- Email: [email protected]
Generation control of a non-diffractive beam in random media by adjusting concentration
Optics Communications 391 (2017) 94–99
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Generation control of a non-diffractive beam in random media by adjusting concentration
MARK
⁎
Ziqi Peng , Tatsuo Shiina Graduate School of Advanced Integration Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Annular beam Non-diffractive beam Scattering Random media Propagation
It takes a long distance of more than 200 m to generate a non-diffractive beam from an annular beam of 40 mm φ (diameter) in space. The authors have worked on the generation, control and optimization of a non-diffractive beam in random media with a short distance (a few tens of centimeters). In this work, the transmittance and the forward scattering waveforms of an annular beam at different propagation distances in random media are presented. Higher intensity of the center peak of a non-diffractive beam can be generated for short-distance propagation and high concentration in random media. The random media under consideration have the same scattering coefficients for all propagation distances. These scattering coefficients depend on the balance between forward scattering and multiple scattering. This balance is defined as the weight parameter α. By using the parameter α, an optimization of a non-diffractive beam in random media is proposed, which is based on adjusting the concentration and the propagation distance in random media.
1. Introduction Non-invasive and non-contact sensing methods such as optical topography and optical coherence tomography, are widely used in medical examination [1–7]. The aforementioned optical sensing technologies have the merit of low risk, high resolution and low cost in comparison with other sensing methods, such as computer tomography, ultrasound imaging and magnetic resonance imaging. However, in strong scattering media (called random media or colloidal suspension), such as the human tissue, which contains scattering particles, the light cannot propagate a long distance because of scattering and absorption. Thus, it is hard to get any information from the deep area inside the human tissue samples [7–12]. Optical sensing could be applied in a variety of random media if the propagation distance and propagation efficiency are improved. Wavefront modulation of transmission beams was suggested in several previous researches [13–17]. This method can help to suppress the diffraction and improve the propagation efficiency. In fact, a modulated light based on the propagation environment can propagate more efficiently in random media [16]. Ideally, a non-diffractive beam is a beam which can keep its original wavefront intensity distribution during its propagation. Moreover, it can suppress beam broadening and have a high tolerance to air fluctuations effectively as compared with a Gaussian beam [18–23]. In this research, we focus on the self-transformation of an annular beam. This beam can be simply created by a pair of Axicon prisms, and
⁎
the polarization of the annular beam remains unchanged after its transformation. An annular beam of 40mmφ transforms into a quasiBessel function beam after it propagates at the distance of 200–300 m in air [24]. The quasi-Bessel beam maintains its waveform in a certain region. In this study, we call it “non-diffractive beam” [25,26]. The main peak diameter of the transformed non-diffractive beam is very narrow in comparison with the transmitting annular beam. It takes advantage of high resolution for sensing works. In our previous work, the non-diffractive beam was generated by using an annular beam in random media at a short range of mere 20 cm. Although only the main peak and sub-peaks on the measured waveform were observed, we confirmed it as having the same features as a non-diffractive beam. As the concentration of the random media changed in a certain range, the waveform variation of the nondiffractive beam showed the same tendency with a self-focusing of the annular beam in air. The non-diffractive beam also kept the same polarization with the incident annular beam [24]. The non-diffractive beam is only transformed at a certain concentration and propagation distance in the random media. In this work, we focus on how to control and optimize the non-diffractive beam in random media. First, the transmittance and the forward scattering waveforms of the annular beam in random media at different propagation distances are presented. Then, a new method to optimize the nondiffractive beam by adjusting the concentration or the propagation distance in random media is proposed. Finally, the following two topics
Corresponding author. E-mail address: [email protected] (Z. Peng).
http://dx.doi.org/10.1016/j.optcom.2017.01.020 Received 24 September 2016; Received in revised form 9 December 2016; Accepted 12 January 2017 Available online 18 January 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Sampling Oscilloscope
Power Supply
Photo Multiplier Tube Optical fiber
Photon Diode
Axicon prisms
Media tank
ND Filter R-r
R r
DPSS Laser Glass Plate Pin Hole
Beam Expander
Diluted milk solution Collimate Lens
Fig. 1. The experiment system to analyze light propagation characteristics in random media. Table 1 Specification of the experiment system. Light source
Receiver
DPSS laser CryLas, 1Q532-1: Wavelength: 532 nm; Pulse width: 2 ns; Peak power: 4.6 kW; Repetition: 15 kHz.
Annular beam converter
Optical elements: Optical Fiber: Multiple mode Diameter.: 50 µm; Collimate lens N.A.:0.55 View angle: 5.5mrad; PMT: Hamamatsu R-636; Response time: 0.78 ns Sampling oscilloscope: Agilent, DCA-J 86100 C with 83484 A module Bandwidth: 50 GHz
Random media
2. Experimental system Fig. 1 and Table 1 show the schematic diagram of the experimental system and the specification of the optical elements in our system, respectively. A high power diode pumped solid state laser (4.6 kW peak power; 2-ns pulse duration) was utilized. The use of a pulsed beam was to increase the efficiency of the non-diffractive beam in random media. As compared with a continuous wave (CW) beam, pulse light interferes only in the time of the pulse width and it can prevent the interference cancelling by multiple scattering. A neutral density filter was used to adjust the optical intensity. A pair of Axicon prisms was used to create the annular beam, and a beam expander was used to control the thickness (the difference between the external diameter and internal diameter) of the annular beam. The intensity of the Gaussian beam can be expressed as in Eq. (1), and the transformation function from an incident beam to an annular beam is expressed as in Eq. (2) [25–28].
1 r ⋅exp[ − ( )2 ] h πh 2
a (r ) =
R−r ⋅ g (R − r ) r
Diameter: 50.8 mm; Annular beam diameter range: 24–42 mm Media tank: Material: Tempax glass; Size: W*H*L 20 cm*20 cm*(10–30) cm; Media: Processed milk with 1.8% fat; Fat size: 1.1 µm; Diluted range: 0.1–1.0%;
the fat particles is about 1.1 µm, which would cause Mie scattering for visible light. The scattering coefficient of milk diluted with water at the concentration of about 30% is close to that of the human tissue [28]. The optical receiver unit consists of a collimated lens and a multiple mode optical fiber, and this combination helps reduce the detection angle to 5.5mrad. The receiving unit was set on an auto mechanical stage with the smallest step of 0.1 mm on horizontal-axis. A photo multiplier tube (PMT) was used as a detector to obtain the weak optical signal through the random medium, and a sampling oscilloscope was used to monitor the pulsed electric signal from PMT.
will be discussed: The relationship between the non-diffractive beam intensity and propagation distance in random media; The correlation between non-diffractive beam self-focusing in air and in random media.
g (r ) =
Axicon prims: Zenith angle: 150° ( ± 10′);
3. Results and discussion 3.1. Transmittance in random media For measuring transmittance accurately, we took off the beam expander and the Axicon prisms from the optical system in Fig. 1, and used the laser beam of a small spot of ＜0.3 mm from laser head, which had a Gaussian intensity distribution. The incident beam was emitted into the media tank; then forward scattering intensity was detected by the PMT at the center optical axis of the back side of the media tank.
