- Email: [email protected]
Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches
ARTICLE IN PRESS
Optik
Optics
Optik 116 (2005) 449–453 www.elsevier.de/ijleo
Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches Pusheng Liua,b, Baida Lu¨a,b, a
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330027, PR China Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, PR China
b
Received 27 October 2004; accepted 18 January 2005
Abstract The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the theory of the boundary diffraction wave. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals. As expected, our results are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory. r 2005 Elsevier GmbH. All rights reserved. Keywords: Boundary diffraction wave; Diffraction at an annular aperture; Spherical wave; Kirchhoff diffraction integral; First and second Rayleigh diffraction integrals
1. Introduction The diffraction is a subject of continuing interest from both theoretical and applicative point of view. There are a variety of approaches to treat diffraction problem in the more strict sense, such as the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals [1,2] and Maggi–Rubinowicz theory of the boundary diffraction wave. A further generalization of the theory of the boundary diffraction wave was made by Wolf et Corresponding author. Fax: +86 28 8540 3260.
E-mail address: [email protected] (B. Lu¨). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.01.037
al. [3–5] by use of the Stokes theorem, which is shown to be consistent with the Kirchhoff diffraction integral. Up to now, the theory of the boundary diffraction wave has been mainly applied to the diffraction at a circular aperture [3–7], and the comparison has been made for axial points [8,9]. The aim of the present paper is to study the diffraction of spherical waves at an annular aperture by using the theory of the boundary diffraction wave. The theoretical formulation is presented in Section 1. Section 2 gives numerical calculation results for axial and transversal intensity distributions of diffracted divergent spherical waves and comparison with the Kirchhoff diffraction integral as well as the first
ARTICLE IN PRESS 450
P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
and second Rayleigh diffraction integrals. Finally, the main results obtained in this paper are summarized in Section 4.
2. Theoretical formulation of spherical waves diffracted at an annular aperture As shown in Fig. 1a, assume that a divergent spherical wave from a point P0 ð0; 0; dÞ ðd40Þ on the z-axis propagates in the half-space z40, and through an annular aperture positioned at the plane z ¼ 0 with inner and outer radii a and b, respectively. The field of the spherical wave reads as U ðiÞ ðP0 Þ ¼
eikjP0 Pj , jP0 Pj
(1)
where k is the wave number related to the wavelength l by k ¼ 2p=l, and jP0 Pj is distance from P0 to the point Pðx; y; zÞ of observation. Now, we construct P a closed surface S bounded by (1) the annular aperture , which is located at the xy plane; (2) the surface S 1 of a truncated cone whose vertex is at P0 and whose generators pass through the inner edge of the annular aperture; (3) the surface S 2 with vertex P0 and generators passing through the outer edge of the annular
S3
A
ZZ 1 q eikR eikR qUðAÞ UðAÞ dS, 4p qn R R qn
z
P0 P S1
0
S
where the integral is performed with respect to the boundary of the annular aperture S. In accordance with the theory of the boundary diffraction wave, the field at the point Pðx; y; zÞ of observation is expressed as [3–7] (4)
(5)
S3
S2
ð3Þ
where U ðPÞ denotes the geometrical wave originating from the source, and Ud(P) denotes the boundary diffraction wave originating from the boundary of the P annular aperture , and 8 ikjP0 Pj > :0 when P is in the geometrical shadow:
S1
(a)
¼
g
n
Σ
U k ðPÞ
U B ðPÞ ¼ U g ðPÞ þ U d ðPÞ,
S2
Σ
aperture; (4) a portion S 3 of a large sphere centered on P. It follows from the Kirchhoff integral theorem that [1] 1 q eikR eikR q a UðAÞ dS UðAÞ 4p SþS1 þS2 þS3 qn R R qn 8 ikjP Pj 0 > : 0; P outside S; P where A denotes a point at the surface S (including , S1, S2 and S3), R is distance from A to the point Pðx; y; zÞ of observation, i.e., R ¼ PA, q/qn denotes differentiation along the unit inward normal to the surface S. The contribution from S3 is negligible, and according to the Kirchhoff diffraction integral the contribution from the annular aperture reads as [1]
From Eqs. (1)–(5), we have U dj ðPÞ D1 C1 A1 B1
Q1´ P0
Γ2 Γ1
Q1
Q1´
(b)
Σ Q2
P0 A1= r1, P0Q1= r1´, P0 A2= r2, P0Q2= r2´, PA1= R1, PQ1= R1´, PA2= R2, PQ2= R2´
ZZ 1 q eikR eikR qUðAÞ UðAÞ dSj , 4p qn R R qn
ð6Þ
Sj
R1´
R1 R2
¼ z
P R2´ A2
B2
Fig. 1. (a) Illustrating notation relating to Eq. (2) and (b) derivation of the boundary diffraction wave.
