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Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere
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Optics Communications 281 (2008) 2337–2341 www.elsevier.com/locate/optcom
Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere Haiyan Wang *, Xiangyin Li Department of Physics, Nanjing University of Science and Technology, Nanjing 210094, PR China Received 2 July 2007; received in revised form 15 November 2007; accepted 23 November 2007
Abstract Based on the extended Huygens–Fresnel principle, the spectrum of twist anisotropic Gaussian Schell-model (TAGSM) beams propagating through turbulent atmosphere is derived analytically by using the partially coherent complex curvature tensor. The relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increasing of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position rc. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Twist anisotropic Gaussian Schell-model beams; Spectral shift; Partially coherent complex curvature tensor; Turbulent atmosphere
1. Introduction In 1986 [1], Wolf pointed out that the spectrum of light which is emitted from a spatially partially coherent source with a wide spectral bandwidth undergoes spectral shift during propagation. This phenomenon is referred to as correlation-induced spectral changes, which take place even if the light propagation through free space [2]. Later on, it was found that the spectrum of the partially coherent beam in the diffracted field will be changed as well [3,4]. This phenomenon is referred to as diffraction-induced spectral changes. The physical mechanism of the spectral changes is different from Doppler effects. On the other hand, the atmospheric turbulence can significantly alert the prosperities of light propagating through it [5–7]. In fact, turbulent atmosphere is a diffracted field. So partially coherent beams propagating through turbulent atmosphere have spectral shift, caused by the correlation of the source and the diffraction of atmospheric turbulence. In the domain of partially coherent *
Corresponding author. Tel.: +86 13182963912. E-mail address: [email protected] (H. Wang).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.069
beams, the twist anisotropic Gaussian Schell-model (TAGSM) beams represents a general type of partially coherent beams. The theory commonly used to study the propagation and transformation of TAGSM beams is based on the Wigner distribution function [8]. However, this method is not appropriate for study spectral shift phenomena. In 2002, Lin et al. introduced a new method called the tensor method [9–12] for treating the propagation and transformation of TAGSM beams, which is more convenient. The purpose of the present paper is to study the spectral changes of TAGSM beams passing through turbulent atmosphere by using tensor method. The influences of the turbulence, beam’s initial parameter and the radial coordinate are studied analytically and numerically. 2. Spectrum of partially coherent TAGSM beams propagating through turbulent atmosphere in terms of the tensor method In this section, we will outline briefly the tensor method for partially coherent TAGSM beams. The cross-spectral density of the TAGSM beams located on the input plane z = 0 takes the following form [2]
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H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
ix T 1 ~r M s ~r ð1Þ 2c where M 1 s is a 4 4 partially coherent complex curvature tensor given by W ð~r; 0; xÞ ¼ S 0 ðxÞ exp
" M 1 s
¼
ic 2 1 ic 2 1 R1 2x ðrI Þ x ðrg Þ ic 2 1 ðrg Þ þ lJ T x
ic
1
ðr2g Þ
þ lJ ic 2 1 ic 2 1 1 R 2x ðrI Þ x ðrg Þ x
#
ð2Þ
r2I
where is a transverse spot width matrix, l is a real-valued constant named the twist factor, r2g is a transverse coherence width matrix, R1 is a wave front curvature matrix. r2I , r2g , R1 are all 2 2 matrices with transposition symmetry, given by " # " # 1=r2g11 1=r2g12 1=r2I11 1=r2I12 2 1 2 1 ðrI Þ ¼ ; ðrg Þ ¼ ; 1=r2I12 1=r2I22 1=r2g12 1=r2g22 1=R11 1=R12 : R1 ¼ 1=R12 1=R22 J is a transposition anti-symmetry matrix given by 0 1 . Based on the extended Huygens–Fresnel J¼ 1 0 integral, we can obtain the following propagation formula for the cross-spectral density of a partially coherent beam through a turbulent atmosphere [11] k2 W 0 ð~ q; z; xÞ ¼ 2 2 4p z
Z
þ1
Z
þ1
Z
þ1
Z
hexp½Wðr1 ; q1 ; z; xÞ þ W ðr2 ; q2 ; z; xÞidr1 dr2
ð3Þ
hidenotes averaging over the ensemble of turbulent media, and can be expressed as hexp½Wðr1 ; q1 ; z; xÞ þ W ðr2 ; q2 ; z; xÞi 1 2 ¼ exp½0:5DW ðr1 r2 Þ ¼ exp 2 ðr1 r2 Þ q0
1
1
þ1
1
ð6Þ
1 e 1 þ B e 1 is the partially coherent where M 1 o ¼ ½ðM s þ P Þ complex curvature tensor in the output plane and eI is a 4 4 unit matrix. Eq. (6) provides a convenient way for analyzing the spectral shift of TAGSM beam in a turbulent atmosphere. The spectrum of the TAGSM beams propagating through turbulent atmosphere can be obtained by setting q1 = q2 in Eq. (6), i.e.