(1)
(r > 0)
(2)
Here, h is the distance from the center to the position where the intensity of the Gaussian beam becomes 1/e2 of the center value; R is the external radius of the annular beam which is determined by the interval between Axicon prisms, and r is the internal radius of the annular beam. Three media tanks with different sizes were used to change the propagation distance (10, 20 and 30 cm). The processed milk with 1.8% fat was chosen as the random medium. The diameter of
Fig. 2. Transmittance in random media at different propagation distance.
95
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Fig. 2 shows experimental and calculation results of light transmittance in random media at propagation distances of 10, 20 and 30 cm. For easy comparison, the x-axis was normalized as the unit of distance×concentration (m%). According to the transmittance, the scattering coefficient and the transport mean free path can be calculated. When the transport mean free path is longer than the propagation distance, it means that light will hit once or never hit the particles in random media. This is the case of the single scattering. When the transport mean free path is shorter than the propagation distance, it means that light will hit the particles more than once in random media averagely. In this case, multiple scattering will occur. The results overlap perfectly in the left part until 0.06 m%. In this part, single scattering is dominant as compared with multiple scattering. In the right part from 0.12 m% higher concentration, multiple scattering is dominant and three curves have the same gradient. When we set the parameter of distance×concentration as the same value, it has higher transmittance when the propagation distance is shorter. As the reason of this result, the shorter the propagation distance is, the higher the concentration is, and the more multiple scattering light increased at the output plane. We also simulated the light transmittance in random media. On the premise of our optical receiver setup, the light transmittance is expressed with the following Eqs. (3) and (4):
I1 = I0⋅exp(−σ1NL ) + I ′
(3)
I ′ = I0⋅[1 − exp(−σ1NL )]⋅α⋅exp(−σ2NL )
(4)
Fig. 3. Non-diffractive beam in random media.
The results of the scattering waveforms are shown in Fig. 4. At propagation distances of (a) 10 cm, (b) 20 cm and (c) 30 cm, the highest intensities of the non-diffractive beams appeared at the concentrations of 1.0%, 0.6%, and 0.4%, respectively. The center intensity ratios were 1.84%, 1.22% and 0.85%, respectively. It means that the annular beam generates the higher intensity non-diffractive beam when the beam propagates at the shorter distance, and the higher intensity non-diffractive beam is generated in the higher concentration of the random media. From Fig. 2, we find that all of the non-diffractive beams appeared at the concentration of the intermediate region between single scattering and multiple scattering (0.07 ~ 0.12 m%). For all of the propagation distances, they have the same process of the waveform transformation, that is, first, main peak and two sub peaks appear near and around the center, then the main peak become stronger with the decline of sub peaks; finally, the main peak become the highest intensity while the sub peaks disappeared. One can find that the width of the main peak is about 6 mm, which is equal to the width of the center peak when the non-diffractive beam is transformed from the annular beam in air. Fig. 5 shows that the annular beam of 40 mm in diameter transforms into a non-diffractive beam after it propagates at 210 m in the air. In the experimental result, the nondiffractive beam transformed from the annular beam should have many sub peaks when it is compared with the non-diffractive beam of 40 mm in air, but here the higher order sub peaks are cancelled by the effect of interference from the multiple scattering.
Here, I0 and I1 are the incident light intensity and the transmitted light intensity, respectively. The transmittance is defined as I1/I0; I1 consists of the directly delivered light without scattering I0·exp(−σ1NL), and the forward scattering light I′. The forward scattering light I′ can be calculated by the Eq. (4). Here, L is the propagation distance; N is the concentration of diluted milk solution; σ1 is the scattering coefficient when single scattering is dominant and σ2 is the scattering coefficient when multiple scattering is dominant; the parameter α is introduced as the weight of the multiple scattering for the forward direction and the properties of α will be discussed in Section 3.3. Usually, single and multiple scattering are dependent on the size, the refractive index of the particles and its concentration. These parameters are reflected in σ1 and σ2 here. Considering the particles block light entirely, σ1 can be derived according to the definition of scattering cross-section (the average diameter of the fat particles is set as 1.1 μm as we mentioned in Section 2). σ2 can be derived from σ1 and the anisotropic parameter g. Here, g is calculated by Mie theory when the size and the refractive index of the particles are given. The absorption in our experiment is quite small to compare with the scattering (the absorption coefficient is 0.0066 cm−1 at the concentration of 40%, while the scattering coefficient is 22 cm−1 at the concentration of 0.6% in our current experiment) [29]. We ignored it in the calculation.
3.3. Optimization of the non-diffractive beam The non-diffractive beam in random media is determined by the shape of the incident beam, the propagation distance, the concentration of the random media and the view angle of the optical receiver. The other parameters, such as particle size, refractive index of the random media are assumed to be as constants when we discussed the relationship between the propagation distance and the media concentrations. Usually, the light will not retain its polarization when multiple scattering is dominant in random media. In our experiment, the nondiffractive beam was observed because of the following two reasons: one is that the light in random media is scattered 4 or 5 times in average at the propagation distance of 20 cm (transport mean free path is about 5–4 cm), which is calculated by diffusion equation from the propagation distance and the concentration of the random media; another reason is that the light received by the receiver must only be the forward scattering light, that is the reason why a receiver with the view angle of 5.5mrad was employed in our experiment. Only the forward scattering light can retain its polarization through the random media after 4 or 5 times scatterings. In fact, the times that scattering events occurred until the beam went out from the random media was determined by the concentration and the propagation distance in the
3.2. Scattering waveform An annular beam with a diameter of 40 mm and thickness of 5 mm was used in our experiments. The receiver unit was scanned in the direction which was perpendicular to the optical axis. Fig. 3 shows a non-diffractive beam obtained in the experiment. The waveform consists of the center part and the surrounding part. The surrounding part has an intensity distribution close to isotropic distribution (dashed line) caused by multiple scattering. Although there are also other scattering components at the output plane of the random media tank, the surrounding part is selected by the narrow FOV receiver at the center area on the optical axis. The surrounding part intensity at the distance of 20 mm with respect to the center is normalized as 1. The non-diffractive beam in the center is evaluated by the intensity ratio between its center peak intensity (Vp) and the center intensity in isotropic distribution (Vs). 96
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
0.8 Annular beam
Normalized Intensity
0.7 Propagation pattern after 210m
0.6 0.5 0.4 0.3 0.2 0.1 0 -30
-20
-10 0 10 Position (mm)
20
30
Fig. 5. Propagation pattern of annular beam in air.
