- Email: [email protected]
Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture
Optics Communications 285 (2012) 4856–4860
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture Jiang Guo a,n, Zao Li b a b
Department of Geophysics, Chengdu University of Technology, Chengdu 610059, China Jincheng College of Sichuan University, Chengdu 611731, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 December 2011 Received in revised form 6 July 2012 Accepted 6 August 2012 Available online 29 August 2012
Based on the vectorial Rayleith–Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs. & 2012 Elsevier B.V. All rights reserved.
Keywords: Hollow Gaussian beams (HGBs) Vectorial Rayleith–Sommerfeld formulae Nonparaxial propagation Vectorial diffraction theory
1. Introduction In recent years, there have been increasing interests in the study of optical beams with zero central intensity called darkhollow beams (DHBs) because of many practical applications in science and technology, including cold atom guiding and optical atom trapping [1–2], laser writing and drilling [3], laser manipulation of microscopic dielectric and metal objects [4]. Many different methods have been used to obtain the hollow beams [5–8]. In recent theoretical investigations, several different mathematical models for describing DHBs have been proposed, such as the TEM01 beam and the higher-order Bessel beam, high-order Laguerre– Gaussian beam, hollow Gaussian beams (HGBs) [9–11]. Followed by the advent of HGBs, a rich amount of investigations about HGBs have been increasingly presented. The paraxial propagation properties of HGBs through aperture have been studied [12], and the analytical vectorial structure of HGBs in the far field has been studied [13], and the free-space nonparaxial propagation properties of the HGBs have been also studied [14]. The description of optical beam propagation properties in the nonparaxial region becomes more and more important with the development of new optical structures (e.g., micro-cavities, photonic crystals, and so on) and near-field optics, characterizing linear dimensions or spatial scales of variation comparable to or even smaller than wavelength l.
n
Corresponding author. Tel.: þ86 28 84079327. E-mail addresses: [email protected], [email protected] (J. Guo).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.08.032
In the near field, the nonparaxial propagation properties and intensity distribution of vector HGBs through an aperture optical system have been rarely reported so far. In this paper, the nonparaxial propagation properties of the HGBs through a circular aperture were studied through the vectorial Rayleith–Sommerfeld formulae in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expressions do not depend on the approximation of the paraxial and far field (i.e., ac l or Rc a, where a R and l are aperture radius, beam propagation distention and wavelength, respectively). And the derived expressions are valid either for the far and near field or for the systems in which the condition a l or ao l was considerate. By using the deduced formula, the propagation properties of HGBs through a circular aperture are numerically calculated and graphically illustrated. Moreover, numerically-calculated results are analyzed and compared with those obtained by using the Collins formulae of scalar approximation. In addition, we also research the influence of the truncation parameter on the near field propagation properties.
2. Nonparaxial vectorial propagation integral expressions of HGBs According to the boundary values E(x0,y0,0)¼Ex(x0,y0,0)i þ Ey(x0,y0,0)j on the plane z¼ 0, the solution of the Helmholtz
J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
4857
(
Eq. (1) is expressed as Eq. (2a)–(2c) in the half-space z 40 [15].
r2 E þ k2 E ¼ 0 Ex ðx,y,zÞ ¼
Ey ðx,y,zÞ ¼
Ez ðx,y,zÞ ¼
z 2p z 2p
ð1Þ Z
þ1 Z 1
Z
þ1
þ1
Ex ðx0 ,y0 ,0Þ
ikR1 R3
1
Z
1
þ1
Ey ðx0 ,y0 ,0Þ
ikR1
1
R3
exp ðikRÞdx0 dy0
ð2aÞ
expðikRÞdx0 dy0
ð2bÞ
Z þ1 Z þ1 1 ½Ex ðx0 ,y0 ,0Þðxx0 Þ þ Ey ðx0 ,y0 ,0Þðyy0 Þ 2p 1 1 ikR1 expðikRÞdx0 dy0 ð2cÞ R3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where k is wave number, R ¼ 9rq0 9 ¼ ðxx0 Þ2 þ ðyy0 Þ2 þ z2 , q0 ¼x0i þy0j is vector on the plane z ¼0, r¼xiþyjþ zk is spatial vector. The Eq. (2) is called the vectorial Rayleigh–Sommerfeld integrals. By using the formulae, the propagation properties of vectorial beams could be studied. Considering the hollow Gaussian beams, Eðx,y,0Þ ¼ Ex ðx0 ,y0 ,0Þiþ Ey ðx0 ,y0 ,0Þj Ex ðx0 ,y0 ,0Þ ¼ A0
r20
!n exp
w20
r20
ð3Þ
was incident on a circular aperture with radius of a at the plane z¼ 0. Where n is the order of the HGBs, A0 is the amplitude constant, w0 is the beam waist width. For n¼1 Eq. (3a) becomes a Gaussian beam with beam waist w0. The field at the plane after aperture may be represented as !n ! w20
Ez ðx,y,zÞ ¼
ð3aÞ
w20
ð3bÞ
r20
Ey ðx,y,zÞ ¼ 0
!
