Scintillation analysis of truncated Bessel beams via numerical turbulence propagation simulation Halil T. Eyyuboğlu,1,* David Voelz,2 and Xifeng Xiao2 1
Electronics and Communications Department, Çankaya University, Eskişehir Yolu 29. km, Yenimahalle, 06810 Ankara, Turkey 2
Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, New Mexico 88003, USA *Corresponding author: [email protected] Received 13 August 2013; accepted 9 October 2013; posted 17 October 2013 (Doc. ID 195720); published 14 November 2013
Scintillation aspects of truncated Bessel beams propagated through atmospheric turbulence are investigated using a numerical wave optics random phase screen simulation method. On-axis, aperture averaged scintillation and scintillation relative to a classical Gaussian beam of equal source power and scintillation per unit received power are evaluated. It is found that in almost all circumstances studied, the zeroth-order Bessel beam will deliver the lowest scintillation. Low aperture averaged scintillation levels are also observed for the fourth-order Bessel beam truncated by a narrower source window. When assessed relative to the scintillation of a Gaussian beam of equal source power, Bessel beams generally have less scintillation, particularly at small receiver aperture sizes and small beam orders. Upon including in this relative performance measure the criteria of per unit received power, this advantageous position of Bessel beams mostly disappears, but zeroth- and first-order Bessel beams continue to offer some advantage for relatively smaller aperture sizes, larger source powers, larger source plane dimensions, and intermediate propagation lengths. © 2013 Optical Society of America OCIS codes: (010.0010) Atmospheric and oceanic optics; (010.3310) Laser beam transmission; (140.3295) Laser beam characterization. http://dx.doi.org/10.1364/AO.52.008032
1. Introduction
Bessel beams are known to constitute a valid solution to the wave equation [1]. Theoretically, they possess a nondiffracting property; however, without any truncation, such beams also represent an infinite amount of energy and are therefore not practicable. The truncation can be achieved either by incorporating a Gaussian apodization function or some other types of apodization apertures [2] or even a phase filter [3]. Many studies have been carried out on the propagation of truncated Bessel beams. The irradiance distribution of the Bessel Gaussian beam as a function of the degree of apodization was investigated in [4]. 1559-128X/13/338032-08$15.00/0 © 2013 Optical Society of America 8032
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
The propagation characteristics of a Bessel Gaussian beam passing through an annual aperture was analyzed with an ABCD matrix treatment in [5,6]. By adopting a quadratic radial dependence in the argument of the Bessel function, the beam propagation factor and the far-field distribution of Bessel Gaussian beams were studied in [7]. In [8], the propagation characteristics of both Bessel and Bessel Gaussian beams passing through apertured and unapertured misaligned paraxial optical systems were examined. Misalignment was also considered in [9], for propagation of higher-order Bessel Gaussian beams. The above works concern free space propagation. Recently, the scope of this problem has been extended to propagation in the turbulent atmosphere as well. This way, a turbulence propagation analysis
of higher-order Bessel Gaussian beams was provided in [10]. A similar analysis was offered in [11] for the partially coherent case of Bessel Gaussian beams. With the help of a well-approximated expansion, the effects of the topological charge in the propagation of Bessel Gaussian beams were explored in [12]. The axial intensity distribution of a sourceplane apertured Bessel Gaussian beam was studied in [13]. For a partially coherent Bessel Gaussian beam propagating in turbulence, the beam quality factor M 2 was derived and its variation as a function of source and propagation parameters was illustrated in [14]. The modeling of beam propagation in turbulence via computational wave optics with a split-step random phase screen approach was proposed quite some time ago [15–17]. The rules for proper numerical implementation, application examples, evaluation of various beam statistics using this approach, and the comparison of outcomes with analytic results have been well documented in numerous publications [18–22]. The wave optics random phase screen method has been utilized in propagation and scintillation studies of a variety of beams. To this end, a propagation simulation approach for partial spatial coherent beams was implemented in [23]. Scintillation aspects of the same type of beam were examined in [24]. In [25], scintillations of a Gaussian beam whose spectral degree of coherence is given by the zeroth-order Bessel function were considered. Scintillation evaluations of vortex beams were made in [26,27]. In another study [28], scintillation properties of the Gaussian Schell model beam were investigated. In this paper, a Bessel beam is considered that is truncated at the boundaries of a square source plane. The scintillation aspects of such a beam are explored using the wave optics random phase screen approach. Although it is possible this analysis could be performed in an analytic manner, it is clear that such an endeavor would involve a number of mathematical difficulties and lead to lengthy derivations and equations as seen in [29]. From a simplicity point of view, the use of the numerical wave optics approach is therefore preferable. Furthermore, the analytic treatment of aperture averaged scintillation is quite involved and complex [30–32]. When the random-phase-screen approach is employed, however, both the calculation of scintillation index at a certain receiver plane location and the aperture averaged scintillation, i.e., the power scintillation over a certain area of the receiver plane, is reduced to the simple arithmetic operations of summations and divisions. The main theme of this study is the scintillations of truncated Bessel beams. The emphasis is placed on aperture-averaged scintillations that also take into account the power-capturing capability. It is worth stating that such a study has not been undertaken so far, and it is envisioned that the results of this study can be useful for future free space optical links.
