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Propagation of vectorial Gaussian beams behind a circular aperture
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 598–604 www.elsevier.com/locate/optlastec
Propagation of vectorial Gaussian beams behind a circular aperture Chongwei Zhenga, Yaoju Zhanga,, Ling Wangb a
Department of Physics and Electric Information Science, Wenzhou University, Wenzhou 325027, China b Department of Technical Physics, Zhengzhou Institute of Light Industry, Zhengzhou 45002, China
Received 23 November 2004; received in revised form 12 September 2005; accepted 13 October 2005 Available online 28 November 2005
Abstract Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum. r 2005 Elsevier Ltd. All rights reserved. PACS: 42.25.Fx; 42.79.Ag; 42.30.Kq Keywords: Vectorial Gaussian beams; Aperture; Vectorial Rayleigh diffraction integral
1. Introduction The analysis of many systems in optical communication and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical mode of Gaussian beam propagation. So far most of the problems involving beam propagation have been studied successfully within the framework of the paraxial approximation. However, the paraxial approximation is invalid for beams with a large divergence angle or a small spot size that is comparable with the wavelength. Many authors have used various approaches to study the corrections to the paraxial approximation for a fundamental Gaussian beam or a high-order Gaussian beam with a complex argument [1–10]. But these studies have been restricted to the unapertured case. In practice, the aperture effect exists, more-or-less. Harvey and Krywonos [11] analyzed the axial intensity distribution of the plane wave throughout the whole space behind an annular aperture. Recently, Duan Corresponding author.
E-mail address: [email protected] (Y. Zhang). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.10.003
and Lu¨ [12] presented nonparaxial analysis of Gaussian beams diffracted at a circular aperture, but they analyzed only the far-field properties using the scalar angular spectrum and the stationary phase methods. Our purpose in the present paper is to deal with apertured vectorial Gaussian beams diffracted in the general case. A simple analytical expression for nonparaxial diffraction of vectorial plane waves at a circular aperture can be easily derived from the Rayleigh diffraction integrals due to the simplicity in mathematics [13]. In order to obtain an analytical expression for nonparaxial diffraction of vectorial Gaussian beams at a circular aperture, we expand the hardedge aperture function as the sum of complex Gaussian functions with finite numbers. Our analytical expression is derived in Section 2. In Section 3, we present numerical simulation and compare it with the results without apertures. Finally, we summarize our main results in Section 4.
2. Analytical expression of the propagation for apertured vectorial Gaussian beams Assume that a Gaussian beam is linearly polarized in the x-direction and illuminates a circular aperture with radius
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
Diffracting aperture
1 E z ðrÞ ¼ 2p
Observation region
x0
x
E x ðr0 Þðx x0 Þ 1
ikjr r0 j 1 jr r0 j3
r r0
O
z
a
y0
x20 þ y20 2xx0 2yy0 , 2r
(6)
(7)
Replacing jr r0 j of the exponential part in Eq. (5) by Eq. (6) and other terms by r, we obtain
z=0 Fig. 1. Geometric relationship between the diffracting aperture and the obervation space.
a at the plane z ¼ 0, as shown in Fig. 1. The field just behind the aperture is, in the Cartesian coordinate system, expressed as x2 þ y2 E x ðx0 ; y0 ; 0Þ ¼ exp 0 2 0 bðx0 ; y0 Þ, (1a) w0
Z Z 1 zðikr 1Þ expðikrÞ E x ðr0 Þ 2pr3 1 h ikðx20 þ y20 Þ xx0 þ yy0 i exp exp ik dx0 dy0 , 2r r ð8aÞ
E x ðrÞ ¼
E z ðrÞ ¼
(1b)
where w0 is the waist width of the Gaussian beams and bðx0 ; y0 Þ the circular window function of the aperture 2 x0 þ y20 bðx0 ; y0 Þ ¼ circ . (2) a2 It is well known that once the boundary condition at the plane z ¼ 0 is given, the vectorial Rayleigh diffraction integrals govern the beam propagation generally. The field in the whole half-space of z40 can be expressed in the form [13,14] Z 1 qGðr; r0 Þ E x ðr0 Þ ds, (3a) E x ðrÞ ¼ 2p qz Z 1 qGðr; r0 Þ E y ðr0 Þ ds, (3b) E y ðrÞ ¼ 2p qz Z 1 qGðr; r0 Þ qGðr; r0 Þ E x ðr0 Þ þ E y ðr0 Þ E z ðrÞ ¼ ds, (3c) 2p qx qy where r0 ¼ x0 i þ y0 j, r ¼ xi þ yj þ zk, i; j; k are unit vectors in the x-, y- and z-directions, respectively, and expðikjr r0 jÞ , jr r0 j
jr r0 j r þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 þ z2 .