Sðq1 ; z; xÞ ¼ W 0 ðq1 ¼ q2 ; z; xÞ e 1 þ B e Pe Þ ¼ S 0 ðxÞ½detðeI þ BM s ik T 1 ~ Mo q ~ exp q 2
1=2
ð7Þ
Eq. (7) shows that S(q1,z;x) depends on the initial spectrum S0(x), position parameters of observation point (q1, z), the atmospheric turbulence Pe and the beams initial parameter M 1 which including wave front curvature, s transverse coherence width, transverse spot width, and twist parameter. We give numerical results in next section. 3. Numerical examples and analysis
k2
1
1=2
ð4Þ
where Dw(r1 r2) is the phase structure function in Rytov’s 3=5 representation and q20 ¼ ð0:545C 2n k 2 zÞ is the coherence length of a spherical wave propagating in the turbulent atmosphere. C 2n is the structure constant of the refraction index, which decribes how strong the turbulence strength. In the derivation of the Eq. (4), we have employed a quadratic approximation for Rytov’s phase structure function. After some rearrangement, we can express Eq. (3) in the following tensor form [11] e 1=2 4p2 ½detð BÞ Z þ1 Z þ1 Z þ1 Z
e 1 þ B e Pe Þ q; z; xÞ ¼ S 0 ðxÞ½detðeI þ BM W 0 ð~ s ik T 1 ~ Mo q ~ exp q 2
þ1
W s ð~r; 0; xÞ 1 1 1 1 ik ik 2 2 exp ðr1 q1 Þ þ ðr2 q2 Þ 2z 2z
W 0 ð~ q; z; xÞ ¼
e ¼ zI 0 ~T ¼ ðqT1 ; qT2 Þ, B , Pe ¼ ikq2 2 where q 0 zI 0 I I , and I is a 2 2 unit matrix. I I Substituting Eq. (1) into Eq. (5), we obtain the following expression for the cross-spectral density of a TAGSM in output plane (after some vector integration and tensor operation)
W s ð~r; 0; xÞ
ik e 1~r 2~rT B e 1 q e 1 q ~þq ~T B ~Þ exp ð~rT B 2 ik T e exp ~r P ~r d~r ð5Þ 2
3.1. The normalized on-axis spectrum Numerical calculations were performed to illustrate how the spectrum of TAGSM beams in turbulent atmosphere. Let us assume the initial spectrum S0(x) is of the Lorentz type, i.e. S 0 ðxÞ ¼
S 0 d2 2
ðx x0 Þ þ d2
ð8Þ
where S0 is a constant, x0 is the central frequency of the initial spectrum, and d is the half-width at half-maximum of the initial spectrum. In the following, the parameters used in the numerical calculation are x0 = 3.2 1015 rad/ s, d = 0.6 1015 rad/s, S0 = 1. Substituting Eq. (8) into Eq. (7), we can obtain the spectrum of the TAGSM beams at any propagation distance in turbulent atmosphere. Fig. 1 shows the on-axis normalized spectrum S(x) at several propagation distances. The parameters used in the calculation are l = 0,
ðr2I Þ1 ¼ 2
1 0:1
0:1 0:5
H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
ðmmÞ2 , ðr2g Þ1 ¼
ðmmÞ , 0:0001 0:0001 1 R1 ¼ ðmmÞ ; 0:0001 0:0001
10 1
1 3:33
2339
C 2n ¼ 1013 m2=3 :
From Fig. 1, we can find that the shape of normalized on-axis spectrum of the TAGSM beams is similar to the original one after propagation through a distance in turbulent atmosphere, but its peak position is blue-shifted. In the following section, we will discuss the quantitative influence of the structure constant, beam’s initial parameter and the radial coordinate on the spectral shift of the TAGSM beams. 3.2. Relative frequency shift The spectral shift of the partially coherent TAGSM beams can be denoted by the relative central frequency shift. Assume xm is the central frequency of the spectrum after propagation, then the relative central frequency shift can be expressed by Dx=x0 ¼ ðxm x0 Þ=x0 The on-axis relative central frequency shift of the TAGSM beams propagating in free space and in turbulent atmosphere versus propagation distance z is depicted in Fig. 2. The parameters used in the calculations are: l = 0, 1 0:1 10 1 2 2 1 2 1 ðrI Þ ¼ ðmmÞ , ðrg Þ ¼ 0:1 0:5 1 3:33 0:0001 0:0001 2 1 1 ðmmÞ , R ¼ ðmmÞ . From 0:0001 0:0001 Fig. 2a, we can find that the on-axis relative frequency shift increase with the increasing of the propagation distance z in the near field both in free space and in turbulent atmosphere. In the far field, however, the relative frequency shift approaches to a constant in free space, and decreases along