case of 30 cm had the intensity ratio of 0.90% when the concentration is 0.433%), and compare it with the weight parameter α in Fig. 6. The center intensity ratio shows the same variation trend with α, which proves that a non-diffractive beam was generated by forward multiple scattering of light. The process of optimization to generate the highest center peak of the non-diffractive beam is shown in Fig. 7. In measurement (ⅰ), the forward multiple scattering weight of light α, the single scattering coefficient σ1 and the multiple scattering coefficient σ2 can be calculated from one set of the transmittance measurement results (propagation distance L) such as Fig. 2. As the second step in simulation (ⅱ), the concentration N for optimizing the non-diffractive beam at distance L can be identified by the optical simulation when σ1 and σ2 are given [24]. Considering the light transmittance intensity obtained in measurement (ⅰ) can be expressed as the function f (n), the scattering coefficient σ in Fig. 7 can be expressed as ∂f(n)/∂n. Here, n in the function concentration. Assigning the value of N obtained in simulation (ⅱ), we can calculate the scattering coefficient σ for optimizing the nondiffractive beam in step (ⅲ). For different propagation distances L’, the new transmittance curve f′(n) can be plotted according to the variation of the parameter α (σ1 and σ2 are constant) in calculation (ⅳ). By using the scattering coefficient σ obtained in (ⅲ), the concentration N’ for optimizing the non-diffractive beam at the propagation distances L’ can be calculated by solving the equation of ∂f′(n)/∂n=σ. The solution of the equation N’ in calculation (ⅴ) is the concentration for optimizing the non-diffractive beam. For investigating the value of α in unknown random media (particle size, refractive index and so on), only one set experimental transmittance measurement is needed. This method is not only used for calculating the concentration of the optimized non-diffractive beam, but also for calculating the other parameters (such as the propagation distance or shape of the incident beam) in order to optimize the non-diffractive beam when some other parameters are defined as constants. For example, when we need to measure the light propagation in human finger or wrist, the shape of the annular beam would be the optimization factor because the concentration of the random media and the propagation distance are defined as constants here.
Fig. 4. Scattering waveforms of annular beam in random media by different propagation distance. (a) 10 cm propagation; (b) 20 cm propagation; (c) 30 cm propagation.
random media, so that the non-diffractive beam could be controlled by adjusting the concentration of the random media. The optimization of the non-diffractive beam is summarized in Table 2. As mentioned in Section 3.2, the highest intensity ratio of the non-diffractive beam appeared in the concentration of 0.4%, 0.6%, and 1.0% when propagation distances were 30, 20 and 10 cm, respectively. At these three ranges, the weight parameter α is calculated from the experimental result of the transmittance as 5×10−5, 2.5×10−5, 1.6×10−5 and all of the scattering coefficients σ are near 22 cm−1. According to the experimental results, we infer the highest ideal center intensity ratio at different propagation distances (the case of 10 cm had the intensity ratio of 2.0% when the concentration is 1.033% and the
3.4. Intensity ratio of the non-diffractive beam Generally, the non-diffractive beam is transformed after the annular beam (40 mm φ) propagates a long distance of a few hundred meters ( > 200 m). In our experiment, the non-diffractive beam appeared in a mere few ten centimeters in random media. Although the particles in random media scatter the annular beam strongly, only the forward scattering light is detected and it interferes at the center of the optical axis. We have considered that the scattering particles in random media act as a lens. In a high concentration random medium, the number of the particles is increased, the intensity of forward 97
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Table 2 Non-diffractive beam optimization in random media. Parameter
10 cm propagation
Concentration Scattering coefficient Intensity ratio α
20 cm propagation
30 cm propagation
Experiment
Optimization
Experiment
Optimization
Experiment
Optimization
1.0% 22.63 cm−1 1.84% 0.00005
1.033% 21.98 cm−1 1.98% 0.00005
0.6% 22.27 cm−1 1.22% 0.000025
0.6% 22.27 cm−1 1.22% 0.000025
0.4% 23.96 cm−1 0.85% 0.000016
0.433% 22.24 cm−1 0.90% 0.000016
Fig. 6. Comparison between center intensity ratio and α.
Fig. 8. Center intensity of non-diffractive beam in air (with lens effect) and center intensity ratio of non-diffractive beam in random media against the propagation distance.
concentrations and the intensity ratio of the non-diffractive beam in random media is another expression of the relationship between the propagation distance and intensity of the non-diffractive beam in air. This result is also the evidence which shows that the center light is the non-diffractive beam. As the difference of the propagation properties of annular beam between propagation in random media and in air, for the case in air, the annular beam just diffracts, and generates the non-diffractive beam, while this process needs a long distance to create the wavefront and the phasefront which can generate a non-diffractive beam. On the other hand, in the case of annular beam propagation in random media, the particles in random media scatter the annular beam. The scattering makes the annular beam get an appropriate wavefront and phasefront distributions in a short distance, which can generate the non-diffractive beam as that in air. However, only the forward scattering can obtain the phasefront which is fit to generate the non-diffractive beam. This is the reason why the narrow detect angle receiver was used in this experiment. The non-diffractive beam can only be created in a certain range of the concentration when the propagation distance is given.
Fig. 7. Process of the optimization.