y
E y ðx0 ; y0 ; 0Þ ¼ 0,
ð5cÞ
In order to obtain the approximate analytical expression of the diffracting field, we expand jr r0 j into a series and keep the first and second terms, i.e.,
|r - r0|
(5b)
Z Z 1 ðikr 1Þ exp ðikrÞ E x ðr0 Þðx x0 Þ 2pr3 1 h ikðx20 þ y20 Þ xx0 þ yy0 i exp exp ik dx0 dy0 . 2r r ð8bÞ
For simplicity, we express Eq. (8) in circular coordinates given by x ¼ r cos f; x0 ¼ r0 cos y; y ¼ r sin f; y0 ¼ r0 sin y, (9) we obtain zðikr 1Þ E x ðrÞ ¼ expðikrÞ 2pr3 2 Z 1 Z 2p 1 ik 2 r exp 2 r0 circ 02 2r a w 0 0 0 ikrr0 cosðy fÞ r0 dr0 dy, exp r E z ðrÞ ¼
(4)
k ¼ 2p=l; l is the illuminated wavelength. Substituting Eq. (4) into Eq. (3) yields Z Z 1 1 ikjr r0 j 1 E x ðrÞ ¼ E x ðr0 Þz 2p jr r0 j3 1 expðikjr r0 jÞ dx0 dy0 , ð5aÞ E y ðrÞ ¼ 0,
1
expðikjr r0 jÞ dx0 dy0 .
E
Gðr; r0 Þ ¼
Z Z
599
ðikr 1Þ expðikrÞ 2pr3 Z 1 Z 2p ðr cos f r0 cos yÞ 0 0 1 ik 2 exp 2 r w0 2r 0 2 r0 ikrr0 cosðy fÞ r0 dr0 dy. ð10bÞ circ 2 exp a r
With the use of the relation Z 2p cos ny expðia cos yÞ dy ¼ 2pin J n ðaÞ, 0
ð10aÞ
(11)
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
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the integration of Eq. (10) over y gives Z 1 zðikr 1Þ 1 ik 2 E x ðrÞ ¼ expðikrÞ exp r r3 w20 2r 0 0 2 r0 krr0 ð12aÞ circ 2 J 0 r0 dr0 , a r Z 1 ðikr 1Þ 1 ik 2 E z ðrÞ ¼ expðikrÞ exp 2 r r3 w0 2r 0 0 2 r krr0 circ 02 r cos fJ 0 a r krr0 ir0 cos fJ 1 ð12bÞ r0 dr0 , r where Jn is the Bessel function of the first kind of the order n. In order to obtain the analytical expression of Eq. (12) and gain intrinsic physical insight, we first expand the hardedge aperture function as the sum of complex Gaussian functions with finite numbers. Thus, the hard edge aperture function can be expressed as 2 X M r Bm circ 02 ¼ Am exp 2 ðr20 Þ ; a a m¼1
(13)
where the complex constants Am and Bm are the expansion and Gaussian coefficients, respectively, which can be obtained by optimization computation directly [15]. Inserting ‘‘the hard edge aperture function’’ into Eq. (12) and applying the Hankel transform, Z 1 xn expðpx2 ÞxJ n ðaxÞ dx 0 an a2 ¼ exp ; n4 1, ð14Þ 4p ð2pÞnþ1
solution and Eq. (15a) alone as the scalar solution. One may have noted that Ex is rotational symmetry and Ez at the boundary z ¼ 0 does not vanish. This is consistent with the theory of electrodynamics: If two boundary field components are known, the third will be fixed automatically. For this reason, unlike a scalar beam, the propagation of a vector beam does not allow a pure scalar Gaussian intensity specification at the boundary. Furthermore, as may be intuitively evident from Eq. (15), the longitudinal component is much weaker in strength than the transverse component for larger z, since, noting that r is dependent on the first power of z (see Eq. (7)), Ez depends on a quadratic power of 1/z but Ex only on the first power of 1/z. In the far-field approximation, Eq. (15) becomes ! M X ikz 1 k 2 r2 , (17a) E x ðrÞ ¼ 2 expðikrÞ Am exp r 2pmf 4pmf r2 m¼1 ! M X ikx 1 k2 r2 E z ðrÞ ¼ 2 expðikrÞ , Am exp r 2pmf 4pmf r2 m¼1 where pmf ¼
1 Bm þ 2. 2 w0 a
E z ðrÞ ¼ A
ðikr 1Þx k k2 r2 expðikrÞ 1 i exp , 2pr3 2pr 4pr2 (19b)
where A is a complex constant and p is denoted as
M X zðikr 1Þ Am k2 r2 expðikrÞ exp , E x ðrÞ ¼ r3 2pm 4pm r2 m¼1
p¼
M X ðikr 1Þx Am 1 k expðikrÞ 1i E z ðrÞ ¼ r3 2pm r 2pm m¼1 k2 r2 exp , 4pm r2
ð15bÞ
1 Bm ik þ . w20 a2 2r
1 ik . w20 2r
For far-field region, Eq. (19) simplifies to izpw20 k2 w20 r2 E x ðrÞ ¼ A expðikrÞ exp , lr2 4r2 ipx k2 w20 r2 E z ðrÞ ¼ A 2 expðikrÞ exp . lr 4r2
(20)
(21a)
(21b)
Apart for an unimportant constant factor A, Eq. (21) is consistent with Eq. (9) in Ref. [7].
