Principle and feasibility of using a 3M reflective film based fold pass laser speckle imaging system for measuring atmospheric optical turbulence

Haiping Mei; Hao Ye; Li Kang; Xianmei Qian; Honghua Huang; Yinbo Huang; Wenyue Zhu; Xiaoqing Wu; Ruizhong Rao

doi:10.1364/OSAC.2.001938

1. Introduction

“Twinkle, twinkle, little star, how I wander what you are”, the famous children’s song notes the phenomenon of scintillation for a light beam transmission through atmosphere. It is well known that the phenomenon is caused by atmospheric optical turbulence (AOT), i.e. the random variations of refractive index due to turbulent movement of air masses. AOT influences light waves in both amplitude and phase, as the result, it limits the performance of ground-based large aperture telescope and satellite-to-ground laser communication systems [1,2]. The evaluation of AOT on the performance of those optoelectronic systems depends on the knowledge of AOT itself [3,4]. Though many instruments have been invented to measure the AOT parameters [5–8], they can’t display the turbulence field over the whole light beam, and are hard to be used to judge whether the turbulence is in the state of homogeneous, well developed or not? So, there need a technique to visualize the optical turbulence field and measure it simultaneously.

We notice that the method of projecting a beam of light onto a white wall to generate a wisp of dynamic shadows has always been used for air flow visualization, which is named shadowgraphy [9]. The sunlight, lamplight and laser can all be used to illuminate the air flow, and a long focal length CCD imaging system is needed to record the shadows. The evolution of shadowgraph technique was reviewed in detail by Settles [10,11]. In his review, a laser illuminated shadowgraph system based on a 3M reflective film was introduced to show the large-scale shock waves. The reflectivity of a 3M film is hundreds of times greater than an even high quality white diffuse reflector. With the help of a large area 3M film, the observed area and distance can be enlarged, the laser source and CCD imaging system can be integrated into one single ended unit.

Enlightened by the concept introduced above, a question of to how extent it can be used for the visualization and detection of AOT arises. Firstly, an experimental system is needed to record the turbulence influenced laser speckle patterns reflected by the 3M film. Then, to estimate the turbulence parameters from the recorded laser speckle images, the modeling of laser beam propagation through turbulent atmosphere is also needed. Many researchers have contributed to the theory. Andrews has analyzed the effect of scintillation enhancement for a Gaussian beam reflected by a finite plane mirror and a corner reflector from the point view of mutual correlation function [12,13]. Yang has established a theory for the imaging of a monochromatic laser beam based on the principle of extended Huygens-Fresnel and paraxial approximation [14]. The angle of arrival, coherent length and scintillation index of the echo beams reflected by kinds of reflectors were also numerically simulated [15–17]. All the researches indicate that the characteristics of fold pass laser beam propagation depend on both the turbulence medium and surface character of reflectors, which make the processes more complicated.

This article investigates the principle and feasibility of using a 3M film based fold pass laser speckle imaging system to visualize and measure AOT. It is divided into five sections. In section 2, the theoretical background is briefly reviewed. In section 3, the processes of fold pass laser transmission by the reflection of a 3M film are numerically simulated. In section 4, experiments were performed to reveal the phenomena and even demonstrate the algorithms of retrieving optical turbulence parameters. In the end, we give a short summarization.

2. Theory

As shown in Fig. 1, considering the scenario of a single mode Gaussian laser beam projecting onto a flat 3M film with the reflected speckles being received by a CCD imaging system. The CCD receiver is placed just next to the laser transmitter. The distance L between laser source and the 3M film should be large enough to get noticeable turbulence effects. The process of fold path laser transmission shall be separated into three different stages. They are respectively the forward transmission of a coherent laser beam, speckle patterns generated on the 3M film and backward transmission of incoherent speckle patterns through the reverse turbulence field.

Fig. 1. Schematic diagram of fold pass laser transmission through the turbulence field

Download Full Size | PDF

Denote $A({\boldsymbol r},z,t)$ and $\psi ({\boldsymbol r},z,t)$ as the complex amplitudes of forward and backward transmitted laser beam respectively, they obey the parabolic wave equations [18]

(1)$$2ik\frac{{\partial A({\boldsymbol r},z,t)}}{{\partial z}} = \nabla _ \bot ^2A({\boldsymbol r},z,t) + 2{k^2}{n_1}({\boldsymbol r},z,t)A({\boldsymbol r},z,t),$$

(2)$$- 2ik\frac{{\partial \psi ({\boldsymbol r},z,t)}}{{\partial z}} = \nabla _ \bot ^2\psi ({\boldsymbol r},z,t) + 2{k^2}{n_1}({\boldsymbol r},z,t)\psi ({\boldsymbol r},z,t).$$

Here $\nabla _ \bot ^2{\ =\ }{\partial ^2}\textrm{/}{\partial ^2}x + {\partial ^2}\textrm{/}{\partial ^2}y$ is the Laplacian operator on transversal coordinates, z is the coordinate along the optical axis, ${n_1}({\boldsymbol r},z,t)$ is a time dependent function describing spatial temporal dynamics of refractive index fluctuations.