scattering light is higher, and the non-diffractive beam is generated at a shorter distance with a high intensity as the generated non-diffractive beam (at a few hundred meters in air) is focused at short-distance with different lenses in air. Fig. 8 shows the variation of the center intensity of the nondiffractive beam when the annular beam is focused in air (adding the lens effect with different focal lengths into the calculation results) and in random media (experimental result). Here, L is the propagation distance in random media, and l is the propagation distance in air. In the calculation, the lens effect is added and it renders the nondiffractive beam focused at a short distance. The center intensity at l=210 m is normalized as 1. The experimental results show that the non-diffractive beam has higher intensity in shorter distance propagation in random media. Variation of the center intensity ratio in the experiment (random media) has the same tendency with the variation of the center intensity of the non-diffractive beam in air (with lens effect) when the propagation distance increases with exponential order. For the same attenuation intensity, the relation between propagation distances in air and in random media can be written as
Irandom(L ) ∝ Iair (ln(l ))
4. Conclusion In our experimental setup, the non-diffractive beam was observed at a short distance in random media. The highest intensity ratio of the non-diffractive beams appeared at the concentration of 0.4%, 0.6%, and 1.0% when propagation distances were 30, 20 and 10 cm in a diluted milk solution. For these three concentrations, the random medium had almost the same scattering coefficient, near 22 cm−1. For the optimized non-diffractive beam at different propagation distances, a shorter random medium channel with a higher concentration generated a higher intensity ratio of the non-diffractive beam. We also proposed a method for optimizing the non-diffractive beam propagation in random media by using a forward multiple scattering
(5)
These two distances are associated with the concentration of the random media. We have considered the relationship between the 98
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
weight α, the scattering coefficient σ and the transmittance curve with different concentrations of random media and certain propagation distances. For optimizing the non-diffractive beam in other types of random media with different particle sizes and materials, we can use this method after we have calculated the transport mean free path (multiple scattering times), which is related to the scattering coefficient in the random media. The relation between the annular beam propagation in air and that in random media was discussed. For the case in random media, the variation of the center intensity ratio would have the same behavior with the center intensity through propagation in air if we set the propagation distance increasing with exponential order in air. However, we did not refer any about the propagation characteristics of the non-diffractive beam (generated from the annular beam) in random media in this study. For applying this propagation technique to practical sensing such as super-resolution sensing in random media, a narrow shape and high intensity of the output light is required. The propagation characteristics of the non-diffractive beam in random media is a quite important problem as the future work [30,31]. Such as for human tissue examination in future, e.g. in some thin tissue such as finger or wrist, the non-diffractive beam can be observed by optimizing the shape of the incident beam. The observed non-diffractive beam can be used as a probe to diagnose the state of human tissue. Here, the absorption by human tissue should be also considered [29].
structure and dynamics, Science 252 (1991) 686–688. [10] D. Voelz, X. Xiao, Metric for optimizing spatially partially coherent beams for propagation through turbulence, Opt. Eng. 48 (2009) 036001. [11] E. Gershnik, E.N. Riba, Light propagation through multilayer atmospheric turbulence, Opt. Commun. 142 (1997) 99–105. [12] A.N. Bogaturov, A.A.D. Canas, J.C. Dainty, A.S. Gurvich, V.A. Myakinin, C.J. Solomon, N.J. Wooder, Observation of the enhancement of coherence by backscattering through turbulence, Opt. Commun. 87 (1992) 1–4. [13] Y. Chen, Y. Cai, H.T. Eyyuboglu, Y. Baykal, Scintillation properties of dark hollow beams in a weak turbulent atmosphere, Appl. Phys. B 90 (2008) 87–92. [14] R. Rao, Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence, Appl. Opt. 47 (2008) 269–276. [15] H.T. Eyyuboglu, Y.E. Yenice, Y. Baykal, Higher order annular Gaussian laser beam propagation in free space, Opt. Eng. 45 (2006) 038002-1-038002-038008. [16] Y. Baykal, Y. Cai, X. Ji, Field correlations of annular beams in extremely strong turbulence, Opt. Commun. 285 (2012) 4171–4174. [17] Y. Baykal, Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium, J. Opt. Soc. Am. A 22 (2005) 672–679. [18] V. Soifer, V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation, Taylor & Francis, 1997. [19] H.T. Eyyuboglu, C.Z. Cil, Beam wander of dark hollow, flat-topped and annular beams, Appl. Phys. B 93 (2008) 595–604. [20] H. Gercekcioglu, Y. Baykal, C. Nakiboglu, Annular beam scintillations in strong turbulence, J. Opt. Soc. Am. A 27 (2010) 1834–1839. [21] T. Shiina, K. Yoshida, M. Ito, Y. Okamura, In-line type micro pulse lidar with annular beam - theoretical approach, Appl. Opt. 44 (2005) 7467–7473. [22] H.T. Eyyuboğlu, S. Altay, Y. Baykal, Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence, Opt. Commun. 264 (2006) 25–34. [23] M.A. Öztan, Y. Baykal, C. Nakiboğlu, Effects of extremely strong turbulent medium on scintillations of partially coherent annular and flat-topped Gaussian beams, Opt. Commun. 285 (2012) 943–946. [24] Z. Peng, T. Shiina, Analysis of Annular Light Propagation Characteristics in Random Media, SPIE-OSA-IEEE Asia Communications and Photonics, 8311 [25] G. Scott, N. McArdle, Efficient generation of nearly diffraction-free beams using an axicon, Opt. Eng. 31 (1992) 2640–2643. [26] T. Aruga, Generation of Long-range nondiffracting narrow light beams, Appl. Opt. 36 (1997) 3762–3768. [27] A. Fedotowsky, K. Lehovec, Optimal filter design for annular imaging, Appl. Opt. 13 (1974) 2919–2923. [28] T. Shiina, Y. Tsuge, T. Honda, Examination of propagation process of annular beam in random media, in: Proceedings SPIE 6585, Optical Sensing Technology and Applications, 658524, 2007. [29] D.A. Boas, Diffuse Photon Probes of Structural and Dynamical Properties of Turbid Media: Theory and Biomedical Applications (Ph.D. Thesis), Univ. of Pennsylvania, 1996. [30] Y. Ismail, N. Khilo, V. Belyi, A. Forbes, Shape invariant higher-order Bessel-like beams carrying orbital angular momentum, J. Opt. 14 (2012) 085703. [31] W. Nelson, J.P. Palastro, C.C. Davis, P. Sprangle, Propagation of Bessel and Airy beams through atmospheric turbulence, J. Opt. Soc. Am. A 31 (2014) 603–609.
References [1] C.F. Bohren, Donald R. Huffman, Absorption and scattering of light by small particles, Wiley Sci. Paperb. (1998). [2] T. Shiina, M. Ito, Y. Okamura, Long range propagation characteristics of annular beam, Opt. Commun. 279 (2007) 159–167. [3] P. Sadoghi, Discrete ordinates method for propagation of light in highly scattering vascular tissues, Opt. Commun. 266 (2006) 359–363. [4] Z. Tong, O. Korotkova, Polarization of random beams scattered from twodimensional bio-tissue slices, Opt. Commun. 322 (2014) 202–204. [5] T. Aruga, S.W. Li, Super high resolution for long-range imaging, Appl. Opt. 38 (1999) 2795–2799. [6] A.E. Profio, Light transport in tissue, Appl. Opt. 28 (1989) 2216–2222. [7] W. Gao, Changes of polarization of light beams on propagation through tissue, Opt. Commun. 260 (2006) 749–754. [8] A. Ishimaru, Diffusion of light in turbid material, Appl. Opt. 28 (1989) 2210–2215. [9] D.J. Durian, D.A. Weitz, D.J. Pine, Multiple light-scattering probes of foam
99
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Generation control of a non-diffractive beam in random media by adjusting concentration
MARK
⁎
Ziqi Peng , Tatsuo Shiina Graduate School of Advanced Integration Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Annular beam Non-diffractive beam Scattering Random media Propagation
It takes a long distance of more than 200 m to generate a non-diffractive beam from an annular beam of 40 mm φ (diameter) in space. The authors have worked on the generation, control and optimization of a non-diffractive beam in random media with a short distance (a few tens of centimeters). In this work, the transmittance and the forward scattering waveforms of an annular beam at different propagation distances in random media are presented. Higher intensity of the center peak of a non-diffractive beam can be generated for short-distance propagation and high concentration in random media. The random media under consideration have the same scattering coefficients for all propagation distances. These scattering coefficients depend on the balance between forward scattering and multiple scattering. This balance is defined as the weight parameter α. By using the parameter α, an optimization of a non-diffractive beam in random media is proposed, which is based on adjusting the concentration and the propagation distance in random media.