2. Modeling Light Beam Propagation
Wave-optics modeling of light beam propagation in the turbulent atmosphere is based on the combined use of Fourier transforms and random phase screens. The Fourier transform provides an efficient approach for simulating the free space propagation, while the phase fluctuations created by turbulence are represented by splitting the total free space propagation path into intervals and inserting random phase screens in between. These processes are briefly described below. The propagation of a light beam transmitted into a free space environment is governed by the Huygens– Fresnel integral. Thus, after propagating a distance of L, such a beam will become ur r; L
−j expj2πL∕λ λL Z∞Z∞ jπ 2 × r − s d2 s; us s exp λL −∞ −∞
(1)
where ur r; L and us s are the receiver and source plane fields, r and s are the transverse coordinates on these planes, and λ is the wavelength of the source. Recognizing that Eq. (1) has the form of a twodimensional Fourier transform, we can write Eq. (1) in the following form [22]: ur r; L F−1 fFus sHfg
(2)
with F denoting the Fourier transform operation and F−1 being the inverse transform. Hf is the free space propagation transfer function written in terms of the spatial frequency variable f and is defined as 2 2 : Hf exp jπL − λjfj λ
(3)
Although it is possible to produce the receiver plane field using Eq. (2) in a single step of distance L, we prefer to split L into several intervals and include the effects of turbulence by inserting random phase screens that are associated with each interval. If a total of M intervals are used, then the field on the m 1th layer can be evaluated from [21] um1 rm1 ; m 1L∕M F−1 Ffum rm ; mL∕M expjϕrm gHf m ;
(4)
where the turbulence induced phase, ϕrm can be obtained from ϕrm F−1 Gf m Φ0.5 ϕ f m
(5)
and Gf m contains Gaussian distributed random spatial frequency values generated with zero mean and unity variance. The parameters of the phase power spectral density for the individual layers Φϕ f m are arranged so that the layered model representation 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8033
matches the continuous link model [17]. Note that the 0.5 power applied to the phase power spectral density is appropriate and consistent with [19,26,27] and arises because the random phase screen operates on the field as opposed to the intensity. In this study, we have employed the von-Karman spectrum, which has a power spectral density of Φϕ f 0.1421LC2n L11∕3 0
exp−1.1265l20 jfj2 ; λ2 L20 jfj2 111∕6
(6)
where l0 , L0 are the inner and outer scales of turbulence. To account for the growing beam size during propagation, a scaling factor is introduced for the sampling intervals of the transmitter and the receiver planes and also for the intermediate interval planes. This scaling is implemented with a multistep form of the propagation algorithm, where the same number of grid points is used on all transverse planes [21]. 3. Representation of Source Beam and Scintillation Expressions
The Bessel beam expression is us s J n aB s expjnφ;
(7)
where the vector s s; φ represents the source plane polar coordinates, J n is the nth order Bessel function of the first kind where n also refers to the order of the beam, and aB is the width parameter that adjusts the decay rate of the beam tails. For a square transverse source plane of side length S, the Bessel beam of Eq. (7) is truncated as −S∕2 ≤ sx ≤ S∕2; s
s2x
s2y ;
− S∕2 ≤ sy ≤ S∕2 φ tan−1 sy ∕sx :
(8)
If ut r; L is the field arriving at the receiver after passing through the series of phase screens and free space intervals, then the scintillation index at a specific location r of the receiver plane is br; L
hut r; Lut r; L2 i hI 2 r; Li −1 t − 1; (9) 2 hut r; Lut r; Li hI t r; Li2
where h i indicates an average and I t is the intensity defined as the multiplication of the receiver field with its complex conjugate. Equation (9) is an accepted measure of scintillation, if the receiver capture area is less than 0.5λL [32]. Otherwise, aperture averaged scintillation occurs, which can be expressed for a circular aperture radius of Ra as DhR R Ra 2π 0
0
rI t r; Ldrdθ
0
0
rI t r; Ldrdθ
bL DR R Ra 2π
8034
i2 E E2 − 1:
(10)
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
The performance of an optical atmospheric link is improved not only by reducing scintillation but also by increasing the power captured by the receiver. This is because capturing more power will increase the signal to noise ratio, leading to receiver performance improvement. Therefore, the amount of scintillation per unit received power represents a useful measure of performance. Such a measure is more meaningful for aperture averaged scintillation, since this type of scintillation applies to an area rather than a single point of the receiver plane. The proposed quantity can be obtained from bL of Eq. (10) in the following way: bP L DR
Ra 0
R 2π 0
bL rI t r; Ldrdθ
E:
(11)
For comparison purposes, it is useful to consider sources with equal power. It is also helpful to assess the scintillation performance of a specific beam relative to the classical Gaussian beam. By letting bB L and bPB L denote the Bessel beam equivalence of Eqs. (10) and (11), respectively, and letting bG L and bPG L be the corresponding parameters for the classical Gaussian beam, then the scintillation performance of the Bessel beam relative to a Gaussian beam can be defined with the following two quantities: bR L
bB L ; bG L
bPR L
bPB L : bPG L
(12)
4. Results and Discussions
To obtain results, the propagation path was sliced into 21 intervals with associated turbulence phase screens of 512 × 512 grid points. Two sizes of square-shaped source planes were employed: one with a side length of S 10 cm and the other S 40 cm. The width parameter was assigned values of aB 1, 2, 3, 4, 5 cm−1 for S 10 cm and aB 0.2, 0.4, 0.6, 0.8, 1 cm−1 for S 40 cm. Beam order was varied over the range n 0, 1, 2, 3, 4, 5. The plots were constructed by taking an average over 500 turbulence realizations. The aim in this work is to assess the scintillation characteristics of truncated Bessel beams for optical communication links. The above selection of side length of source planes and the width parameter settings are based on this requirement. It is also intended that the selected source plane dimensions and width parameters accommodate most of the power contained in the beam. It is known from the literature that the scintillation index associated with each propagation interval should have a value less than 0.1 and not exceed 10% of the scintillation in the entire propagation length [19]. The choice of 21 propagation intervals for our study parameters more than meets these criteria. There is a general consensus in the literature that 500 turbulence realizations will deliver reliable scintillation index results [20,22,26], although in one [27], 100 realizations
1
S = 10 cm L = 0 km
0.5
0.2
L = 1 km
0.15 0.1 0.05
0
-6
-4
-2
0
2
4
6
s (in cm) 0.01
L = 3 km
n=0 n=4
0 -15
-10
-5
0
5
10
15
r (in cm) -3
x 10
aB = 2 cm-14
L = 5 km
3 0.005
2 1
0
-30
-20
-10
0
10
20
30
r (in cm)
0
-40
-20
0
20
40
r (in cm)
Fig. 1. Intensity evolution of two truncated Bessel beams with S 10 cm, n 0, 4, aB 2 cm−1 at L 0, 1, 3, 5 km.