Fig. 2. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different C 2n : C 2n ¼ 0 (in free space - - - - -); C 2n ¼ 1013 m2=3 (in turbulent atmosphere - - -). (a) In near field; (b) in far field.
Fig. 1. The normalized on-axis spectrum S(x) on the propagation distance in turbulent atmosphere: z = 0 (—–); z = 5 m (- - -); z = 50 m ( -).
with the increasing of propagation distance z in turbulent atmosphere as Fig. 2b showed. From Fig. 2, we know that in free space, in the near field the on-axis frequency shift is always blue shift, and in the far field the on-axis frequency shift approaches to a constant which means that there is not frequency shift. However, in turbulent atmosphere with the increasing of the propagation distance z, the on-axis spectral shift increases firstly and then decreases. In turbulent atmosphere, the on-axis spectral shift changes form blue shift in the near field to red shift in the far field. For a certain value of the propagation distance in turbulent atmosphere, the spectral shift is zero. The reason of the difference is that when TAGSM beams propagating through turbulent atmosphere, turbulent atmosphere have important effect on it. As before extended, turbulent atmosphere is a diffracted field, so partially coherent TAGSM beams propagating through turbulent atmosphere have spectral shift,
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H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
caused by both the correlation of the source and the diffraction of atmospheric turbulence. However, when TAGSM beams propagating in free space, there is only correlation-induced spectral changes. The on-axis relative central frequency shift of the TAGSM beams with different twist factors versus the propagation axis z is depicted in Fig. 3. Theparameters used in 1 0:1 2 1 2 1 the calculations are ðrI Þ ¼ ðmmÞ , ðr2g Þ ¼ 0:1 0:5 10 1 2 ðmmÞ 1 3:33 0:0001 0:0001 1 R ¼ ðmmÞ1 ; C 2n ¼ 1013 m2=3 : 0:0001 0:0001 From Fig. 3, we also find that the relative frequency shift is closely related with the twist factor, with the increasing of the twist factor, the relative frequency shift decreases. Fig. 4 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different coherence width matrix elements r2g11 . The param1 eters used in the calculation are l = 0, ðr2I Þ ¼ 1 0:1 2 ðmmÞ , 0:1 0:5 2
Fig. 4. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different coherence width matrix element r2g11 : r2g11 ¼ 0:1 ðmmÞ2 ( – – – -); r2g11 ¼ 1 ðmmÞ2 (- - - -); r2g11 ¼ 1:5 ðmmÞ2 (– – – -).
2
r2g12 ¼ 1 ðmmÞ ; r2g22 ¼ 0:3 ðmmÞ ; 0:0001 0:0001 1 ðmmÞ1 ; C 2n ¼ 1013 m2=3 : R ¼ 0:0001 0:0001 From Fig. 4, we can find that the relative frequency shift is closely related with the coherence width matrix element r2g11 . With the increase of r2g11 , the relative frequency shift decreases. That means for TAGSM beams with higher degree of coherence, the spectral shift is smaller. Fig. 5 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different transverse spot width matrix elements r2I11 . The parameters used in the calculation are l = 0, r2I12 ¼
Fig. 5. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different transverse spot width matrix element r2I11 : r2I11 ¼ 1 ðmmÞ2 (– –); r2I11 ¼ 0:1 ðmmÞ2 (- - - -); r2I11 ¼ 0:01 ðmmÞ2 (– – –).