1. Introduction Non-invasive and non-contact sensing methods such as optical topography and optical coherence tomography, are widely used in medical examination [1–7]. The aforementioned optical sensing technologies have the merit of low risk, high resolution and low cost in comparison with other sensing methods, such as computer tomography, ultrasound imaging and magnetic resonance imaging. However, in strong scattering media (called random media or colloidal suspension), such as the human tissue, which contains scattering particles, the light cannot propagate a long distance because of scattering and absorption. Thus, it is hard to get any information from the deep area inside the human tissue samples [7–12]. Optical sensing could be applied in a variety of random media if the propagation distance and propagation efficiency are improved. Wavefront modulation of transmission beams was suggested in several previous researches [13–17]. This method can help to suppress the diffraction and improve the propagation efficiency. In fact, a modulated light based on the propagation environment can propagate more efficiently in random media [16]. Ideally, a non-diffractive beam is a beam which can keep its original wavefront intensity distribution during its propagation. Moreover, it can suppress beam broadening and have a high tolerance to air fluctuations effectively as compared with a Gaussian beam [18–23]. In this research, we focus on the self-transformation of an annular beam. This beam can be simply created by a pair of Axicon prisms, and
⁎
the polarization of the annular beam remains unchanged after its transformation. An annular beam of 40mmφ transforms into a quasiBessel function beam after it propagates at the distance of 200–300 m in air [24]. The quasi-Bessel beam maintains its waveform in a certain region. In this study, we call it “non-diffractive beam” [25,26]. The main peak diameter of the transformed non-diffractive beam is very narrow in comparison with the transmitting annular beam. It takes advantage of high resolution for sensing works. In our previous work, the non-diffractive beam was generated by using an annular beam in random media at a short range of mere 20 cm. Although only the main peak and sub-peaks on the measured waveform were observed, we confirmed it as having the same features as a non-diffractive beam. As the concentration of the random media changed in a certain range, the waveform variation of the nondiffractive beam showed the same tendency with a self-focusing of the annular beam in air. The non-diffractive beam also kept the same polarization with the incident annular beam [24]. The non-diffractive beam is only transformed at a certain concentration and propagation distance in the random media. In this work, we focus on how to control and optimize the non-diffractive beam in random media. First, the transmittance and the forward scattering waveforms of the annular beam in random media at different propagation distances are presented. Then, a new method to optimize the nondiffractive beam by adjusting the concentration or the propagation distance in random media is proposed. Finally, the following two topics
Corresponding author. E-mail address: [email protected] (Z. Peng).
http://dx.doi.org/10.1016/j.optcom.2017.01.020 Received 24 September 2016; Received in revised form 9 December 2016; Accepted 12 January 2017 Available online 18 January 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Sampling Oscilloscope
Power Supply
Photo Multiplier Tube Optical fiber
Photon Diode
Axicon prisms
Media tank
ND Filter R-r
R r
DPSS Laser Glass Plate Pin Hole
Beam Expander
Diluted milk solution Collimate Lens
Fig. 1. The experiment system to analyze light propagation characteristics in random media. Table 1 Specification of the experiment system. Light source
Receiver
DPSS laser CryLas, 1Q532-1: Wavelength: 532 nm; Pulse width: 2 ns; Peak power: 4.6 kW; Repetition: 15 kHz.
Annular beam converter
Optical elements: Optical Fiber: Multiple mode Diameter.: 50 µm; Collimate lens N.A.:0.55 View angle: 5.5mrad; PMT: Hamamatsu R-636; Response time: 0.78 ns Sampling oscilloscope: Agilent, DCA-J 86100 C with 83484 A module Bandwidth: 50 GHz
Random media
2. Experimental system Fig. 1 and Table 1 show the schematic diagram of the experimental system and the specification of the optical elements in our system, respectively. A high power diode pumped solid state laser (4.6 kW peak power; 2-ns pulse duration) was utilized. The use of a pulsed beam was to increase the efficiency of the non-diffractive beam in random media. As compared with a continuous wave (CW) beam, pulse light interferes only in the time of the pulse width and it can prevent the interference cancelling by multiple scattering. A neutral density filter was used to adjust the optical intensity. A pair of Axicon prisms was used to create the annular beam, and a beam expander was used to control the thickness (the difference between the external diameter and internal diameter) of the annular beam. The intensity of the Gaussian beam can be expressed as in Eq. (1), and the transformation function from an incident beam to an annular beam is expressed as in Eq. (2) [25–28].
1 r ⋅exp[ − ( )2 ] h πh 2
a (r ) =
R−r ⋅ g (R − r ) r
Diameter: 50.8 mm; Annular beam diameter range: 24–42 mm Media tank: Material: Tempax glass; Size: W*H*L 20 cm*20 cm*(10–30) cm; Media: Processed milk with 1.8% fat; Fat size: 1.1 µm; Diluted range: 0.1–1.0%;
the fat particles is about 1.1 µm, which would cause Mie scattering for visible light. The scattering coefficient of milk diluted with water at the concentration of about 30% is close to that of the human tissue [28]. The optical receiver unit consists of a collimated lens and a multiple mode optical fiber, and this combination helps reduce the detection angle to 5.5mrad. The receiving unit was set on an auto mechanical stage with the smallest step of 0.1 mm on horizontal-axis. A photo multiplier tube (PMT) was used as a detector to obtain the weak optical signal through the random medium, and a sampling oscilloscope was used to monitor the pulsed electric signal from PMT.
will be discussed: The relationship between the non-diffractive beam intensity and propagation distance in random media; The correlation between non-diffractive beam self-focusing in air and in random media.
g (r ) =
Axicon prims: Zenith angle: 150° ( ± 10′);
3. Results and discussion 3.1. Transmittance in random media For measuring transmittance accurately, we took off the beam expander and the Axicon prisms from the optical system in Fig. 1, and used the laser beam of a small spot of ＜0.3 mm from laser head, which had a Gaussian intensity distribution. The incident beam was emitted into the media tank; then forward scattering intensity was detected by the PMT at the center optical axis of the back side of the media tank.