U dj ðPÞ
where denotes the contribution from the surface Sj (j ¼ 1; 2 unless otherwise stated). As shown in Fig. 1b, letting Aj be a point at the surface Sj and P0 Aj ¼ rj , the spheres rj ¼ constant cut orthogonally the truncated cone, then we have qUðAj Þ q eikrj ¼ ¼ 0. (7) qnj rj qnj We take as the element dS j the area Aj Bj C j Dj bounded by segments of two neighboring generators and by the arcs of circles in which the spheres rj ¼
ARTICLE IN PRESS P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
constant and rj þ drj ¼ constant intersect the cone. Let Qj and Q0j be the points of intersection of the two P generators with the edge Gj of the annular aperture , and dlj be the length of the element of Gj between these two points, hence q eikRj ik 1 ikRj cosðnj ; Rj Þ, (8) ¼ e qnj Rj Rj Rj dS j ¼
rj drj dl j , r0j
(9)
Rj cosðnj ; Rj Þ ¼ R0j cosðnj ; R0j Þ,
(10)
where r0j ¼ P0 Qj , Rj ¼ PAj , R0j ¼ PQj . The substitution from Eqs. (7)–(10) into Eq. (6) yields Z R0j 1 dl j 0 cosðnj ; R0j Þ U dj ðPÞ ¼ 4p Gj rj ! Z 1 1 ikðrj þRj Þ ik e ð11Þ drj . R2j R3j r0j Tedious but straightforward integral calculations lead to ! Z 1 1 ikðrj þRj Þ ik e drj: R2j R3j r0j 0
0
eikðrj þRj Þ ¼ 02 . Rj ½1 þ cosðR0j ; r0j Þ
ð12Þ
From Eqs. (11)–(12) it turns out that I ikðr0 þR0 Þ 1 e 1 1 cosðn1 ; R01 Þ dl 1 , U d1 ðPÞ ¼ 4p r01 R01 1 þ cosðR01 ; r1 Þ
(13a)
G1
U d2 ðPÞ ¼
1 4p
I G2
0
0
eikðr2 þR2 Þ cosðn2 ; R02 Þ dl 2 . r02 R02 1 þ cosðR02 ; r2 Þ
(13b)
The contour integral is performed along the inner edge G1 and outer edge G2 of the annular aperture S, respectively. Therefore, the field at the point Pðx; y; zÞ of observation is finally expressed as
I þ G2
0
the contributions U d1 ðPÞ þ U d2 ðPÞ from the inner and outer edges of the annular aperture. There is the discontinuity in U d1 ðPÞ and U d2 ðPÞ due to factors 1 þ cosðR0j ; r0j Þ in the denominators of Eq. (14). In addition, because cosðn1 ; R01 Þ40, cosðn2 ; R02 Þo0, we have U d1 ðPÞ40, U d2 ðPÞo0, namely, the contributions from the inner and outer edges of the aperture are positive and negative, respectively. Some special cases of Eq. (14) are of interest. (1) Diffraction at a circular aperture: Making use of a ¼ 0, Eq. (14) reduces to U B ðPÞ ¼ U g ðPÞ I ikðr0 þR0 Þ 1 e 2 2 cosðn2 ; R02 Þ þ dl 2 . 4p r02 R02 1 þ cosðR02 ; r02 Þ
Eq. (15) is consistent with the result in Ref. [1]. (2) Diffraction at a circular disk: Letting b ! 1, Eq. (14) leads to U B ðPÞ ¼ U g ðPÞ I ikðr0 þR0 Þ 1 e 1 1 cosðn1 ; R01 Þ þ dl 1 . 4p r01 R01 1 þ cosðR01 ; r01 Þ
(3) Axial field expression: On placing x ¼ y ¼ 0 into Eq. (14), we obtain d þz a2 eikðr1 þt1 Þ U B ðPÞ ¼ 2 r21 t1 þ a2 z dz r1 t1 ! b2 eikðr2 þt2 Þ , ð17Þ 2 r2 t2 þ b2 z d 1 z r2 t2 where U g ðPÞ ¼ 0, and r1 ¼ ða2 þ d 2 Þ1=2 , r2 ¼ ðb2 þ d 2 Þ1=2 , t1 ¼ ða2 þ z2 Þ1=2 , t2 ¼ ðb2 þ z2 Þ1=2 . (4) Diffraction of plane waves at an annular aperture: By letting d ! 1, the field of plane waves diffracted at an annular aperture is expressed as U B ðPÞ
0
I þ G2
0
eikðr2 þR2 Þ cosðn2 ; R02 Þ dl 2 C A. r02 R02 1 þ cosðR02 ; r02 Þ
ð14Þ
Eq. (14) indicates that the field of the spherical wave diffracted at an annular aperture includes the contribution Ug(P) from the geometrical wave and
ð16Þ
G1
I G1
1
ð15Þ
G2
1 B ¼ U g ðPÞ þ @ 4p
U B ðPÞ ¼ U g ðPÞ þ U d1 ðPÞ þ U d2 ðPÞ 0 I 0 0 1 B eikðr1 þR1 Þ cosðn1 ; R01 Þ dl 1 g ¼ U ðPÞ þ @ 4p r01 R01 1 þ cosðR01 ; r01 Þ G1
451
0