Fig. 3. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different twist factors: l = 0 (– –); l = 0.0002 (mm)1 (- - - - - -); l = 0.0008 (mm)1 (– – – -).
10 1 2 2 1 10 ðmmÞ , r2I22 ¼ 2 ðmmÞ ,ðr2g Þ ¼ 1 3:33 0:0001 0:0001 ðmmÞ2 , R1 ¼ ðmmÞ1 , C 2n ¼ 0:0001 0:0001 1013 m2=3 . From Fig. 5, we can find that the relative frequency shift is closely related with the transverse spot width matrix element r2I11 . With the decreasing of r2I11 , the relative frequency shift increases. Fig. 6 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance for different wave front curvature matrix elements R11. The parameters used in the calculation are l = 0, 1 0:1 10 1 1 2 2 2 1 ðmmÞ , ðrg Þ ¼ ðrI Þ ¼ 0:1 0:5 1 3:33
H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
2341
1 0:1 ðmmÞ2 , ðr2g Þ1 ¼ l = 0.0002 (mm)1, ðr2I Þ1 ¼ 0:1 0:5 10 1 0:0001 0:0001 2 1 1 ðmmÞ , R ¼ ðmmÞ . 1 3:33 0:0001 0:0001 Fig. 7 indicates that the for the TAGSM beams propagating through turbulent atmosphere and free space there exists a rapid transition of the spectrum at the critical position rc = 5100 mm (in turbulent atmosphere) and rc = 4200 mm (in free space), and the transition of the relative spectral shift is 0.6617 (in turbulent atmosphere) and 0.4969 (in free space). From Fig. 7, we also know that the turbulence strength is strong, the critical position and the transition of the relative spectral shift are larger. For r < rc, the absolute value of relative spectral shift is small, but for r > rc, relative spectral shift is large due to rapid transition of the spectrum. Fig. 6. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different wave front curvature matrix element R11: R11 = 1000 mm (– –); R11 = 4000 mm (- - - -); R11 = 8000 mm (– – -).
Fig. 7. Relative spectral shift of TAGSM beams propagating through turbulent atmosphere and free space versus radial coordinate r. C 2n ¼ 1013 m2=3 (– –); C 2n ¼ 0 (- - - -).
ðmmÞ2 ,
R12 = 10,000 mm, R22 = 10,000 mm,
C 2n ¼
1013 m2=3 . From Fig. 6, we can find that the relative frequency shift is closely related with the wave front curvature matrix element R11. In the near field, with the increase of R11, the relative frequency shift increases, but after a certain propagation distance z, with the increase of R11, the relative frequency shift decrease. Fig. 7 shows the relative central frequency shift of the TAGSM beams propagating through turbulent atmosphere and in free space versus radial coordinate r. The parameters used in the calculation are z = 5,000,000 mm,
4. Conclusions In this study, the expression for the spectrum of TAGSM beams propagation through turbulent atmosphere has been derived by using the partially coherent complex curvature tensor and used to study spectral changes of TAGSM. The correlation of the source and the diffraction of turbulent atmosphere cause these kind of spectral changes. It has been shown that, the normalized spectrum S(x) of TAGSM beams propagating through turbulent atmosphere is the same as the normalized source spectrum S0(x). On the other hand, the relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increase of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position rc. This method used in this paper can avoid numerical integral that is time consuming and commonly used in the previous study of spectral shift problems. The result obtained in this paper would be useful for studying the spectrum of partially coherent beams in propagating through turbulent atmosphere. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
E. Wolf, Phys. Rev. Lett. 56 (1986) 1370. Y. Cai, Q. Lin, Opt. Commun. 204 (2002) 17. J. Pu, H. Zhang, S. Nemoto, Opt. Commun. 162 (1999) 57. B. Lu¨, L. Pan, IEEE J. Quantum Electron. 38 (2002) 340. Ji. Xiaoling, Zhang. Entao, Lu. Baida, Opt. Commun. 259 (2006) 1. A. Dogariu, S. Amarande, Opt. Lett. 28 (2003) 10. T. Shirai, A. Dogariu, E. Wolf, J. Opt. Soc. Am. A 20 (2003) 1094. R. Simon, N. Mukunda, J. Opt. Soc. Am. A 17 (2000) 2440. Q. Lin, Y. Cai, Opt. Lett. 27 (2002) 216. Y. Cai, L. Hu, Opt. Lett. 31 (2006) 685. Y. Cai, H. Sailing, Appl. Phys. Lett. 89 (2006) 041117. Y. Cai, D. Ge, Opt. Commun. 271 (2007) 509.