where pm ¼
(18)
when a ! 1, i.e., for the unpertured case, Eq. (15) reduces to zðikr 1Þ k2 r2 expðikrÞ exp E x ðrÞ ¼ A , (19a) 2pr3 4pr2
we obtain
(15a)
(17b)
(16)
The analytical formula Eq. (15) is our main result. Although it is the approximate analytical expression, it provides a more convenient method for studying the propagation of apertured vectorial Gaussian beams than that using the diffraction integral formula (Eq. (5)) directly. We refer to Eqs. (15a) and (15b) together as the vector
3. Numerical calculation results and analyses To compare the results obtained by the use of approximate analytical Eq. (15) with those by the use of the diffraction integral formula (Eq. (5)) directly, some numerical simulations were performed in the environment of Matlab. It should be pointed out that the numerical calculation efficiency using the approximate analytical
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
formula is over 1000 times that obtained using the numerical integral calculation directly. Fig. 2 shows the normalized transverse intensity distributions of the Gaussian beams diffracted by a circular aperture at the plane of (a) z ¼ 4a and (b) z ¼ 10a when a ¼ 10l and a=w0 ¼ 1, whereas Fig. 3 shows the axial intensity distribution. The solid curve represents the case by using the diffraction integral formula (Eq. (5)) directly and the dotted curve denotes the case by using the approximate analytical formula (Eq. (15)) (where the optimized 16-term expansion coefficients Am and Bm in Table 1 are used). We find from Figs. 2 and 3 that the results of the two approaches are in quite a good consistency when z46a. For zo6a, the declination between the approximate results by the use of the approximate analytical formula and the rigorous results by the use of the numerical integral calculation comes into existence, which increases as the diffraction distance decreases. However, it should be noted that Eq. (15) is
601
Fig. 3. Axial intensity distribution of the Gaussian beam diffracted a circular aperture. The calculation parameters are the same as those in Fig. 2. The solid curve represents the case by using the diffraction integral formula directly and the dotted curve denotes the case by using the approximate analytical formula (where M ¼ 16).
Table 1 Coefficients used in evaluating the Gaussian beam description of the field of a hard-edge aperture
Fig. 2. Normalized transverse intensity distributions of the Gaussian beams diffracted by a circular aperture at the plane of (a) z ¼ 4a and (b) z ¼ 10a. The calculation parameters are w0 ¼ a ¼ 10l. The solid curve represents the case by using the diffraction integral formula directly and the dotted curve denotes the case by using the approximate analytical formula (where M ¼ 16).
m
Am
Bm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1.3372053.5907175i 143.89418207.30815i 748.708701617.3281i 5233.9118+1781.262i 4785.7959+7985.7539i 4815.73077447.1249i 4490.5754+333.15631i 726.74624+609.86461i 1.3362521+3.5910719i 143.83871+207.34664i 749.14144+1617.1275i 5233.43361782.6615i 4787.92897984.4714i 4813.7382+7448.4085i 4490.4842334.35596i 726.90874609.67017i
pð1 þ iÞ 2pð1 þ iÞ 3pð1 þ iÞ 4pð1 þ iÞ 5pð1 þ iÞ 6pð1 þ iÞ 7pð1 þ iÞ 8pð1 þ iÞ pð1 iÞ 2pð1 iÞ 3pð1 iÞ 4pð1 iÞ 5pð1 iÞ 6pð1 iÞ 7pð1 iÞ 8pð1 iÞ
only an approximate analytical expression for the apertured Gaussion beams. Fig. 4 shows the intensity of the field components and the total intensity along x direction for different propagation distance. When z is large, the z component is very small compared with the x component and hence the intensity is dominated by the x component (see Fig. 4(b)). The z component is completely independent of the y variable and its importance increases with decreasing the size of aperture and the propagation distance. The z component has important effect on the shape of spot in the near field. As z and a decrease, the shape of spot changes from circle into ellipse (compare Figs. 5(a)–(c)) which major axis is in the x direction (that is, the polarized
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C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
Fig. 4. Transverse intensity distributions of a vectorial Gaussian beams diffracted a circular aperture at the plane of (a) z ¼ 3a and (b) z ¼ 10a. The calculation parameters are the same as those in Fig. 2. The dashed and the dotted curves denote, respectively, the x and the z components of the electric field. The solid curve is the total intensity.
direction of circular Gaussian beams incident on the aperture). Fig. 6 shows the axial intensity distributions of an apertured Gaussion beam with the initial beam waist w0 ¼ 10l. For convenience of comparison, the result for unapertured case (solid curve) is also calculated. Obviously, the axial intensity oscillates due to the aperture. The oscillations are due to the interference between the edge-diffracted wave and the direct illumination. The strength of the oscillation increases as the truncation parameter dð¼ a=w0 Þ decreases. And the maximum of intensity shift to the aperture as d decreases. An additional bit of insight into the above behavior is obtained if we recall that the light from adjacent Fresnel zones in a circular aperture interferes destructively, whereas light from either even or odd Fresnel zones interferes constructively [11]. Considering the number of Fresnel zones subtended by a circular aperture of fixed size as we move along the optical axis, we can assign a unique Fresnel number N to the above diffracting aperture for
Fig. 5. Contour lines of intensity distributions of a Gaussian beam without aperture (a) and with the aperture (b) and (c). The calculation parameters are the same as those in Fig. 2. (a) and (b) at the plane z ¼ 2a, (c) at the plane z ¼ 10a.
each axial position in Fig. 6: N¼
a2 . lz
(22)
Note that the Fraunhofer criterion is expressed in the inequality as [16]. zb
ðx20 þ y20 Þmax . l
(23)
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
Fig. 6. Axial intensity distributions of the Gaussian beams for w0 ¼ 10l. The solid curve is the unapertured case. The dashed and dotted curves are the apertured cases with the truncation parameters of d ¼ 1 and 0.8, respectively.