Ey ðx0 ,y0 ,0Þ ¼ 0
Ex ðx0 ,y0 ,0Þ ¼ tðx0 ,y0 ÞA0
1 x20 þ y20 r a2 where tðx0 ,y0 Þ ¼ is the transmission function of 0 else aperture. Replacing R in the exponent part with [16,17], ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xx0 þ yy0 2 2 R ¼ 9rq0 90 r þ r0 1 2 ð5Þ r þ r20 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and R everywhere else with R r 2 þ r20 in Eq. (2) (it requires that ðxx0 þyy0 =r 2 þ r20 Þ{1, these are the only approximation conditions required in this paper), and using Eq. (4) and cylindrical coordinates, the Eq. (2a)–(2c) are transformed as: !n Z i2pzA0 1 a2 r2 a2 r i Ex ðx,y,zÞ ¼ 1þ 2 2 kg l g w0 0 ! 2 2 kabr a r exp ikg 2 dr J0 ð6aÞ g w0
exp
r20
ð4aÞ
w20
Ey ðx0 ,y0 ,0Þ ¼ 0
ð4bÞ
1
2
2
2
2
Iðx,y,zÞ ¼ 9Eðx,y,zÞ9 ¼ 9Ex ðx,y,zÞ9 þ9Ey ðx,y,zÞ9 þ9Ez ðx,y,zÞ9
Normalized intensity
Normalized intensity
Z
ð7Þ
1.2
0.9 0.6 0.3 0 -0.36 -0.24
-0.12
0 x/mm
0.12
0.24
0.8
0.4
0 -0.45
0.36
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
1.2 Normalized intensity
1.2 Normalized intensity
!n a2 r2 a2 r i 1 þ kg l g2 w20 0 ! kabr xra kabr a2 r2 ixJ 0 J exp ikg 2 dr b 1 g g w0 2pA0
ð6cÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 where b¼ x þy , g ¼ b þ z þ ðarÞ , J0 (kabr/g) and J1 (kabr/ g) is Bessel function of the zero- and first-order, respectively, r0 ¼ar, r is a normalized variable (0 r r r1). Eq. (6) is the main result in this paper, which is general propagation integral expression for nonparaxial vectorial HGBs through a circular aperture. We note that the propagation field owns both the longitudinal and the transversal components. However, the longitudinal components is asymmetric along x and y directions. Thus the propagation field is asymmetric. The intensity distribution is
1.2
1 0.8 0.6 0.4 0.2 0 -0.45
ð6bÞ
-0.3
-0.15
0 0.15 x/mm
0.3
0.45
0.9 0.6 0.3 0 -0.45
-0.3
-0.15
0
0.15
0.3
0.45
x/mm
Fig. 1. (Color online) Normalized transversal intensity distributions along x direction for the different propagation distances z. (a) z¼ 2.4 mm; (b) z ¼ 3 mm; (c) z ¼15 mm; (d) z ¼50 mm. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the data from Collins formula. The calculated parameters are a ¼0.35 mm, w0 ¼ 0.1 mm.
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J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
In the course of deducing Eq. (6) without the use of the approximations Rba, and ab l, Eq. (6) is valid for far field, near field and for the optical systems in which the size of the aperture is comparable with or smaller than the wavelength (i.e., a l or ao l). Eq. (6) is general propagation integral expression for nonparaxial vectorial HGBs through a circular aperture. In addition, for the observatqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ion point on axis R ¼ 9rq0 9 r 2 þ r20 , Eq. (6) is a strict integral formula.