are reported to be satisfactory. In another case [28], the number 2000 is quoted. Here we have chosen 500 realizations as a compromise between reliability of results and run times. In our simulation runs, the wavelength, refractive index structure constant, and inner and outer scales are fixed at λ 1.55 μm, C2n 10−15 m−2∕3 , l0 → 0, L0 → ∞. These parameters represent a typical optical communications wavelength and Kolmogorov spectrum turbulence that is relatively “weak.” According to sampling criteria, it is impossible to lower the inner scale below twice the sample interval size as also pointed out in [20]. From our runs, we have found no visible difference between setting the inner scale to twice the sample interval size and zero. No such limitation exists for outer scale. To illustrate the typical beam evolution, we display in Figs. 1 and 2 profiles of the Bessel beam as a function of propagation length. These figures indicate (as do analyses of other cases not illustrated here) that higher beam orders create a widening hollowness at the source plane and this hollowness has more persistence along the propagation axis at smaller width values. Similarly, smaller width values also allow the
Bessel beam of n 0 to propagate for longer distances with less profile changes. Moreover, placing a Bessel beam of the same width on a larger source plane reveals more of the nondiffractive property of this beam during propagation. This effect was clearly observed for the pair of settings, S 10 cm, aB 1 cm−1 and S 40 cm, aB 1 cm−1 . Figures 3 and 4 are based on Eq. (9) and show the variation of on-axis scintillation as a function of propagation distance for the two source plane side length settings, the five width parameters, and for two values of the beam order, namely, n 0 and n 1. The variations in the curves, particularly in Fig. 4, are due to the limited number of turbulence realizations; however, the trends for the results are still apparent. Generally the scintillation of the zeroth beam order seems to be lower than the scintillation of the first beam order. Additionally, the lowest scintillation curve for n 0 is that of S 10 cm, aB 1 cm−1 in Fig. 3, while in Fig. 4, the lowest scintillation is exhibited by the curve of S 40 cm, aB 1 cm−1 . Note that with the change of the beam order to n 1, the respective positions of the curves become almost the opposite. The trend
Fig. 2. Intensity evolution of two truncated Bessel beams with S 40 cm, n 0, 4, aB 0.4 cm−1 at L 0, 1, 3, 5 km. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8035
S = 10 cm , aB = 4 cm-1
n=0
S = 40 cm , aB = 0.8 cm-1
0.4
S = 40 cm , aB = 1 cm-1
0.35
S = 40 cm , aB = 0.6 cm-1
0.3
-1 S = 40 cm , aB = 0.4 cm
Onaxis scintillation - b ( r , L )
0.5 0.45
S = 10 cm , aB = 2 cm-1
0.25
S = 10 cm , aB = 3 cm-1
0.2 0.15
S = 40 cm , aB = 0.2 cm-1
0.1 0.05
S = 10 cm , aB = 1 cm-1
0.5
1
1.5
2 2.5 3 3.5 4 Propagation distance ( L in km )
4.5
S = 10 cm , aB = 5 cm-1
5
5.5
Fig. 3. On-axis scintillation of various Bessel beams against propagation distance at n 0.
Fig. 5. Aperture averaged scintillation of various Bessel beams against aperture radius at S 10 cm, L 3 km.
against the width parameter is somewhat oscillatory. That is, if the previous rise in value of aB caused a drop in scintillation; then the next rise of aB mostly leads to a fall. Now we turn to aperture averaged scintillation. The data collected from the simulation runs demonstrate that beam orders of n 1, 2, 3 produce higher scintillations, which is consistent with the trends seen in the comparisons of Figs. 3 and 4. This was particularly true near the on-axis limit of the receiver aperture radius. A beam order of n 4 offered a different behavior, however. Based on Eq. (10), Figs. 5 and 6 display the variation of aperture averaged scintillation as a function of receiver aperture size for beams orders of n 0, 4 at the two source plane side lengths of S 10, 40 cm and at a propagation length of L 3 km. The joint examination of Figs. 5 and 6 reveals that the use of smaller truncation, i.e., S 10 cm, leads to less scintillation. At S 10 cm, both beam orders, that is n 0, 4, possess more or less the same amount of aperture averaged scintillation as exhibited by Fig. 5. This changes, however, when S 40 cm. There, as observed from Fig. 6, the aperture-averaged scintillation of beams with n 0 becomes substantially less than that of n 4. As pointed out in Section 3, one quantitative measure for the performance of an optical link is the ratio of the scintillation of a specific beam relative to the classical Gaussian beam on an equal source power basis. The other measure is the same ratio of scintillations but including normalization with respect to per unit captured power. These two quantities are
respectively the first and the second expressions defined in Eq. (12). To facilitate the computation of Eq. (12), we have designed the equal power beams listed in Table 1. As seen from this table, the source power of the beam is adjusted to the numeric value written on the first row, by changing aB , the width parameter in the case of Bessel beams, and by changing αs, the source size in the case of Gaussian beam. The results compiled by utilizing Eq. (12) and the beams of Table 1 are displayed in Figs. 7 and 8, at two different propagation lengths of L 3, 5 km. In these figures, the unity value for the respective measure is marked by dashed green lines. Curves below the green lines indicate regions where the Bessel beams are more advantageous than the corresponding Gaussian beams in terms of scintillation performance as defined in Eq. (12). Such regions are present throughout the subfigures of Fig 7, which means that if the received power criteria is disregarded, then with the selection of Bessel beam parameters as shown in Fig. 7, it is always possible to obtain better scintillation performance than the corresponding equal source power Gaussian beam. In Fig. 7, the advantageous regions generally incorporate lower-order Bessel beams and smaller aperture radius sizes. The broadest region of advantage in Fig. 7 is found at S 40 cm, Ps 10.8 mW, L 5 km (middle panel, bottom row), where all beams except n 5 can be manipulated to lie within this region. When scintillation per unit received power is taken into account, the advantageous
Fig. 4. On-axis scintillation of various Bessel beams against propagation distance at n 1.
Fig. 6. Aperture averaged scintillation of various Bessel beams against aperture radius at S 40 cm, L 3 km.
8036
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
Table 1.
List of Equal Power Beams
Equal Source Power Ps (mW)
0.3
10.8
21.1
S (cm)
10
40
40
n aB cm−1 αs (cm)
Bessel beams Gaussian beams
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 3.76 3.8 3.74 3.77 3.69 3.71 0.75 0.73 0.75 0.73 0.74 0.72 0.38 0.39 0.37 0.39 0.36 0.37 1.38 8.29 11.59
S = 40 cm
S = 40 cm S = 10 cm
10
P = 10.8 mW
2 s
10
n=5
2
10
n=3
0
10
bR ( L )
bR ( L )
bR ( L )
n=4
n=1
10
1
10
0
n=2
5 10 Aperture radius ( Ra in cm )
n=2
n=3
0
n=0
n=0
n=4
L = 3 km
n=4
n=2
1
s
2
n=5
n=3
L = 3 km
L = 3 km
10
10
s
P = 0.3 mW
n=5
P = 21.1 mW
n=0 n=1
n=1
15
5 10 Aperture radius ( Ra in cm )
15
5 10 Aperture radius ( Ra in cm )
15
S = 40 cm S = 10 cm
10
P = 10.8 mW s
n=5
10
1
10
L = 5 km
bR ( L )
bR ( L )
s
n=2
n=3
10
1
n=4
L = 5 km
P = 21.1 mW
n=3
0
10
10
n=1
n=2 0
n=2
-1
n=1
n=1
5 10 15 20 Aperture radius ( Ra in cm )
n=4
L = 5 km
1
n=3
10
n=0
0
n=5
s
n=4
10
S = 40 cm
n=5
bR ( L )
P = 0.3 mW
2
n=0
5 10 Aperture radius ( Ra in cm )
n=0
15
5 10 15 Aperture radius ( Ra in cm )
Fig. 7. Aperture averaged scintillation of Bessel beams relative to a Gaussian beam of equal source power.