We substitute an equal sign the unequal sign in inequality (23) to sign the boundary between the Fraunhofer region and the Fresnel region. The vertical line in Fig. 3 specifies this critical axial position. From Fig. 3, we can see that the axial intensity decreases steadily as 1/r2 in the Fraunhofer region. But in the Fresnel region the axial intensity oscillates. As we move closer to the aperture, passing from the Fraunhofer region into the Fresnel region, the axial intensity increases, reaching a maximum at a Fresnel number of exact unity. A further reduction in z causes the aperture to consist of more than one Fresnel zone, and destructive interference starts to diminish the axial intensity. It reaches a minimum at the axial position that corresponds to two Fresnel zones. The oscillatory behavior in Fig. 3 is thus completely consistent with a qualitative discussion of destructive (constructive) interference of an even (odd) number of Fresnel zones. Fig. 3 illustrates a portion of above intensity distribution with a few discrete Fresnel numbers indicated. For convenience the beam width is defined as the width at which the intensity is reduced to the half of its maximum value [5], namely, for beam widths wx ðzÞ; wy ðzÞ in the x and y directions, we have Ið0; 0; zÞ ¼ 2Iðwx ðzÞ; 0; zÞ,
(24a)
Ið0; 0; zÞ ¼ 2Iðwy ðzÞ; 0; zÞ.
(24b)
Fig. 7 gives the beam widths of an apertured Gaussian beam at the positions of the odd number of Fresnel zones in the Fresnel region, whereas Fig. 8 shows the beam widths in the Fraunhofer region. The calculation parameters are the same as those in Fig. 2. For convenience of comparison, the unapertured results calculated by use of Eq. (19) are compiled together. We find wx 4wy in the Fresnel region and wx ¼ wy in the Fraunhofer region. The beam widths with the aperture are smaller than that
603
Fig. 7. Relative beam widths wx ðzÞ=l and wy ðzÞ=l of an apertured vectorial Gaussian beam at the positions of the odd number of Fresnel zones versus normalized propagation distance z/a. As comparison, the results for the unapertured case are also given. The calculation parameters are the same as those in Fig. 2.
Fig. 8. Relative beam widths in the Fraunhofer region versus normalized propagation distance z/a.
without aperture, which illustrates that the aperture has an ability of focusing light. When N ¼ 1, the beam becomes circular in shape and the beam width reaches the minimum value. The beam width with the aperture, which increases as the propagation distance increases, is the similar to that without aperture throughout the whole Fraunhofer region. 4. Conclusions In summary, we have derived an analytical expression of vectorial Gaussian beams diffracted at a circular aperture using the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finiteterm complex Gaussian functions. The apertured Gaussian beam propagation formula in far field and the unapertured Gaussian beam propagation formula have been given as
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special cases of our general result. The comparison of the numerical results using our analytical expression (Eq. (15)) and those using the numerical integral formula (Eq. (5)) have been given, showing the validity of our method. We find that an aperture can produce focusing effect and the focused spot becomes ellipse in the Fresnel region. These effects become more obvious when the truncation parameter is smaller. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum. It should be pointed out that in spite of the advantages of a compact analytical formalism, enabling rapid modeling of apertured or unapertured Gaussian beam systems, this approach has a notable drawback. When zoa, this approach is invalid since the approximate relation (Eq. (6)) and the Rayleigh diffraction integral formula (Eq. (3)) is inadequate for very near field [17].
References [1] Carter WH. Electromagnetic field of a Gaussian beam with an elliptical cross section. J Opt Soc Am 1972;62:1195–201. [2] Agrawal GP, Pattanayak DN. Gaussian beam propagation beyond the paraxial approximation. J Opt Soc Am 1979;69:575–8. [3] Zhang Y, Zheng C. Axial intensity distribution behind a Fresnel zone plate. Opt Laser Technol 2004;37:77–80. [4] Wu¨nsche A. Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams. J Opt Soc Am A 1992;9:765–74.
