- Email: [email protected]
Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere
ARTICLE IN PRESS
Optics and Lasers in Engineering 46 (2008) 1–5 www.elsevier.com/locate/optlaseng
Review
Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere M. Alavinejad, B. Ghafary, F.D. Kashani Photonics Laboratory, Physics Department, Iran University of Science and Technology, Tehran, Iran Received 19 May 2007; received in revised form 2 July 2007; accepted 16 July 2007 Available online 20 August 2007
Abstract Propagation of flat-topped beam with circular symmetry in a turbulent atmosphere is investigated and has led to the development of an analytical formula for the average intensity. Detailed analyses of flat-topped beams demonstrate that higher order flat-topped beams are less affected from turbulence in the media. Also an analytical formula for beam width of flat-topped beams propagating through turbulent atmosphere is derived which shows that higher order flat-topped beams are less broadened. We also compute Strehl ratio which evidently proves validation of the above discussion for any propagation distance. r 2007 Elsevier Ltd. All rights reserved. Keyword: Flat-topped beam; Propagation; Turbulence and order of flat-topped beam
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of flat-topped beam with circular symmetry at source and receiver plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere . . . . . . . . . . . . . . . . . . . . . Beam width and Strehl ratio for flat-topped beam propagated through turbulent media . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Recently, light beams with flat-topped profile have attracted more attention because of its wide application in material thermal processing by laser beam, inertial confinement fusion, etc. During past years, several theoretical models have been developed for describing flat-topped beams [1,2]. Among these models it can be referred to the super-Gaussian beam (SGB) model and flattened Gaussian beam (FGB) model, Corresponding author.
E-mail address: [email protected] (M. Alavinejad). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.07.003
1 2 4 4 5 5 5
proposed firstly by Gori in 1994. FGB can be expressed as a sum of finite Laguerre–Gaussian modes or Hermite– Gaussian modes [3,4]. More recently, Li proposed a new theoretical model to describe a flat-topped light beam of circular or non-circular symmetry by expressing its electric field as a finite sum of fundamental Gaussian beams [5–8]. On the other side, propagation of laser beam through random media is a topic that has been of considerable theoretical and practical interest for a long time, as is evident from the number of books and papers written on the subject [9,10]. In many practical applications such as the remote sensing, tracking, and long-distance optical communication, etc., the propagation properties of laser
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
2
beam through atmospheric turbulence are of an outstanding importance and atmospheric turbulence has a significant role in such applications. Much work has been carried out concerning the spreading of a laser beam in a turbulent atmosphere. Propagation and spreading of partially coherent beams in a turbulent atmosphere have been studied [11–14]. More recently, Eyyuboglu and Baykal investigated the properties of cos-Gaussian, cosh-Gaussian, Hermitesinusoidal-Gaussian and Hermite-cosine-Gaussian laser beams in a turbulent atmosphere [15–17]; Cai and He have studied the spreading properties of an elliptical Gaussian beam in a turbulent atmosphere [18]; and Cai investigated the propagation of various flat-topped beams in a turbulent atmosphere [19–20]. Eyyubog˘lu et al. investigated the flat-topped beams and their characteristics in turbulent media [21]. In this paper, we have focused on the average intensity of flat-toped beams propagating in turbulent media and have analyzed the effect of order on it. Also beam width of such beams is computed analytically and finally some comparisons are conducted, which reveal important facts about flat-topped beams properties propagating in turbulent and free space media. 2. Intensity of flat-topped beam with circular symmetry at source and receiver plans Fig. 1 illustrates the propagation geometry exploited in our discussion. Transverse source and receiver plans face the axis of propagations at z ¼ 0 and z ¼ L, respectively, thus ‘‘L’’ measures the propagation distance, r ¼ (x, y) and r ¼ (rx, ry) are the source and receiver plans coordinates, respectively. Field distribution of a typical flat-topped beam in the source plan can be presented as follows [5–7]: N X ð1Þn1 N nðx2 þ y2 Þ E N ðx; y; z ¼ 0Þ ¼ exp , N w20 n n¼1
distribution in receiver plan under strong fluctuation condition: ZZ ik expðikzÞ ðr rÞ2 Eðr; wÞ exp ik Eðr; z; wÞ ¼ 2pz 2z exp½cðr; r; zÞ d2 r,
ð2Þ
where E(r, w) and E ¼ (r, z, w) represent field distribution in source and receiver plan, respectively, and c denotes the phase function that depends on the properties of the medium. The average intensity at the output plan is given by hIðr; zÞi ¼ hEðr; z; wÞE n ðr; z; wÞi where hi denotes the ensemble average over the medium statistic. From Eq. (2), we obtain 2 ZZ ZZ k Iðr; zÞ ¼ d2 r d2 r0 Eðr; wÞ 2pz ðr rÞ2 ðr r0 Þ2 E n ðr0 ; wÞ exp ik 2z hexp½cðr; r; zÞ þ cn ðr; r0 ; zÞi.