3. Numerical calculation results and analysis The normalized intensity distributions of nonparaxial vectorial HGBs through a circular aperture are calculated by using Eq. (6), and the calculated results are compared in detail with those obtained by the generalized Collins formula. Within the framework of the paraxial approximation, under rotationally symmetric cylindrical coordinate condition, Collins formula can be expressed as [18]
4.E-03
Z a ik ik ik kbr0 2 E0 ðr0 ,0Þexp r0 dr0 exp ikz Db Ar20 J0 B B 2B 2B 0
Normalized intensity
Eðx,y,zÞ ¼
between Eq. (6) and the Collins formulae in the situation of paraxial approximation. Therefore, the accuracy of Eq. (6) is well confirmed. When the paraxial approximation condition is not satisfied, i.e., z a or z oa, it can be seen from Fig. 2a–d that the numerical errors of different formulae gradually increase with the decreasing of z, such as the maximal difference of intensities transversal along the x direction is 0.077, 0.195, 0.43, 0.95 when z is 5a, 3a, 1.72a, and 0.857a, respectively. From Fig. 2a, the high-spatialfrequency oscillation appears in the calculated results using Collins formula under the condition of z ¼0.857a. In contrast, the calculated beam profile using the Eq. (6) are smooth. From Fig. 3, it can be seen that the on-axis intensity by using the strict integral formula (i.e., Eq. (6)) is not in agreement with those by using the Collins formula when z oa, and the on-axis intensity calculated using Collins formula tends to infinity. Only when the propagation distance is larger than a (i.e., far field), the results obtained from two different formulae are in agreement. It
ð8Þ where A, B, C and D are the transition matrix elements of the HGBs in paraxial ABCD optical system. The calculated results are shown in Figs. 1–5. During the calculation, we choose the parameters A¼1, B ¼z, C ¼0, D ¼1, l ¼1060 nm, n ¼5. If R ba, i.e., when the paraxial approximation condition is satisfied, the propagation field of HGBs is symmetric around the propagating axis, and the transversal intensities distributions by using Eq. (6) are in agreement with those by using the Collins formula Eq. (8), as shown in Fig. 1a–d. And the maximal errors are less than 0.048, 0.025, 9.7 10 4, 1.29 10 5 when z is 2.4, 3, 15 and 50 mm, respectively. It is evident that there is no difference
0.2 -0.45
-0.3
-0.15
0
0 x/mm
0.15
0.3
4
6 z /mm
8
10
12
0.4
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
-0.24
-0.12
0 x/mm
0.12
0.24
0.36
1.2 Normalized intensity
Normalized intensity
2
0.8
0 -0.45
0.45
1.2
0.8
0.4
0 -0.45
0
Fig. 3. (Color online) On-axis normalized intensity distribution. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red dot line’’ represents the data from Collins formula. The calculation parameters are a ¼0.35 mm, w0 ¼ 0.1 mm.
Normalized intensity
0.4
1.E-03
1.2 Normalized intensity
0.6
2.E-03
0.E+00
1 0.8
3.E-03
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
0.8
0.4
0 -0.36
Fig. 2. (Color online) Normalized transversal intensity distributions along x direction for the different propagation distances z. (a) z ¼ 0.872a, (b) z ¼ 1.72a, (c) z ¼3a and (d) z ¼5a. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the data from Collins formula. The calculated parameters are a¼0.35 mm, w0 ¼ 0.1 mm.
J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
4859
a 0.04
0.02
1 0.75 0.5 0.25 0
0
0.05 0.025
-0.05 -0.025
-0.025
0
x
-0.02
y
0
0.025 0.05
-0.04
-0.05
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
b 0.04
0.02
1 0.75 0.5 0.25 0 -0.05 -0.025
0.05
0
0.025
y
0 -0.025
0
x
-0.02
-0.04
-0.05
0.025 0.05
c 0.04
0.02
1 0.75 0.5 0.25 0 -0.05
0.05
0
0.025
y
0 -0.025
-0.025
0 0.025
x
0.05
-0.02
-0.04
-0.05
d 0.04
0.02
1 0.75 0.5 0.25 0 -0.05 -0.025
0.05
0
0.025 0 -0.025
0
x
y
0.025 0.05
-0.02
-0.04
-0.05 -0.04
-0.02
0
0.02
0.04
Fig. 4. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different truncation parameter a/w0 at z ¼0.2a plane. (a) a/ w0 ¼ 30, (b) a/w0 ¼ 10, (c) a/w0 ¼8 and (d) a/w0 ¼ 5. The calculated parameters are w0 ¼ 0.01 mm.