[5] Zeng X, Liang C, An Y. Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation. Appl Opt 1997;36:2042–7. [6] Cao Q, Deng X. Corrections to the paraxial approximation of an arbitrary free-propagation beam. J Opt Soc Am A 1998;15:1144–9. [7] Zeng X, Liang C, An Y. Far-field propagation of an off-axis Gaussian wave. Appl Opt 1999;38:6253–6. [8] Cai Y, Lu X, Lin Q. Hollow Gaussian beams and their propagation properties. Opt Lett 2003;28:1084–6. [9] Chen CG, Konkola PT, Ferrera J, Heilmann RK, Schattenburg MJ. Analyses of vector Gaussian beam propagation and validity of paraxial and spherical approximation. J Opt Soc Am A 2002;19: 404–12. [10] Duan K, Lu¨ B. Far-field properties of non-paraxial elliptical Gaussian beams. Opt Quant Electron 2003;35:179–82. [11] Harvey JE, Krywonos A. Axial irradiance distribution throughout the whole space behind an annular aperture. Appl Opt 2002;41:3790–5. [12] Duan K, Lu¨ B. Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture. Opt Exp 2003;11:1474–80. [13] Duan K, Lu¨ B. Nonparaxial diffraction of vectorial plane waves at a small aperture. Opt Laser Technol 2004;36:489–96. [14] Luneberg RK. Mathematical theory of optics. Berkely and Los, Angeles: University of California Press; 1964. [15] Zhang Y. Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture. Opt Commun 2005;248:317–26. [16] Born M, Wolf E. Principles of optics (Chapt. 8), 7th ed. Cambridge,: Cambridge University; 1999. pp. 425–7. [17] Totzeck M. Validity of the scalar Kirchhoff and Rayleigh Sommerfield diffraction theories in the near field of small phase objects. J Opt Soc Am A 1991;8:27–32.
Optics & Laser Technology 39 (2007) 598–604 www.elsevier.com/locate/optlastec
Propagation of vectorial Gaussian beams behind a circular aperture Chongwei Zhenga, Yaoju Zhanga,, Ling Wangb a
Department of Physics and Electric Information Science, Wenzhou University, Wenzhou 325027, China b Department of Technical Physics, Zhengzhou Institute of Light Industry, Zhengzhou 45002, China
Received 23 November 2004; received in revised form 12 September 2005; accepted 13 October 2005 Available online 28 November 2005
Abstract Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum. r 2005 Elsevier Ltd. All rights reserved. PACS: 42.25.Fx; 42.79.Ag; 42.30.Kq Keywords: Vectorial Gaussian beams; Aperture; Vectorial Rayleigh diffraction integral
1. Introduction The analysis of many systems in optical communication and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical mode of Gaussian beam propagation. So far most of the problems involving beam propagation have been studied successfully within the framework of the paraxial approximation. However, the paraxial approximation is invalid for beams with a large divergence angle or a small spot size that is comparable with the wavelength. Many authors have used various approaches to study the corrections to the paraxial approximation for a fundamental Gaussian beam or a high-order Gaussian beam with a complex argument [1–10]. But these studies have been restricted to the unapertured case. In practice, the aperture effect exists, more-or-less. Harvey and Krywonos [11] analyzed the axial intensity distribution of the plane wave throughout the whole space behind an annular aperture. Recently, Duan Corresponding author.
E-mail address: [email protected] (Y. Zhang). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.10.003
and Lu¨ [12] presented nonparaxial analysis of Gaussian beams diffracted at a circular aperture, but they analyzed only the far-field properties using the scalar angular spectrum and the stationary phase methods. Our purpose in the present paper is to deal with apertured vectorial Gaussian beams diffracted in the general case. A simple analytical expression for nonparaxial diffraction of vectorial plane waves at a circular aperture can be easily derived from the Rayleigh diffraction integrals due to the simplicity in mathematics [13]. In order to obtain an analytical expression for nonparaxial diffraction of vectorial Gaussian beams at a circular aperture, we expand the hardedge aperture function as the sum of complex Gaussian functions with finite numbers. Our analytical expression is derived in Section 2. In Section 3, we present numerical simulation and compare it with the results without apertures. Finally, we summarize our main results in Section 4.
2. Analytical expression of the propagation for apertured vectorial Gaussian beams Assume that a Gaussian beam is linearly polarized in the x-direction and illuminates a circular aperture with radius
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
Diffracting aperture
1 E z ðrÞ ¼ 2p
Observation region
x0
x
E x ðr0 Þðx x0 Þ 1
ikjr r0 j 1 jr r0 j3
r r0
O
z
a
y0
x20 þ y20 2xx0 2yy0 , 2r
(6)
(7)
Replacing jr r0 j of the exponential part in Eq. (5) by Eq. (6) and other terms by r, we obtain
z=0 Fig. 1. Geometric relationship between the diffracting aperture and the obervation space.
a at the plane z ¼ 0, as shown in Fig. 1. The field just behind the aperture is, in the Cartesian coordinate system, expressed as x2 þ y2 E x ðx0 ; y0 ; 0Þ ¼ exp 0 2 0 bðx0 ; y0 Þ, (1a) w0
Z Z 1 zðikr 1Þ expðikrÞ E x ðr0 Þ 2pr3 1 h ikðx20 þ y20 Þ xx0 þ yy0 i exp exp ik dx0 dy0 , 2r r ð8aÞ
E x ðrÞ ¼
E z ðrÞ ¼
(1b)
where w0 is the waist width of the Gaussian beams and bðx0 ; y0 Þ the circular window function of the aperture 2 x0 þ y20 bðx0 ; y0 Þ ¼ circ . (2) a2 It is well known that once the boundary condition at the plane z ¼ 0 is given, the vectorial Rayleigh diffraction integrals govern the beam propagation generally. The field in the whole half-space of z40 can be expressed in the form [13,14] Z 1 qGðr; r0 Þ E x ðr0 Þ ds, (3a) E x ðrÞ ¼ 2p qz Z 1 qGðr; r0 Þ E y ðr0 Þ ds, (3b) E y ðrÞ ¼ 2p qz Z 1 qGðr; r0 Þ qGðr; r0 Þ E x ðr0 Þ þ E y ðr0 Þ E z ðrÞ ¼ ds, (3c) 2p qx qy where r0 ¼ x0 i þ y0 j, r ¼ xi þ yj þ zk, i; j; k are unit vectors in the x-, y- and z-directions, respectively, and expðikjr r0 jÞ , jr r0 j
jr r0 j r þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 þ z2 .
