1 Introduction

Atmospheric optics are one of the branches of optics that describe light transmission, absorption, emission, refraction and reflection by the atmosphere [1, 2]. This includes an understanding of various optical phenomena such as the blue colour of the sky, rainbows, twinkling of stars and mirages involving visible and near-visible radiation as well as the propagation and distortion of electromagnetic signals through the air [2].

The propagation of light through the atmosphere depends to large extents on the physical composition of the atmosphere several optical interaction phenomena and the frequent changes in the atmosphere due to turbulent air and various other factors [2,3,4]. Atmospheric turbulence is attributed to slight spatial and temporal fluctuations in temperature, velocity and refractive index on a rapid scale which in turn causes the air particles in the atmosphere to randomly mix together [4, 5].

The turbulent atmosphere causes the beam intensity of a propagating wave to fluctuate or scintillate, leading to distortion of the beam and occurrences of phenomena such as beam wander, beam spreading and random displacement of images [6,7,8,9].

Although there have been many investigations and models on the propagation properties of various coherent sources such as laser beams in a turbulent atmosphere [10,11,12,13,14], the characteristics of the optical wave transmitted in different atmospheric propagation conditions have to date not been extensively studied. This knowledge represents an important task and can assist in the adequate evaluation of atmospheric turbulence effects on optical system performance and development.

In this paper, following our recent work [11], we provide the findings from our investigation into the effects of thermal turbulence on a propagating laser beam in the air that was obtained through analysing the positional impacts of a heat source. Several heat sources in the earth’s atmosphere can significantly create an inhomogeneous rise in air temperature. This rise in air temperature increases the convection air current mixing turbulently with the upper cooler air layers [15, 16].

The turbulence source was a proportional integral derivative (PID) controlled heated plate positioned above (bottom-up) and beneath a He–Ne 532 nm Gaussian laser beam with a constant plate to beam separation. The thermal turbulence effect on the beam was measured using a point diffraction interferometer (PDI) capable of precise detection and measurement of wave-front irregularities [17].

The interferograms and turbulence-induced intensity at different temperatures were imaged through cameras placed in the focal-planes with the interference patterns displayed simultaneously on a connected monitor. The data were analysed by image processing software wherein various characteristics such as the temperature, phase and refractive index structure functions, the Rytov variance and Fried’s parameter (coherence diameter) of the turbulent model were determined. The findings are discussed in detail.

2 Theory

When a laser beam propagates through a turbulent medium, there exists random changes or fluctuations in the irradiance (or spatial power density) received [15]. The degree of fluctuating irradiance is quantified by the scintillation index calculated by the Rytov variance \(\sigma_{R}^{2}\) represented in terms of the refractive-index structure coefficient \(C_{n}^{2}\) as [15, 18]:

$$\sigma_{R}^{2} = 1.23C_{n}^{2} k^{{\frac{7}{6}}} L^{{\frac{11}{6}}}$$
(1)

Here k and L are the wavenumber and propagation length, respectively. The Rytov variance values are \(\sigma_{R}^{2} \ll 1\), \(\sigma_{R}^{2} \sim1\) and \(\sigma_{R}^{2} \gg 1\) for weak, moderate and strong degrees of scintillation [19].

The optical effects of atmospheric turbulence depend primarily on the refractive-index structure coefficient \(C_{n}^{2}\) which can be expressed for dry air in the visible and infrared wavebands as [6, 18]:

$$C_{n}^{2} = \left[ {79.0 \times 10^{ - 6} \left( {\frac{P}{{T^{2} }}} \right)} \right]^{2} C_{T}^{2}$$
(2)

with

$$C_{T}^{2} = \frac{{\left\langle {\left( {T_{1} - T_{2} } \right)^{2} } \right\rangle }}{{r^{{\frac{2}{3}}} }}$$
(3)

The temperature structure–function \(C_{T}^{2}\) in (K2 m−2/3) defined as the temperature difference between two reference points, separated by an optical distance r. The atmospheric pressure is (P) in millibars, and T is the ambient temperature in kelvin within the turbulent region [6].

The intensity redistribution caused as the laser beam propagates in the turbulent medium can be described using the phase structure–function \(D_{s}\) measured at a given position along the propagation path as [20, 21]:

$$I\left( {\rho_{c} ,z} \right) = A_{0}^{2} \frac{{2W_{0}^{2} }}{{W^{2} }}\exp \left( { - \alpha_{E} z} \right)\mathop \int \limits_{0}^{\infty } tJ_{0} \left( {t\rho^{{\prime }} } \right) \times \exp \left[ { - t^{2} - \left( {\frac{1}{2}} \right)D_{s} t^{{\frac{5}{3}}} } \right]{\text{d}}t$$
(4)

where A0 is the uniform amplitude of the plane laser wave, W0 is the Gaussian beam size, J0 is the first-order Bessel function, \(\rho\) is the transverse distance from the beam, and W is the beam size at z the propagation path length within the turbulent region.