ð3Þ
The last term in the integrand of Eq. (3) can be expressed as [11–13,20] hexp½cðr; r; zÞ þ cn ðr; r0 ; zÞi ¼ exp½0:5Dc ðr r0 Þ ¼ exp½Fðr r0 Þ2 ,
ð4Þ
0
where Dc ðr r Þ is the phase structure function in Ratove’s representation and F ¼ ð0:545C 2n k2 zÞ6=5 is the coherence length of a spherical wave propagating in the turbulent medium. Finally, C 2n is the structure constant. Quadratic approximation (see, e.g., Eqs. (14) and (15) in [20]) is used for Rytov’s phase structure function, this quadratic approximation has been shown to be reliable and has been widely investigated [11–13].
(1) N where w0 is the beam waist of the Gaussian beam, n denotes the binomial coefficient, and ‘‘N’’ is the order of the circular flat-topped beams. When N ¼ 1, Eq. (1) reduces to a Gaussian beam evidently. In the next step, using the paraxial form of the extended Huygence–Fresnel principle [10], we start calculating field
r
Beam
x
x y
y Turbulent atmosphere
(0,0,0)
(0,0, L) Fig. 1. Propagation geometry.
Fig. 2. Normalized average intensity of a circular flat-topped beam at several propagation distances in a turbulent atmosphere under condition: w0 ¼ 5 cm, N ¼ 8, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
To evaluate Eq. (3), it is convenient to introduce the new variable of integration, u¼
r þ r0 2
and
v ¼ r r0 .
(5)
Substituting Eq. (5) into Eq. (3) yields following form for intensity: 2 ZZ ZZ k v hIðr; zÞi ¼ d2 r d2 vE u þ ; w 2pz 2 v k E n u ; w exp i uv 2 z k exp i rv exp½Fv2 . ð6Þ z
3
In our case, this equation serves as the basic formula for studying propagation of flat-topped beams through atmospheric turbulence. We can express Eðu þ ðv=2Þ; wÞE n ðu ðv=2Þ; wÞ in Eq. (6) in the following form:
N X N v v X ð1Þnþm2 E u þ ; w En u ; w ¼ 2 2 N2 n¼1 m¼1
vy 2 n vx 2 exp 2 ux þ þ uy þ 2 2 w0
2 vy 2 m vx . exp 2 ux þ uy 2 2 w0
N n
!
N
!
m
ð7Þ
Fig. 3. Normalized intensity distortion for flat-topped beam in free space and turbulent media under condition that w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 : (a) N ¼ 1, (b) N ¼ 4, (c) N ¼ 8, and (d) N ¼ 12.