Denote $U({\boldsymbol r},z,t)$ as the optical field of laser beam, the boundary condition for the outgoing laser beam at the transceiver plain is

(3)$$U({\boldsymbol r},z = 0,t) = {A_0}({\boldsymbol r},t)\exp [i{\varphi _0}({\boldsymbol r},t)],$$

Here ${A_0}({\boldsymbol r},t)$ and ${\varphi _0}({\boldsymbol r},t)$ are respectively the original amplitude and wave-front phase. For a Gaussian beam with the unit amplitude on axis, ${A_0}$ has the expression

(4)$${A_0}({\boldsymbol r}) = \exp [ - {r^2}/W_0^2 - ik{r^2}/(2{F_0})].$$

Here ${W_0}$ is the beam radius corresponding to ${e^{ - 1}}$ fall off in intensity, ${F_0}$ is the curvature radius of the wave-front.

The beam radius and intensity distribution projected onto the target film are approximated by the theory of vacuum diffraction. The beam parameters of the outgoing laser beam at the transceiver plane are respectively ${{\varTheta }_0} = 1 - L/{F_0}$ and ${{\varLambda }_0} = 2L/kW_0^2$, where k is the wave number. When it arrives at the target plain, they become [19]

(5)$${\Theta _1} = 1 - L/{F_1} = {\Theta _0}/(\Theta _0^2 + \Lambda _0^2),$$

(6)$${\Lambda _1} = 2L/kW_1^2 = {\Lambda _0}/(\Theta _0^2 + \Lambda _0^2).$$

Then, the amplitude of the laser beam at the 3M film becomes

(7)$$A({\boldsymbol r},z = L) = \frac{1}{{{\Theta _1} + i{\Lambda _1}}}\exp \left( { - \frac{{{r^2}}}{{W_1^2}} - i\frac{{k{r^2}}}{{2{F_1}}}} \right)\exp ({ikL} ),$$

If turbulence effects are considered, by using the extended Huygens-Fresnel principle, the complex amplitude on the target film just before refracting can be written as [20]

(8)$$A({{\boldsymbol r},z = L} )= -\frac{{ik}}{{2\pi L}}\exp ({ikL} )\int_{ - \infty }^\infty d {\boldsymbol \rho }{A_0}({\boldsymbol \rho } )\cdot \exp \left( {\frac{{ik{{|{{\boldsymbol r} - {\boldsymbol \rho }} |}^2}}}{{2L}} + {\varphi_1}({{\boldsymbol r},{\boldsymbol \rho }} )} \right).$$

Here ${\boldsymbol \rho }$ is the coordinate on transceiver plain, ${\varphi _1}({\boldsymbol \rho },{\boldsymbol r})$ is the random complex phase caused by the AOT.

When the laser spot described by the Eq. (8) encounters a large enough 3M film, it spreads on the film and most of the energy will be reflected backward. The 3M film names after the company (Minnesota Mining and Manufacturing) who firstly invented it. As shown in Fig. 2, it is mainly consisted by adhesive, cushion layer, cubic crystal layer, air layer and polycarbonate material. The crystal is in essence the low precision corner prism. A photo of the film will be used in the following experiment shows about sixty crystals per mm². The 3M film has the advantages of high reflectivity, good uniformity, light in weight and easy to be jointed into large area. It has already been widely used in highway or billboard for notification.

Fig. 2. The (a) composition, (b) mechanism and (c), (d) photograph of prisms in a 3M film

Download Full Size | PDF

The complex amplitude of echo beam just after the reflection has the expression [18]

(9)$$\psi ({\boldsymbol r},z = L,t) = T({\boldsymbol r},t)A({\boldsymbol r},z = L,t),$$

Here $T({\boldsymbol r},t) = V({\boldsymbol r},t)\exp [i\varphi ({\boldsymbol r},t)]$ is the reflecting coefficient. If the reflectivity of nth crystal prism is ${V_n}$ and whose phase difference is ${\varphi _n}$, $T({\boldsymbol r},t)$ can be discreted as

(10)$$T({\boldsymbol r},t) = \sum\limits_{n = 1}^N {{V_n}({{\boldsymbol r}_{\boldsymbol n}},t)\exp ({i{\varphi_n}({{\boldsymbol r}_{\boldsymbol n}},t)} )} .$$

The reflected laser speckle is composed by numbers of bright dots reflected by the prisms embedded in the film. The reflectivity within a certain area, taken 1mm² for example, will be averaged by numbers of prisms, which ensures the uniformity of reflectivity on the whole film. While, in reference to the wavelength of visible laser, the displacement and roughness of each prism are much greater, and the coherence of laser spot must be severely degraded. So, the backward transmitted laser speckle shall be considered as non-coherent. The laser speckles are ultimately focused onto a CCD surface for recording. By eliminating the dark noises and sky background radiance, the digital numbers of recorded images are proportional to the entranced laser intensity. The dynamic shadows in the images are the result of scintillation effect, and the bright dots in the images are the result of speckle effect.