indicates that Collins formula of scalar paraxial approximation is invalid for near field, and a vectorial diffraction theory should be used in near field. Fig. 4a–d show the numerically-calculated results of normalized 3-D intensity distribution and the intensity contour lines of HGBs through a circular aperture for the different truncation parameter a/w0 at z ¼0.2a plane by using Eq. (6). From Fig. 4a–d, we find that the propagation field of HGBs gradually becomes asymmetric around the propagating axis in the near field with the decrease of truncation parameter. The maximal difference of transversal intensities between along the x direction and that of the y direction is 0.003, 0.016, 0.146, 0.237, 0.488 when
a/w0 ¼30, 20, 10, 8 and 5, respectively. Moreover, when truncation parameter a/w0 ¼constant, with the increase of propagation distance the propagation field of HGBs gradually becomes symmetric. For example, when a/w0 ¼5, the calculated results of normalized 3-D intensity distribution and the intensity contour lines of HGBs by using Eq. (6) are shown in Fig. 4d and Fig. 5a–b at the different propagation distance z. The maximal difference of intensities transversal between along the x direction and that of the y direction is 0.488, 0.146, 0.016 when z¼0.2a, 0.4a, 0.8a, respectively. The asymmetric propagation distance is about 0.8a for a/w0 ¼5. In addition, the asymmetric propagation distance is about 1.5a, a, 0.5a, 0.4a, 0.2a when a/w0 ¼3, 4, 8, 10 and 20,
4860
J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
a
0.04
0.02
1 0.75 0.5 0.25 0
0.05
0
0.025
y
0
-0.05 -0.025
-0.025
0
x
0.025
-0.02
-0.04
-0.05 0.05
-0.04
-0.02
0
0.02
0.04
0.04
0.02
1 0.75 0.5 0.25 0
0.05
0
0.025 0
-0.05 -0.025
-0.02
-0.025
0
x
y
-0.04
0.025
-0.05 0.05
-0.04
-0.02
0
0.02
0.04
Fig. 5. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different propagation distances z when a/w0 ¼5. (a) z¼ 0.4a, (b) z ¼0.8a. The calculated parameters are w0 ¼0.01 mm.
respectively. We can draw a conclusion that the asymmetric propagation distance gradually decreases with the increase of truncation parameter. Therefore, the asymmetric property of HGBs through a circular aperture in near field should be considered in practical applications, especially for those small a truncation parameter a/w0. For example, we often encounter optical beams with very small spot size (spot size is comparable with the wavelength of light) on the micro-cavities lasers, photonic crystals lasers. Then the propagation properties of these beams cannot be described by scalar diffraction theory, but described by the vector diffraction theory. For scalar diffraction theory and vector diffraction theory, the nonparaxial propagation properties and the asymmetric propagation properties of the beams exist some different in the near field by which would have potential influence on designing the micro lasers.
4. Conclusion From the vectorial Rayleigh–Sommerfild integrals, the general propagation integral expressions for nonparaxial vectorial hollow Gaussian beams through a circular aperture are derived. And the expressions do not depend on the far-field paraxial approximation Rb a, or scalar approximation a/l b 1. Thus, it is valid for either far-field or near-field, and for the systems in which the size of the aperture is comparable with or smaller than the wavelength. And it provides the strict integral formula for the light field on the axis. The derived formulae consist with the Collins formulae of the scalar diffraction theory in the situation of paraxial approximation or scalar approximation. Therefore, it is the general propagation integral expressions for nonparaxial vectorial hollow Gaussian
beams through a circular aperture. The calculated results from Eq. (6) show that the propagation field of HGBs through a circular aperture is asymmetric in near field, and the propagation field is symmetric in far field. These research can shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and will play a guiding role in the future research and practical application of HGBs.
References [1] F.K. Fatemi, M. Bashkansky, Optics Express 14 (2006) 1368. [2] Y. Song, D. Milam, W.T. Hill III, Optics Letters 24 (1999) 1805. [3] R.B. Charters, B. Luther-Davies, F. Ladouceur, IEEE Photonics Technology Letters 11 (1999) 1617. [4] L. Paterson, M.P. Macdonald, J. Arit, et al., Science 292 (2001) 912. [5] Zhengjun Liu, Haifa Zhao, Jianlong Liu, Jie Lin, et al., Optics Letters 32 (2007) 2076. [6] F.K. Fatemi, M. Bashkansky, Optics Letters 31 (2006) 864. [7] C. Paterson, R. Smith, Optics Communication 124 (1996) 121. [8] M.L. Hu, C.Y. Wang, Y.J. Song, Y.F. Li, L. Chai, E.E. Serebryannikov, A.M. Zheltikov, Optics Express 14 (2006) 4128. [9] J. Arlt, K. Dholakia, Optics Communication 177 (2000) 297. [10] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Physical Review Letters 78 (1997) 4713. [11] Y.J. Cai, X.H. Lu, Q. Lin, Optics Letters 28 (2003) 1084. [12] Y. Cai, L. Zhang, Optics Communication 265 (2006) 607. [13] Y. Zhang, B. Wang, X. Li, J. Zhu, Optika 121 (2010) 652. [14] D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, Journal of the Optical Society of America B25 (2008) 83. [15] R.K. Luneberg, Mathematical Theory of Optics, University of California Press, Berkeley, California, 1964, Eq. (45.76). [16] J. Guo, X. Zhao, M. Yan, Optics Communication 282 (2009) 1511. ¨ K.L. Duan, Optics Communication 260 (2006) 57. [17] J. Guo, B.D. Lu, ¨ Propagation and Control of High-Power Lasers, National Defence [18] B.D. Lu, Industry Press, Beijing, 1999, Eq. (1.4-9).