S = 40 cm
n=5
S = 10 cm
10
10
P = 0.3 mW
3
10
bPR ( L )
bPR ( L )
2
n=3
n=0
10
5 10 Aperture radius ( Ra in cm )
n=3
5 10 n=0 Aperture radius ( Ra in cm )
S = 10 cm
S = 40 cm
P = 0.3 mW
P = 10.8 mW
n=2
0
15
5 10 n=0 Aperture radius ( Ra in cm )
10
4
15
S = 40 cm n=5 P = 21.1 mW s
s
n=4 n=5
L = 5 km
10
4
L = 5 km
n=4
n=5
bPR ( L )
bPR ( L ) 10 n=1
n=3
2
n=1
15
L = 5 km n=4
10
n=2
0
s
4
L = 3 km
n=1 n=2
n=1
10
10
3
10
bPR ( L )
bPR ( L )
10
4
n=3
n=5
4
s
10
L = 3 km
n=4
L = 3 km n=4
10
P = 21.1 mW
s
n=5
s
5
S = 40 cm
n=4
P = 10.8 mW
4
2
n=3
n=2
n=3
10
2
n=2
n=2 n=0
n=0 n=1
5 10 15 20 Aperture radius ( Ra in cm )
n=0
5 10 15 Aperture radius ( Ra in cm )
n=1
5 10 15 Aperture radius ( Ra in cm )
Fig. 8. Aperture averaged scintillation of Bessel beams per unit received power relative to a Gaussian beam of equal source power. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8037
regions of Fig. 7 are mostly removed and the situation becomes as depicted in Fig. 8. From Fig. 8, we see that there are two advantageous regions, namely at S 40 cm, Ps 10.8 mW, L 3 km and S 40 cm, Ps 21.1 mW, L 3 km, where the Bessel beams of n 0, 1 are included. 5. Conclusion
We have studied the scintillation properties of truncated Bessel beams with the help of a numerical wave optics simulation and random turbulence screens. Square-shaped truncation at the source plane with side lengths of 10 and 40 cm was applied. The width parameter of the Bessel beam was varied from 1 cm−1 to 5 cm−1 for the 10 cm source plane-side length and from 0.2 cm−1 to 1 cm−1 for the 40 cm source plane-side length. Beam order was varied in the range of 0 up to 5. Propagation lengths up to 5.5 km were covered. Against the variations of these parameters, the dependence of on-axis and aperture averaged scintillation was examined. It was found that the zeroth-order Bessel beam offered the lowest scintillation in almost all cases. In other words, the higher-order Bessel beams were generally found to have unfavorable scintillation performance. A possible explanation for this is that the higher orders tend toward a beam pattern that is more like an annulus with relatively low on-axis intensity. A normalized variance measure such as the scintillation index is much more sensitive to small amounts of intensity change when the average intensity value is small to begin with. At propagation lengths of 3 and 5 km, aperture averaged scintillation of Bessel beams were also evaluated relative to classical Gaussian beams under the conditions of equal source power. Bessel beams were found to have lower scintillation at small receiver aperture sizes and smaller beam orders. It seems reasonable to expect the reduced diffractive properties of the low-order Bessel beams to have the most impact with smaller receiver apertures. However, the specific mechanisms that result in reduced scintillation for the small apertures are not clear at this time. When the criterion of per-unit received power was included in this assessment, however, these desirable features of the Bessel beams were lost to a great extent, and only the zeroth- and first-order Bessel beams could offer some improvement over the Gaussian beams. Further investigation is needed in this case to clarify the specific interactions of the scintillation and the received power. We plan to bring more insight to this matter in future studies by evaluating the probability of error performance of optical links exploiting the above revealed advantages of truncated Bessel beams. It is foreseen that our findings can be useful for future practical optical links. References 1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 8038
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
2. R. M. Herman and T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000). 3. A. G. Sedukhin, “Marginal phase correction of truncated Bessel beams,” J. Opt. Soc. Am. A 17, 1059–1066 (2000). 4. Y. Li, V. Gurevich, M. Kirchever, J. Katz, and E. Marom, “Propagation of anisotropic Bessel–Gaussian beams: sidelobe control, mode selection, and field depth,” Appl. Opt. 40, 2709–2721 (2001). 5. J. Gu, D. Zhao, Z. Mei, H. Mao, and H. Xu, “Propagation characteristics of linearly polarized Bessel–Gaussian beams through an annular aperture paraxial ABCD optical system,” Optik 115, 529–532 (2004). 6. Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005). 7. X. Wang and B. Lü, “The beam propagation factor and far-field distribution of Bessel-modulated Gaussian beams,” Opt. Quantum Electron. 34, 1071–1077 (2002). 8. Y. Cai and X. Lü, “Propagation of Bessel and Bessel– Gaussian beams through an unapertured and aperture misaligned paraxial optical systems,” Opt. Commun. 274, 1–7 (2007). 9. C. Zhao, L. Wang, X. Lu, and H. Chen, “Propagation of high-order Bessel–Gaussian beam through a misaligned firstorder optical system,” Opt. Laser Technol. 39, 1199–1203 (2007). 10. H. T. Eyyuboğlu, “Propagation of higher order Bessel– Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007). 11. B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008). 12. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel–Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16, 21315–21320 (2008). 13. J. Cang and Y. Zhang, “Axial intensity distribution of truncated Bessel–Gauss beams in turbulent atmosphere,” Optik 121, 239–245 (2010). 14. K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel–Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29, 251–257 (2012). 15. C. Macaskill and T. E. Ewart, “Computer simulation of twodimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984). 16. J. M. Martin and S. M. Flatte, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). 17. M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995). 18. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995). 19. D. H. Nelson, D. L. Walters, E. P. MacKerrow, M. J. Schmitt, C. R. Quick, W. M. Porch, and R. R. Petrin, “Wave optics simulation of atmospheric turbulence and reflective speckle effects in CO2 lidar,” Appl. Opt. 39, 1857–1871 (2000). 20. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). 21. J. D. Schmidt, in Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010), Chaps. 6–9. 22. D. Voelz, in Computational Fourier Optics a MATLAB Tutorial (SPIE, 2011), Chaps. 4 and 5. 23. X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006). 24. X. M. Qian, W. Y. Zhu, A. T. Wang, C. Gu, and R. Z. Rao, “Numeric simulation for coherent and partially coherent
25. 26. 27. 28.
beam propagation through atmospheric turbulence,” Chin. Phys. Lett. 27, 044214 (2010). Y. Gu and G. Gbur, “Scintillation of pseudo-Bessel correlated beams in atmospheric turbulence,” J. Opt. Soc. Am. A 27, 2621–2629 (2010). W. Cheng, J. H. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009). X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19, 26444–26450 (2011). X. Qian, W. Zhu, and R. Rao, “Numerical investigation on propagation effects of pseudo-partially coherent Gaussian
29.
30. 31. 32.
Schell-model beams in atmospheric turbulence,” Opt. Express 17, 3782–3791 (2009). H. T. Eyyuboğlu, E. Sermtlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605–611 (2008). D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967). J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991). L. C. Andrews and R. L. Phillips, in Laser Beam Propagation through Random Media (SPIE, 2005), Chap. 10.