Now, we’ve got the basic understanding of the fold pass laser transmission and imaging. It is easy to understand that the forward transmission of a laser beam onto a 3M film meets the principle of shadowgraphy, and the backward transmission of the laser speckle patterns to the CCD focus plane meets the principle of schilieren [9–11]. But the reality is always more complicated and even more attractive, which is investigated in detail in the following sections.

3. Simulation

For the complexity of optical turbulence, it is hard to get the analytic solution of laser speckle patterns from Eq. (1) and (2). Numerical simulation is the most practical way to uncover the details of fold pass laser transmission. The algorithm of step Fourier transformation based on multilayer phase screen is the widely accepted way to realize the simulation [21]. Under the condition of weak perturbation, the transmission of laser beam can be separated into two independent processes: the diffraction in a vacuum section and the modulation of wave front by a thin phase screens. The procedure is expressed as

(11)$$U(\rho ,{z_{i + 1}}) = {f_2}^{ - 1}\{{\exp (i\Delta z{k_z}){f_2}} {[{[{U(\rho^{\prime},{z_i})\exp [i{S_i}(\rho^{\prime})} ]} ]} \}.$$

Here ${z_i}$ is the location of $i$th phase screen, ${z_{i + 1}}$ is the location of $i + 1$th phase screen, ${f_2}( \cdot )$ and ${f_2}^{ - 1}( \cdot )$ are respectively two dimensional Fourier and reverse Fourier transformation.

3.1 Phase screen

The key of numerical simulation is to construct the phase screen faith to the turbulence field. The method of spectral transformation is the mostly accepted one to generate phase screen, whose principle is to get the phase screen from the reverse Fourier transform of a two-dimensional matrix of Gaussian random number filtered by the power spectrum of turbulence induced phase fluctuation, that is

(12)$$\varphi (x,y) = \sqrt {\Delta {\kappa _x}\Delta {\kappa _y}} \sum {{\kappa _x}} \sum {{\kappa _y}} R({\kappa _x},{\kappa _y})\sqrt {{F_\varphi }({\kappa _x},{\kappa _y})} {e^{j({\kappa _x}x + {\kappa _y}y)}}.$$

Here, $\Delta {\kappa _x}$ and $\Delta {\kappa _y}$ are the sampling interval of x and y coordinates, ${\kappa _x}$ and ${\kappa _y}$ are spatial wave numbers, $R({\kappa _x},{\kappa _y})$ is the matrix of Gaussian number, ${F_\varphi }({\kappa _x},{\kappa _y})$ is the phase power spectrum. For the homogeneous Kolmogorov turbulence, ${F_\varphi }({\kappa _x},{\kappa _y})$ has the form

(13)$${F_\varphi }({\kappa _x},{\kappa _y}) = 2\pi {k^2}\Delta zC_n^2{(\kappa _x^2 + \kappa _y^2)^{ - 11/6}}$$

If sampling theorem is satisfied and sampling number of each coordinate is set as $N$, the discretized phase screen has the form

(14)$$\begin{array}{l} {\varphi _{high}}(m,n) = \frac{{2\pi }}{N}{\left( {\frac{{0.033\pi \Delta zC_n^2}}{{\Delta x\Delta y}}} \right)^{1/2}}\\ \times \sum\limits_{m^{\prime} = - N/2}^{N/2 - 1} {\sum\limits_{n^{\prime} = - N/2}^{N/2 - 1} {\exp \left( {\frac{{2\pi jmm^{\prime}}}{N} + \frac{{2\pi jnn^{\prime}}}{N}} \right) \times \frac{{R(m^{\prime},n^{\prime})}}{{{{\left[ {{{\left( {\frac{{2\pi m^{\prime}}}{{N\Delta x}}} \right)}^2}{\ +\ }{{\left( {\frac{{2\pi n^{\prime}}}{{N\Delta y}}} \right)}^2}} \right]}^{11/12}}}}} } \end{array}$$

Here, m, $m^{\prime}$, n and $n^{\prime}$ are integers, $\Delta x$ and $\Delta y$ are respectively the sampling interval of x and y coordinates in the spatial domain. The area of phase screen is $N\Delta x \times N\Delta y$, the spatial wave number intervals are $\Delta {\kappa _x} = 2\pi /(N\Delta x)$ and $\Delta {\kappa _y} = 2\pi /(N\Delta y)$, and the minimal spatial frequency intervals are $\Delta {f_x} = 1/(N\Delta x)$ and $\Delta {f_y} = 1/(N\Delta y)$. The subscript high in ${\varphi _{high}}$ means that the low frequency components relate to the aperture of phase screen during the range of $( - \Delta {f_x}/2,\Delta {f_x}/2)$ and $( - \Delta {f_y}/2,\Delta {f_y}/2)$ is excluded from the formula. To compensate the low frequency components, a subharmonic algorithm was proposed by lane [21], which is