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture Jiang Guo a,n, Zao Li b a b
Department of Geophysics, Chengdu University of Technology, Chengdu 610059, China Jincheng College of Sichuan University, Chengdu 611731, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 December 2011 Received in revised form 6 July 2012 Accepted 6 August 2012 Available online 29 August 2012
Based on the vectorial Rayleith–Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs. & 2012 Elsevier B.V. All rights reserved.
Keywords: Hollow Gaussian beams (HGBs) Vectorial Rayleith–Sommerfeld formulae Nonparaxial propagation Vectorial diffraction theory
1. Introduction In recent years, there have been increasing interests in the study of optical beams with zero central intensity called darkhollow beams (DHBs) because of many practical applications in science and technology, including cold atom guiding and optical atom trapping [1–2], laser writing and drilling [3], laser manipulation of microscopic dielectric and metal objects [4]. Many different methods have been used to obtain the hollow beams [5–8]. In recent theoretical investigations, several different mathematical models for describing DHBs have been proposed, such as the TEM01 beam and the higher-order Bessel beam, high-order Laguerre– Gaussian beam, hollow Gaussian beams (HGBs) [9–11]. Followed by the advent of HGBs, a rich amount of investigations about HGBs have been increasingly presented. The paraxial propagation properties of HGBs through aperture have been studied [12], and the analytical vectorial structure of HGBs in the far field has been studied [13], and the free-space nonparaxial propagation properties of the HGBs have been also studied [14]. The description of optical beam propagation properties in the nonparaxial region becomes more and more important with the development of new optical structures (e.g., micro-cavities, photonic crystals, and so on) and near-field optics, characterizing linear dimensions or spatial scales of variation comparable to or even smaller than wavelength l.
n
Corresponding author. Tel.: þ86 28 84079327. E-mail addresses: [email protected], [email protected] (J. Guo).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.08.032
In the near field, the nonparaxial propagation properties and intensity distribution of vector HGBs through an aperture optical system have been rarely reported so far. In this paper, the nonparaxial propagation properties of the HGBs through a circular aperture were studied through the vectorial Rayleith–Sommerfeld formulae in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expressions do not depend on the approximation of the paraxial and far field (i.e., ac l or Rc a, where a R and l are aperture radius, beam propagation distention and wavelength, respectively). And the derived expressions are valid either for the far and near field or for the systems in which the condition a l or ao l was considerate. By using the deduced formula, the propagation properties of HGBs through a circular aperture are numerically calculated and graphically illustrated. Moreover, numerically-calculated results are analyzed and compared with those obtained by using the Collins formulae of scalar approximation. In addition, we also research the influence of the truncation parameter on the near field propagation properties.
2. Nonparaxial vectorial propagation integral expressions of HGBs According to the boundary values E(x0,y0,0)¼Ex(x0,y0,0)i þ Ey(x0,y0,0)j on the plane z¼ 0, the solution of the Helmholtz
J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
4857
(
Eq. (1) is expressed as Eq. (2a)–(2c) in the half-space z 40 [15].
r2 E þ k2 E ¼ 0 Ex ðx,y,zÞ ¼
Ey ðx,y,zÞ ¼
Ez ðx,y,zÞ ¼
z 2p z 2p
ð1Þ Z
þ1 Z 1
Z
þ1
þ1
Ex ðx0 ,y0 ,0Þ
ikR1 R3
1
Z
1
þ1
Ey ðx0 ,y0 ,0Þ
ikR1
1
R3
exp ðikRÞdx0 dy0
ð2aÞ
expðikRÞdx0 dy0
ð2bÞ
Z þ1 Z þ1 1 ½Ex ðx0 ,y0 ,0Þðxx0 Þ þ Ey ðx0 ,y0 ,0Þðyy0 Þ 2p 1 1 ikR1 expðikRÞdx0 dy0 ð2cÞ R3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where k is wave number, R ¼ 9rq0 9 ¼ ðxx0 Þ2 þ ðyy0 Þ2 þ z2 , q0 ¼x0i þy0j is vector on the plane z ¼0, r¼xiþyjþ zk is spatial vector. The Eq. (2) is called the vectorial Rayleigh–Sommerfeld integrals. By using the formulae, the propagation properties of vectorial beams could be studied. Considering the hollow Gaussian beams, Eðx,y,0Þ ¼ Ex ðx0 ,y0 ,0Þiþ Ey ðx0 ,y0 ,0Þj Ex ðx0 ,y0 ,0Þ ¼ A0
r20
!n exp
w20
r20
ð3Þ
was incident on a circular aperture with radius of a at the plane z¼ 0. Where n is the order of the HGBs, A0 is the amplitude constant, w0 is the beam waist width. For n¼1 Eq. (3a) becomes a Gaussian beam with beam waist w0. The field at the plane after aperture may be represented as !n ! w20
Ez ðx,y,zÞ ¼
ð3aÞ
w20
ð3bÞ
r20
Ey ðx,y,zÞ ¼ 0
!