eikR1 cosðn1 ; R01 Þ dl 1 R01 1 þ cosðR01 ; r01 Þ 1
ikR02
e cosðn2 ; R02 Þ dl 2 C A, 0 R2 1 þ cosðR02 ; r02 Þ
ð18Þ
where U g ðPÞ ( ¼
eikz
when P is in the direct beam;
0
when P is in the geometrical shadow:
ð19Þ
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(×10−4) 9
3. Numerical calculation results and comparison
8
1 U 2 ðPÞ ¼ 2p
qUðx; y; zÞ eikR dS qz S z¼0 R
U 2 ðPÞ ¼
d 2p
0
2p
Z
b ikðLþRÞ
e
a
L3 R
ðikL 1Þr dr dy,
(21b)
where L ¼ ðr2 þ d 2 Þ1=2 . By letting x ¼ y ¼ 0 and integrating Eq. (21), the axial field can be expressed in closed form U 1 ðPÞ ¼
z eikðr1 þt1 Þ z eikðr2 þt2 Þ , t1 r1 þ t1 t2 r2 þ t2
(22a)
U 2 ðPÞ ¼
d eikðr1 þt1 Þ d eikðr2 þt2 Þ . r1 r1 þ t 1 r2 r2 þ t 2
(22b)
It is noted that in Eq. (21), ð1 ikRÞ is not approximately replaced by 2ikR to ensure the calculation accuracy [2]. In addition, Eq. (22) is in agreement with Eqs. (8), (9) in Ref. [8]. From Eqs. (17), and (22) it follows that U B ðPÞ ¼ 12½U 1 ðPÞ þ U 2 ðPÞ ¼ U k ðPÞ.
5 4 3 2 1 0
20
40
(20b)
P are used, where R is distance from a point at to the point Pðx; y; zÞ of observation. In the cylindrical coordinate system Eq. (20) is rewritten as Z 2p Z b ikðLþRÞ z e U 1 ðPÞ ¼ ðikR 1Þr dr dy, (21a) 2p 0 LR3 a Z
6
0
Z Z
I2(0,0,z) IB(0,0,z) IK(0,0,z) I1(0,0,z)
7 I(0,0,z)
Numerical calculations were made to confirm the validity of Eq. (14) and to compare different integral approaches. The calculation parameters are d ¼ 50l, b ¼ 20l, a ¼ 10l. For comparison the first and second Rayleigh diffraction integrals of the form Z Z 1 q eikR U 1 ðPÞ ¼ Uðx; y; 0Þ dS, (20a) 2p qz R S
(23)
Eq. (23) indicates that the mean value of the axial field in the use of the two Rayleigh diffraction integrals is equal to the result in the use of Eq. (17). Furthermore, if the source and observation points are equal distant from the aperture, i.e., d ¼ z, from Eqs. (17) and (22), we have U B ðPÞ ¼ U 1 ðPÞ ¼ U 2 ðPÞ ¼ U k ðPÞ. Fig. 2 gives the axial intensity distribution IðPÞ ¼ jUðPÞj2 of a divergent spherical wave diffracted at an annular aperture. It is seen that I B ð0; 0; zÞ ¼ I k ð0; 0; zÞ ¼ ½I 1 ð0; 0; zÞ þ I 2 ð0; 0; zÞ =2, and I B ð0; 0; 50lÞ ¼ I k ð0; 0; 50lÞ ¼ I 1 ð0; 0; 50lÞ ¼ I 2 ð0; 0; 50lÞ, which is consistent with the above theoretical predication. The difference between I B ð0; 0; zÞ and I 1 ð0; 0; zÞ, I 2 ð0; 0; zÞ increases as the point of observation is closer to the aperture. But, if zX110l, we have I B ð0; 0; zÞ ¼ I k ð0; 0; zÞ ¼ I 1 ð0; 0; zÞ ¼ I 2 ð0; 0; zÞ. The transversal intensity distributions Iðx; 0; zÞ
60 z/λ
80
100
120
Fig. 2. Axial intensity distribution Ið0; 0; zÞ of a divergent spherical wave diffracted at an annular aperture, _________: I B ð0; 0; zÞ, I k ð0; 0; zÞ; - – - – : I 1 ð0; 0; zÞ; - - - - - -: I 2 ð0; 0; zÞ.
of a divergent spherical wave diffracted at an annular aperture for different values of z ¼ 10l, 50l and 110l are plotted in Fig. 3a–c, respectively, from which we see that at z ¼ d ¼ 50l we have I B ðx; 0; zÞ ¼ I k ðx; 0; zÞ ¼ I 1 ðx; 0; zÞ ¼ I 2 ðx; 0; zÞ, whereas at z ¼ 10lo50l there is the difference between I B ðx; 0; zÞ and I 1 ðx; 0; zÞ, I 2 ðx; 0; zÞ. But as shown in Fig. 3c the difference between them vanishes if the point of observation is positioned enough far from the aperture, e.g., z ¼ 110l. As expected, in all cases, we have I B ðx; 0; zÞ ¼ I k ðx; 0; zÞ, because Eq. (14) is consistent with the Kirchhoff diffraction integral.
