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Propagation properties of anomalous hollow beams in a turbulent atmosphere
Optics Communications 281 (2008) 5291–5297
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation properties of anomalous hollow beams in a turbulent atmosphere Yangjian Cai a,b,*, Halil T. Eyyubog˘lu c, Yahya Baykal c a b c
College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of Erlangen, Staudtstr. 7/B2, D-91058 Erlangen, Germany Department of Electronic and Communication Engineering, Çankaya University, Ög˘retmenler Cad. 14, Yüzüncüyıl 06530 Balgat Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 7 February 2008 Received in revised form 4 June 2008 Accepted 25 July 2008
PACS: 42.25.Bs 41.85.Ew .
a b s t r a c t Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In the past several years, conventional dark-hollow beams (DHBs) with zero central intensity have been widely investigated both theoretically and experimentally due to their wide applications in atomic optics, free-space optical communications, binary optics, optical trapping of particles and medical sciences [1–6]. Propagation of various DHBs in free space, paraxial optical system, turbulent atmosphere have been widely studied [6–22], and it has been found that the scintillation index of a DHB can be smaller than that of a Gaussian beam, annular beam and flat-topped beam under certain conditions, which are useful in long-distance freespace optical communication [21,22]. However up to now, only little work seems to have been done on anomalous hollow beams [23–25]. In 2005, for the first time, Wu et al. demonstrated experimentally an anomalous hollow beam of elliptical symmetry with an elliptical solid core, which can be used for studying the transverse instability and provides a powerful tool for studying the linear and nonlinear particle dynamics in the storage ring [23]. More recently, Cai proposed a theoretical model to describe an anomalous hollow beam of circular and elliptical symmetry with a solid core [24]. More recently, Cai and Wang extended anomalous hollow beam to the partially coherent case, and study the propagation of partially coherent anomalous hollow beam through paraxial
* Corresponding author. Address: Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of Erlangen, Staudtstr. 7/B2, D91058 Erlangen, Germany. E-mail address: [email protected] (Y. Cai). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.080
optical system [25]. In this paper, we study the propagation properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere. Analytical propagation formula is derived and some numerical examples are given. Comparisons of the results with those of free space are also carried out. 2. Average irradiance of an anomalous hollow beam in a turbulent atmosphere In the past decades, much work has been carried out concerning the propagation of various coherent and partially coherent laser beams in a turbulent atmosphere because of their wide applications in e.g. free-space optical communications, imaging system and remote sensing [26–53]. Within the framework of paraxial approximation, the propagation of a laser beam in a turbulent atmosphere can be treated with the following extended Huygens–Fresnel integral formula [28–31]
hIðq; zÞi ¼
k
2
Z
1
Z
1
Z
1
Z
1
Eðr 1 ; 0ÞE ðr 2 ; 0Þ ik ik exp ðr 1 qÞ2 þ ðr 2 qÞ2 2z 2z
4p
2 z2
1
1
1
1
ð1Þ
hexp½Wðr 1 ; qÞ þ W ðr2 ; qÞidr1 dr2 where h i denotes ensemble average over the turbulent media. r = (x, y) and q = (qx, qy) are the position vectors at the input plane (z = 0) and output plane (z), respectively. hIðq; zÞi is the average irradiance at the output plane at z, E(r1, 0) is the electric field of the laser beam at the source plane (z = 0), and dr1dr2 = dx1dy1dx2dy2. The ensemble average term in Eq. (1) can be expressed as [28–31]
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hexp½Wðr 1 ; qÞ þ W ðr2 ; qÞi ¼ exp ½0:5DW ðr1 r2 Þ 1 ¼ exp 2 ðr 1 r2 Þ2 ;
q0
ð2Þ
DW(r1 r2) is the wave structure function approximated by the phase structure function in Rytov’s representation and q0 ¼ ð0:545C 2n k2 zÞ3=5 is the coherence length of a spherical wave propagating in the turbulent medium with C 2n being the structure constant. In derivation of Eq. (2), we have employed Kolmogorov spectrum and a quadratic approximation for Rytov’s phase structure function [28–31]. The electric field of an anomalous hollow beam of elliptical symmetry with an elliptical solid core at z = 0 is expressed as follows [24]
!
Eðx; y; 0Þ ¼ exp
x2 y2 H2 w20x w20y
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2x2 2y2 ; þ w20x w20y
Eðx; y; 0Þ ¼
hIðq; zÞi ¼ I1 I2 I3 I4 I5 þ I6 þ I7 þ I8 þ I9 ;
ð3Þ
ð4Þ
Fig. 1 shows the contour graph of the 3D-normalized irradiance and cross line (y = 0) of an anomalous hollow beam for two different sets of w0x and w0y. One sees that Eq. (3) provides a model for describing an anomalous hollow beam demonstrated in Ref. [23].