20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8039
Electronics and Communications Department, Çankaya University, Eskişehir Yolu 29. km, Yenimahalle, 06810 Ankara, Turkey 2
Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, New Mexico 88003, USA *Corresponding author: [email protected] Received 13 August 2013; accepted 9 October 2013; posted 17 October 2013 (Doc. ID 195720); published 14 November 2013
Scintillation aspects of truncated Bessel beams propagated through atmospheric turbulence are investigated using a numerical wave optics random phase screen simulation method. On-axis, aperture averaged scintillation and scintillation relative to a classical Gaussian beam of equal source power and scintillation per unit received power are evaluated. It is found that in almost all circumstances studied, the zeroth-order Bessel beam will deliver the lowest scintillation. Low aperture averaged scintillation levels are also observed for the fourth-order Bessel beam truncated by a narrower source window. When assessed relative to the scintillation of a Gaussian beam of equal source power, Bessel beams generally have less scintillation, particularly at small receiver aperture sizes and small beam orders. Upon including in this relative performance measure the criteria of per unit received power, this advantageous position of Bessel beams mostly disappears, but zeroth- and first-order Bessel beams continue to offer some advantage for relatively smaller aperture sizes, larger source powers, larger source plane dimensions, and intermediate propagation lengths. © 2013 Optical Society of America OCIS codes: (010.0010) Atmospheric and oceanic optics; (010.3310) Laser beam transmission; (140.3295) Laser beam characterization. http://dx.doi.org/10.1364/AO.52.008032
1. Introduction
Bessel beams are known to constitute a valid solution to the wave equation [1]. Theoretically, they possess a nondiffracting property; however, without any truncation, such beams also represent an infinite amount of energy and are therefore not practicable. The truncation can be achieved either by incorporating a Gaussian apodization function or some other types of apodization apertures [2] or even a phase filter [3]. Many studies have been carried out on the propagation of truncated Bessel beams. The irradiance distribution of the Bessel Gaussian beam as a function of the degree of apodization was investigated in [4]. 1559-128X/13/338032-08$15.00/0 © 2013 Optical Society of America 8032
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
The propagation characteristics of a Bessel Gaussian beam passing through an annual aperture was analyzed with an ABCD matrix treatment in [5,6]. By adopting a quadratic radial dependence in the argument of the Bessel function, the beam propagation factor and the far-field distribution of Bessel Gaussian beams were studied in [7]. In [8], the propagation characteristics of both Bessel and Bessel Gaussian beams passing through apertured and unapertured misaligned paraxial optical systems were examined. Misalignment was also considered in [9], for propagation of higher-order Bessel Gaussian beams. The above works concern free space propagation. Recently, the scope of this problem has been extended to propagation in the turbulent atmosphere as well. This way, a turbulence propagation analysis
of higher-order Bessel Gaussian beams was provided in [10]. A similar analysis was offered in [11] for the partially coherent case of Bessel Gaussian beams. With the help of a well-approximated expansion, the effects of the topological charge in the propagation of Bessel Gaussian beams were explored in [12]. The axial intensity distribution of a sourceplane apertured Bessel Gaussian beam was studied in [13]. For a partially coherent Bessel Gaussian beam propagating in turbulence, the beam quality factor M 2 was derived and its variation as a function of source and propagation parameters was illustrated in [14]. The modeling of beam propagation in turbulence via computational wave optics with a split-step random phase screen approach was proposed quite some time ago [15–17]. The rules for proper numerical implementation, application examples, evaluation of various beam statistics using this approach, and the comparison of outcomes with analytic results have been well documented in numerous publications [18–22]. The wave optics random phase screen method has been utilized in propagation and scintillation studies of a variety of beams. To this end, a propagation simulation approach for partial spatial coherent beams was implemented in [23]. Scintillation aspects of the same type of beam were examined in [24]. In [25], scintillations of a Gaussian beam whose spectral degree of coherence is given by the zeroth-order Bessel function were considered. Scintillation evaluations of vortex beams were made in [26,27]. In another study [28], scintillation properties of the Gaussian Schell model beam were investigated. In this paper, a Bessel beam is considered that is truncated at the boundaries of a square source plane. The scintillation aspects of such a beam are explored using the wave optics random phase screen approach. Although it is possible this analysis could be performed in an analytic manner, it is clear that such an endeavor would involve a number of mathematical difficulties and lead to lengthy derivations and equations as seen in [29]. From a simplicity point of view, the use of the numerical wave optics approach is therefore preferable. Furthermore, the analytic treatment of aperture averaged scintillation is quite involved and complex [30–32]. When the random-phase-screen approach is employed, however, both the calculation of scintillation index at a certain receiver plane location and the aperture averaged scintillation, i.e., the power scintillation over a certain area of the receiver plane, is reduced to the simple arithmetic operations of summations and divisions. The main theme of this study is the scintillations of truncated Bessel beams. The emphasis is placed on aperture-averaged scintillations that also take into account the power-capturing capability. It is worth stating that such a study has not been undertaken so far, and it is envisioned that the results of this study can be useful for future free space optical links.