(15)$${\varphi _{low}}(m,n) = \sum\limits_{p = - 1}^{{N_p}} {\sum\limits_{m^{\prime} = - 1}^1 {\sum\limits_{n^{\prime} = - 1}^1 {\exp [2\pi j{3^{ - p}}(\frac{{mm^{\prime}}}{N} + \frac{{nn^{\prime}}}{N})]} } } \times R(m^{\prime},n^{\prime}) \times f(m^{\prime},n^{\prime})$$

Here ${N_p}$ is the orders of subharmonics, $f(m^{\prime},n^{\prime})$ is the function relates to the power spectral density of phase. Add it to the high frequency components, we get

(16)$$\varphi (m,n) = {\varphi _{high}}(m,n) + {\varphi _{low}}(m,n)$$

Figure 3 gives two examples of the phase screen generated by the synthesized algorithm.

Fig. 3. Kolmogorov phase screen simulated under the condition (a) $C_n^2 = {10^{ - 15}}{m^{ - 2/3}}$ and (b) $C_n^2 = {10^{ - 14}}{m^{ - 2/3}}$, the distance $L = 250m$

Download Full Size | PDF

3.2 Reflective film

Different from turbulence induced phase screen, the phase differences caused by 3M film is time invariant. For a Gaussian distributed rough surface with the correlation length ${\rho _h}$ and root mean square $\sigma _h^2$ of height variation, its height correlation function has the form [22]

(17)$${B_h}({\rho _1},{\rho _2}) = \langle{h({\rho_1})h({\rho_2})} \rangle = \sigma _h^2\exp (\Delta {\rho ^2}/\rho _h^2),$$

Where $h(\rho )$ is the height distribution function, $\Delta \rho = {\rho _1} - {\rho _2}$ is the displacement of any two points on the film. Then, the power spectrum density of height variation can be written as

(18)$${\Phi _h}(\kappa ) = \sqrt \pi \sigma _h^2\exp ( - {\kappa ^2}\rho _h^2/4).$$

Figure 4 shows an example of the simulated $h(\rho )$ generated from the function of power spectrum by the algorithm of spectral inversion. In which, $\sigma _h^2$ is set as 0.01 mm and ${\rho _h}$ is set as 50 mm. The former is about ten percent of the dimension of each crystal prism and the latter is about ten percent of the dimension of simulated target screen.

Fig. 4. An example of simulated height distribution for the Gaussian distributed rough surface

Download Full Size | PDF

Suppose that the averaged reflectivity on the film is ${R_d}$, the scattered optical field by the film will has the expression

(19)$${U_s}(\rho ) = {R_d}W(\rho )U(\rho )\exp [i2kh(\rho )]$$

Here $U(\rho )$ and ${U_s}(\rho )$ are respectively the incident and scattered optical field, $\rho $ is the transverse coordinate on the film. $W(\rho )$ is the aperture function whose value is $W(\rho ) = 1$ when its dimension is larger than the area of numerical simulation.

3.3 Speckle pattern

Numerical simulation is operated to reveal the speckle pattern of backscattered optical field. The parameters used in this simulation are listed in Table 1. The original laser beam is ideally collimated and the curvature radius ${F_0}$ is supposed to be infinite. The results of single trip and fold pass propagation of a laser beam in vacuum, weak turbulence and moderate strength turbulence field are shown in Fig. 5. Without consideration of the divergence caused by reflection, the radius of returned laser speckle (b), (d) and (f) back to the transceiver plain is about twice the radius of the laser spot (a), (c), (e) at the reflective plain. The speckle pattern (b) in vacuum is determined by the surface character of the reflective film and is time invariant. It is also the basis of the turbulence influenced speckle pattern (d) and (e). The influence of optical turbulence on the speckle pattern is hard to be recognized by naked eyes, but it is easy to be identified by the algorithm of digital image processing. After binaryzation (g) and xor operation, the differences appear in (h) and (i), which show that the deviation increases significantly with the increasing of turbulence intensity.

Fig. 5. The laser speckle patterns of single trip and fold pass transmitted laser beam: (a), (b) in vacuum; (c), (d) in $C_n^2 = {10^{ - 15}}{m^{ - 2/3}}$; (e), (f) in $C_n^2 = {10^{ - 14}}{m^{ - 2/3}}$; (g) the binary speckle in vacuum; (h) the difference of speckle pattern between turbulence in $C_n^2 = {10^{ - 15}}{m^{ - 2/3}}$ and in vacuum; (i) the difference of speckle pattern between turbulence in $C_n^2 = {10^{ - 14}}{m^{ - 2/3}}$ and in vacuum

Download Full Size | PDF

Table 1. Parameters of numerical simulation

View Table

4. Experiments

Though the numerical simulation provides the characteristics of laser speckles reflected by a 3M film, it is just similar to but not same as the reality. The model of phase screen is simpler than the reality and the surface height of a 3M film is hard to be measured exactly. Experiments are essential and helpful for the understanding of varied laser speckles and exploring the potential of retrieving turbulence parameters from the laser speckle patterns.