Optics Communications 281 (2008) 2337–2341 www.elsevier.com/locate/optcom
Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere Haiyan Wang *, Xiangyin Li Department of Physics, Nanjing University of Science and Technology, Nanjing 210094, PR China Received 2 July 2007; received in revised form 15 November 2007; accepted 23 November 2007
Abstract Based on the extended Huygens–Fresnel principle, the spectrum of twist anisotropic Gaussian Schell-model (TAGSM) beams propagating through turbulent atmosphere is derived analytically by using the partially coherent complex curvature tensor. The relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increasing of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position rc. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Twist anisotropic Gaussian Schell-model beams; Spectral shift; Partially coherent complex curvature tensor; Turbulent atmosphere
1. Introduction In 1986 [1], Wolf pointed out that the spectrum of light which is emitted from a spatially partially coherent source with a wide spectral bandwidth undergoes spectral shift during propagation. This phenomenon is referred to as correlation-induced spectral changes, which take place even if the light propagation through free space [2]. Later on, it was found that the spectrum of the partially coherent beam in the diffracted field will be changed as well [3,4]. This phenomenon is referred to as diffraction-induced spectral changes. The physical mechanism of the spectral changes is different from Doppler effects. On the other hand, the atmospheric turbulence can significantly alert the prosperities of light propagating through it [5–7]. In fact, turbulent atmosphere is a diffracted field. So partially coherent beams propagating through turbulent atmosphere have spectral shift, caused by the correlation of the source and the diffraction of atmospheric turbulence. In the domain of partially coherent *
Corresponding author. Tel.: +86 13182963912. E-mail address: [email protected] (H. Wang).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.069
beams, the twist anisotropic Gaussian Schell-model (TAGSM) beams represents a general type of partially coherent beams. The theory commonly used to study the propagation and transformation of TAGSM beams is based on the Wigner distribution function [8]. However, this method is not appropriate for study spectral shift phenomena. In 2002, Lin et al. introduced a new method called the tensor method [9–12] for treating the propagation and transformation of TAGSM beams, which is more convenient. The purpose of the present paper is to study the spectral changes of TAGSM beams passing through turbulent atmosphere by using tensor method. The influences of the turbulence, beam’s initial parameter and the radial coordinate are studied analytically and numerically. 2. Spectrum of partially coherent TAGSM beams propagating through turbulent atmosphere in terms of the tensor method In this section, we will outline briefly the tensor method for partially coherent TAGSM beams. The cross-spectral density of the TAGSM beams located on the input plane z = 0 takes the following form [2]
2338
H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
ix T 1 ~r M s ~r ð1Þ 2c where M 1 s is a 4 4 partially coherent complex curvature tensor given by W ð~r; 0; xÞ ¼ S 0 ðxÞ exp
" M 1 s
¼
ic 2 1 ic 2 1 R1 2x ðrI Þ x ðrg Þ ic 2 1 ðrg Þ þ lJ T x
ic
1
ðr2g Þ
þ lJ ic 2 1 ic 2 1 1 R 2x ðrI Þ x ðrg Þ x
#
ð2Þ
r2I
where is a transverse spot width matrix, l is a real-valued constant named the twist factor, r2g is a transverse coherence width matrix, R1 is a wave front curvature matrix. r2I , r2g , R1 are all 2 2 matrices with transposition symmetry, given by " # " # 1=r2g11 1=r2g12 1=r2I11 1=r2I12 2 1 2 1 ðrI Þ ¼ ; ðrg Þ ¼ ; 1=r2I12 1=r2I22 1=r2g12 1=r2g22 1=R11 1=R12 : R1 ¼ 1=R12 1=R22 J is a transposition anti-symmetry matrix given by 0 1 . Based on the extended Huygens–Fresnel J¼ 1 0 integral, we can obtain the following propagation formula for the cross-spectral density of a partially coherent beam through a turbulent atmosphere [11] k2 W 0 ð~ q; z; xÞ ¼ 2 2 4p z
Z
þ1
Z
þ1
Z
þ1
Z
hexp½Wðr1 ; q1 ; z; xÞ þ W ðr2 ; q2 ; z; xÞidr1 dr2
ð3Þ
hidenotes averaging over the ensemble of turbulent media, and can be expressed as hexp½Wðr1 ; q1 ; z; xÞ þ W ðr2 ; q2 ; z; xÞi 1 2 ¼ exp½0:5DW ðr1 r2 Þ ¼ exp 2 ðr1 r2 Þ q0
1
1
þ1
1
ð6Þ
1 e 1 þ B e 1 is the partially coherent where M 1 o ¼ ½ðM s þ P Þ complex curvature tensor in the output plane and eI is a 4 4 unit matrix. Eq. (6) provides a convenient way for analyzing the spectral shift of TAGSM beam in a turbulent atmosphere. The spectrum of the TAGSM beams propagating through turbulent atmosphere can be obtained by setting q1 = q2 in Eq. (6), i.e.