2. Modeling Light Beam Propagation
Wave-optics modeling of light beam propagation in the turbulent atmosphere is based on the combined use of Fourier transforms and random phase screens. The Fourier transform provides an efficient approach for simulating the free space propagation, while the phase fluctuations created by turbulence are represented by splitting the total free space propagation path into intervals and inserting random phase screens in between. These processes are briefly described below. The propagation of a light beam transmitted into a free space environment is governed by the Huygens– Fresnel integral. Thus, after propagating a distance of L, such a beam will become ur r; L
−j expj2πL∕λ λL Z∞Z∞ jπ 2 × r − s d2 s; us s exp λL −∞ −∞
(1)
where ur r; L and us s are the receiver and source plane fields, r and s are the transverse coordinates on these planes, and λ is the wavelength of the source. Recognizing that Eq. (1) has the form of a twodimensional Fourier transform, we can write Eq. (1) in the following form [22]: ur r; L F−1 fFus sHfg
(2)
with F denoting the Fourier transform operation and F−1 being the inverse transform. Hf is the free space propagation transfer function written in terms of the spatial frequency variable f and is defined as 2 2 : Hf exp jπL − λjfj λ
(3)
Although it is possible to produce the receiver plane field using Eq. (2) in a single step of distance L, we prefer to split L into several intervals and include the effects of turbulence by inserting random phase screens that are associated with each interval. If a total of M intervals are used, then the field on the m 1th layer can be evaluated from [21] um1 rm1 ; m 1L∕M F−1 Ffum rm ; mL∕M expjϕrm gHf m ;
(4)
where the turbulence induced phase, ϕrm can be obtained from ϕrm F−1 Gf m Φ0.5 ϕ f m
(5)
and Gf m contains Gaussian distributed random spatial frequency values generated with zero mean and unity variance. The parameters of the phase power spectral density for the individual layers Φϕ f m are arranged so that the layered model representation 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8033
matches the continuous link model [17]. Note that the 0.5 power applied to the phase power spectral density is appropriate and consistent with [19,26,27] and arises because the random phase screen operates on the field as opposed to the intensity. In this study, we have employed the von-Karman spectrum, which has a power spectral density of Φϕ f 0.1421LC2n L11∕3 0
exp−1.1265l20 jfj2 ; λ2 L20 jfj2 111∕6
(6)
where l0 , L0 are the inner and outer scales of turbulence. To account for the growing beam size during propagation, a scaling factor is introduced for the sampling intervals of the transmitter and the receiver planes and also for the intermediate interval planes. This scaling is implemented with a multistep form of the propagation algorithm, where the same number of grid points is used on all transverse planes [21]. 3. Representation of Source Beam and Scintillation Expressions
The Bessel beam expression is us s J n aB s expjnφ;
(7)
where the vector s s; φ represents the source plane polar coordinates, J n is the nth order Bessel function of the first kind where n also refers to the order of the beam, and aB is the width parameter that adjusts the decay rate of the beam tails. For a square transverse source plane of side length S, the Bessel beam of Eq. (7) is truncated as −S∕2 ≤ sx ≤ S∕2; s
s2x
s2y ;
− S∕2 ≤ sy ≤ S∕2 φ tan−1 sy ∕sx :
(8)
If ut r; L is the field arriving at the receiver after passing through the series of phase screens and free space intervals, then the scintillation index at a specific location r of the receiver plane is br; L
hut r; Lut r; L2 i hI 2 r; Li −1 t − 1; (9) 2 hut r; Lut r; Li hI t r; Li2
where h i indicates an average and I t is the intensity defined as the multiplication of the receiver field with its complex conjugate. Equation (9) is an accepted measure of scintillation, if the receiver capture area is less than 0.5λL [32]. Otherwise, aperture averaged scintillation occurs, which can be expressed for a circular aperture radius of Ra as DhR R Ra 2π 0
0
rI t r; Ldrdθ
0
0
rI t r; Ldrdθ
bL DR R Ra 2π
8034
i2 E E2 − 1:
(10)
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
The performance of an optical atmospheric link is improved not only by reducing scintillation but also by increasing the power captured by the receiver. This is because capturing more power will increase the signal to noise ratio, leading to receiver performance improvement. Therefore, the amount of scintillation per unit received power represents a useful measure of performance. Such a measure is more meaningful for aperture averaged scintillation, since this type of scintillation applies to an area rather than a single point of the receiver plane. The proposed quantity can be obtained from bL of Eq. (10) in the following way: bP L DR
Ra 0
R 2π 0
bL rI t r; Ldrdθ
E:
(11)
For comparison purposes, it is useful to consider sources with equal power. It is also helpful to assess the scintillation performance of a specific beam relative to the classical Gaussian beam. By letting bB L and bPB L denote the Bessel beam equivalence of Eqs. (10) and (11), respectively, and letting bG L and bPG L be the corresponding parameters for the classical Gaussian beam, then the scintillation performance of the Bessel beam relative to a Gaussian beam can be defined with the following two quantities: bR L
bB L ; bG L
bPR L
bPB L : bPG L
(12)
4. Results and Discussions
To obtain results, the propagation path was sliced into 21 intervals with associated turbulence phase screens of 512 × 512 grid points. Two sizes of square-shaped source planes were employed: one with a side length of S 10 cm and the other S 40 cm. The width parameter was assigned values of aB 1, 2, 3, 4, 5 cm−1 for S 10 cm and aB 0.2, 0.4, 0.6, 0.8, 1 cm−1 for S 40 cm. Beam order was varied over the range n 0, 1, 2, 3, 4, 5. The plots were constructed by taking an average over 500 turbulence realizations. The aim in this work is to assess the scintillation characteristics of truncated Bessel beams for optical communication links. The above selection of side length of source planes and the width parameter settings are based on this requirement. It is also intended that the selected source plane dimensions and width parameters accommodate most of the power contained in the beam. It is known from the literature that the scintillation index associated with each propagation interval should have a value less than 0.1 and not exceed 10% of the scintillation in the entire propagation length [19]. The choice of 21 propagation intervals for our study parameters more than meets these criteria. There is a general consensus in the literature that 500 turbulence realizations will deliver reliable scintillation index results [20,22,26], although in one [27], 100 realizations
1
S = 10 cm L = 0 km
0.5
0.2
L = 1 km
0.15 0.1 0.05
0
-6
-4
-2
0
2
4
6
s (in cm) 0.01
L = 3 km
n=0 n=4
0 -15
-10
-5
0
5
10
15
r (in cm) -3
x 10
aB = 2 cm-14
L = 5 km
3 0.005
2 1
0
-30
-20
-10
0
10
20
30
r (in cm)
0
-40
-20
0
20
40
r (in cm)
Fig. 1. Intensity evolution of two truncated Bessel beams with S 10 cm, n 0, 4, aB 2 cm−1 at L 0, 1, 3, 5 km.
are reported to be satisfactory. In another case [28], the number 2000 is quoted. Here we have chosen 500 realizations as a compromise between reliability of results and run times. In our simulation runs, the wavelength, refractive index structure constant, and inner and outer scales are fixed at λ 1.55 μm, C2n 10−15 m−2∕3 , l0 → 0, L0 → ∞. These parameters represent a typical optical communications wavelength and Kolmogorov spectrum turbulence that is relatively “weak.” According to sampling criteria, it is impossible to lower the inner scale below twice the sample interval size as also pointed out in [20]. From our runs, we have found no visible difference between setting the inner scale to twice the sample interval size and zero. No such limitation exists for outer scale. To illustrate the typical beam evolution, we display in Figs. 1 and 2 profiles of the Bessel beam as a function of propagation length. These figures indicate (as do analyses of other cases not illustrated here) that higher beam orders create a widening hollowness at the source plane and this hollowness has more persistence along the propagation axis at smaller width values. Similarly, smaller width values also allow the
Bessel beam of n 0 to propagate for longer distances with less profile changes. Moreover, placing a Bessel beam of the same width on a larger source plane reveals more of the nondiffractive property of this beam during propagation. This effect was clearly observed for the pair of settings, S 10 cm, aB 1 cm−1 and S 40 cm, aB 1 cm−1 . Figures 3 and 4 are based on Eq. (9) and show the variation of on-axis scintillation as a function of propagation distance for the two source plane side length settings, the five width parameters, and for two values of the beam order, namely, n 0 and n 1. The variations in the curves, particularly in Fig. 4, are due to the limited number of turbulence realizations; however, the trends for the results are still apparent. Generally the scintillation of the zeroth beam order seems to be lower than the scintillation of the first beam order. Additionally, the lowest scintillation curve for n 0 is that of S 10 cm, aB 1 cm−1 in Fig. 3, while in Fig. 4, the lowest scintillation is exhibited by the curve of S 40 cm, aB 1 cm−1 . Note that with the change of the beam order to n 1, the respective positions of the curves become almost the opposite. The trend
Fig. 2. Intensity evolution of two truncated Bessel beams with S 40 cm, n 0, 4, aB 0.4 cm−1 at L 0, 1, 3, 5 km. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8035
S = 10 cm , aB = 4 cm-1
n=0
S = 40 cm , aB = 0.8 cm-1
0.4
S = 40 cm , aB = 1 cm-1
0.35
S = 40 cm , aB = 0.6 cm-1
0.3
-1 S = 40 cm , aB = 0.4 cm
Onaxis scintillation - b ( r , L )
0.5 0.45
S = 10 cm , aB = 2 cm-1
0.25
S = 10 cm , aB = 3 cm-1
0.2 0.15
S = 40 cm , aB = 0.2 cm-1
0.1 0.05
S = 10 cm , aB = 1 cm-1
0.5
1
1.5
2 2.5 3 3.5 4 Propagation distance ( L in km )
4.5
S = 10 cm , aB = 5 cm-1
5
5.5
Fig. 3. On-axis scintillation of various Bessel beams against propagation distance at n 0.