(1)
(r > 0)
(2)
Here, h is the distance from the center to the position where the intensity of the Gaussian beam becomes 1/e2 of the center value; R is the external radius of the annular beam which is determined by the interval between Axicon prisms, and r is the internal radius of the annular beam. Three media tanks with different sizes were used to change the propagation distance (10, 20 and 30 cm). The processed milk with 1.8% fat was chosen as the random medium. The diameter of
Fig. 2. Transmittance in random media at different propagation distance.
95
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Fig. 2 shows experimental and calculation results of light transmittance in random media at propagation distances of 10, 20 and 30 cm. For easy comparison, the x-axis was normalized as the unit of distance×concentration (m%). According to the transmittance, the scattering coefficient and the transport mean free path can be calculated. When the transport mean free path is longer than the propagation distance, it means that light will hit once or never hit the particles in random media. This is the case of the single scattering. When the transport mean free path is shorter than the propagation distance, it means that light will hit the particles more than once in random media averagely. In this case, multiple scattering will occur. The results overlap perfectly in the left part until 0.06 m%. In this part, single scattering is dominant as compared with multiple scattering. In the right part from 0.12 m% higher concentration, multiple scattering is dominant and three curves have the same gradient. When we set the parameter of distance×concentration as the same value, it has higher transmittance when the propagation distance is shorter. As the reason of this result, the shorter the propagation distance is, the higher the concentration is, and the more multiple scattering light increased at the output plane. We also simulated the light transmittance in random media. On the premise of our optical receiver setup, the light transmittance is expressed with the following Eqs. (3) and (4):
I1 = I0⋅exp(−σ1NL ) + I ′
(3)
I ′ = I0⋅[1 − exp(−σ1NL )]⋅α⋅exp(−σ2NL )
(4)
Fig. 3. Non-diffractive beam in random media.
The results of the scattering waveforms are shown in Fig. 4. At propagation distances of (a) 10 cm, (b) 20 cm and (c) 30 cm, the highest intensities of the non-diffractive beams appeared at the concentrations of 1.0%, 0.6%, and 0.4%, respectively. The center intensity ratios were 1.84%, 1.22% and 0.85%, respectively. It means that the annular beam generates the higher intensity non-diffractive beam when the beam propagates at the shorter distance, and the higher intensity non-diffractive beam is generated in the higher concentration of the random media. From Fig. 2, we find that all of the non-diffractive beams appeared at the concentration of the intermediate region between single scattering and multiple scattering (0.07 ~ 0.12 m%). For all of the propagation distances, they have the same process of the waveform transformation, that is, first, main peak and two sub peaks appear near and around the center, then the main peak become stronger with the decline of sub peaks; finally, the main peak become the highest intensity while the sub peaks disappeared. One can find that the width of the main peak is about 6 mm, which is equal to the width of the center peak when the non-diffractive beam is transformed from the annular beam in air. Fig. 5 shows that the annular beam of 40 mm in diameter transforms into a non-diffractive beam after it propagates at 210 m in the air. In the experimental result, the nondiffractive beam transformed from the annular beam should have many sub peaks when it is compared with the non-diffractive beam of 40 mm in air, but here the higher order sub peaks are cancelled by the effect of interference from the multiple scattering.
Here, I0 and I1 are the incident light intensity and the transmitted light intensity, respectively. The transmittance is defined as I1/I0; I1 consists of the directly delivered light without scattering I0·exp(−σ1NL), and the forward scattering light I′. The forward scattering light I′ can be calculated by the Eq. (4). Here, L is the propagation distance; N is the concentration of diluted milk solution; σ1 is the scattering coefficient when single scattering is dominant and σ2 is the scattering coefficient when multiple scattering is dominant; the parameter α is introduced as the weight of the multiple scattering for the forward direction and the properties of α will be discussed in Section 3.3. Usually, single and multiple scattering are dependent on the size, the refractive index of the particles and its concentration. These parameters are reflected in σ1 and σ2 here. Considering the particles block light entirely, σ1 can be derived according to the definition of scattering cross-section (the average diameter of the fat particles is set as 1.1 μm as we mentioned in Section 2). σ2 can be derived from σ1 and the anisotropic parameter g. Here, g is calculated by Mie theory when the size and the refractive index of the particles are given. The absorption in our experiment is quite small to compare with the scattering (the absorption coefficient is 0.0066 cm−1 at the concentration of 40%, while the scattering coefficient is 22 cm−1 at the concentration of 0.6% in our current experiment) [29]. We ignored it in the calculation.
3.3. Optimization of the non-diffractive beam The non-diffractive beam in random media is determined by the shape of the incident beam, the propagation distance, the concentration of the random media and the view angle of the optical receiver. The other parameters, such as particle size, refractive index of the random media are assumed to be as constants when we discussed the relationship between the propagation distance and the media concentrations. Usually, the light will not retain its polarization when multiple scattering is dominant in random media. In our experiment, the nondiffractive beam was observed because of the following two reasons: one is that the light in random media is scattered 4 or 5 times in average at the propagation distance of 20 cm (transport mean free path is about 5–4 cm), which is calculated by diffusion equation from the propagation distance and the concentration of the random media; another reason is that the light received by the receiver must only be the forward scattering light, that is the reason why a receiver with the view angle of 5.5mrad was employed in our experiment. Only the forward scattering light can retain its polarization through the random media after 4 or 5 times scatterings. In fact, the times that scattering events occurred until the beam went out from the random media was determined by the concentration and the propagation distance in the
3.2. Scattering waveform An annular beam with a diameter of 40 mm and thickness of 5 mm was used in our experiments. The receiver unit was scanned in the direction which was perpendicular to the optical axis. Fig. 3 shows a non-diffractive beam obtained in the experiment. The waveform consists of the center part and the surrounding part. The surrounding part has an intensity distribution close to isotropic distribution (dashed line) caused by multiple scattering. Although there are also other scattering components at the output plane of the random media tank, the surrounding part is selected by the narrow FOV receiver at the center area on the optical axis. The surrounding part intensity at the distance of 20 mm with respect to the center is normalized as 1. The non-diffractive beam in the center is evaluated by the intensity ratio between its center peak intensity (Vp) and the center intensity in isotropic distribution (Vs). 96
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
0.8 Annular beam
Normalized Intensity
0.7 Propagation pattern after 210m
0.6 0.5 0.4 0.3 0.2 0.1 0 -30
-20
-10 0 10 Position (mm)
20
30
Fig. 5. Propagation pattern of annular beam in air.