ð5Þ
where
I1 ðq; zÞ ¼
" # 2 2 2 k k k 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp q q 4ax z2 x 4ay z2 y z2 ax bx ay by " 2 2 # 2 2 k 1 k 1 2 1 q 1 q2y ; exp x 4bx z2 4by z2 ax q20 ay q20
2 # k k 1 2bx 2 1 q2x I2 ðq; zÞ ¼ 5=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ax q20 ax ay by z2 w20x bx " 2 2 2 # 2 k 1 k qx 2 q 1 2 exp 4ax z2 x ax q0 4z2 bx " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 " 2 # 2 2 k k 1 I3 ðq; zÞ ¼ 2b 1 q2x p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x z2 ax q20 z2 w20x ðbx Þ5=2 ax ay by " 2 2 2 # 2 1 k 1 k qx exp 2 q2x 1 2 4ax z ax q0 4z2 bx " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 2
where w0x and w0y are the beam widths of an astigmatic Gaussian beam in x and y directions, respectively. H2 is the second order Hermite polynomial. When w0x = w0y, Eq. (3) reduces to a circular anomalous hollow beam with a circular solid core (see Fig. 1a). Eq. (3) can be expanded as superposition of astigmatic Gaussian modes and astigmatic doughnut modes as follows [24]
! ! 8x2 8y2 x2 y2 2 þ 2 þ 2 exp 2 2 : w0x w0y w0x w0y
Substituting Eq. (4) into Eq. (1), after tedious but straightforward integration, we obtain the following expression for the average irradiance of an anomalous hollow beam propagating in a turbulent atmosphere
"
ð6Þ
2
ð7Þ
ð8Þ
Fig. 1. Contour graph of the 3D-normalized irradiance and cross line (y = 0) of an anomalous hollow beam for two different sets of w0x and w0y (a) w0x = w0y = 1 cm, (b) w0 = 2.5 cm, w0y = 1 cm.
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"
#
2 2 k 1 1 q2y 2b p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi y 5=2 z2 ay q20 z2 w20y by ay ax bx " 2 2 2 # 2 k qy k 1 2 q 1 2 exp 4ay z2 y ay q0 4z2 by " 2 # 2 2 k k 1 2 q 1 q2x ; exp 4ax z2 x 4bx z2 ax q20 2 !2 3 2 2 k k 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 42by 2 1 2 q2y 5 I5 ðq; zÞ ¼ z ay q0 z2 w2 ðb Þ5=2 a ax bx k
I4 ðq; zÞ ¼
2
y
0y
ð9Þ
4k
ð10Þ
2
pffiffiffiffiffiffiffiffiffi ðA1x þ A2x þ A3x Þ ay by " 2 # 2 2 k k 1 2 q 1 q2x exp 4ax z2 x 4z2 bx ax q20 " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 z2 w40x ax5=2
ð11Þ
2
I7 ðq; zÞ ¼
I8 ðq; zÞ ¼
4k
pffiffiffiffiffiffiffiffiffi ðA1y þ A2y þ A3y Þ ax bx " 2 # 2 2 k k 1 2 exp q 1 q2y 4ay z2 y 4z2 by ay q20 " 2 # 2 2 k k 1 2 q 1 q2x ; exp 4ax z2 x 4bx z2 ax q20 z2 w40y ay5=2
k
ð12Þ
ð13Þ
2
I9 ðq; zÞ ¼
k
pffiffiffiffiffiffiffiffiffi 5=2 z2 w20x w20y by ðbx Þ5=2 ax ay
2 # 2 k 1 1 q2x z2 ax q20 " 2 # 2 k 1 q2y 2by 2 1 z ay q20 " 2 2 2 # 2 1 k 1 k qx exp 2 q2x 1 2 4ax z ax q0 4z2 bx " 2 2 2 # 2 k qy k 1 2 q 1 2 exp ; 4ay z2 y ay q0 4z2 by
"
2bx
with
1 1 ik 1 þ þ ; w20x q20 2z ax q40
ax ¼
1 1 ik þ þ ; w20x q20 2z
1 1 ik þ ; w20y q20 2z
by ¼ ay ¼
1 1 ik 1 þ þ w20y q20 2z ay q40
1 1 ik þ þ w20y q20 2z
1 1 ik 1 1 1 ik 1 þ ; by ¼ 2 þ 2 4 w20x q20 2z ax q40 w0y q0 2z ay q0 !" 2 # 2 2 1 k 2 k 1 A1x ¼ 5=2 2ax 2 qx 2bx 2 1 q2x ; z z ax q20 4bx " 2 4 # 2 4 1 k 1 k 1 2 2 A2x ¼ 9=2 12bx 12bx 2 1 q þ 1 q4x ; x z z4 ax q20 ax q20 4bx q40 " 2 # 2 2 k 1 k 1 2 1 q 6b 1 q2x ; A3x ¼ x 7=2 z2 ax q20 x ax q20 2q20 z2 bx !" 2 # 2 2 1 k k 1 A1y ¼ 5=2 2ay 2 q2y 2by 2 1 q2y ; z z ay q20 4by " 2 4 # 2 4 1 k 1 k 1 2 2 12by 12by 2 1 qy þ 4 1 2 q4y ; A2y ¼ 9=2 z z ay q20 ay q0 4by q40 " # 2 2 2 k 1 k 1 A3y ¼ 1 q2y 6by 2 1 2 q2y : ð15Þ 2 2 2 7=2 z ay q0 ay q0 2q 0 z b y Eqs. (5)–(15) are the main analytical results of this paper. They provide the formulae to calculate and analyze the propagation properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere (C 2n > 0). Under the condition of C 2n ¼ 0, Eqs. (5)–(15) reduce to the formulae for an anomalous hollow beam propagating in free space, which are consistent with the formula derived in Ref. [24]. 3. Results and discussion