Ey ðx0 ,y0 ,0Þ ¼ 0
Ex ðx0 ,y0 ,0Þ ¼ tðx0 ,y0 ÞA0
1 x20 þ y20 r a2 where tðx0 ,y0 Þ ¼ is the transmission function of 0 else aperture. Replacing R in the exponent part with [16,17], ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xx0 þ yy0 2 2 R ¼ 9rq0 90 r þ r0 1 2 ð5Þ r þ r20 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and R everywhere else with R r 2 þ r20 in Eq. (2) (it requires that ðxx0 þyy0 =r 2 þ r20 Þ{1, these are the only approximation conditions required in this paper), and using Eq. (4) and cylindrical coordinates, the Eq. (2a)–(2c) are transformed as: !n Z i2pzA0 1 a2 r2 a2 r i Ex ðx,y,zÞ ¼ 1þ 2 2 kg l g w0 0 ! 2 2 kabr a r exp ikg 2 dr J0 ð6aÞ g w0
exp
r20
ð4aÞ
w20
Ey ðx0 ,y0 ,0Þ ¼ 0
ð4bÞ
1
2
2
2
2
Iðx,y,zÞ ¼ 9Eðx,y,zÞ9 ¼ 9Ex ðx,y,zÞ9 þ9Ey ðx,y,zÞ9 þ9Ez ðx,y,zÞ9
Normalized intensity
Normalized intensity
Z
ð7Þ
1.2
0.9 0.6 0.3 0 -0.36 -0.24
-0.12
0 x/mm
0.12
0.24
0.8
0.4
0 -0.45
0.36
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
1.2 Normalized intensity
1.2 Normalized intensity
!n a2 r2 a2 r i 1 þ kg l g2 w20 0 ! kabr xra kabr a2 r2 ixJ 0 J exp ikg 2 dr b 1 g g w0 2pA0
ð6cÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 where b¼ x þy , g ¼ b þ z þ ðarÞ , J0 (kabr/g) and J1 (kabr/ g) is Bessel function of the zero- and first-order, respectively, r0 ¼ar, r is a normalized variable (0 r r r1). Eq. (6) is the main result in this paper, which is general propagation integral expression for nonparaxial vectorial HGBs through a circular aperture. We note that the propagation field owns both the longitudinal and the transversal components. However, the longitudinal components is asymmetric along x and y directions. Thus the propagation field is asymmetric. The intensity distribution is
1.2
1 0.8 0.6 0.4 0.2 0 -0.45
ð6bÞ
-0.3
-0.15
0 0.15 x/mm
0.3
0.45
0.9 0.6 0.3 0 -0.45
-0.3
-0.15
0
0.15
0.3
0.45
x/mm
Fig. 1. (Color online) Normalized transversal intensity distributions along x direction for the different propagation distances z. (a) z¼ 2.4 mm; (b) z ¼ 3 mm; (c) z ¼15 mm; (d) z ¼50 mm. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the data from Collins formula. The calculated parameters are a ¼0.35 mm, w0 ¼ 0.1 mm.
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J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
In the course of deducing Eq. (6) without the use of the approximations Rba, and ab l, Eq. (6) is valid for far field, near field and for the optical systems in which the size of the aperture is comparable with or smaller than the wavelength (i.e., a l or ao l). Eq. (6) is general propagation integral expression for nonparaxial vectorial HGBs through a circular aperture. In addition, for the observatqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ion point on axis R ¼ 9rq0 9 r 2 þ r20 , Eq. (6) is a strict integral formula.