4.1 Qualitative experiments

For the qualitative research, a general experimental system was setup. In which, a diode laser head works at the wavelength of 650 nm with the maximum power of 200 mW was used as the light source. Its output power and divergence were adjustable. A CMOS camera (resolution 1936×1096 pixels) coupled with a telescope (Φ=100 mm, F = 1400 mm) were used to record the laser speckle images. A 10 nm bandwidth optical filter was setup in front of the camera to reduce the background sky radiance. All the above units were mounted on a sturdy tripod. A 1 m square 3M film pasted on a flat carbon fiber board was placed at a certain distance perpendicular to the laser beam. The divergence of laser beam was adjusted larger enough to illuminate the whole film.

Two experiments were performed (see Fig. 6) and part of the area on the film (about as large as an A4 paper) was recorded. From the movies, we can see the shadows of the hot air flow (b) and natural turbulence field (d) move dynamically on the background of randomly distributed bright dots. And, the turbulence information is contained in the variation of the brightness, shape and position of the dots.

Fig. 6. Experimental setup for air flow visualization, the sampling frequency was 170fps, (a) experiment performed in a floor corridor within 42 m, a cup of hot water was placed beneath the laser beam in the middle; (b) the shadow of hot air flow on the random dot speckle pattern (see Visualization 1); (c) experiment performed 1.5 m above the grass land within 100 m; (d) the shadow of natural turbulence field added on the random dots (see Visualization 2), the upward, downward, left, right are respectively sky, ground, south and north

Download Full Size | PDF

From the point view of digital image processing, random dots in an image are generally considered as high frequency white noise. After low pass and high pass filtering, the images are decomposed into low frequency shadows and high frequency bright dots (see Fig. 7). If the turbulence is strong enough to produce notable effects, the movement of shadows is easy to be estimated from adjacent shadows with the algorithm of cross correlation [23], and the vector vortices are the representative of path integrated transversal wind field. As for the dots, results show that their shapes are maintained in weak perturbation but much distorted in strong perturbation. So, theoretically we can get the two-dimensional displacement matrix and path integrated air density distribution in weak turbulence, but unfortunately, it is hard to get a reference image in the air without turbulence, and the problem can’t be solved in this article.

Fig. 7. Image of laser speckle patterns estimated by the algorithm of frequency decomposition and cross correlation, (a) the original images; (b) the low frequency component of shadows; (c) the binarized high frequency component of the bright dots; (d) the movement of shadows in weak turbulence, its averaged speeds were -3.0 mm/s horizontally and -5.7 mm/s vertically; (e) the movement of shadows in strong turbulence, the averaged speeds were 14.0 mm/s horizontally and -15.5 mm/s vertically, the vector vortexes noted as the arrow length and pseudo color was caused by the transversal wind field, and the size in the image was calibrated as 0.22 mm/pixel

Download Full Size | PDF

4.2 Quantitative experiments

For the quantitative research, a professional experimental system was setup with the configuration of two AOT monitoring instruments (see Fig. 8). The light source was a 671 nm single mode solid state laser (the maximum power is 200 mW, the stability is better than 1%). A high speed 12bit CMOS camera (Model PCO.dimax HD, the maximum sampling frequency is 1603fps@1920 ×1440 full resolution) was used with the cooperation of a long focal length Cassegrain telescope (the aperture is 350 mm, the focal length is 3.55 m). A 1.2 m square 3M film pasted on a flat aluminum frame was arranged just 1000 m away perpendicular to the laser beam. The AOT monitoring instruments were respectively a large aperture laser scintillometer (Model BLS450, the receiver’s aperture is Φ=140 mm) and an atmospheric coherence meter (Model ATCM-Meade, the receiver’s aperture is Φ=350 mm, its sub-aperture is Φ=126 mm).

Fig. 8. Schematic diagram of the experimental setup for the quantitative experimentation

Download Full Size | PDF

The transceiver components were placed on an optical table in the laboratory located at the Science Island, Hefei, China. The 3M film was placed near the window of a house located at the edge of the lake. The underlying surface was about 100 m grassland near the laser source and 900 m water surface near the film. The medium height of the laser beam to the ground was about 12 m. The receiver of scintillometer was placed close to the reflective film and the atmospheric coherence meter was placed close to the laser source. The experiment was conducted in an extremely sunny day whose visibility was always larger than 40 km.