2
pffiffiffiffiffiffiffiffiffi 5=2 z2 w20x w20y bx ðby Þ5=2 ax ay " 2 # 2 k 1 2bx 2 1 q2x z ax q20 2 !2 3 2 k 1 4 2by 2 1 2 q2y 5 z ay q0 " 2 2 2 # 2 k 1 k qx 2 q 1 2 exp 4ax z2 x ax q0 4z2 bx 2 !2 2 3 2 k q2y 1 k 2 1 4 5 exp 2 qy 1 2 ; 4ay z ay q0 4z2 by
bx ¼
ay ¼
2
I6 ðq; zÞ ¼
1 1 ik þ ; w20x q20 2z
bx ¼
y
!2 2 3 2 k q2y 1 k 1 2 5 exp 4 2 qy 1 2 4ay z ay q0 4z2 by " 2 # 2 2 k k 1 2 exp q 1 q2x ; 4ax z2 x 4bx z2 ax q20
ax ¼
ð14Þ
In this section, we study the propagation properties (i.e, irradiance and spreading properties) of circular and elliptical anomalous hollow beam in a turbulent atmosphere, and compare the results to those of free space using the formulae derived in Section 2. Fig. 2 shows the cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in a turbulent atmosphere with k ¼ 632:8 nm w0x = w0y = 2 cm and C 2n ¼ 5 1014 m2=3 . For the convenience of comparison, Fig. 3 shows the cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in free space (C 2n ¼ 0) with other parameters remaining as given for Fig. 2. From Figs. 2a,b and 3a,b, we find that a circular anomalous hollow beam in turbulent atmosphere has similar propagation properties to those of free space at short propagation distances, namely, the beam profile remains almost invariant although the beam spot spreads. As the propagation distance increases, the propagation properties of a circular anomalous hollow beam in a turbulent atmosphere gradually becomes different from that in free space (see Figs. 2c–f and 3c–f). In free space, the central irradiance of a circular anomalous hollow beam rises rapidly as the propagation distance increases, in the far field, the central irradiance becomes the maximum and the total beam profile becomes a circular solid beam spot with a small bright ring around the brightest center (see Fig. 3f). While in a turbulent atmosphere, the central irradiance of a circular anomalous hollow beam grows slowly as the propagation distance increases although the total beam spot spreads rapidly, at intermediate propagation distances, a flat-topped beam profile can be observed, in the far field, the circular anomalous hollow beam eventually becomes a Gauss-
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ian beam under the influence of the turbulent atmosphere (see Fig. 2f). Fig. 4 shows the 3D-normalized irradiance distribution of an elliptical anomalous hollow beam and contour graph at several propagation distances in a turbulent atmosphere with w0x = 2.5 cm, w0y = 1 cm, k ¼ 632:8 nm and C 2n ¼ 5 1014 m2=3 .
For comparison, Fig. 5 shows the 3D-normalized irradiance distribution of an elliptical anomalous hollow beam at several propagation distances in free space (C 2n ¼ 0) with other parameters remaining as given for Fig. 4. From Figs. 4 and 5, one sees that an elliptical anomalous hollow beam in turbulent atmosphere also
Fig. 2. Cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in a turbulent atmosphere (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 0.8 km, (e) z = 5km, (f) z = 15 km.
Fig. 3. Cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in free space (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 1.5 km, (e) z = 5 km, (f) z = 15 km.