4. Concluding remarks In this study, the diffraction integral expression for divergent spherical waves diffracted at an annular aperture has been presented, which is expressed in terms of a sum of the geometrical wave and two boundary diffraction waves. The contributions from the inner and outer boundary diffraction waves are positive and negative, respectively. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the closed-form axial field expression have been obtained as the special cases of our general one. Numerical calculation results for the axial and transversal intensity distributions of diffracted divergent spherical waves by using the theory of the boundary diffraction wave, Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals have been given. It has been shown that the results in the use of Eq. (14) are the same with those in the use Eq. (3). This is an expected conclusion, because the representation of the boundary diffraction wave is consistent with the Kirchhoff diffraction integral. Apart from providing
ARTICLE IN PRESS P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
(×10−4) 5 4
I(x,0,z)
3 IB(x,0,z), Ik(x,0,z) I2(x,0,z)
2
I1(x,0,z)
1 0 -35 -30 -25 -20 -15 -10 -5
0
5
10 15 20 25 30 35
x/λ
(a) (×10−4) 2.4 2
IB(x,0,z) IK(x,0,z), I1(x,0,z), I2(x,0,z)
I(x,0,z)
1.6 1.2
453
in Eq. (14), its main drawback is that, the boundary waves U d1 ðPÞ and U d2 ðPÞ are discontinuous across the edges of the geometrical shadow, e.g., the discontinuity exists at (a) y ¼ 12l, 24l; (b) y ¼ 20l, 40l; (c) y ¼ 32l, 64l in Fig. 3a–c, respectively. Although the Rayleigh diffraction integrals have found many applications in optics, they could not give the exact calculation result in the very near field. The four diffraction formulae have been shown to be consistent for axial and transversal intensity distribution of divergent spherical waves diffracted at an annular aperture if the source and observation point is equally distant from the aperture, and the observation points are located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory. Finally, we would like to mention that, although the study has been performed for divergent spherical waves, the result for convergent spherical waves diffracted at an annular aperture can be obtained in a similar way, where the initial field U ðiÞ ðP0 Þ ¼ expðikjP0 Pj=jP0 PjÞ has to be replaced by U ðiÞ ðP0 Þ ¼ expðikjP0 Pj=jP0 PjÞ.
0.8 0.4 0 -50 -40 -30 -20 -10
Acknowledgements 0
( b)
1 x/λ
0
20
30
40
50
This work was supported by the Nature Science Foundation of Jiangxi Province.
(×10−4) 1
References
I(x,0,z)
0.8 0.6 0.4
IB(x,0,z) IK(x,0,z), I1(x,0,z), I2(x,0,z)
0.2 0 -90 -75 -60 -45 -30 -15 (c)
0 15 30 45 60 75 90 x/λ
Fig. 3. Transversal intensity distributions Iðx; 0; zÞ in the x direction of a divergent spherical wave diffracted at an annular aperture,________: I B ðx; 0; zÞ, I k ðx; 0; zÞ; - – - –: I 1 ðx; 0; zÞ; - - - - - -: I 2 ðx; 0; zÞ. (a) z ¼ 10l; (b) z ¼ 50l; (c) z ¼ 110l.
an intuitive physical insight into the diffraction phenomenon, the advantage of the boundary diffraction wave theory is that it saves the computer time greatly as compared with the direct integration of the Kirchoff diffraction formula. For example, in Fig. 3a–c the ratios of the computer time in the use of Eqs. (14) and (3) are 1:4326; 1:3546; 1:2387, respectively. However, as shown
[1] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambridge University Press, England, 1999, pp. 412–514. [2] A.S. Marathay, J.F. McCalmont, On the usual approximation used in the Rayleigh–Sommerfeld diffraction theory, J. Opt. Soc. Am. A 21 (2004) 510–516. [3] K. Miyamoto, E. Wolf, Generalization of the Maggi– Rubinowicz theory of the boundary diffraction wave—part I, J. Opt. Soc. Am. A 52 (1962) 615–625. [4] K. Miyamoto, E. Wolf, Generalization of the Maggi– Rubinowicz theory of the boundary diffraction wave—part I, J. Opt. Soc. A 52 (1962) 626–637. [5] E.W. Marchand, E. Wolf, Consistent formulation of Kirchhoff’s diffraction theory, J. Opt. Soc. A 56 (1966) 1712–1722. [6] K. Miyamoto, E. Wolf, Boundary diffraction wave in the domain of the Rayleigh–Kirchholf diffraction theory, J. Opt. Soc. A 52 (1962) 761–767. [7] E.W. Marchand, E. Wolf, Diffraction at small apertures black screens, J. Opt. Soc. A 59 (1969) 79–90. [8] J.C. Heurtley, Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points, J. Opt. Soc. A 63 (1973) 1003–1008. [9] H. Osterberg, L.W. Smith, Closed solutions of Rayleigh’s diffraction integral for axial points, J. Opt. Soc. A 51 (1961) 1050–1054.