ARTICLE IN PRESS 4
M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
Substituting Eq. (7) into Eq. (6) and calculating the related integral we can obtain ! ! 2 X N X N N k ð1Þnþm2 N p2 Iðr; z; wÞ ¼ 2pz n¼1 m¼1 N2 n m a1 a2 k2 2 exp ðr þ r2y Þ , ð8Þ 4a2 z2 x
beam width: w2N ðzÞ 2
ð4z2 =k Þ
PN PN n¼1
¼ 2
ðp=N Þ
m¼1 ðð1Þ
PN PN n¼1
where n þ m b21 a2 ¼ þ F; 4a1 4w20
n m ik b1 ¼ þ ; z w20
nþm a1 ¼ . w20
This equation is the base of the results, presented in the next section. 3. Propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere In this section, we study the propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere, using the analytical formulas derived in the previous section (Eq (8)). In Fig. 2, normalized average intensity of a circular flat-topped beam is plotted at several propagation distances in a turbulent atmosphere. It can be seen that the evolution properties of the intensity of the circular flat-topped beam in a turbulent atmosphere are similar to those in free space [5,6]. Fig. 3 shows the behavior of the normalized intensity distribution of such a flat-topped beam through atmospheric turbulence and also in free space. Fig. 3 clearly demonstrates that, increasing the order of circular flattopped beam (N) will decrease the amount of difference between the peak intensity of turbulent propagated beam and free space propagated one. These results indicate that, in a well-defined sense, flattopped beams with bigger order of circular flat-topped are less affected than flat-topped beam with smaller order by atmospheric turbulence. Fig. 4 shows the behavior for the ratio of the maximum intensity in turbulence media to maximum intensity of free space vs. order of circular flat-topped which evidently provide validation of the above discussion for any propagation distance.
nþm2
m¼1 ðð1Þ
=N 2 Þ
nþm2
N
=a1 Þ
!
N
N
m !
n n
! ða2 =a1 Þ N
!
.
m ð10Þ
Obviously, it is enough to conduct two sums to achieve beam width in any arbitrary distance and order. Fig. 5 originates from implementation of Eq. (10). It shows the beam width wN ðzÞ as a function of the propagation distance ‘‘z’’, for selected value of the order of circular flat-topped beams. It is seen that, for a given propagation distance, higher order circular flat-topped beams are less broadened. Furthermore it can be seen that although in near field higher order circular flat-topped beam have higher beam width, in the far field beam width of various flat-topped beams, with any arbitrary N value, approach to a definite number. The reason is that the flat-topped beams with a high value of ‘‘N’’ are less affected by the turbulence in the atmosphere in comparison with the smaller order values of ‘‘N’’. The Strehl ratio is defined as [22] SR ¼
I max turbulence , I max free
(11)
where I max turbulence and I max free are peak intensity of flattopped beam propagating through turbulence media and free space, respectively. The large value of SR means the higher peak intensity of the flat-topped beam in the turbulent media.
4. Beam width and Strehl ratio for flat-topped beam propagated through turbulent media Beam width can be calculated by the first- and the second-order moment of the squared modulus for the amplitude distribution. Therefore, the beam width is defined as [22] R 2p R 1 2 r Iðr; z; wÞr dr dy 2 hwðzÞ i ¼ 0 R 2p0 R 1 . (9) 0 0 Iðr; zÞr dr dy Substituting Eq. (8) into Eq. (9) and calculating the related integral, one can acquire following relation for
Fig. 4. Relative maximum intensity in turbulent media with respect to maximum intensity in free space vs. order of circular flat-topped beam.