Y. Cai et al. / Optics Communications 281 (2008) 5291–5297
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Fig. 4. 3D-normalized irradiance distribution of an elliptical anomalous hollow beam and contour graph at several propagation distances in a turbulent atmosphere (a) z = 0, (b) z = 0.05 km, (c) z = 0.4 km, (d) z = 0.8 km, (e) z = 5 km, (f) z = 15 km.
has similar propagation properties with that in free space at short propagation distances. As the propagation distance increases, the central irradiance of an elliptical anomalous hollow beam rises gradually, and the beam profile becomes non-elliptical symmetric both in a turbulent atmosphere and in free space. In the far field, the elliptical anomalous hollow beam retains its elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot in free space (see Fig. 5f), note the long axis and short axis of the elliptical beam spot in the far field are
in y- and x-directions, respectively, while the long axis and short axis of the elliptical beam spot in the near field are in x- and ydirections, respectively (see Fig. 5a). In a turbulent atmosphere, the elliptical anomalous hollow beam finally becomes a circular Gaussian beam under the influence of the turbulent atmosphere in the far field (see Fig. 4f). Numerical results also show that the conversion from circular and elliptical anomalous hollow beams to circular Gaussian beams becomes quicker for a larger structure constant, a shorter wavelength, and a smaller waist size of the ini-
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Fig. 5. 3D-normalized irradiance distribution of an elliptical anomalous hollow beam at several propagation distances in free space (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 1.5 km, (e) z = 5 km, (f) z = 15 km.
tial beam. The propagation properties of an anomalous hollow beam in free space calculated by Eqs. (5)–(15) are also consistent with those reported in Ref. [24]. To learn about the spreading properties of anomalous hollow beam, we study evolution properties of the effective beam spot of an elliptical anomalous hollow beam on propagation. By use of twice the variance of x or y, the effective beam size of an elliptical anomalous hollow beam at plane z can be defined as [54]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 R1 2 1 1 s2 hIðx; y; zÞidxdy R1 R1 W sz ¼ ðs ¼ x; yÞ; hIðx; y; zÞidxdy 1 1
ð16Þ
Here Wxz and Wyz are the effective beam sizes of an elliptical anomalous hollow beam in the x and y directions, respectively. At z p = 0, for an elliptical anomalous hollow beam, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi W zx ¼ 17=10w0x and W zy ¼ 17=10w0y [24]. Substituting Eq. (5) into Eq. (16), we can calculate the effective beam spot size of an elliptical anomalous hollow beam in a turbulent atmosphere numerically. Fig. 6 shows the dependence of the effective beam spot sizes Wzx and Wzy of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the structure constant C 2n with w0x = 2.5 cm, w0y = 1 cm and k ¼ 632:8 nm. Fig. 7 shows the dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in turbulent atmosphere for two different sets of w0x and w0y with C 2n ¼ 5 1014 m2=3 and k ¼ 632:8 nm. Fig. 8 shows the dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the wavelength k with w0x = 2.5 cm, w0y = 1 cm and C 2n ¼ 5 1014 m2=3 . From Fig. 6, one finds that the effective beam spot sizes gradually increase on propagation both in free space and in a turbulent atmosphere, Wzx finally becomes smaller than Wzy in the far field in free space, and Wzx finally becomes equal to Wzy in the far field in a turbulent atmosphere, which means the beam spot eventually becomes circular under the influence of turbulent atmosphere. From Figs. 6–8, we also find that the beam spot of the anomalous hollow beam spreads more rapidly for a larger structure constant, a shorter wavelength, and a smaller waist size of the initial beam.
Fig. 6. Dependence of the effective beam spot sizes Wzx and Wzy of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the structure constant C 2n .
Y. Cai et al. / Optics Communications 281 (2008) 5291–5297
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sphere. Our analytical formulae provide an effective and convenient way for analyzing the propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere. Further investigation on intensity fluctuation and phase fluctuation of anomalous hollow beams in a turbulent atmosphere will be carried out based on the main results of this paper. Acknowledgements Cai gratefully acknowledges the supports from the Alexander von Humboldt Foundation and the Ministry of Science and Technology of China (Grant Nos. 2006CB921403 and 2006AA06A204). References
Fig. 7. Dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in turbulent atmosphere for two different sets of w0x and w0y.
Fig. 8. Dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the wavelength k.
4. Conclusions In conclusion, we have derived some analytical formulae of the average irradiance for circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere based on the extended Huygens–Fresnel integral. The irradiance and spreading properties of circular and elliptical anomalous hollow beams have been investigated numerically. We have found that the evolution properties of the irradiance distribution of circular and elliptical anomalous hollow beams are similar to those of free space in the near field. However, in the far field, circular and elliptical anomalous hollow beams become circular Gaussian beams spot under the influence of the atmosphere turbulence. This phenomenon is quite different from the propagation properties of circular and elliptical anomalous hollow beams in free space. The irradiance and spreading properties are closely related with the parameters of the beam and the structure constant of the turbulent atmo-
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation properties of anomalous hollow beams in a turbulent atmosphere Yangjian Cai a,b,*, Halil T. Eyyubog˘lu c, Yahya Baykal c a b c
College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of Erlangen, Staudtstr. 7/B2, D-91058 Erlangen, Germany Department of Electronic and Communication Engineering, Çankaya University, Ög˘retmenler Cad. 14, Yüzüncüyıl 06530 Balgat Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 7 February 2008 Received in revised form 4 June 2008 Accepted 25 July 2008
PACS: 42.25.Bs 41.85.Ew .