Optik
Optics
Optik 116 (2005) 449–453 www.elsevier.de/ijleo
Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches Pusheng Liua,b, Baida Lu¨a,b, a
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330027, PR China Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, PR China
b
Received 27 October 2004; accepted 18 January 2005
Abstract The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the theory of the boundary diffraction wave. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals. As expected, our results are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory. r 2005 Elsevier GmbH. All rights reserved. Keywords: Boundary diffraction wave; Diffraction at an annular aperture; Spherical wave; Kirchhoff diffraction integral; First and second Rayleigh diffraction integrals
1. Introduction The diffraction is a subject of continuing interest from both theoretical and applicative point of view. There are a variety of approaches to treat diffraction problem in the more strict sense, such as the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals [1,2] and Maggi–Rubinowicz theory of the boundary diffraction wave. A further generalization of the theory of the boundary diffraction wave was made by Wolf et Corresponding author. Fax: +86 28 8540 3260.
E-mail address: [email protected] (B. Lu¨). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.01.037
al. [3–5] by use of the Stokes theorem, which is shown to be consistent with the Kirchhoff diffraction integral. Up to now, the theory of the boundary diffraction wave has been mainly applied to the diffraction at a circular aperture [3–7], and the comparison has been made for axial points [8,9]. The aim of the present paper is to study the diffraction of spherical waves at an annular aperture by using the theory of the boundary diffraction wave. The theoretical formulation is presented in Section 1. Section 2 gives numerical calculation results for axial and transversal intensity distributions of diffracted divergent spherical waves and comparison with the Kirchhoff diffraction integral as well as the first
ARTICLE IN PRESS 450
P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
and second Rayleigh diffraction integrals. Finally, the main results obtained in this paper are summarized in Section 4.
2. Theoretical formulation of spherical waves diffracted at an annular aperture As shown in Fig. 1a, assume that a divergent spherical wave from a point P0 ð0; 0; dÞ ðd40Þ on the z-axis propagates in the half-space z40, and through an annular aperture positioned at the plane z ¼ 0 with inner and outer radii a and b, respectively. The field of the spherical wave reads as U ðiÞ ðP0 Þ ¼
eikjP0 Pj , jP0 Pj
(1)
where k is the wave number related to the wavelength l by k ¼ 2p=l, and jP0 Pj is distance from P0 to the point Pðx; y; zÞ of observation. Now, we construct P a closed surface S bounded by (1) the annular aperture , which is located at the xy plane; (2) the surface S 1 of a truncated cone whose vertex is at P0 and whose generators pass through the inner edge of the annular aperture; (3) the surface S 2 with vertex P0 and generators passing through the outer edge of the annular
S3
A
ZZ 1 q eikR eikR qUðAÞ UðAÞ dS, 4p qn R R qn
z
P0 P S1
0
S
where the integral is performed with respect to the boundary of the annular aperture S. In accordance with the theory of the boundary diffraction wave, the field at the point Pðx; y; zÞ of observation is expressed as [3–7] (4)
(5)
S3
S2
ð3Þ
where U ðPÞ denotes the geometrical wave originating from the source, and Ud(P) denotes the boundary diffraction wave originating from the boundary of the P annular aperture , and 8 ikjP0 Pj >
S1
(a)
¼
g
n
Σ
U k ðPÞ
U B ðPÞ ¼ U g ðPÞ þ U d ðPÞ,
S2
Σ
aperture; (4) a portion S 3 of a large sphere centered on P. It follows from the Kirchhoff integral theorem that [1] 1 q eikR eikR q a UðAÞ dS UðAÞ 4p SþS1 þS2 þS3 qn R R qn 8 ikjP Pj 0 >
From Eqs. (1)–(5), we have U dj ðPÞ D1 C1 A1 B1
Q1´ P0
Γ2 Γ1
Q1
Q1´
(b)
Σ Q2
P0 A1= r1, P0Q1= r1´, P0 A2= r2, P0Q2= r2´, PA1= R1, PQ1= R1´, PA2= R2, PQ2= R2´
ZZ 1 q eikR eikR qUðAÞ UðAÞ dSj , 4p qn R R qn
ð6Þ
Sj
R1´
R1 R2
¼ z
P R2´ A2
B2
Fig. 1. (a) Illustrating notation relating to Eq. (2) and (b) derivation of the boundary diffraction wave.