3. Numerical calculation results and analysis The normalized intensity distributions of nonparaxial vectorial HGBs through a circular aperture are calculated by using Eq. (6), and the calculated results are compared in detail with those obtained by the generalized Collins formula. Within the framework of the paraxial approximation, under rotationally symmetric cylindrical coordinate condition, Collins formula can be expressed as [18]
4.E-03
Z a ik ik ik kbr0 2 E0 ðr0 ,0Þexp r0 dr0 exp ikz Db Ar20 J0 B B 2B 2B 0
Normalized intensity
Eðx,y,zÞ ¼
between Eq. (6) and the Collins formulae in the situation of paraxial approximation. Therefore, the accuracy of Eq. (6) is well confirmed. When the paraxial approximation condition is not satisfied, i.e., z a or z oa, it can be seen from Fig. 2a–d that the numerical errors of different formulae gradually increase with the decreasing of z, such as the maximal difference of intensities transversal along the x direction is 0.077, 0.195, 0.43, 0.95 when z is 5a, 3a, 1.72a, and 0.857a, respectively. From Fig. 2a, the high-spatialfrequency oscillation appears in the calculated results using Collins formula under the condition of z ¼0.857a. In contrast, the calculated beam profile using the Eq. (6) are smooth. From Fig. 3, it can be seen that the on-axis intensity by using the strict integral formula (i.e., Eq. (6)) is not in agreement with those by using the Collins formula when z oa, and the on-axis intensity calculated using Collins formula tends to infinity. Only when the propagation distance is larger than a (i.e., far field), the results obtained from two different formulae are in agreement. It
ð8Þ where A, B, C and D are the transition matrix elements of the HGBs in paraxial ABCD optical system. The calculated results are shown in Figs. 1–5. During the calculation, we choose the parameters A¼1, B ¼z, C ¼0, D ¼1, l ¼1060 nm, n ¼5. If R ba, i.e., when the paraxial approximation condition is satisfied, the propagation field of HGBs is symmetric around the propagating axis, and the transversal intensities distributions by using Eq. (6) are in agreement with those by using the Collins formula Eq. (8), as shown in Fig. 1a–d. And the maximal errors are less than 0.048, 0.025, 9.7 10 4, 1.29 10 5 when z is 2.4, 3, 15 and 50 mm, respectively. It is evident that there is no difference
0.2 -0.45
-0.3
-0.15
0
0 x/mm
0.15
0.3
4
6 z /mm
8
10
12
0.4
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
-0.24
-0.12
0 x/mm
0.12
0.24
0.36
1.2 Normalized intensity
Normalized intensity
2
0.8
0 -0.45
0.45
1.2
0.8
0.4
0 -0.45
0
Fig. 3. (Color online) On-axis normalized intensity distribution. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red dot line’’ represents the data from Collins formula. The calculation parameters are a ¼0.35 mm, w0 ¼ 0.1 mm.
Normalized intensity
0.4
1.E-03
1.2 Normalized intensity
0.6
2.E-03
0.E+00
1 0.8
3.E-03
-0.3
-0.15
0 x/mm
0.15
0.3
0.45
0.8
0.4
0 -0.36
Fig. 2. (Color online) Normalized transversal intensity distributions along x direction for the different propagation distances z. (a) z ¼ 0.872a, (b) z ¼ 1.72a, (c) z ¼3a and (d) z ¼5a. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the data from Collins formula. The calculated parameters are a¼0.35 mm, w0 ¼ 0.1 mm.
J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
4859
a 0.04
0.02
1 0.75 0.5 0.25 0
0
0.05 0.025
-0.05 -0.025
-0.025
0
x
-0.02
y
0
0.025 0.05
-0.04
-0.05
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
b 0.04
0.02
1 0.75 0.5 0.25 0 -0.05 -0.025
0.05
0
0.025
y
0 -0.025
0
x
-0.02
-0.04
-0.05
0.025 0.05
c 0.04
0.02
1 0.75 0.5 0.25 0 -0.05
0.05
0
0.025
y
0 -0.025
-0.025
0 0.025
x
0.05
-0.02
-0.04
-0.05
d 0.04
0.02
1 0.75 0.5 0.25 0 -0.05 -0.025
0.05
0
0.025 0 -0.025
0
x
y
0.025 0.05
-0.02
-0.04
-0.05 -0.04
-0.02
0
0.02
0.04
Fig. 4. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different truncation parameter a/w0 at z ¼0.2a plane. (a) a/ w0 ¼ 30, (b) a/w0 ¼ 10, (c) a/w0 ¼8 and (d) a/w0 ¼ 5. The calculated parameters are w0 ¼ 0.01 mm.