a b s t r a c t Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In the past several years, conventional dark-hollow beams (DHBs) with zero central intensity have been widely investigated both theoretically and experimentally due to their wide applications in atomic optics, free-space optical communications, binary optics, optical trapping of particles and medical sciences [1–6]. Propagation of various DHBs in free space, paraxial optical system, turbulent atmosphere have been widely studied [6–22], and it has been found that the scintillation index of a DHB can be smaller than that of a Gaussian beam, annular beam and flat-topped beam under certain conditions, which are useful in long-distance freespace optical communication [21,22]. However up to now, only little work seems to have been done on anomalous hollow beams [23–25]. In 2005, for the first time, Wu et al. demonstrated experimentally an anomalous hollow beam of elliptical symmetry with an elliptical solid core, which can be used for studying the transverse instability and provides a powerful tool for studying the linear and nonlinear particle dynamics in the storage ring [23]. More recently, Cai proposed a theoretical model to describe an anomalous hollow beam of circular and elliptical symmetry with a solid core [24]. More recently, Cai and Wang extended anomalous hollow beam to the partially coherent case, and study the propagation of partially coherent anomalous hollow beam through paraxial
* Corresponding author. Address: Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of Erlangen, Staudtstr. 7/B2, D91058 Erlangen, Germany. E-mail address: [email protected] (Y. Cai). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.080
optical system [25]. In this paper, we study the propagation properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere. Analytical propagation formula is derived and some numerical examples are given. Comparisons of the results with those of free space are also carried out. 2. Average irradiance of an anomalous hollow beam in a turbulent atmosphere In the past decades, much work has been carried out concerning the propagation of various coherent and partially coherent laser beams in a turbulent atmosphere because of their wide applications in e.g. free-space optical communications, imaging system and remote sensing [26–53]. Within the framework of paraxial approximation, the propagation of a laser beam in a turbulent atmosphere can be treated with the following extended Huygens–Fresnel integral formula [28–31]
hIðq; zÞi ¼
k
2
Z
1
Z
1
Z
1
Z
1
Eðr 1 ; 0ÞE ðr 2 ; 0Þ ik ik exp ðr 1 qÞ2 þ ðr 2 qÞ2 2z 2z
4p
2 z2
1
1
1
1
ð1Þ
hexp½Wðr 1 ; qÞ þ W ðr2 ; qÞidr1 dr2 where h i denotes ensemble average over the turbulent media. r = (x, y) and q = (qx, qy) are the position vectors at the input plane (z = 0) and output plane (z), respectively. hIðq; zÞi is the average irradiance at the output plane at z, E(r1, 0) is the electric field of the laser beam at the source plane (z = 0), and dr1dr2 = dx1dy1dx2dy2. The ensemble average term in Eq. (1) can be expressed as [28–31]
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hexp½Wðr 1 ; qÞ þ W ðr2 ; qÞi ¼ exp ½0:5DW ðr1 r2 Þ 1 ¼ exp 2 ðr 1 r2 Þ2 ;
q0
ð2Þ
DW(r1 r2) is the wave structure function approximated by the phase structure function in Rytov’s representation and q0 ¼ ð0:545C 2n k2 zÞ3=5 is the coherence length of a spherical wave propagating in the turbulent medium with C 2n being the structure constant. In derivation of Eq. (2), we have employed Kolmogorov spectrum and a quadratic approximation for Rytov’s phase structure function [28–31]. The electric field of an anomalous hollow beam of elliptical symmetry with an elliptical solid core at z = 0 is expressed as follows [24]
!
Eðx; y; 0Þ ¼ exp
x2 y2 H2 w20x w20y
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2x2 2y2 ; þ w20x w20y
Eðx; y; 0Þ ¼
hIðq; zÞi ¼ I1 I2 I3 I4 I5 þ I6 þ I7 þ I8 þ I9 ;
ð3Þ
ð4Þ
Fig. 1 shows the contour graph of the 3D-normalized irradiance and cross line (y = 0) of an anomalous hollow beam for two different sets of w0x and w0y. One sees that Eq. (3) provides a model for describing an anomalous hollow beam demonstrated in Ref. [23].