U dj ðPÞ
where denotes the contribution from the surface Sj (j ¼ 1; 2 unless otherwise stated). As shown in Fig. 1b, letting Aj be a point at the surface Sj and P0 Aj ¼ rj , the spheres rj ¼ constant cut orthogonally the truncated cone, then we have qUðAj Þ q eikrj ¼ ¼ 0. (7) qnj rj qnj We take as the element dS j the area Aj Bj C j Dj bounded by segments of two neighboring generators and by the arcs of circles in which the spheres rj ¼
ARTICLE IN PRESS P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
constant and rj þ drj ¼ constant intersect the cone. Let Qj and Q0j be the points of intersection of the two P generators with the edge Gj of the annular aperture , and dlj be the length of the element of Gj between these two points, hence q eikRj ik 1 ikRj cosðnj ; Rj Þ, (8) ¼ e qnj Rj Rj Rj dS j ¼
rj drj dl j , r0j
(9)
Rj cosðnj ; Rj Þ ¼ R0j cosðnj ; R0j Þ,
(10)
where r0j ¼ P0 Qj , Rj ¼ PAj , R0j ¼ PQj . The substitution from Eqs. (7)–(10) into Eq. (6) yields Z R0j 1 dl j 0 cosðnj ; R0j Þ U dj ðPÞ ¼ 4p Gj rj ! Z 1 1 ikðrj þRj Þ ik e ð11Þ drj . R2j R3j r0j Tedious but straightforward integral calculations lead to ! Z 1 1 ikðrj þRj Þ ik e drj: R2j R3j r0j 0
0
eikðrj þRj Þ ¼ 02 . Rj ½1 þ cosðR0j ; r0j Þ
ð12Þ
From Eqs. (11)–(12) it turns out that I ikðr0 þR0 Þ 1 e 1 1 cosðn1 ; R01 Þ dl 1 , U d1 ðPÞ ¼ 4p r01 R01 1 þ cosðR01 ; r1 Þ
(13a)
G1
U d2 ðPÞ ¼
1 4p
I G2
0
0
eikðr2 þR2 Þ cosðn2 ; R02 Þ dl 2 . r02 R02 1 þ cosðR02 ; r2 Þ
(13b)
The contour integral is performed along the inner edge G1 and outer edge G2 of the annular aperture S, respectively. Therefore, the field at the point Pðx; y; zÞ of observation is finally expressed as
I þ G2
0
the contributions U d1 ðPÞ þ U d2 ðPÞ from the inner and outer edges of the annular aperture. There is the discontinuity in U d1 ðPÞ and U d2 ðPÞ due to factors 1 þ cosðR0j ; r0j Þ in the denominators of Eq. (14). In addition, because cosðn1 ; R01 Þ40, cosðn2 ; R02 Þo0, we have U d1 ðPÞ40, U d2 ðPÞo0, namely, the contributions from the inner and outer edges of the aperture are positive and negative, respectively. Some special cases of Eq. (14) are of interest. (1) Diffraction at a circular aperture: Making use of a ¼ 0, Eq. (14) reduces to U B ðPÞ ¼ U g ðPÞ I ikðr0 þR0 Þ 1 e 2 2 cosðn2 ; R02 Þ þ dl 2 . 4p r02 R02 1 þ cosðR02 ; r02 Þ
Eq. (15) is consistent with the result in Ref. [1]. (2) Diffraction at a circular disk: Letting b ! 1, Eq. (14) leads to U B ðPÞ ¼ U g ðPÞ I ikðr0 þR0 Þ 1 e 1 1 cosðn1 ; R01 Þ þ dl 1 . 4p r01 R01 1 þ cosðR01 ; r01 Þ
(3) Axial field expression: On placing x ¼ y ¼ 0 into Eq. (14), we obtain d þz a2 eikðr1 þt1 Þ U B ðPÞ ¼ 2 r21 t1 þ a2 z dz r1 t1 ! b2 eikðr2 þt2 Þ , ð17Þ 2 r2 t2 þ b2 z d 1 z r2 t2 where U g ðPÞ ¼ 0, and r1 ¼ ða2 þ d 2 Þ1=2 , r2 ¼ ðb2 þ d 2 Þ1=2 , t1 ¼ ða2 þ z2 Þ1=2 , t2 ¼ ðb2 þ z2 Þ1=2 . (4) Diffraction of plane waves at an annular aperture: By letting d ! 1, the field of plane waves diffracted at an annular aperture is expressed as U B ðPÞ
0
I þ G2
0
eikðr2 þR2 Þ cosðn2 ; R02 Þ dl 2 C A. r02 R02 1 þ cosðR02 ; r02 Þ
ð14Þ
Eq. (14) indicates that the field of the spherical wave diffracted at an annular aperture includes the contribution Ug(P) from the geometrical wave and
ð16Þ
G1
I G1
1
ð15Þ
G2
1 B ¼ U g ðPÞ þ @ 4p
U B ðPÞ ¼ U g ðPÞ þ U d1 ðPÞ þ U d2 ðPÞ 0 I 0 0 1 B eikðr1 þR1 Þ cosðn1 ; R01 Þ dl 1 g ¼ U ðPÞ þ @ 4p r01 R01 1 þ cosðR01 ; r01 Þ G1
451
0