case of 30 cm had the intensity ratio of 0.90% when the concentration is 0.433%), and compare it with the weight parameter α in Fig. 6. The center intensity ratio shows the same variation trend with α, which proves that a non-diffractive beam was generated by forward multiple scattering of light. The process of optimization to generate the highest center peak of the non-diffractive beam is shown in Fig. 7. In measurement (ⅰ), the forward multiple scattering weight of light α, the single scattering coefficient σ1 and the multiple scattering coefficient σ2 can be calculated from one set of the transmittance measurement results (propagation distance L) such as Fig. 2. As the second step in simulation (ⅱ), the concentration N for optimizing the non-diffractive beam at distance L can be identified by the optical simulation when σ1 and σ2 are given [24]. Considering the light transmittance intensity obtained in measurement (ⅰ) can be expressed as the function f (n), the scattering coefficient σ in Fig. 7 can be expressed as ∂f(n)/∂n. Here, n in the function concentration. Assigning the value of N obtained in simulation (ⅱ), we can calculate the scattering coefficient σ for optimizing the nondiffractive beam in step (ⅲ). For different propagation distances L’, the new transmittance curve f′(n) can be plotted according to the variation of the parameter α (σ1 and σ2 are constant) in calculation (ⅳ). By using the scattering coefficient σ obtained in (ⅲ), the concentration N’ for optimizing the non-diffractive beam at the propagation distances L’ can be calculated by solving the equation of ∂f′(n)/∂n=σ. The solution of the equation N’ in calculation (ⅴ) is the concentration for optimizing the non-diffractive beam. For investigating the value of α in unknown random media (particle size, refractive index and so on), only one set experimental transmittance measurement is needed. This method is not only used for calculating the concentration of the optimized non-diffractive beam, but also for calculating the other parameters (such as the propagation distance or shape of the incident beam) in order to optimize the non-diffractive beam when some other parameters are defined as constants. For example, when we need to measure the light propagation in human finger or wrist, the shape of the annular beam would be the optimization factor because the concentration of the random media and the propagation distance are defined as constants here.
Fig. 4. Scattering waveforms of annular beam in random media by different propagation distance. (a) 10 cm propagation; (b) 20 cm propagation; (c) 30 cm propagation.
random media, so that the non-diffractive beam could be controlled by adjusting the concentration of the random media. The optimization of the non-diffractive beam is summarized in Table 2. As mentioned in Section 3.2, the highest intensity ratio of the non-diffractive beam appeared in the concentration of 0.4%, 0.6%, and 1.0% when propagation distances were 30, 20 and 10 cm, respectively. At these three ranges, the weight parameter α is calculated from the experimental result of the transmittance as 5×10−5, 2.5×10−5, 1.6×10−5 and all of the scattering coefficients σ are near 22 cm−1. According to the experimental results, we infer the highest ideal center intensity ratio at different propagation distances (the case of 10 cm had the intensity ratio of 2.0% when the concentration is 1.033% and the
3.4. Intensity ratio of the non-diffractive beam Generally, the non-diffractive beam is transformed after the annular beam (40 mm φ) propagates a long distance of a few hundred meters ( > 200 m). In our experiment, the non-diffractive beam appeared in a mere few ten centimeters in random media. Although the particles in random media scatter the annular beam strongly, only the forward scattering light is detected and it interferes at the center of the optical axis. We have considered that the scattering particles in random media act as a lens. In a high concentration random medium, the number of the particles is increased, the intensity of forward 97
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
Table 2 Non-diffractive beam optimization in random media. Parameter
10 cm propagation
Concentration Scattering coefficient Intensity ratio α
20 cm propagation
30 cm propagation
Experiment
Optimization
Experiment
Optimization
Experiment
Optimization
1.0% 22.63 cm−1 1.84% 0.00005
1.033% 21.98 cm−1 1.98% 0.00005
0.6% 22.27 cm−1 1.22% 0.000025
0.6% 22.27 cm−1 1.22% 0.000025
0.4% 23.96 cm−1 0.85% 0.000016
0.433% 22.24 cm−1 0.90% 0.000016
Fig. 6. Comparison between center intensity ratio and α.
Fig. 8. Center intensity of non-diffractive beam in air (with lens effect) and center intensity ratio of non-diffractive beam in random media against the propagation distance.
concentrations and the intensity ratio of the non-diffractive beam in random media is another expression of the relationship between the propagation distance and intensity of the non-diffractive beam in air. This result is also the evidence which shows that the center light is the non-diffractive beam. As the difference of the propagation properties of annular beam between propagation in random media and in air, for the case in air, the annular beam just diffracts, and generates the non-diffractive beam, while this process needs a long distance to create the wavefront and the phasefront which can generate a non-diffractive beam. On the other hand, in the case of annular beam propagation in random media, the particles in random media scatter the annular beam. The scattering makes the annular beam get an appropriate wavefront and phasefront distributions in a short distance, which can generate the non-diffractive beam as that in air. However, only the forward scattering can obtain the phasefront which is fit to generate the non-diffractive beam. This is the reason why the narrow detect angle receiver was used in this experiment. The non-diffractive beam can only be created in a certain range of the concentration when the propagation distance is given.
Fig. 7. Process of the optimization.