Sðq1 ; z; xÞ ¼ W 0 ðq1 ¼ q2 ; z; xÞ e 1 þ B e Pe Þ ¼ S 0 ðxÞ½detðeI þ BM s ik T 1 ~ Mo q ~ exp q 2
1=2
ð7Þ
Eq. (7) shows that S(q1,z;x) depends on the initial spectrum S0(x), position parameters of observation point (q1, z), the atmospheric turbulence Pe and the beams initial parameter M 1 which including wave front curvature, s transverse coherence width, transverse spot width, and twist parameter. We give numerical results in next section. 3. Numerical examples and analysis
k2
1
1=2
ð4Þ
where Dw(r1 r2) is the phase structure function in Rytov’s 3=5 representation and q20 ¼ ð0:545C 2n k 2 zÞ is the coherence length of a spherical wave propagating in the turbulent atmosphere. C 2n is the structure constant of the refraction index, which decribes how strong the turbulence strength. In the derivation of the Eq. (4), we have employed a quadratic approximation for Rytov’s phase structure function. After some rearrangement, we can express Eq. (3) in the following tensor form [11] e 1=2 4p2 ½detð BÞ Z þ1 Z þ1 Z þ1 Z
e 1 þ B e Pe Þ q; z; xÞ ¼ S 0 ðxÞ½detðeI þ BM W 0 ð~ s ik T 1 ~ Mo q ~ exp q 2
þ1
W s ð~r; 0; xÞ 1 1 1 1 ik ik 2 2 exp ðr1 q1 Þ þ ðr2 q2 Þ 2z 2z
W 0 ð~ q; z; xÞ ¼
e ¼ zI 0 ~T ¼ ðqT1 ; qT2 Þ, B , Pe ¼ ikq2 2 where q 0 zI 0 I I , and I is a 2 2 unit matrix. I I Substituting Eq. (1) into Eq. (5), we obtain the following expression for the cross-spectral density of a TAGSM in output plane (after some vector integration and tensor operation)
W s ð~r; 0; xÞ
ik e 1~r 2~rT B e 1 q e 1 q ~þq ~T B ~Þ exp ð~rT B 2 ik T e exp ~r P ~r d~r ð5Þ 2
3.1. The normalized on-axis spectrum Numerical calculations were performed to illustrate how the spectrum of TAGSM beams in turbulent atmosphere. Let us assume the initial spectrum S0(x) is of the Lorentz type, i.e. S 0 ðxÞ ¼
S 0 d2 2
ðx x0 Þ þ d2
ð8Þ
where S0 is a constant, x0 is the central frequency of the initial spectrum, and d is the half-width at half-maximum of the initial spectrum. In the following, the parameters used in the numerical calculation are x0 = 3.2 1015 rad/ s, d = 0.6 1015 rad/s, S0 = 1. Substituting Eq. (8) into Eq. (7), we can obtain the spectrum of the TAGSM beams at any propagation distance in turbulent atmosphere. Fig. 1 shows the on-axis normalized spectrum S(x) at several propagation distances. The parameters used in the calculation are l = 0,
ðr2I Þ1 ¼ 2
1 0:1
0:1 0:5
H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
ðmmÞ2 , ðr2g Þ1 ¼
ðmmÞ , 0:0001 0:0001 1 R1 ¼ ðmmÞ ; 0:0001 0:0001
10 1
1 3:33
2339
C 2n ¼ 1013 m2=3 :
From Fig. 1, we can find that the shape of normalized on-axis spectrum of the TAGSM beams is similar to the original one after propagation through a distance in turbulent atmosphere, but its peak position is blue-shifted. In the following section, we will discuss the quantitative influence of the structure constant, beam’s initial parameter and the radial coordinate on the spectral shift of the TAGSM beams. 