eikR1 cosðn1 ; R01 Þ dl 1 R01 1 þ cosðR01 ; r01 Þ 1
ikR02
e cosðn2 ; R02 Þ dl 2 C A, 0 R2 1 þ cosðR02 ; r02 Þ
ð18Þ
where U g ðPÞ ( ¼
eikz
when P is in the direct beam;
0
when P is in the geometrical shadow:
ð19Þ
ARTICLE IN PRESS 452
P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
(×10−4) 9
3. Numerical calculation results and comparison
8
1 U 2 ðPÞ ¼ 2p
qUðx; y; zÞ eikR dS qz S z¼0 R
U 2 ðPÞ ¼
d 2p
0
2p
Z
b ikðLþRÞ
e
a
L3 R
ðikL 1Þr dr dy,
(21b)
where L ¼ ðr2 þ d 2 Þ1=2 . By letting x ¼ y ¼ 0 and integrating Eq. (21), the axial field can be expressed in closed form U 1 ðPÞ ¼
z eikðr1 þt1 Þ z eikðr2 þt2 Þ , t1 r1 þ t1 t2 r2 þ t2
(22a)
U 2 ðPÞ ¼
d eikðr1 þt1 Þ d eikðr2 þt2 Þ . r1 r1 þ t 1 r2 r2 þ t 2
(22b)
It is noted that in Eq. (21), ð1 ikRÞ is not approximately replaced by 2ikR to ensure the calculation accuracy [2]. In addition, Eq. (22) is in agreement with Eqs. (8), (9) in Ref. [8]. From Eqs. (17), and (22) it follows that U B ðPÞ ¼ 12½U 1 ðPÞ þ U 2 ðPÞ ¼ U k ðPÞ.
5 4 3 2 1 0
20
40
(20b)
P are used, where R is distance from a point at to the point Pðx; y; zÞ of observation. In the cylindrical coordinate system Eq. (20) is rewritten as Z 2p Z b ikðLþRÞ z e U 1 ðPÞ ¼ ðikR 1Þr dr dy, (21a) 2p 0 LR3 a Z
6
0
Z Z
I2(0,0,z) IB(0,0,z) IK(0,0,z) I1(0,0,z)
7 I(0,0,z)
Numerical calculations were made to confirm the validity of Eq. (14) and to compare different integral approaches. The calculation parameters are d ¼ 50l, b ¼ 20l, a ¼ 10l. For comparison the first and second Rayleigh diffraction integrals of the form Z Z 1 q eikR U 1 ðPÞ ¼ Uðx; y; 0Þ dS, (20a) 2p qz R S
(23)
Eq. (23) indicates that the mean value of the axial field in the use of the two Rayleigh diffraction integrals is equal to the result in the use of Eq. (17). Furthermore, if the source and observation points are equal distant from the aperture, i.e., d ¼ z, from Eqs. (17) and (22), we have U B ðPÞ ¼ U 1 ðPÞ ¼ U 2 ðPÞ ¼ U k ðPÞ. Fig. 2 gives the axial intensity distribution IðPÞ ¼ jUðPÞj2 of a divergent spherical wave diffracted at an annular aperture. It is seen that I B ð0; 0; zÞ ¼ I k ð0; 0; zÞ ¼ ½I 1 ð0; 0; zÞ þ I 2 ð0; 0; zÞ =2, and I B ð0; 0; 50lÞ ¼ I k ð0; 0; 50lÞ ¼ I 1 ð0; 0; 50lÞ ¼ I 2 ð0; 0; 50lÞ, which is consistent with the above theoretical predication. The difference between I B ð0; 0; zÞ and I 1 ð0; 0; zÞ, I 2 ð0; 0; zÞ increases as the point of observation is closer to the aperture. But, if zX110l, we have I B ð0; 0; zÞ ¼ I k ð0; 0; zÞ ¼ I 1 ð0; 0; zÞ ¼ I 2 ð0; 0; zÞ. The transversal intensity distributions Iðx; 0; zÞ
60 z/λ
80
100
120
Fig. 2. Axial intensity distribution Ið0; 0; zÞ of a divergent spherical wave diffracted at an annular aperture, _________: I B ð0; 0; zÞ, I k ð0; 0; zÞ; - – - – : I 1 ð0; 0; zÞ; - - - - - -: I 2 ð0; 0; zÞ.
of a divergent spherical wave diffracted at an annular aperture for different values of z ¼ 10l, 50l and 110l are plotted in Fig. 3a–c, respectively, from which we see that at z ¼ d ¼ 50l we have I B ðx; 0; zÞ ¼ I k ðx; 0; zÞ ¼ I 1 ðx; 0; zÞ ¼ I 2 ðx; 0; zÞ, whereas at z ¼ 10lo50l there is the difference between I B ðx; 0; zÞ and I 1 ðx; 0; zÞ, I 2 ðx; 0; zÞ. But as shown in Fig. 3c the difference between them vanishes if the point of observation is positioned enough far from the aperture, e.g., z ¼ 110l. As expected, in all cases, we have I B ðx; 0; zÞ ¼ I k ðx; 0; zÞ, because Eq. (14) is consistent with the Kirchhoff diffraction integral.