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
5
derived which clarifies that higher order circular flattopped beams are less broadened in turbulent media. We also numerically analyzed the Strehl ratio for various order flat-topped beams, which demonstrate that the increasing N causes decreasing effectiveness of turbulence. Acknowledgment The authors are thankful of Mr. D. Razzaghi for his intimate cooperation in completing this article. References
Fig. 5. Beam width vs. propagation distance under condition: w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
Fig. 6. Strehl ratio vs. propagation distance under condition: w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
The Strehl ratio SR vs. propagation distance ‘‘z’’ is plotted in Fig. 6. It follows that SR increases as N is increased, i.e., maximum intensity of flat-topped beam propagating through atmosphere decrease with decreasing ‘‘N’’. Therefore, the flat-topped beams with large value of ‘‘N’’ are less affected by turbulence in comparison with the lower order flat-topped beams. 5. Conclusion In this article, we have derived an analytical formula for average intensity of flat-topped beam propagating in turbulent media. It is found that higher order flat-topped beam is less affected in comparison with the lower order beam. Another analytical formula for beam width of flat-topped beam propagating through turbulent media is
[1] Bowers MS. Diffractive analysis of unstable optical resonator with super-Gaussian mirrors. Opt Lett 1992;19:1319. [2] Perrone MR, Piegari A. On the super-Gaussian unstable resonator for high-gain, short-pulse laser media. IEEE J Quantum Electron 1993;29:1423. [3] Gori F. Flattened Gaussian beams. Opt Commun 1994;107:335. [4] Bagini V, Borghi GRF, Pacileo AM, Santarsiero MJ. Propagation of axially symmetric flattened Gaussian beams. Opt Soc Am A 1996; 13:1385. [5] Li Y. Light beam with flat-topped profiles. Opt Lett 2002;27:1007. [6] Li Y. Flat-topped beam with non-circular cross-sections. J. Mod Opt 2003;50:1957. [7] Li Y. New expressions for flat-topped beams. Opt Commun 2002; 206:225–34. [8] Cai Y, Lin Q. Light beam with elliptical flat-topped profiles. J Opt A: Pure Appl Opt 2004;6:390–5. [9] Fante RL. Wave propagation in random media: a system approach. In: Wolf E, editor. Progress in optics, vol. 22. Amesterdam: Elsevier; 1985. p. 341–98. [10] Andrews LC, Phillips RL. Laser beam propagation through random media. Bellingham, WA: PIE Press; 1998. [11] Wang SCH, Aplonus M. Optical beam propagation for a partially coherent source in the turbulent atmosphere. J Opt Soc Am 1979; 69:1297–304. [12] Leader JC. Atmospheric propagation of partially coherent radiation. J Opt Soc Am 1978;68:175–85. [13] Shirai T, Dogariu A, Wolf E. Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence. J Opt Soc Am A 2003;20:1094–102. [14] Cai Y, He S. Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere. Appl Phys Lett 2006;89(4):041117. [15] Eyyuboglu HT, Baykal Y. Reciprocity of cos-Gaussian and coshGaussian laser beams in turbulent atmosphere. Opt Express 2004; 12:4659–74. [16] Eyyuboglu HT, Baykal Y. Average intensity and spreading of coshGaussian beams in the turbulent atmosphere. Appl Opt 2005;44: 976–83. [17] Baykal Y. Correlation and structure functions of Hermite-sinusoidalGaussian laser beams in a turbulent atmosphere. J Opt Soc Am A 2004;21:1290–9. [18] Cai Y, He S. Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere. Opt Lett 2006;31:568–70. [19] Cai Y. Propagation of various flat-topped beams in a turbulent atmosphere. J Opt A: Pure Appl Opt 2006;8:537–45. [20] Eyyubog˘lu HT, Caglar A, Baykal Y. Flat-topped beams and their characteristics in turbulent media. Opt Express 2006;14(10):4196. [21] Young CY, Gilchrest YV, Macon BR. Turbulence-induced beam spreading of higher-order mode. Opt Waves Opt Eng 2002;41:1097–103. [22] Siegman AE. How to (maybe) measure laser beam quality. OSA TOPS 1998;17:184.