A movie made of the time series of laser speckle images is shown in Fig. 9. In which, the area of 231×231 pixels are corresponding to about 1m×1 m on the film (4.3 mm/pixel). Affected by the 1 km turbulence medium, the centroid of each speckle varies randomly with time, and each speckle’s centroid can be estimated by the equation

(20)$$\begin{array}{l} {x_c} = \sum\limits_{i = 1}^M {\left( {i \cdot \sum\limits_{j = 1}^M {{I_{i,j}}} } \right)} /\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^M {{I_{i,j}}} } ,\\ {y_c} = \sum\limits_{j = 1}^M {\left( {j \cdot \sum\limits_{i = 1}^M {{I_{i,j}}} } \right)} /\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^M {{I_{i,j}}} } . \end{array}$$

Here, the intensity of incident laser speckle ${I_{i,j}}$ is proportional to the effective digital number $DN$ estimated by the subtraction of the digital number of dark pixels. So, using these images, laser speckle intensities on the reflective film can be statistically analyzed, and some of the optical turbulence parameters can be retrieved.

Fig. 9. Movie of recorded laser speckle images (see Visualization 3), the sampling frequency was 1000 fps

Download Full Size | PDF

4.3 Probability density function and spatial correlation function

Experiments have shown kinds of probability density function (PDF) of light intensity for the propagation of a laser beam in the turbulence atmosphere [24]. But how it appears on the laser speckle images is still a mystery. To get the PDFs, 6000 adjacent images are counted to get the effective digital numbers in different turbulence intensities, and the results are shown in Fig. 10. In which, (a) and (c) indicate that the PDF at any single pixel well obeys the log-normal distribution, (b) and (d) indicate that the PDF of averaged laser intensity in the centered rectangle area varies obviously with the side length. It confirms that the PDF varies with the area of receiver aperture.

Fig. 10. The probability density function of laser intensity represented by digital number, (a) (c): the single pixel at different axis offset d; (b) (d): the centered rectangle area with different side length L

Download Full Size | PDF

Taken the centroid of laser speckles as the original point and define ${C_I}(r)$ and ${\sigma _I}( \cdot )$ as the correlation coefficient and variance of light intensity respectively, the normalized spatial correlation coefficient has the expression

(21)$${B_I}(r) = {C_I}(r)/[{{\sigma_I}(0) \cdot {\sigma_I}(r)} ],$$

${B_I}(r)$ at different turbulence intensities are shown in Fig. 11, which indicate that it decreases with the increasing of r and tends to 0 when reach 15∼20 pixels. If r at the value of ${B_I}(r) = 1/e$ (the line in figure) is defined as the correlation distance of laser speckle patterns, it tends to rise with the increasing of turbulence intensity. The values of correlation distance agree well with Tatarskii’s theoretical expectation of spherical beam propagation in weak turbulence, which is about 1∼3 times of $\sqrt {\lambda L} = 25.6mm$.

Fig. 11. Variation of normalized correlation coefficient with axis offset r

Download Full Size | PDF

4.4 Scintillation index and refractive index structure constant

Scintillation index $\beta _I^2$ defined as the normalized variance of light intensity is one of the most important parameters represent the effect of turbulence on the laser beam. To disclose how it appears on the laser speckle images, $\beta _I^2$ in centered circular with different diameters D are calculated. Figure 12 (a) shows the result of $\beta _I^2(D)$ at five different turbulence intensities, which proves the property of aperture smoothing. In princile, $\beta _I^2(0)$ is corresponding to none pixel, so it isn’t estimated directly from the images but from the fitted function of $\beta _I^2(D > 0)$. Here, $\beta _I^2(D > 0)$ is well fitted by four orders of polynomial fitting.

Fig. 12. (a) variation of scintillation index with the diameter D and Fried number ${r_0}$; (b) comparing of refractive index structure constant measured by three methods, the x coordinate is the Beijing time from 14:30 (T1) to the same time next day (T49), T10∼T36 refers to the night time from 19:00 to 7:00 next day, the sampling time of scintillometer, coherence meter and laser speckle imaging system are respectively 1 min, 20s, and 6s, they are determined by the instruments

Download Full Size | PDF

Refractive index structure constant $C_n^2$ is the measure of optical turbulence intensity. It is usually estimated from the point scintillation index. For the forward transmission from the laser source to the film, the scintillation index meets the theory of spherical wave propagation through turbulence atmosphere, i.e. $\beta _I^2(0) = 0.496C_n^2{k^{7/6}}{L^{11/6}}$ [25]. While for the optical imaging of the backward transmitted laser speckle patterns, the scintillation effect is well smoothed by 0.35 m receiver aperture and can be neglected. Figure 12 (b) compares the estimated $C_n^2$ with the monitored ones at the wavelength of 671 nm. Though all the differences of principle, sampling time, noise and the non-uniform underlying topography make the values of $C_n^2$ difference, the similar trends and overlapped values of $C_n^2$ in the whole day still makes the comparison meaningful. The fold pass laser speckle imaging system is proved to be a novel instrument to get the scintillation index, aperture smoothing function and turbulence intensity simultaneously.