4. Concluding remarks In this study, the diffraction integral expression for divergent spherical waves diffracted at an annular aperture has been presented, which is expressed in terms of a sum of the geometrical wave and two boundary diffraction waves. The contributions from the inner and outer boundary diffraction waves are positive and negative, respectively. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the closed-form axial field expression have been obtained as the special cases of our general one. Numerical calculation results for the axial and transversal intensity distributions of diffracted divergent spherical waves by using the theory of the boundary diffraction wave, Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals have been given. It has been shown that the results in the use of Eq. (14) are the same with those in the use Eq. (3). This is an expected conclusion, because the representation of the boundary diffraction wave is consistent with the Kirchhoff diffraction integral. Apart from providing
ARTICLE IN PRESS P. Liu, B. Lu¨ / Optik 116 (2005) 449–453
(×10−4) 5 4
I(x,0,z)
3 IB(x,0,z), Ik(x,0,z) I2(x,0,z)
2
I1(x,0,z)
1 0 -35 -30 -25 -20 -15 -10 -5
0
5
10 15 20 25 30 35
x/λ
(a) (×10−4) 2.4 2
IB(x,0,z) IK(x,0,z), I1(x,0,z), I2(x,0,z)
I(x,0,z)
1.6 1.2
453
in Eq. (14), its main drawback is that, the boundary waves U d1 ðPÞ and U d2 ðPÞ are discontinuous across the edges of the geometrical shadow, e.g., the discontinuity exists at (a) y ¼ 12l, 24l; (b) y ¼ 20l, 40l; (c) y ¼ 32l, 64l in Fig. 3a–c, respectively. Although the Rayleigh diffraction integrals have found many applications in optics, they could not give the exact calculation result in the very near field. The four diffraction formulae have been shown to be consistent for axial and transversal intensity distribution of divergent spherical waves diffracted at an annular aperture if the source and observation point is equally distant from the aperture, and the observation points are located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory. Finally, we would like to mention that, although the study has been performed for divergent spherical waves, the result for convergent spherical waves diffracted at an annular aperture can be obtained in a similar way, where the initial field U ðiÞ ðP0 Þ ¼ expðikjP0 Pj=jP0 PjÞ has to be replaced by U ðiÞ ðP0 Þ ¼ expðikjP0 Pj=jP0 PjÞ.
0.8 0.4 0 -50 -40 -30 -20 -10
Acknowledgements 0
( b)
1 x/λ
0
20
30
40
50
This work was supported by the Nature Science Foundation of Jiangxi Province.
(×10−4) 1
References
I(x,0,z)
0.8 0.6 0.4
IB(x,0,z) IK(x,0,z), I1(x,0,z), I2(x,0,z)
0.2 0 -90 -75 -60 -45 -30 -15 (c)
0 15 30 45 60 75 90 x/λ
Fig. 3. Transversal intensity distributions Iðx; 0; zÞ in the x direction of a divergent spherical wave diffracted at an annular aperture,________: I B ðx; 0; zÞ, I k ðx; 0; zÞ; - – - –: I 1 ðx; 0; zÞ; - - - - - -: I 2 ðx; 0; zÞ. (a) z ¼ 10l; (b) z ¼ 50l; (c) z ¼ 110l.
an intuitive physical insight into the diffraction phenomenon, the advantage of the boundary diffraction wave theory is that it saves the computer time greatly as compared with the direct integration of the Kirchoff diffraction formula. For example, in Fig. 3a–c the ratios of the computer time in the use of Eqs. (14) and (3) are 1:4326; 1:3546; 1:2387, respectively. However, as shown
[1] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambridge University Press, England, 1999, pp. 412–514. [2] A.S. Marathay, J.F. McCalmont, On the usual approximation used in the Rayleigh–Sommerfeld diffraction theory, J. Opt. Soc. Am. A 21 (2004) 510–516. [3] K. Miyamoto, E. Wolf, Generalization of the Maggi– Rubinowicz theory of the boundary diffraction wave—part I, J. Opt. Soc. Am. A 52 (1962) 615–625. [4] K. Miyamoto, E. Wolf, Generalization of the Maggi– Rubinowicz theory of the boundary diffraction wave—part I, J. Opt. Soc. A 52 (1962) 626–637. [5] E.W. Marchand, E. Wolf, Consistent formulation of Kirchhoff’s diffraction theory, J. Opt. Soc. A 56 (1966) 1712–1722. [6] K. Miyamoto, E. Wolf, Boundary diffraction wave in the domain of the Rayleigh–Kirchholf diffraction theory, J. Opt. Soc. A 52 (1962) 761–767. [7] E.W. Marchand, E. Wolf, Diffraction at small apertures black screens, J. Opt. Soc. A 59 (1969) 79–90. [8] J.C. Heurtley, Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points, J. Opt. Soc. A 63 (1973) 1003–1008. [9] H. Osterberg, L.W. Smith, Closed solutions of Rayleigh’s diffraction integral for axial points, J. Opt. Soc. A 51 (1961) 1050–1054.