y
E y ðx0 ; y0 ; 0Þ ¼ 0,
ð5cÞ
In order to obtain the approximate analytical expression of the diffracting field, we expand jr r0 j into a series and keep the first and second terms, i.e.,
|r - r0|
(5b)
Z Z 1 ðikr 1Þ exp ðikrÞ E x ðr0 Þðx x0 Þ 2pr3 1 h ikðx20 þ y20 Þ xx0 þ yy0 i exp exp ik dx0 dy0 . 2r r ð8bÞ
For simplicity, we express Eq. (8) in circular coordinates given by x ¼ r cos f; x0 ¼ r0 cos y; y ¼ r sin f; y0 ¼ r0 sin y, (9) we obtain zðikr 1Þ E x ðrÞ ¼ expðikrÞ 2pr3 2 Z 1 Z 2p 1 ik 2 r exp 2 r0 circ 02 2r a w 0 0 0 ikrr0 cosðy fÞ r0 dr0 dy, exp r E z ðrÞ ¼
(4)
k ¼ 2p=l; l is the illuminated wavelength. Substituting Eq. (4) into Eq. (3) yields Z Z 1 1 ikjr r0 j 1 E x ðrÞ ¼ E x ðr0 Þz 2p jr r0 j3 1 expðikjr r0 jÞ dx0 dy0 , ð5aÞ E y ðrÞ ¼ 0,
1
expðikjr r0 jÞ dx0 dy0 .
E
Gðr; r0 Þ ¼
Z Z
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ðikr 1Þ expðikrÞ 2pr3 Z 1 Z 2p ðr cos f r0 cos yÞ 0 0 1 ik 2 exp 2 r w0 2r 0 2 r0 ikrr0 cosðy fÞ r0 dr0 dy. ð10bÞ circ 2 exp a r
With the use of the relation Z 2p cos ny expðia cos yÞ dy ¼ 2pin J n ðaÞ, 0
ð10aÞ
(11)
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the integration of Eq. (10) over y gives Z 1 zðikr 1Þ 1 ik 2 E x ðrÞ ¼ expðikrÞ exp r r3 w20 2r 0 0 2 r0 krr0 ð12aÞ circ 2 J 0 r0 dr0 , a r Z 1 ðikr 1Þ 1 ik 2 E z ðrÞ ¼ expðikrÞ exp 2 r r3 w0 2r 0 0 2 r krr0 circ 02 r cos fJ 0 a r krr0 ir0 cos fJ 1 ð12bÞ r0 dr0 , r where Jn is the Bessel function of the first kind of the order n. In order to obtain the analytical expression of Eq. (12) and gain intrinsic physical insight, we first expand the hardedge aperture function as the sum of complex Gaussian functions with finite numbers. Thus, the hard edge aperture function can be expressed as 2 X M r Bm circ 02 ¼ Am exp 2 ðr20 Þ ; a a m¼1
(13)
where the complex constants Am and Bm are the expansion and Gaussian coefficients, respectively, which can be obtained by optimization computation directly [15]. Inserting ‘‘the hard edge aperture function’’ into Eq. (12) and applying the Hankel transform, Z 1 xn expðpx2 ÞxJ n ðaxÞ dx 0 an a2 ¼ exp ; n4 1, ð14Þ 4p ð2pÞnþ1
solution and Eq. (15a) alone as the scalar solution. One may have noted that Ex is rotational symmetry and Ez at the boundary z ¼ 0 does not vanish. This is consistent with the theory of electrodynamics: If two boundary field components are known, the third will be fixed automatically. For this reason, unlike a scalar beam, the propagation of a vector beam does not allow a pure scalar Gaussian intensity specification at the boundary. Furthermore, as may be intuitively evident from Eq. (15), the longitudinal component is much weaker in strength than the transverse component for larger z, since, noting that r is dependent on the first power of z (see Eq. (7)), Ez depends on a quadratic power of 1/z but Ex only on the first power of 1/z. In the far-field approximation, Eq. (15) becomes ! M X ikz 1 k 2 r2 , (17a) E x ðrÞ ¼ 2 expðikrÞ Am exp r 2pmf 4pmf r2 m¼1 ! M X ikx 1 k2 r2 E z ðrÞ ¼ 2 expðikrÞ , Am exp r 2pmf 4pmf r2 m¼1 where pmf ¼
1 Bm þ 2. 2 w0 a
E z ðrÞ ¼ A
ðikr 1Þx k k2 r2 expðikrÞ 1 i exp , 2pr3 2pr 4pr2 (19b)
where A is a complex constant and p is denoted as
M X zðikr 1Þ Am k2 r2 expðikrÞ exp , E x ðrÞ ¼ r3 2pm 4pm r2 m¼1
p¼
M X ðikr 1Þx Am 1 k expðikrÞ 1i E z ðrÞ ¼ r3 2pm r 2pm m¼1 k2 r2 exp , 4pm r2
ð15bÞ
1 Bm ik þ . w20 a2 2r
1 ik . w20 2r
For far-field region, Eq. (19) simplifies to izpw20 k2 w20 r2 E x ðrÞ ¼ A expðikrÞ exp , lr2 4r2 ipx k2 w20 r2 E z ðrÞ ¼ A 2 expðikrÞ exp . lr 4r2
(20)
(21a)
(21b)
Apart for an unimportant constant factor A, Eq. (21) is consistent with Eq. (9) in Ref. [7].