ð5Þ
where
I1 ðq; zÞ ¼
" # 2 2 2 k k k 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp q q 4ax z2 x 4ay z2 y z2 ax bx ay by " 2 2 # 2 2 k 1 k 1 2 1 q 1 q2y ; exp x 4bx z2 4by z2 ax q20 ay q20
2 # k k 1 2bx 2 1 q2x I2 ðq; zÞ ¼ 5=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ax q20 ax ay by z2 w20x bx " 2 2 2 # 2 k 1 k qx 2 q 1 2 exp 4ax z2 x ax q0 4z2 bx " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 " 2 # 2 2 k k 1 I3 ðq; zÞ ¼ 2b 1 q2x p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x z2 ax q20 z2 w20x ðbx Þ5=2 ax ay by " 2 2 2 # 2 1 k 1 k qx exp 2 q2x 1 2 4ax z ax q0 4z2 bx " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 2
where w0x and w0y are the beam widths of an astigmatic Gaussian beam in x and y directions, respectively. H2 is the second order Hermite polynomial. When w0x = w0y, Eq. (3) reduces to a circular anomalous hollow beam with a circular solid core (see Fig. 1a). Eq. (3) can be expanded as superposition of astigmatic Gaussian modes and astigmatic doughnut modes as follows [24]
! ! 8x2 8y2 x2 y2 2 þ 2 þ 2 exp 2 2 : w0x w0y w0x w0y
Substituting Eq. (4) into Eq. (1), after tedious but straightforward integration, we obtain the following expression for the average irradiance of an anomalous hollow beam propagating in a turbulent atmosphere
"
ð6Þ
2
ð7Þ
ð8Þ
Fig. 1. Contour graph of the 3D-normalized irradiance and cross line (y = 0) of an anomalous hollow beam for two different sets of w0x and w0y (a) w0x = w0y = 1 cm, (b) w0 = 2.5 cm, w0y = 1 cm.
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"
#
2 2 k 1 1 q2y 2b p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi y 5=2 z2 ay q20 z2 w20y by ay ax bx " 2 2 2 # 2 k qy k 1 2 q 1 2 exp 4ay z2 y ay q0 4z2 by " 2 # 2 2 k k 1 2 q 1 q2x ; exp 4ax z2 x 4bx z2 ax q20 2 !2 3 2 2 k k 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 42by 2 1 2 q2y 5 I5 ðq; zÞ ¼ z ay q0 z2 w2 ðb Þ5=2 a ax bx k
I4 ðq; zÞ ¼
2
y
0y
ð9Þ
4k
ð10Þ
2
pffiffiffiffiffiffiffiffiffi ðA1x þ A2x þ A3x Þ ay by " 2 # 2 2 k k 1 2 q 1 q2x exp 4ax z2 x 4z2 bx ax q20 " 2 # 2 2 k k 1 2 q 1 q2y ; exp 4ay z2 y 4by z2 ay q20 z2 w40x ax5=2
ð11Þ
2
I7 ðq; zÞ ¼
I8 ðq; zÞ ¼
4k
pffiffiffiffiffiffiffiffiffi ðA1y þ A2y þ A3y Þ ax bx " 2 # 2 2 k k 1 2 exp q 1 q2y 4ay z2 y 4z2 by ay q20 " 2 # 2 2 k k 1 2 q 1 q2x ; exp 4ax z2 x 4bx z2 ax q20 z2 w40y ay5=2
k
ð12Þ
ð13Þ
2
I9 ðq; zÞ ¼
k
pffiffiffiffiffiffiffiffiffi 5=2 z2 w20x w20y by ðbx Þ5=2 ax ay
2 # 2 k 1 1 q2x z2 ax q20 " 2 # 2 k 1 q2y 2by 2 1 z ay q20 " 2 2 2 # 2 1 k 1 k qx exp 2 q2x 1 2 4ax z ax q0 4z2 bx " 2 2 2 # 2 k qy k 1 2 q 1 2 exp ; 4ay z2 y ay q0 4z2 by
"
2bx
with
1 1 ik 1 þ þ ; w20x q20 2z ax q40
ax ¼
1 1 ik þ þ ; w20x q20 2z
1 1 ik þ ; w20y q20 2z
by ¼ ay ¼
1 1 ik 1 þ þ w20y q20 2z ay q40
1 1 ik þ þ w20y q20 2z
1 1 ik 1 1 1 ik 1 þ ; by ¼ 2 þ 2 4 w20x q20 2z ax q40 w0y q0 2z ay q0 !" 2 # 2 2 1 k 2 k 1 A1x ¼ 5=2 2ax 2 qx 2bx 2 1 q2x ; z z ax q20 4bx " 2 4 # 2 4 1 k 1 k 1 2 2 A2x ¼ 9=2 12bx 12bx 2 1 q þ 1 q4x ; x z z4 ax q20 ax q20 4bx q40 " 2 # 2 2 k 1 k 1 2 1 q 6b 1 q2x ; A3x ¼ x 7=2 z2 ax q20 x ax q20 2q20 z2 bx !" 2 # 2 2 1 k k 1 A1y ¼ 5=2 2ay 2 q2y 2by 2 1 q2y ; z z ay q20 4by " 2 4 # 2 4 1 k 1 k 1 2 2 12by 12by 2 1 qy þ 4 1 2 q4y ; A2y ¼ 9=2 z z ay q20 ay q0 4by q40 " # 2 2 2 k 1 k 1 A3y ¼ 1 q2y 6by 2 1 2 q2y : ð15Þ 2 2 2 7=2 z ay q0 ay q0 2q 0 z b y Eqs. (5)–(15) are the main analytical results of this paper. They provide the formulae to calculate and analyze the propagation properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere (C 2n > 0). Under the condition of C 2n ¼ 0, Eqs. (5)–(15) reduce to the formulae for an anomalous hollow beam propagating in free space, which are consistent with the formula derived in Ref. [24]. 3. Results and discussion