scattering light is higher, and the non-diffractive beam is generated at a shorter distance with a high intensity as the generated non-diffractive beam (at a few hundred meters in air) is focused at short-distance with different lenses in air. Fig. 8 shows the variation of the center intensity of the nondiffractive beam when the annular beam is focused in air (adding the lens effect with different focal lengths into the calculation results) and in random media (experimental result). Here, L is the propagation distance in random media, and l is the propagation distance in air. In the calculation, the lens effect is added and it renders the nondiffractive beam focused at a short distance. The center intensity at l=210 m is normalized as 1. The experimental results show that the non-diffractive beam has higher intensity in shorter distance propagation in random media. Variation of the center intensity ratio in the experiment (random media) has the same tendency with the variation of the center intensity of the non-diffractive beam in air (with lens effect) when the propagation distance increases with exponential order. For the same attenuation intensity, the relation between propagation distances in air and in random media can be written as
Irandom(L ) ∝ Iair (ln(l ))
4. Conclusion In our experimental setup, the non-diffractive beam was observed at a short distance in random media. The highest intensity ratio of the non-diffractive beams appeared at the concentration of 0.4%, 0.6%, and 1.0% when propagation distances were 30, 20 and 10 cm in a diluted milk solution. For these three concentrations, the random medium had almost the same scattering coefficient, near 22 cm−1. For the optimized non-diffractive beam at different propagation distances, a shorter random medium channel with a higher concentration generated a higher intensity ratio of the non-diffractive beam. We also proposed a method for optimizing the non-diffractive beam propagation in random media by using a forward multiple scattering
(5)
These two distances are associated with the concentration of the random media. We have considered the relationship between the 98
Optics Communications 391 (2017) 94–99
Z. Peng, T. Shiina
weight α, the scattering coefficient σ and the transmittance curve with different concentrations of random media and certain propagation distances. For optimizing the non-diffractive beam in other types of random media with different particle sizes and materials, we can use this method after we have calculated the transport mean free path (multiple scattering times), which is related to the scattering coefficient in the random media. The relation between the annular beam propagation in air and that in random media was discussed. For the case in random media, the variation of the center intensity ratio would have the same behavior with the center intensity through propagation in air if we set the propagation distance increasing with exponential order in air. However, we did not refer any about the propagation characteristics of the non-diffractive beam (generated from the annular beam) in random media in this study. For applying this propagation technique to practical sensing such as super-resolution sensing in random media, a narrow shape and high intensity of the output light is required. The propagation characteristics of the non-diffractive beam in random media is a quite important problem as the future work [30,31]. Such as for human tissue examination in future, e.g. in some thin tissue such as finger or wrist, the non-diffractive beam can be observed by optimizing the shape of the incident beam. The observed non-diffractive beam can be used as a probe to diagnose the state of human tissue. Here, the absorption by human tissue should be also considered [29].
structure and dynamics, Science 252 (1991) 686–688. [10] D. Voelz, X. Xiao, Metric for optimizing spatially partially coherent beams for propagation through turbulence, Opt. Eng. 48 (2009) 036001. [11] E. Gershnik, E.N. Riba, Light propagation through multilayer atmospheric turbulence, Opt. Commun. 142 (1997) 99–105. [12] A.N. Bogaturov, A.A.D. Canas, J.C. Dainty, A.S. Gurvich, V.A. Myakinin, C.J. Solomon, N.J. Wooder, Observation of the enhancement of coherence by backscattering through turbulence, Opt. Commun. 87 (1992) 1–4. [13] Y. Chen, Y. Cai, H.T. Eyyuboglu, Y. Baykal, Scintillation properties of dark hollow beams in a weak turbulent atmosphere, Appl. Phys. B 90 (2008) 87–92. [14] R. Rao, Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence, Appl. Opt. 47 (2008) 269–276. [15] H.T. Eyyuboglu, Y.E. Yenice, Y. Baykal, Higher order annular Gaussian laser beam propagation in free space, Opt. Eng. 45 (2006) 038002-1-038002-038008. [16] Y. Baykal, Y. Cai, X. Ji, Field correlations of annular beams in extremely strong turbulence, Opt. Commun. 285 (2012) 4171–4174. [17] Y. Baykal, Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium, J. Opt. Soc. Am. A 22 (2005) 672–679. [18] V. Soifer, V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation, Taylor & Francis, 1997. [19] H.T. Eyyuboglu, C.Z. Cil, Beam wander of dark hollow, flat-topped and annular beams, Appl. Phys. B 93 (2008) 595–604. [20] H. Gercekcioglu, Y. Baykal, C. Nakiboglu, Annular beam scintillations in strong turbulence, J. Opt. Soc. Am. A 27 (2010) 1834–1839. [21] T. Shiina, K. Yoshida, M. Ito, Y. Okamura, In-line type micro pulse lidar with annular beam - theoretical approach, Appl. Opt. 44 (2005) 7467–7473. [22] H.T. Eyyuboğlu, S. Altay, Y. Baykal, Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence, Opt. Commun. 264 (2006) 25–34. [23] M.A. Öztan, Y. Baykal, C. Nakiboğlu, Effects of extremely strong turbulent medium on scintillations of partially coherent annular and flat-topped Gaussian beams, Opt. Commun. 285 (2012) 943–946. [24] Z. Peng, T. Shiina, Analysis of Annular Light Propagation Characteristics in Random Media, SPIE-OSA-IEEE Asia Communications and Photonics, 8311 [25] G. Scott, N. McArdle, Efficient generation of nearly diffraction-free beams using an axicon, Opt. Eng. 31 (1992) 2640–2643. [26] T. Aruga, Generation of Long-range nondiffracting narrow light beams, Appl. Opt. 36 (1997) 3762–3768. [27] A. Fedotowsky, K. Lehovec, Optimal filter design for annular imaging, Appl. Opt. 13 (1974) 2919–2923. [28] T. Shiina, Y. Tsuge, T. Honda, Examination of propagation process of annular beam in random media, in: Proceedings SPIE 6585, Optical Sensing Technology and Applications, 658524, 2007. [29] D.A. Boas, Diffuse Photon Probes of Structural and Dynamical Properties of Turbid Media: Theory and Biomedical Applications (Ph.D. Thesis), Univ. of Pennsylvania, 1996. [30] Y. Ismail, N. Khilo, V. Belyi, A. Forbes, Shape invariant higher-order Bessel-like beams carrying orbital angular momentum, J. Opt. 14 (2012) 085703. [31] W. Nelson, J.P. Palastro, C.C. Davis, P. Sprangle, Propagation of Bessel and Airy beams through atmospheric turbulence, J. Opt. Soc. Am. A 31 (2014) 603–609.
References [1] C.F. Bohren, Donald R. Huffman, Absorption and scattering of light by small particles, Wiley Sci. Paperb. (1998). [2] T. Shiina, M. Ito, Y. Okamura, Long range propagation characteristics of annular beam, Opt. Commun. 279 (2007) 159–167. [3] P. Sadoghi, Discrete ordinates method for propagation of light in highly scattering vascular tissues, Opt. Commun. 266 (2006) 359–363. [4] Z. Tong, O. Korotkova, Polarization of random beams scattered from twodimensional bio-tissue slices, Opt. Commun. 322 (2014) 202–204. [5] T. Aruga, S.W. Li, Super high resolution for long-range imaging, Appl. Opt. 38 (1999) 2795–2799. [6] A.E. Profio, Light transport in tissue, Appl. Opt. 28 (1989) 2216–2222. [7] W. Gao, Changes of polarization of light beams on propagation through tissue, Opt. Commun. 260 (2006) 749–754. [8] A. Ishimaru, Diffusion of light in turbid material, Appl. Opt. 28 (1989) 2210–2215. [9] D.J. Durian, D.A. Weitz, D.J. Pine, Multiple light-scattering probes of foam
99