Optics and Lasers in Engineering 46 (2008) 1–5 www.elsevier.com/locate/optlaseng
Review
Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere M. Alavinejad, B. Ghafary, F.D. Kashani Photonics Laboratory, Physics Department, Iran University of Science and Technology, Tehran, Iran Received 19 May 2007; received in revised form 2 July 2007; accepted 16 July 2007 Available online 20 August 2007
Abstract Propagation of flat-topped beam with circular symmetry in a turbulent atmosphere is investigated and has led to the development of an analytical formula for the average intensity. Detailed analyses of flat-topped beams demonstrate that higher order flat-topped beams are less affected from turbulence in the media. Also an analytical formula for beam width of flat-topped beams propagating through turbulent atmosphere is derived which shows that higher order flat-topped beams are less broadened. We also compute Strehl ratio which evidently proves validation of the above discussion for any propagation distance. r 2007 Elsevier Ltd. All rights reserved. Keyword: Flat-topped beam; Propagation; Turbulence and order of flat-topped beam
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of flat-topped beam with circular symmetry at source and receiver plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere . . . . . . . . . . . . . . . . . . . . . Beam width and Strehl ratio for flat-topped beam propagated through turbulent media . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Recently, light beams with flat-topped profile have attracted more attention because of its wide application in material thermal processing by laser beam, inertial confinement fusion, etc. During past years, several theoretical models have been developed for describing flat-topped beams [1,2]. Among these models it can be referred to the super-Gaussian beam (SGB) model and flattened Gaussian beam (FGB) model, Corresponding author.
E-mail address: [email protected] (M. Alavinejad). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.07.003
1 2 4 4 5 5 5
proposed firstly by Gori in 1994. FGB can be expressed as a sum of finite Laguerre–Gaussian modes or Hermite– Gaussian modes [3,4]. More recently, Li proposed a new theoretical model to describe a flat-topped light beam of circular or non-circular symmetry by expressing its electric field as a finite sum of fundamental Gaussian beams [5–8]. On the other side, propagation of laser beam through random media is a topic that has been of considerable theoretical and practical interest for a long time, as is evident from the number of books and papers written on the subject [9,10]. In many practical applications such as the remote sensing, tracking, and long-distance optical communication, etc., the propagation properties of laser
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
2
beam through atmospheric turbulence are of an outstanding importance and atmospheric turbulence has a significant role in such applications. Much work has been carried out concerning the spreading of a laser beam in a turbulent atmosphere. Propagation and spreading of partially coherent beams in a turbulent atmosphere have been studied [11–14]. More recently, Eyyuboglu and Baykal investigated the properties of cos-Gaussian, cosh-Gaussian, Hermitesinusoidal-Gaussian and Hermite-cosine-Gaussian laser beams in a turbulent atmosphere [15–17]; Cai and He have studied the spreading properties of an elliptical Gaussian beam in a turbulent atmosphere [18]; and Cai investigated the propagation of various flat-topped beams in a turbulent atmosphere [19–20]. Eyyubog˘lu et al. investigated the flat-topped beams and their characteristics in turbulent media [21]. In this paper, we have focused on the average intensity of flat-toped beams propagating in turbulent media and have analyzed the effect of order on it. Also beam width of such beams is computed analytically and finally some comparisons are conducted, which reveal important facts about flat-topped beams properties propagating in turbulent and free space media. 2. Intensity of flat-topped beam with circular symmetry at source and receiver plans Fig. 1 illustrates the propagation geometry exploited in our discussion. Transverse source and receiver plans face the axis of propagations at z ¼ 0 and z ¼ L, respectively, thus ‘‘L’’ measures the propagation distance, r ¼ (x, y) and r ¼ (rx, ry) are the source and receiver plans coordinates, respectively. Field distribution of a typical flat-topped beam in the source plan can be presented as follows [5–7]: N X ð1Þn1 N nðx2 þ y2 Þ E N ðx; y; z ¼ 0Þ ¼ exp , N w20 n n¼1
distribution in receiver plan under strong fluctuation condition: ZZ ik expðikzÞ ðr rÞ2 Eðr; wÞ exp ik Eðr; z; wÞ ¼ 2pz 2z exp½cðr; r; zÞ d2 r,
ð2Þ
where E(r, w) and E ¼ (r, z, w) represent field distribution in source and receiver plan, respectively, and c denotes the phase function that depends on the properties of the medium. The average intensity at the output plan is given by hIðr; zÞi ¼ hEðr; z; wÞE n ðr; z; wÞi where hi denotes the ensemble average over the medium statistic. From Eq. (2), we obtain 2 ZZ ZZ k Iðr; zÞ ¼ d2 r d2 r0 Eðr; wÞ 2pz ðr rÞ2 ðr r0 Þ2 E n ðr0 ; wÞ exp ik 2z hexp½cðr; r; zÞ þ cn ðr; r0 ; zÞi.