where pm ¼
(18)
when a ! 1, i.e., for the unpertured case, Eq. (15) reduces to zðikr 1Þ k2 r2 expðikrÞ exp E x ðrÞ ¼ A , (19a) 2pr3 4pr2
we obtain
(15a)
(17b)
(16)
The analytical formula Eq. (15) is our main result. Although it is the approximate analytical expression, it provides a more convenient method for studying the propagation of apertured vectorial Gaussian beams than that using the diffraction integral formula (Eq. (5)) directly. We refer to Eqs. (15a) and (15b) together as the vector
3. Numerical calculation results and analyses To compare the results obtained by the use of approximate analytical Eq. (15) with those by the use of the diffraction integral formula (Eq. (5)) directly, some numerical simulations were performed in the environment of Matlab. It should be pointed out that the numerical calculation efficiency using the approximate analytical
ARTICLE IN PRESS C. Zheng et al. / Optics & Laser Technology 39 (2007) 598–604
formula is over 1000 times that obtained using the numerical integral calculation directly. Fig. 2 shows the normalized transverse intensity distributions of the Gaussian beams diffracted by a circular aperture at the plane of (a) z ¼ 4a and (b) z ¼ 10a when a ¼ 10l and a=w0 ¼ 1, whereas Fig. 3 shows the axial intensity distribution. The solid curve represents the case by using the diffraction integral formula (Eq. (5)) directly and the dotted curve denotes the case by using the approximate analytical formula (Eq. (15)) (where the optimized 16-term expansion coefficients Am and Bm in Table 1 are used). We find from Figs. 2 and 3 that the results of the two approaches are in quite a good consistency when z46a. For zo6a, the declination between the approximate results by the use of the approximate analytical formula and the rigorous results by the use of the numerical integral calculation comes into existence, which increases as the diffraction distance decreases. However, it should be noted that Eq. (15) is
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Fig. 3. Axial intensity distribution of the Gaussian beam diffracted a circular aperture. The calculation parameters are the same as those in Fig. 2. The solid curve represents the case by using the diffraction integral formula directly and the dotted curve denotes the case by using the approximate analytical formula (where M ¼ 16).
Table 1 Coefficients used in evaluating the Gaussian beam description of the field of a hard-edge aperture
Fig. 2. Normalized transverse intensity distributions of the Gaussian beams diffracted by a circular aperture at the plane of (a) z ¼ 4a and (b) z ¼ 10a. The calculation parameters are w0 ¼ a ¼ 10l. The solid curve represents the case by using the diffraction integral formula directly and the dotted curve denotes the case by using the approximate analytical formula (where M ¼ 16).
m
Am
Bm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1.3372053.5907175i 143.89418207.30815i 748.708701617.3281i 5233.9118+1781.262i 4785.7959+7985.7539i 4815.73077447.1249i 4490.5754+333.15631i 726.74624+609.86461i 1.3362521+3.5910719i 143.83871+207.34664i 749.14144+1617.1275i 5233.43361782.6615i 4787.92897984.4714i 4813.7382+7448.4085i 4490.4842334.35596i 726.90874609.67017i
pð1 þ iÞ 2pð1 þ iÞ 3pð1 þ iÞ 4pð1 þ iÞ 5pð1 þ iÞ 6pð1 þ iÞ 7pð1 þ iÞ 8pð1 þ iÞ pð1 iÞ 2pð1 iÞ 3pð1 iÞ 4pð1 iÞ 5pð1 iÞ 6pð1 iÞ 7pð1 iÞ 8pð1 iÞ
only an approximate analytical expression for the apertured Gaussion beams. Fig. 4 shows the intensity of the field components and the total intensity along x direction for different propagation distance. When z is large, the z component is very small compared with the x component and hence the intensity is dominated by the x component (see Fig. 4(b)). The z component is completely independent of the y variable and its importance increases with decreasing the size of aperture and the propagation distance. The z component has important effect on the shape of spot in the near field. As z and a decrease, the shape of spot changes from circle into ellipse (compare Figs. 5(a)–(c)) which major axis is in the x direction (that is, the polarized
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Fig. 4. Transverse intensity distributions of a vectorial Gaussian beams diffracted a circular aperture at the plane of (a) z ¼ 3a and (b) z ¼ 10a. The calculation parameters are the same as those in Fig. 2. The dashed and the dotted curves denote, respectively, the x and the z components of the electric field. The solid curve is the total intensity.