Fig. 5. Aperture averaged scintillation of various Bessel beams against aperture radius at S 10 cm, L 3 km.
against the width parameter is somewhat oscillatory. That is, if the previous rise in value of aB caused a drop in scintillation; then the next rise of aB mostly leads to a fall. Now we turn to aperture averaged scintillation. The data collected from the simulation runs demonstrate that beam orders of n 1, 2, 3 produce higher scintillations, which is consistent with the trends seen in the comparisons of Figs. 3 and 4. This was particularly true near the on-axis limit of the receiver aperture radius. A beam order of n 4 offered a different behavior, however. Based on Eq. (10), Figs. 5 and 6 display the variation of aperture averaged scintillation as a function of receiver aperture size for beams orders of n 0, 4 at the two source plane side lengths of S 10, 40 cm and at a propagation length of L 3 km. The joint examination of Figs. 5 and 6 reveals that the use of smaller truncation, i.e., S 10 cm, leads to less scintillation. At S 10 cm, both beam orders, that is n 0, 4, possess more or less the same amount of aperture averaged scintillation as exhibited by Fig. 5. This changes, however, when S 40 cm. There, as observed from Fig. 6, the aperture-averaged scintillation of beams with n 0 becomes substantially less than that of n 4. As pointed out in Section 3, one quantitative measure for the performance of an optical link is the ratio of the scintillation of a specific beam relative to the classical Gaussian beam on an equal source power basis. The other measure is the same ratio of scintillations but including normalization with respect to per unit captured power. These two quantities are
respectively the first and the second expressions defined in Eq. (12). To facilitate the computation of Eq. (12), we have designed the equal power beams listed in Table 1. As seen from this table, the source power of the beam is adjusted to the numeric value written on the first row, by changing aB , the width parameter in the case of Bessel beams, and by changing αs, the source size in the case of Gaussian beam. The results compiled by utilizing Eq. (12) and the beams of Table 1 are displayed in Figs. 7 and 8, at two different propagation lengths of L 3, 5 km. In these figures, the unity value for the respective measure is marked by dashed green lines. Curves below the green lines indicate regions where the Bessel beams are more advantageous than the corresponding Gaussian beams in terms of scintillation performance as defined in Eq. (12). Such regions are present throughout the subfigures of Fig 7, which means that if the received power criteria is disregarded, then with the selection of Bessel beam parameters as shown in Fig. 7, it is always possible to obtain better scintillation performance than the corresponding equal source power Gaussian beam. In Fig. 7, the advantageous regions generally incorporate lower-order Bessel beams and smaller aperture radius sizes. The broadest region of advantage in Fig. 7 is found at S 40 cm, Ps 10.8 mW, L 5 km (middle panel, bottom row), where all beams except n 5 can be manipulated to lie within this region. When scintillation per unit received power is taken into account, the advantageous
Fig. 4. On-axis scintillation of various Bessel beams against propagation distance at n 1.
Fig. 6. Aperture averaged scintillation of various Bessel beams against aperture radius at S 40 cm, L 3 km.
8036
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
Table 1.
List of Equal Power Beams
Equal Source Power Ps (mW)
0.3
10.8
21.1
S (cm)
10
40
40
n aB cm−1 αs (cm)
Bessel beams Gaussian beams
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 3.76 3.8 3.74 3.77 3.69 3.71 0.75 0.73 0.75 0.73 0.74 0.72 0.38 0.39 0.37 0.39 0.36 0.37 1.38 8.29 11.59
S = 40 cm
S = 40 cm S = 10 cm
10
P = 10.8 mW
2 s
10
n=5
2
10
n=3
0
10
bR ( L )
bR ( L )
bR ( L )
n=4
n=1
10
1
10
0
n=2
5 10 Aperture radius ( Ra in cm )
n=2
n=3
0
n=0
n=0
n=4
L = 3 km
n=4
n=2
1
s
2
n=5
n=3
L = 3 km
L = 3 km
10
10
s
P = 0.3 mW
n=5
P = 21.1 mW
n=0 n=1
n=1
15
5 10 Aperture radius ( Ra in cm )
15
5 10 Aperture radius ( Ra in cm )
15
S = 40 cm S = 10 cm
10
P = 10.8 mW s
n=5
10
1
10
L = 5 km
bR ( L )
bR ( L )
s
n=2
n=3
10
1
n=4
L = 5 km
P = 21.1 mW
n=3
0
10
10
n=1
n=2 0
n=2
-1
n=1
n=1
5 10 15 20 Aperture radius ( Ra in cm )
n=4
L = 5 km
1
n=3
10
n=0
0
n=5
s
n=4
10
S = 40 cm
n=5
bR ( L )
P = 0.3 mW
2
n=0
5 10 Aperture radius ( Ra in cm )
n=0
15
5 10 15 Aperture radius ( Ra in cm )
Fig. 7. Aperture averaged scintillation of Bessel beams relative to a Gaussian beam of equal source power.