4.5 Inner and outer scales

Inner scale ${l_0}$ and outer scale ${L_0}$ are two spatial structure parameters describing the low and up boundaries of the turbulence inertial subrange. ${l_0}$ is the dimension where the dissipation of kinetic energy occurs and ${L_0}$ is the dimension where the energy is injected into the turbulent medium. Numerical simulation has revealed that ${l_0}$ plays a noticeable role while ${L_0}$ has only negligible effect on scintillation [26]. According to Churnside’s theory of aperture averaging for the spherical wave propagation in weak turbulence [27], when ${r_0} > \sqrt {L/k} $ and ${l_0} > 1.5\sqrt {L/k} $, ${l_0}$ is related to the aperture averaging factor by $A = {[1 + 0.109{(D/{l_0})^{7/3}}]^{ - 1}}$. Otherwise when ${r_0} > \sqrt {L/k} $ and ${l_0} \le 1.5\sqrt {L/k} $, $A = {[1 + 0.214{(k{D^2}/4L)^{7/6}}]^{ - 1}}$, that is the small-inner-scale approximation.

In our experiment, the aperture averaging factor is estimated by the definition $A = \beta _I^2(D)/\beta _I^2(0)$ from the data shown in Fig. 12 (a), and the results are shown in Fig. 13 as the scattered dots with the lines fitted by to get the ${l_0}$. It can be seen from the figure that when D is smaller than 120 mm, A deviates the theory of weak-turbulence approximation, which is in consistence with what have been given by Churnside himself (Fig. 14 and 15 in ref. [27]). Just as mentioned in his article, six aperture diameters smaller than 50 mm were insufficient to verify his theory. Now, it is for luckily that his expectation is verified by our experiment for that when D is larger than 120 mm, almost all the experimental results meet well with the fitted lines, and when the inner scale is small enough, taken ${r_0} = 25.72cm$ for example, the experimental results are also very close to the expression of small-inner-scale approximation.

Fig. 13. Comparison of measured and fitted aperture averaging factor with the inner scales retrieved from the fittings, the same turbulence strength is noted with the same color, they match well when the aperture is larger than 120mm

Download Full Size | PDF

As for the outer scale, it has little effect on the scintillation index and the aperture averaging factor, so it can’t be retrieved from the data shown above. The problem of how the outer scale influences the image of reflected laser speckle pattern is still a mystery, which needs to be investigated in the next stage.

5. Conclusion

The process of 3M film based fold pass laser speckle imaging through atmospheric optical turbulence shall be separated into three stages: the forward transmission of a coherent laser beam from the laser source to the 3M film, the reflected laser speckle pattern generated on the 3M film, and the backward transmission of incoherent laser speckle pattern to the laser source. Numerical simulation reveals the mechanism of random bright dots scattered on the reflected laser speckle pattern. It is time invariant in vacuum and varies dynamically in turbulence field. Qualitative experiments show the dynamical images of the laser speckle pattern. From the point view of digital image processing, the images of laser speckle pattern can be separated into high frequency bright dots and low frequency shadows, they are the good indicator to visualize the optical turbulence field. Statistical characters of the far field laser speckle, such as probability distribution function, scintillation index, spatial correlation coefficient and aperture smoothing function are easy to be estimated from the time series of laser speckle images. If the condition is satisfied, the refractive index structure constant and inner scale of optical turbulence can be retrieved with the usage of appropriate models. Our work is just the beginning of exploring the potency of a laser speckle imaging system in optical turbulence visualization and detection. For further research, more sophisticated digital image processing algorithms are needed to retrieve the parameters of AOT.

Funding

Chinese Academy of Sciences (CAS) (XDA17010104); National Natural Science Foundation of China (NSFC) (91752103).

Acknowledgments

This research was supported in part by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA17010104) and National Science Foundation (NSF) (91752103). And, we thank William Thielicke and Eize J. Stamhuis for their free software Digital Particle Image Velocimetry Tool for MATLAB (PIVlab).

References

1. T. S. McKechnie, “Atmospheric turbulence and the resolution limits of large ground-based telescopes,” J. Opt. Soc. Am. A 9(11), 1937–1954 (1992). [CrossRef]

2. L. C. Andrews, R. L. Phillips, and P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34(33), 7742–7751 (1995). [CrossRef]

3. https://www.researchgate.net/profile/Elena_Masciadri2/publication/252209187_OPTICAL_TURBULENCE_Astronomy_Meets_Meteorology/links/551acd480cf251c35b4fe46c/OPTICAL-TURBULENCE-Astronomy-Meets-Meteorology.pdf

4. M. Chen, C. Liu, D. Rui, and H. Xian, “Performance verification of adaptive optics for satellite-to-ground coherent optical communications at large zenith angle,” Opt. Express 26(4), 4230–4242 (2018). [CrossRef]