3.2. Relative frequency shift The spectral shift of the partially coherent TAGSM beams can be denoted by the relative central frequency shift. Assume xm is the central frequency of the spectrum after propagation, then the relative central frequency shift can be expressed by Dx=x0 ¼ ðxm x0 Þ=x0 The on-axis relative central frequency shift of the TAGSM beams propagating in free space and in turbulent atmosphere versus propagation distance z is depicted in Fig. 2. The parameters used in the calculations are: l = 0, 1 0:1 10 1 2 2 1 2 1 ðrI Þ ¼ ðmmÞ , ðrg Þ ¼ 0:1 0:5 1 3:33 0:0001 0:0001 2 1 1 ðmmÞ , R ¼ ðmmÞ . From 0:0001 0:0001 Fig. 2a, we can find that the on-axis relative frequency shift increase with the increasing of the propagation distance z in the near field both in free space and in turbulent atmosphere. In the far field, however, the relative frequency shift approaches to a constant in free space, and decreases along
Fig. 2. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different C 2n : C 2n ¼ 0 (in free space - - - - -); C 2n ¼ 1013 m2=3 (in turbulent atmosphere - - -). (a) In near field; (b) in far field.
Fig. 1. The normalized on-axis spectrum S(x) on the propagation distance in turbulent atmosphere: z = 0 (—–); z = 5 m (- - -); z = 50 m ( -).
with the increasing of propagation distance z in turbulent atmosphere as Fig. 2b showed. From Fig. 2, we know that in free space, in the near field the on-axis frequency shift is always blue shift, and in the far field the on-axis frequency shift approaches to a constant which means that there is not frequency shift. However, in turbulent atmosphere with the increasing of the propagation distance z, the on-axis spectral shift increases firstly and then decreases. In turbulent atmosphere, the on-axis spectral shift changes form blue shift in the near field to red shift in the far field. For a certain value of the propagation distance in turbulent atmosphere, the spectral shift is zero. The reason of the difference is that when TAGSM beams propagating through turbulent atmosphere, turbulent atmosphere have important effect on it. As before extended, turbulent atmosphere is a diffracted field, so partially coherent TAGSM beams propagating through turbulent atmosphere have spectral shift,
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H. Wang, X. Li / Optics Communications 281 (2008) 2337–2341
caused by both the correlation of the source and the diffraction of atmospheric turbulence. However, when TAGSM beams propagating in free space, there is only correlation-induced spectral changes. The on-axis relative central frequency shift of the TAGSM beams with different twist factors versus the propagation axis z is depicted in Fig. 3. Theparameters used in 1 0:1 2 1 2 1 the calculations are ðrI Þ ¼ ðmmÞ , ðr2g Þ ¼ 0:1 0:5 10 1 2 ðmmÞ 1 3:33 0:0001 0:0001 1 R ¼ ðmmÞ1 ; C 2n ¼ 1013 m2=3 : 0:0001 0:0001 From Fig. 3, we also find that the relative frequency shift is closely related with the twist factor, with the increasing of the twist factor, the relative frequency shift decreases. Fig. 4 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different coherence width matrix elements r2g11 . The param1 eters used in the calculation are l = 0, ðr2I Þ ¼ 1 0:1 2 ðmmÞ , 0:1 0:5 2
Fig. 4. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different coherence width matrix element r2g11 : r2g11 ¼ 0:1 ðmmÞ2 ( – – – -); r2g11 ¼ 1 ðmmÞ2 (- - - -); r2g11 ¼ 1:5 ðmmÞ2 (– – – -).