S = 40 cm
n=5
S = 10 cm
10
10
P = 0.3 mW
3
10
bPR ( L )
bPR ( L )
2
n=3
n=0
10
5 10 Aperture radius ( Ra in cm )
n=3
5 10 n=0 Aperture radius ( Ra in cm )
S = 10 cm
S = 40 cm
P = 0.3 mW
P = 10.8 mW
n=2
0
15
5 10 n=0 Aperture radius ( Ra in cm )
10
4
15
S = 40 cm n=5 P = 21.1 mW s
s
n=4 n=5
L = 5 km
10
4
L = 5 km
n=4
n=5
bPR ( L )
bPR ( L ) 10 n=1
n=3
2
n=1
15
L = 5 km n=4
10
n=2
0
s
4
L = 3 km
n=1 n=2
n=1
10
10
3
10
bPR ( L )
bPR ( L )
10
4
n=3
n=5
4
s
10
L = 3 km
n=4
L = 3 km n=4
10
P = 21.1 mW
s
n=5
s
5
S = 40 cm
n=4
P = 10.8 mW
4
2
n=3
n=2
n=3
10
2
n=2
n=2 n=0
n=0 n=1
5 10 15 20 Aperture radius ( Ra in cm )
n=0
5 10 15 Aperture radius ( Ra in cm )
n=1
5 10 15 Aperture radius ( Ra in cm )
Fig. 8. Aperture averaged scintillation of Bessel beams per unit received power relative to a Gaussian beam of equal source power. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8037
regions of Fig. 7 are mostly removed and the situation becomes as depicted in Fig. 8. From Fig. 8, we see that there are two advantageous regions, namely at S 40 cm, Ps 10.8 mW, L 3 km and S 40 cm, Ps 21.1 mW, L 3 km, where the Bessel beams of n 0, 1 are included. 5. Conclusion
We have studied the scintillation properties of truncated Bessel beams with the help of a numerical wave optics simulation and random turbulence screens. Square-shaped truncation at the source plane with side lengths of 10 and 40 cm was applied. The width parameter of the Bessel beam was varied from 1 cm−1 to 5 cm−1 for the 10 cm source plane-side length and from 0.2 cm−1 to 1 cm−1 for the 40 cm source plane-side length. Beam order was varied in the range of 0 up to 5. Propagation lengths up to 5.5 km were covered. Against the variations of these parameters, the dependence of on-axis and aperture averaged scintillation was examined. It was found that the zeroth-order Bessel beam offered the lowest scintillation in almost all cases. In other words, the higher-order Bessel beams were generally found to have unfavorable scintillation performance. A possible explanation for this is that the higher orders tend toward a beam pattern that is more like an annulus with relatively low on-axis intensity. A normalized variance measure such as the scintillation index is much more sensitive to small amounts of intensity change when the average intensity value is small to begin with. At propagation lengths of 3 and 5 km, aperture averaged scintillation of Bessel beams were also evaluated relative to classical Gaussian beams under the conditions of equal source power. Bessel beams were found to have lower scintillation at small receiver aperture sizes and smaller beam orders. It seems reasonable to expect the reduced diffractive properties of the low-order Bessel beams to have the most impact with smaller receiver apertures. However, the specific mechanisms that result in reduced scintillation for the small apertures are not clear at this time. When the criterion of per-unit received power was included in this assessment, however, these desirable features of the Bessel beams were lost to a great extent, and only the zeroth- and first-order Bessel beams could offer some improvement over the Gaussian beams. Further investigation is needed in this case to clarify the specific interactions of the scintillation and the received power. We plan to bring more insight to this matter in future studies by evaluating the probability of error performance of optical links exploiting the above revealed advantages of truncated Bessel beams. It is foreseen that our findings can be useful for future practical optical links. References 1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 8038
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
2. R. M. Herman and T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000). 3. A. G. Sedukhin, “Marginal phase correction of truncated Bessel beams,” J. Opt. Soc. Am. A 17, 1059–1066 (2000). 4. Y. Li, V. Gurevich, M. Kirchever, J. Katz, and E. Marom, “Propagation of anisotropic Bessel–Gaussian beams: sidelobe control, mode selection, and field depth,” Appl. Opt. 40, 2709–2721 (2001). 5. J. Gu, D. Zhao, Z. Mei, H. Mao, and H. Xu, “Propagation characteristics of linearly polarized Bessel–Gaussian beams through an annular aperture paraxial ABCD optical system,” Optik 115, 529–532 (2004). 6. Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005). 7. X. Wang and B. Lü, “The beam propagation factor and far-field distribution of Bessel-modulated Gaussian beams,” Opt. Quantum Electron. 34, 1071–1077 (2002). 8. Y. Cai and X. Lü, “Propagation of Bessel and Bessel– Gaussian beams through an unapertured and aperture misaligned paraxial optical systems,” Opt. Commun. 274, 1–7 (2007). 9. C. Zhao, L. Wang, X. Lu, and H. Chen, “Propagation of high-order Bessel–Gaussian beam through a misaligned firstorder optical system,” Opt. Laser Technol. 39, 1199–1203 (2007). 10. H. T. Eyyuboğlu, “Propagation of higher order Bessel– Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007). 11. B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008). 12. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel–Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16, 21315–21320 (2008). 13. J. Cang and Y. Zhang, “Axial intensity distribution of truncated Bessel–Gauss beams in turbulent atmosphere,” Optik 121, 239–245 (2010). 14. K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel–Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29, 251–257 (2012). 15. C. Macaskill and T. E. Ewart, “Computer simulation of twodimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984). 16. J. M. Martin and S. M. Flatte, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). 17. M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995). 18. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995). 19. D. H. Nelson, D. L. Walters, E. P. MacKerrow, M. J. Schmitt, C. R. Quick, W. M. Porch, and R. R. Petrin, “Wave optics simulation of atmospheric turbulence and reflective speckle effects in CO2 lidar,” Appl. Opt. 39, 1857–1871 (2000). 20. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). 21. J. D. Schmidt, in Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010), Chaps. 6–9. 22. D. Voelz, in Computational Fourier Optics a MATLAB Tutorial (SPIE, 2011), Chaps. 4 and 5. 23. X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006). 24. X. M. Qian, W. Y. Zhu, A. T. Wang, C. Gu, and R. Z. Rao, “Numeric simulation for coherent and partially coherent
25. 26. 27. 28.
beam propagation through atmospheric turbulence,” Chin. Phys. Lett. 27, 044214 (2010). Y. Gu and G. Gbur, “Scintillation of pseudo-Bessel correlated beams in atmospheric turbulence,” J. Opt. Soc. Am. A 27, 2621–2629 (2010). W. Cheng, J. H. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009). X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19, 26444–26450 (2011). X. Qian, W. Zhu, and R. Rao, “Numerical investigation on propagation effects of pseudo-partially coherent Gaussian
29.
30. 31. 32.
Schell-model beams in atmospheric turbulence,” Opt. Express 17, 3782–3791 (2009). H. T. Eyyuboğlu, E. Sermtlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605–611 (2008). D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967). J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991). L. C. Andrews and R. L. Phillips, in Laser Beam Propagation through Random Media (SPIE, 2005), Chap. 10.
20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
8039