2
pffiffiffiffiffiffiffiffiffi 5=2 z2 w20x w20y bx ðby Þ5=2 ax ay " 2 # 2 k 1 2bx 2 1 q2x z ax q20 2 !2 3 2 k 1 4 2by 2 1 2 q2y 5 z ay q0 " 2 2 2 # 2 k 1 k qx 2 q 1 2 exp 4ax z2 x ax q0 4z2 bx 2 !2 2 3 2 k q2y 1 k 2 1 4 5 exp 2 qy 1 2 ; 4ay z ay q0 4z2 by
bx ¼
ay ¼
2
I6 ðq; zÞ ¼
1 1 ik þ ; w20x q20 2z
bx ¼
y
!2 2 3 2 k q2y 1 k 1 2 5 exp 4 2 qy 1 2 4ay z ay q0 4z2 by " 2 # 2 2 k k 1 2 exp q 1 q2x ; 4ax z2 x 4bx z2 ax q20
ax ¼
ð14Þ
In this section, we study the propagation properties (i.e, irradiance and spreading properties) of circular and elliptical anomalous hollow beam in a turbulent atmosphere, and compare the results to those of free space using the formulae derived in Section 2. Fig. 2 shows the cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in a turbulent atmosphere with k ¼ 632:8 nm w0x = w0y = 2 cm and C 2n ¼ 5 1014 m2=3 . For the convenience of comparison, Fig. 3 shows the cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in free space (C 2n ¼ 0) with other parameters remaining as given for Fig. 2. From Figs. 2a,b and 3a,b, we find that a circular anomalous hollow beam in turbulent atmosphere has similar propagation properties to those of free space at short propagation distances, namely, the beam profile remains almost invariant although the beam spot spreads. As the propagation distance increases, the propagation properties of a circular anomalous hollow beam in a turbulent atmosphere gradually becomes different from that in free space (see Figs. 2c–f and 3c–f). In free space, the central irradiance of a circular anomalous hollow beam rises rapidly as the propagation distance increases, in the far field, the central irradiance becomes the maximum and the total beam profile becomes a circular solid beam spot with a small bright ring around the brightest center (see Fig. 3f). While in a turbulent atmosphere, the central irradiance of a circular anomalous hollow beam grows slowly as the propagation distance increases although the total beam spot spreads rapidly, at intermediate propagation distances, a flat-topped beam profile can be observed, in the far field, the circular anomalous hollow beam eventually becomes a Gauss-
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ian beam under the influence of the turbulent atmosphere (see Fig. 2f). Fig. 4 shows the 3D-normalized irradiance distribution of an elliptical anomalous hollow beam and contour graph at several propagation distances in a turbulent atmosphere with w0x = 2.5 cm, w0y = 1 cm, k ¼ 632:8 nm and C 2n ¼ 5 1014 m2=3 .
For comparison, Fig. 5 shows the 3D-normalized irradiance distribution of an elliptical anomalous hollow beam at several propagation distances in free space (C 2n ¼ 0) with other parameters remaining as given for Fig. 4. From Figs. 4 and 5, one sees that an elliptical anomalous hollow beam in turbulent atmosphere also
Fig. 2. Cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in a turbulent atmosphere (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 0.8 km, (e) z = 5km, (f) z = 15 km.
Fig. 3. Cross line (y = 0) of the normalized irradiance distribution of a circular anomalous hollow beam at several propagation distances in free space (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 1.5 km, (e) z = 5 km, (f) z = 15 km.
