dielectric films, are made of quasitruncated plane waves and they differ from the non-diffractive localized solutions of the Helmholtz equation, object of many works starting with Bateman [3] and still flourishing [4,5]. An excellent discussion is given in [6] where further references on this class of solutions can be found. But, wave scattering from infinite rough surfaces requires boundary conditions easier to get from a simple generalization of the Idemen technique [7] with quasi-plane waves than with localized waves.
References [1] A.A. Maradudin, T. Michel, A.R. McGurn, E.R. Mendez, Ann. Phys. 203 (1990) 255. [2] G.A. Korn, T.A. Korn, Mathematical Handbook for Scientists and Engineers, MacGraw-Hill, New York, 1968. [3] H. Bateman, Electrical and Optical Wave Motion, University Press, Cambridge, 1915. [4] A.P. Kiselev, J. Phys. A 36 (2003) L145. [5] P. Hillion, Eur. Phys. J. B 30 (2002) 527. [6] J. Salo, Propagation-Invariant Waves in Acoustic, Optical and Radiowave Fields, University Press, Helsinki, 2003. [7] H. Idemen, Essays on the Formal Aspects of Electromagnetic Theory, World Scientific, Singapore, 1993.