2
r2g12 ¼ 1 ðmmÞ ; r2g22 ¼ 0:3 ðmmÞ ; 0:0001 0:0001 1 ðmmÞ1 ; C 2n ¼ 1013 m2=3 : R ¼ 0:0001 0:0001 From Fig. 4, we can find that the relative frequency shift is closely related with the coherence width matrix element r2g11 . With the increase of r2g11 , the relative frequency shift decreases. That means for TAGSM beams with higher degree of coherence, the spectral shift is smaller. Fig. 5 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different transverse spot width matrix elements r2I11 . The parameters used in the calculation are l = 0, r2I12 ¼
Fig. 5. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different transverse spot width matrix element r2I11 : r2I11 ¼ 1 ðmmÞ2 (– –); r2I11 ¼ 0:1 ðmmÞ2 (- - - -); r2I11 ¼ 0:01 ðmmÞ2 (– – –).
Fig. 3. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different twist factors: l = 0 (– –); l = 0.0002 (mm)1 (- - - - - -); l = 0.0008 (mm)1 (– – – -).
10 1 2 2 1 10 ðmmÞ , r2I22 ¼ 2 ðmmÞ ,ðr2g Þ ¼ 1 3:33 0:0001 0:0001 ðmmÞ2 , R1 ¼ ðmmÞ1 , C 2n ¼ 0:0001 0:0001 1013 m2=3 . From Fig. 5, we can find that the relative frequency shift is closely related with the transverse spot width matrix element r2I11 . With the decreasing of r2I11 , the relative frequency shift increases. Fig. 6 shows the on-axis relative central frequency shift of the TAGSM beams versus propagation distance for different wave front curvature matrix elements R11. The parameters used in the calculation are l = 0, 1 0:1 10 1 1 2 2 2 1 ðmmÞ , ðrg Þ ¼ ðrI Þ ¼ 0:1 0:5 1 3:33
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1 0:1 ðmmÞ2 , ðr2g Þ1 ¼ l = 0.0002 (mm)1, ðr2I Þ1 ¼ 0:1 0:5 10 1 0:0001 0:0001 2 1 1 ðmmÞ , R ¼ ðmmÞ . 1 3:33 0:0001 0:0001 Fig. 7 indicates that the for the TAGSM beams propagating through turbulent atmosphere and free space there exists a rapid transition of the spectrum at the critical position rc = 5100 mm (in turbulent atmosphere) and rc = 4200 mm (in free space), and the transition of the relative spectral shift is 0.6617 (in turbulent atmosphere) and 0.4969 (in free space). From Fig. 7, we also know that the turbulence strength is strong, the critical position and the transition of the relative spectral shift are larger. For r < rc, the absolute value of relative spectral shift is small, but for r > rc, relative spectral shift is large due to rapid transition of the spectrum. Fig. 6. The on-axis relative central frequency shift of the TAGSM beams versus propagation distance z for different wave front curvature matrix element R11: R11 = 1000 mm (– –); R11 = 4000 mm (- - - -); R11 = 8000 mm (– – -).
Fig. 7. Relative spectral shift of TAGSM beams propagating through turbulent atmosphere and free space versus radial coordinate r. C 2n ¼ 1013 m2=3 (– –); C 2n ¼ 0 (- - - -).
ðmmÞ2 ,
R12 = 10,000 mm, R22 = 10,000 mm,
C 2n ¼
1013 m2=3 . From Fig. 6, we can find that the relative frequency shift is closely related with the wave front curvature matrix element R11. In the near field, with the increase of R11, the relative frequency shift increases, but after a certain propagation distance z, with the increase of R11, the relative frequency shift decrease. Fig. 7 shows the relative central frequency shift of the TAGSM beams propagating through turbulent atmosphere and in free space versus radial coordinate r. The parameters used in the calculation are z = 5,000,000 mm,
4. Conclusions In this study, the expression for the spectrum of TAGSM beams propagation through turbulent atmosphere has been derived by using the partially coherent complex curvature tensor and used to study spectral changes of TAGSM. The correlation of the source and the diffraction of turbulent atmosphere cause these kind of spectral changes. It has been shown that, the normalized spectrum S(x) of TAGSM beams propagating through turbulent atmosphere is the same as the normalized source spectrum S0(x). On the other hand, the relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increase of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position rc. This method used in this paper can avoid numerical integral that is time consuming and commonly used in the previous study of spectral shift problems. The result obtained in this paper would be useful for studying the spectrum of partially coherent beams in propagating through turbulent atmosphere. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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