indicates that Collins formula of scalar paraxial approximation is invalid for near field, and a vectorial diffraction theory should be used in near field. Fig. 4a–d show the numerically-calculated results of normalized 3-D intensity distribution and the intensity contour lines of HGBs through a circular aperture for the different truncation parameter a/w0 at z ¼0.2a plane by using Eq. (6). From Fig. 4a–d, we find that the propagation field of HGBs gradually becomes asymmetric around the propagating axis in the near field with the decrease of truncation parameter. The maximal difference of transversal intensities between along the x direction and that of the y direction is 0.003, 0.016, 0.146, 0.237, 0.488 when
a/w0 ¼30, 20, 10, 8 and 5, respectively. Moreover, when truncation parameter a/w0 ¼constant, with the increase of propagation distance the propagation field of HGBs gradually becomes symmetric. For example, when a/w0 ¼5, the calculated results of normalized 3-D intensity distribution and the intensity contour lines of HGBs by using Eq. (6) are shown in Fig. 4d and Fig. 5a–b at the different propagation distance z. The maximal difference of intensities transversal between along the x direction and that of the y direction is 0.488, 0.146, 0.016 when z¼0.2a, 0.4a, 0.8a, respectively. The asymmetric propagation distance is about 0.8a for a/w0 ¼5. In addition, the asymmetric propagation distance is about 1.5a, a, 0.5a, 0.4a, 0.2a when a/w0 ¼3, 4, 8, 10 and 20,
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J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860
a
0.04
0.02
1 0.75 0.5 0.25 0
0.05
0
0.025
y
0
-0.05 -0.025
-0.025
0
x
0.025
-0.02
-0.04
-0.05 0.05
-0.04
-0.02
0
0.02
0.04
0.04
0.02
1 0.75 0.5 0.25 0
0.05
0
0.025 0
-0.05 -0.025
-0.02
-0.025
0
x
y
-0.04
0.025
-0.05 0.05
-0.04
-0.02
0
0.02
0.04
Fig. 5. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different propagation distances z when a/w0 ¼5. (a) z¼ 0.4a, (b) z ¼0.8a. The calculated parameters are w0 ¼0.01 mm.
respectively. We can draw a conclusion that the asymmetric propagation distance gradually decreases with the increase of truncation parameter. Therefore, the asymmetric property of HGBs through a circular aperture in near field should be considered in practical applications, especially for those small a truncation parameter a/w0. For example, we often encounter optical beams with very small spot size (spot size is comparable with the wavelength of light) on the micro-cavities lasers, photonic crystals lasers. Then the propagation properties of these beams cannot be described by scalar diffraction theory, but described by the vector diffraction theory. For scalar diffraction theory and vector diffraction theory, the nonparaxial propagation properties and the asymmetric propagation properties of the beams exist some different in the near field by which would have potential influence on designing the micro lasers.
4. Conclusion From the vectorial Rayleigh–Sommerfild integrals, the general propagation integral expressions for nonparaxial vectorial hollow Gaussian beams through a circular aperture are derived. And the expressions do not depend on the far-field paraxial approximation Rb a, or scalar approximation a/l b 1. Thus, it is valid for either far-field or near-field, and for the systems in which the size of the aperture is comparable with or smaller than the wavelength. And it provides the strict integral formula for the light field on the axis. The derived formulae consist with the Collins formulae of the scalar diffraction theory in the situation of paraxial approximation or scalar approximation. Therefore, it is the general propagation integral expressions for nonparaxial vectorial hollow Gaussian
beams through a circular aperture. The calculated results from Eq. (6) show that the propagation field of HGBs through a circular aperture is asymmetric in near field, and the propagation field is symmetric in far field. These research can shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and will play a guiding role in the future research and practical application of HGBs.
References [1] F.K. Fatemi, M. Bashkansky, Optics Express 14 (2006) 1368. [2] Y. Song, D. Milam, W.T. Hill III, Optics Letters 24 (1999) 1805. [3] R.B. Charters, B. Luther-Davies, F. Ladouceur, IEEE Photonics Technology Letters 11 (1999) 1617. [4] L. Paterson, M.P. Macdonald, J. Arit, et al., Science 292 (2001) 912. [5] Zhengjun Liu, Haifa Zhao, Jianlong Liu, Jie Lin, et al., Optics Letters 32 (2007) 2076. [6] F.K. Fatemi, M. Bashkansky, Optics Letters 31 (2006) 864. [7] C. Paterson, R. Smith, Optics Communication 124 (1996) 121. [8] M.L. Hu, C.Y. Wang, Y.J. Song, Y.F. Li, L. Chai, E.E. Serebryannikov, A.M. Zheltikov, Optics Express 14 (2006) 4128. [9] J. Arlt, K. Dholakia, Optics Communication 177 (2000) 297. [10] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Physical Review Letters 78 (1997) 4713. [11] Y.J. Cai, X.H. Lu, Q. Lin, Optics Letters 28 (2003) 1084. [12] Y. Cai, L. Zhang, Optics Communication 265 (2006) 607. [13] Y. Zhang, B. Wang, X. Li, J. Zhu, Optika 121 (2010) 652. [14] D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, Journal of the Optical Society of America B25 (2008) 83. [15] R.K. Luneberg, Mathematical Theory of Optics, University of California Press, Berkeley, California, 1964, Eq. (45.76). [16] J. Guo, X. Zhao, M. Yan, Optics Communication 282 (2009) 1511. ¨ K.L. Duan, Optics Communication 260 (2006) 57. [17] J. Guo, B.D. Lu, ¨ Propagation and Control of High-Power Lasers, National Defence [18] B.D. Lu, Industry Press, Beijing, 1999, Eq. (1.4-9).