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Fig. 4. 3D-normalized irradiance distribution of an elliptical anomalous hollow beam and contour graph at several propagation distances in a turbulent atmosphere (a) z = 0, (b) z = 0.05 km, (c) z = 0.4 km, (d) z = 0.8 km, (e) z = 5 km, (f) z = 15 km.
has similar propagation properties with that in free space at short propagation distances. As the propagation distance increases, the central irradiance of an elliptical anomalous hollow beam rises gradually, and the beam profile becomes non-elliptical symmetric both in a turbulent atmosphere and in free space. In the far field, the elliptical anomalous hollow beam retains its elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot in free space (see Fig. 5f), note the long axis and short axis of the elliptical beam spot in the far field are
in y- and x-directions, respectively, while the long axis and short axis of the elliptical beam spot in the near field are in x- and ydirections, respectively (see Fig. 5a). In a turbulent atmosphere, the elliptical anomalous hollow beam finally becomes a circular Gaussian beam under the influence of the turbulent atmosphere in the far field (see Fig. 4f). Numerical results also show that the conversion from circular and elliptical anomalous hollow beams to circular Gaussian beams becomes quicker for a larger structure constant, a shorter wavelength, and a smaller waist size of the ini-
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Fig. 5. 3D-normalized irradiance distribution of an elliptical anomalous hollow beam at several propagation distances in free space (a) z = 0, (b) z = 0.05 km, (c) z = 0.5 km, (d) z = 1.5 km, (e) z = 5 km, (f) z = 15 km.
tial beam. The propagation properties of an anomalous hollow beam in free space calculated by Eqs. (5)–(15) are also consistent with those reported in Ref. [24]. To learn about the spreading properties of anomalous hollow beam, we study evolution properties of the effective beam spot of an elliptical anomalous hollow beam on propagation. By use of twice the variance of x or y, the effective beam size of an elliptical anomalous hollow beam at plane z can be defined as [54]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 R1 2 1 1 s2 hIðx; y; zÞidxdy R1 R1 W sz ¼ ðs ¼ x; yÞ; hIðx; y; zÞidxdy 1 1
ð16Þ
Here Wxz and Wyz are the effective beam sizes of an elliptical anomalous hollow beam in the x and y directions, respectively. At z p = 0, for an elliptical anomalous hollow beam, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi W zx ¼ 17=10w0x and W zy ¼ 17=10w0y [24]. Substituting Eq. (5) into Eq. (16), we can calculate the effective beam spot size of an elliptical anomalous hollow beam in a turbulent atmosphere numerically. Fig. 6 shows the dependence of the effective beam spot sizes Wzx and Wzy of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the structure constant C 2n with w0x = 2.5 cm, w0y = 1 cm and k ¼ 632:8 nm. Fig. 7 shows the dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in turbulent atmosphere for two different sets of w0x and w0y with C 2n ¼ 5 1014 m2=3 and k ¼ 632:8 nm. Fig. 8 shows the dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the wavelength k with w0x = 2.5 cm, w0y = 1 cm and C 2n ¼ 5 1014 m2=3 . From Fig. 6, one finds that the effective beam spot sizes gradually increase on propagation both in free space and in a turbulent atmosphere, Wzx finally becomes smaller than Wzy in the far field in free space, and Wzx finally becomes equal to Wzy in the far field in a turbulent atmosphere, which means the beam spot eventually becomes circular under the influence of turbulent atmosphere. From Figs. 6–8, we also find that the beam spot of the anomalous hollow beam spreads more rapidly for a larger structure constant, a shorter wavelength, and a smaller waist size of the initial beam.
Fig. 6. Dependence of the effective beam spot sizes Wzx and Wzy of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the structure constant C 2n .
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sphere. Our analytical formulae provide an effective and convenient way for analyzing the propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere. Further investigation on intensity fluctuation and phase fluctuation of anomalous hollow beams in a turbulent atmosphere will be carried out based on the main results of this paper. Acknowledgements Cai gratefully acknowledges the supports from the Alexander von Humboldt Foundation and the Ministry of Science and Technology of China (Grant Nos. 2006CB921403 and 2006AA06A204). References
Fig. 7. Dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in turbulent atmosphere for two different sets of w0x and w0y.
Fig. 8. Dependence of the effective beam spot size Wzx of an elliptical anomalous hollow beam on propagation distance z in a turbulent atmosphere for different values of the wavelength k.
4. Conclusions In conclusion, we have derived some analytical formulae of the average irradiance for circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere based on the extended Huygens–Fresnel integral. The irradiance and spreading properties of circular and elliptical anomalous hollow beams have been investigated numerically. We have found that the evolution properties of the irradiance distribution of circular and elliptical anomalous hollow beams are similar to those of free space in the near field. However, in the far field, circular and elliptical anomalous hollow beams become circular Gaussian beams spot under the influence of the atmosphere turbulence. This phenomenon is quite different from the propagation properties of circular and elliptical anomalous hollow beams in free space. The irradiance and spreading properties are closely related with the parameters of the beam and the structure constant of the turbulent atmo-
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