5. F. D. Eaton, “Recent developments of optical turbulence measurement techniques,” Proc. SPIE 5793, 68–72 (2005).

6. S. M. Xiao, H. P. Mei, and R. Z. Rao, “Fiber optic turbulence sensing system,” Proc. SPIE 8417, 841733 (2012). [CrossRef]

7. J. Ko and C.C. Davis, “Comparison of the plenoptic sensor and the Shack–Hartmann sensor,” Appl. Opt. 56(13), 3689–3698 (2017). [CrossRef]

8. L. Y. Liu, Y. Q. Yao, J. Vernin, H. S. Wang, J. Yin, and X. Qian, “Multi-instrument characterization of optical turbulence at the Ali observatory,” J. Phys.: Conf. Ser. 595, 012019 (2015). [CrossRef]

9. M. J. Hargather and G. S. Settles, “Retroreflective shadowgraph technique for large-scale flow visualization,” Appl. Opt. 48(22), 4449–4457 (2009). [CrossRef]

10. G. S. Settles, “Schlieren and shadowgraph techniques,” (Springer-Verlag, 2001).

11. G. S. Settles and M. J. Hargather, “A review of recent developments in schlieren and shadowgraph techniques,” Meas. Sci. Technol. 28(4), 042001 (2017). [CrossRef]

12. L. C. Andrews and W. B. Miller, “The mutual coherence function and the backscatter amplification effect for a reflected Gaussian-beam wave in atmosphere turbulence,” Waves in Random Media 5(2), 167–182 (1995). [CrossRef]

13. L. C. Andrews, R. L. Phillips, and W. B. Miller, “Mutual coherence function for a double-passage retroreflected optical wave in atmospheric turbulence,” Appl. Opt. 36(3), 698–708 (1997). [CrossRef]

14. C. C. Yang, B. H. Elsebelgy, and M. A. Plonus, “Imaging after double passage through a turbulence medium,” Opt. Lett. 18(24), 2087–2089 (1993). [CrossRef]

15. Y. X. Zhang, “Angle-of-arrival fluctuation of reflected laser beam in atmospheric turbulence,” Laser Technol. 21(1), 25–29 (1997).

16. X. Z. Ke and J. Wang, “Intensity of reflected wave from corner reflector illuminated by partially coherent beam in the atmospheric turbulence,” Acta Optical Sinica 35(10), 1001001 (2015). [CrossRef]

17. H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta Physica Sinica 57(10), 6666–6672 (2008).

18. V. V. Dudorov, “Speckle-field propagation in frozen turbulence: brightness function approach,” J. Opt. Soc. Am. A 23(8), 1924–1936 (2006). [CrossRef]

19. H. Y. Wei, “Laser beam propagation on the slant path through the turbulent atmosphere and the characteristics of returned waves by targets”, (Paper for Doctor’s degree, Xidian University, Xi’an, 2009).

20. L. C. Andrews and R. L. Phillips, “Laser beam propagation through random media, second edition,” (SPIE press, 2005), P.159.

21. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in Random and Complex Media 2(3), 209–224 (1992). [CrossRef]

22. A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136(2), 313–327 (1990). [CrossRef]

23. W. Thielicke and E. J. Stamhuis, “PIVlab – Towards User-friendly, Aﬀordable and Accurate Digital Particle Image Velocimetry in MATLAB,” Journal of Open Research Software 2(1), e30 (2014). [CrossRef]

24. V. S. Rao Gudimetla and J. Fred Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72(9), 1213–1218 (1982). [CrossRef]

25. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999). [CrossRef]

26. X. Yi, Z. Liu, and P. Yue, “Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012). [CrossRef]

27. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991). [CrossRef]

Name	Description
Visualization 1	The shadow of hot air flow on the random dot speckle pattern.
Visualization 2	The shadow of natural turbulence field added on the random dots.
Visualization 3	Movie of recorded laser speckle images, the sampling frequency was 1000fps.

Parameter	Symbol	Value
Wavelength	$λ$	671nm
Beam radius	$W_{0}$	3mm
Refractive index structure constant	$C_{n}^{2}$	$10^{- 15} / 10^{- 14} m^{- 2 / 3}$
Transmission Distance	$L$	1000m
Distance of phase screen	$Δ z$	250m
Film reflectivity	$R_{d}$	0.8
Grid width	$Δ x$	1mm
Grid Number	$N$	$512 \times 512$

Principle and feasibility of using a 3M reflective film based fold pass laser speckle imaging system for measuring atmospheric optical turbulence

Abstract

1. Introduction

2. Theory

3. Simulation

3.1 Phase screen

3.2 Reflective film

3.3 Speckle pattern

4. Experiments

4.1 Qualitative experiments

4.2 Quantitative experiments

4.3 Probability density function and spatial correlation function

4.4 Scintillation index and refractive index structure constant

4.5 Inner and outer scales

5. Conclusion

Funding

Acknowledgments

References

Supplementary Material (3)

Cited By

Figures (13)

Tables (1)

Equations (21)

OSA Continuum