direction of circular Gaussian beams incident on the aperture). Fig. 6 shows the axial intensity distributions of an apertured Gaussion beam with the initial beam waist w0 ¼ 10l. For convenience of comparison, the result for unapertured case (solid curve) is also calculated. Obviously, the axial intensity oscillates due to the aperture. The oscillations are due to the interference between the edge-diffracted wave and the direct illumination. The strength of the oscillation increases as the truncation parameter dð¼ a=w0 Þ decreases. And the maximum of intensity shift to the aperture as d decreases. An additional bit of insight into the above behavior is obtained if we recall that the light from adjacent Fresnel zones in a circular aperture interferes destructively, whereas light from either even or odd Fresnel zones interferes constructively [11]. Considering the number of Fresnel zones subtended by a circular aperture of fixed size as we move along the optical axis, we can assign a unique Fresnel number N to the above diffracting aperture for
Fig. 5. Contour lines of intensity distributions of a Gaussian beam without aperture (a) and with the aperture (b) and (c). The calculation parameters are the same as those in Fig. 2. (a) and (b) at the plane z ¼ 2a, (c) at the plane z ¼ 10a.
each axial position in Fig. 6: N¼
a2 . lz
(22)
Note that the Fraunhofer criterion is expressed in the inequality as [16]. zb
ðx20 þ y20 Þmax . l
(23)
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Fig. 6. Axial intensity distributions of the Gaussian beams for w0 ¼ 10l. The solid curve is the unapertured case. The dashed and dotted curves are the apertured cases with the truncation parameters of d ¼ 1 and 0.8, respectively.
We substitute an equal sign the unequal sign in inequality (23) to sign the boundary between the Fraunhofer region and the Fresnel region. The vertical line in Fig. 3 specifies this critical axial position. From Fig. 3, we can see that the axial intensity decreases steadily as 1/r2 in the Fraunhofer region. But in the Fresnel region the axial intensity oscillates. As we move closer to the aperture, passing from the Fraunhofer region into the Fresnel region, the axial intensity increases, reaching a maximum at a Fresnel number of exact unity. A further reduction in z causes the aperture to consist of more than one Fresnel zone, and destructive interference starts to diminish the axial intensity. It reaches a minimum at the axial position that corresponds to two Fresnel zones. The oscillatory behavior in Fig. 3 is thus completely consistent with a qualitative discussion of destructive (constructive) interference of an even (odd) number of Fresnel zones. Fig. 3 illustrates a portion of above intensity distribution with a few discrete Fresnel numbers indicated. For convenience the beam width is defined as the width at which the intensity is reduced to the half of its maximum value [5], namely, for beam widths wx ðzÞ; wy ðzÞ in the x and y directions, we have Ið0; 0; zÞ ¼ 2Iðwx ðzÞ; 0; zÞ,
(24a)
Ið0; 0; zÞ ¼ 2Iðwy ðzÞ; 0; zÞ.
(24b)
Fig. 7 gives the beam widths of an apertured Gaussian beam at the positions of the odd number of Fresnel zones in the Fresnel region, whereas Fig. 8 shows the beam widths in the Fraunhofer region. The calculation parameters are the same as those in Fig. 2. For convenience of comparison, the unapertured results calculated by use of Eq. (19) are compiled together. We find wx 4wy in the Fresnel region and wx ¼ wy in the Fraunhofer region. The beam widths with the aperture are smaller than that
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Fig. 7. Relative beam widths wx ðzÞ=l and wy ðzÞ=l of an apertured vectorial Gaussian beam at the positions of the odd number of Fresnel zones versus normalized propagation distance z/a. As comparison, the results for the unapertured case are also given. The calculation parameters are the same as those in Fig. 2.
Fig. 8. Relative beam widths in the Fraunhofer region versus normalized propagation distance z/a.
without aperture, which illustrates that the aperture has an ability of focusing light. When N ¼ 1, the beam becomes circular in shape and the beam width reaches the minimum value. The beam width with the aperture, which increases as the propagation distance increases, is the similar to that without aperture throughout the whole Fraunhofer region. 4. Conclusions In summary, we have derived an analytical expression of vectorial Gaussian beams diffracted at a circular aperture using the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finiteterm complex Gaussian functions. The apertured Gaussian beam propagation formula in far field and the unapertured Gaussian beam propagation formula have been given as
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special cases of our general result. The comparison of the numerical results using our analytical expression (Eq. (15)) and those using the numerical integral formula (Eq. (5)) have been given, showing the validity of our method. We find that an aperture can produce focusing effect and the focused spot becomes ellipse in the Fresnel region. These effects become more obvious when the truncation parameter is smaller. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum. It should be pointed out that in spite of the advantages of a compact analytical formalism, enabling rapid modeling of apertured or unapertured Gaussian beam systems, this approach has a notable drawback. When zoa, this approach is invalid since the approximate relation (Eq. (6)) and the Rayleigh diffraction integral formula (Eq. (3)) is inadequate for very near field [17].
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