ð3Þ
The last term in the integrand of Eq. (3) can be expressed as [11–13,20] hexp½cðr; r; zÞ þ cn ðr; r0 ; zÞi ¼ exp½0:5Dc ðr r0 Þ ¼ exp½Fðr r0 Þ2 ,
ð4Þ
0
where Dc ðr r Þ is the phase structure function in Ratove’s representation and F ¼ ð0:545C 2n k2 zÞ6=5 is the coherence length of a spherical wave propagating in the turbulent medium. Finally, C 2n is the structure constant. Quadratic approximation (see, e.g., Eqs. (14) and (15) in [20]) is used for Rytov’s phase structure function, this quadratic approximation has been shown to be reliable and has been widely investigated [11–13].
(1) N where w0 is the beam waist of the Gaussian beam, n denotes the binomial coefficient, and ‘‘N’’ is the order of the circular flat-topped beams. When N ¼ 1, Eq. (1) reduces to a Gaussian beam evidently. In the next step, using the paraxial form of the extended Huygence–Fresnel principle [10], we start calculating field
r
Beam
x
x y
y Turbulent atmosphere
(0,0,0)
(0,0, L) Fig. 1. Propagation geometry.
Fig. 2. Normalized average intensity of a circular flat-topped beam at several propagation distances in a turbulent atmosphere under condition: w0 ¼ 5 cm, N ¼ 8, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
To evaluate Eq. (3), it is convenient to introduce the new variable of integration, u¼
r þ r0 2
and
v ¼ r r0 .
(5)
Substituting Eq. (5) into Eq. (3) yields following form for intensity: 2 ZZ ZZ k v hIðr; zÞi ¼ d2 r d2 vE u þ ; w 2pz 2 v k E n u ; w exp i uv 2 z k exp i rv exp½Fv2 . ð6Þ z
3
In our case, this equation serves as the basic formula for studying propagation of flat-topped beams through atmospheric turbulence. We can express Eðu þ ðv=2Þ; wÞE n ðu ðv=2Þ; wÞ in Eq. (6) in the following form:
N X N v v X ð1Þnþm2 E u þ ; w En u ; w ¼ 2 2 N2 n¼1 m¼1
vy 2 n vx 2 exp 2 ux þ þ uy þ 2 2 w0
2 vy 2 m vx . exp 2 ux þ uy 2 2 w0
N n
!
N
!
m
ð7Þ
Fig. 3. Normalized intensity distortion for flat-topped beam in free space and turbulent media under condition that w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 : (a) N ¼ 1, (b) N ¼ 4, (c) N ¼ 8, and (d) N ¼ 12.
ARTICLE IN PRESS 4
M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
Substituting Eq. (7) into Eq. (6) and calculating the related integral we can obtain ! ! 2 X N X N N k ð1Þnþm2 N p2 Iðr; z; wÞ ¼ 2pz n¼1 m¼1 N2 n m a1 a2 k2 2 exp ðr þ r2y Þ , ð8Þ 4a2 z2 x
beam width: w2N ðzÞ 2
ð4z2 =k Þ
PN PN n¼1
¼ 2
ðp=N Þ
m¼1 ðð1Þ
PN PN n¼1
where n þ m b21 a2 ¼ þ F; 4a1 4w20
n m ik b1 ¼ þ ; z w20
nþm a1 ¼ . w20
This equation is the base of the results, presented in the next section. 3. Propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere In this section, we study the propagation properties of a flat-topped beam with circular symmetry in a turbulent atmosphere, using the analytical formulas derived in the previous section (Eq (8)). In Fig. 2, normalized average intensity of a circular flat-topped beam is plotted at several propagation distances in a turbulent atmosphere. It can be seen that the evolution properties of the intensity of the circular flat-topped beam in a turbulent atmosphere are similar to those in free space [5,6]. Fig. 3 shows the behavior of the normalized intensity distribution of such a flat-topped beam through atmospheric turbulence and also in free space. Fig. 3 clearly demonstrates that, increasing the order of circular flattopped beam (N) will decrease the amount of difference between the peak intensity of turbulent propagated beam and free space propagated one. These results indicate that, in a well-defined sense, flattopped beams with bigger order of circular flat-topped are less affected than flat-topped beam with smaller order by atmospheric turbulence. Fig. 4 shows the behavior for the ratio of the maximum intensity in turbulence media to maximum intensity of free space vs. order of circular flat-topped which evidently provide validation of the above discussion for any propagation distance.
nþm2
m¼1 ðð1Þ
=N 2 Þ
nþm2
N
=a1 Þ
!
N
N
m !
n n
! ða2 =a1 Þ N
!
.
m ð10Þ
Obviously, it is enough to conduct two sums to achieve beam width in any arbitrary distance and order. Fig. 5 originates from implementation of Eq. (10). It shows the beam width wN ðzÞ as a function of the propagation distance ‘‘z’’, for selected value of the order of circular flat-topped beams. It is seen that, for a given propagation distance, higher order circular flat-topped beams are less broadened. Furthermore it can be seen that although in near field higher order circular flat-topped beam have higher beam width, in the far field beam width of various flat-topped beams, with any arbitrary N value, approach to a definite number. The reason is that the flat-topped beams with a high value of ‘‘N’’ are less affected by the turbulence in the atmosphere in comparison with the smaller order values of ‘‘N’’. The Strehl ratio is defined as [22] SR ¼
I max turbulence , I max free
(11)
where I max turbulence and I max free are peak intensity of flattopped beam propagating through turbulence media and free space, respectively. The large value of SR means the higher peak intensity of the flat-topped beam in the turbulent media.
4. Beam width and Strehl ratio for flat-topped beam propagated through turbulent media Beam width can be calculated by the first- and the second-order moment of the squared modulus for the amplitude distribution. Therefore, the beam width is defined as [22] R 2p R 1 2 r Iðr; z; wÞr dr dy 2 hwðzÞ i ¼ 0 R 2p0 R 1 . (9) 0 0 Iðr; zÞr dr dy Substituting Eq. (8) into Eq. (9) and calculating the related integral, one can acquire following relation for
Fig. 4. Relative maximum intensity in turbulent media with respect to maximum intensity in free space vs. order of circular flat-topped beam.
ARTICLE IN PRESS M. Alavinejad et al. / Optics and Lasers in Engineering 46 (2008) 1–5
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derived which clarifies that higher order circular flattopped beams are less broadened in turbulent media. We also numerically analyzed the Strehl ratio for various order flat-topped beams, which demonstrate that the increasing N causes decreasing effectiveness of turbulence. Acknowledgment The authors are thankful of Mr. D. Razzaghi for his intimate cooperation in completing this article. References
Fig. 5. Beam width vs. propagation distance under condition: w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
Fig. 6. Strehl ratio vs. propagation distance under condition: w0 ¼ 5 cm, k ¼ 107 m1, l0 ¼ 0.01 m, and C 2n ¼ 1014 .
The Strehl ratio SR vs. propagation distance ‘‘z’’ is plotted in Fig. 6. It follows that SR increases as N is increased, i.e., maximum intensity of flat-topped beam propagating through atmosphere decrease with decreasing ‘‘N’’. Therefore, the flat-topped beams with large value of ‘‘N’’ are less affected by turbulence in comparison with the lower order flat-topped beams. 5. Conclusion In this article, we have derived an analytical formula for average intensity of flat-topped beam propagating in turbulent media. It is found that higher order flat-topped beam is less affected in comparison with the lower order beam. Another analytical formula for beam width of flat-topped beam propagating through turbulent media is
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