The phase structure–function \(D_{s}\), a measure of the random variations in the atmospheric refraction index, is often expressed as [21]:

$$D_{s} = 2.91C_{n}^{2} r^{{\frac{5}{3}}} z\left[ {1 - 0.8\left( {\frac{2\pi r}{{L_{0} }}} \right)^{{\frac{1}{3}}} } \right]$$
(5)

where L0 relates to the scale size of the turbulent eddies and r the difference between the inner and outer scales.

Another parameter vital to understanding the integrated strength of atmospheric turbulence on optical imaging systems is the atmospheric coherence diameter r0 (also called the Fried parameter). The Fried parameter depends on the turbulence profile C 2n (z), the zenith angle ζ and the wavelength λ as follows [20]:

$$r_{0} = 0.185\left[ {\frac{{\lambda^{2} }}{{\mathop \smallint \nolimits_{0}^{z} C_{n}^{2} \left( \xi \right){\text{d}}\xi }}} \right]^{{\frac{3}{5}}}$$
(6)

Typically, r0 values under good observation conditions range between 5 cm to 20 cm. This is entirely dependent on the seeing conditions at the time.

3 Experimental details

The experimental setup (Fig. 1), as adapted from our previous work, is built in such a way as to allow the laser beam to propagate through the turbulent region with a controlled turbulence model. A detailed description of the whole optical train and associated components can be found in the work of Augustine et al. [11]. The turbulence source in this study is a heated aluminium panel, 1-cm thick with dimensions 20 cm × 20 cm. An insulating material is introduced between the heating plate and electronics panel to maintain an even heat distribution and minimise heat loss to the surroundings. The homogeneous distribution of heat is due to the calculative structure of the turbulence panel resistors (Fig. 2).

Fig. 1
figure 1

2D front view of the experimental setup

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Fig. 2
figure 2

Schematic representation of the turbulence resistance panel

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In our quest to determine the positional impact of thermal turbulence on a propagating laser beam, the PID temperature-controlled heated plate used to generate the uniform thermal turbulence was placed at a distance of 15 mm above (bottom-up) and below the He–Ne 532 nm Gaussian beam centre. Negligible turbulence effect of the laser was observed beyond that point. The alignment of the laser was checked using a monitoring screen for a perfect interferogram, after which adequate time was given to the laser to stabilize before each set of readings.

The air temperatures (T1 and T2) in Kelvin at two reference points from the centre of the heated plate (centre of the beam and 14 cm to the left) and pressure difference in kPa in and out of the turbulent region was measured simultaneously using a thermocouple positioned in very close proximity to the laser beam, and a differential pressure sensor at a sensitive setting of 0–10 kPa.

The resultant perturbed interferogram at characteristic air temperature and pressure were captured by a digital single-lens reflex (DSLR) camera. This was chosen for its ease of operation, low cost, and high resolution and later transferred to a high definition (HD) monitor for proper visualization.

The interferogram images were then analysed by extracting the wavefront information from the phase shifts due to thermal perturbations on the propagating laser beam using MATLAB.

Since pressure is pivotal in the determination of thermal turbulence effects, we further introduced a GLX XPLORER to accurately measure the atmospheric pressure inside the experiment room. The GLX Xplorer is a digital easier to manage data measuring and analysing device. It incorporates a pressure sensor with an improved resolution to detect the smallest changes or differences in the atmospheric pressure.

4 Result and discussion

The positional impacts of a turbulence source on a propagating laser beam were studied experimentally. Analysis was restricted to comparisons between results obtained at the same temperature as exhaustive discussion and comparisons on the unperturbed system at varying temperatures have been made in our previous study [11]. The measurements were carried out for five different temperatures of the heated plate from (80 °C to 180 °C) and (90 °C to 240 °C) for the upwards and downwards heat flow direction, respectively. The temperature of the heated plate and the air temperatures for both positions are presented in Tables 1 and 2. For each reading, the value of air temperature did not correlate with the apparent temperature of the plate probably due to heat being transferred only through convection [11].

Table 1 Actual air temperature and temperature of the heated plate for upwards heat flow
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Table 2 Actual air temperature and temperature of the heated plate for downwards heat flow
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Figures 3, 4, 5, 6 and 7 showed the interferogram images with intensity profiles and relevant atmospheric turbulence parameters obtained at equal temperatures for the two positions. The results indicate that the beam deviations (centroid wander), beam size (spreading), and intensity distribution within the beam (scintillation) can be strongly affected by the position of turbulence source. Specifically, for experimental runs where the turbulence source was below the beam, the degrees of energy redistribution, blurriness, defocus and beam jitter increased. However, for runs where the heated plate was rotated 180˚ about the beam axis, an increase in temperature primarily intensified stretching of the upper half of the beam with the lower half maintaining its original shape, since the region of higher kinetic energy is naturally upwards, thus hindering its approach to the lower half.

Fig. 3
figure 3

Interferogram and intensity profile of readings 1 and 1†

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Fig. 4
figure 4

Interferogram and intensity profile of reading 2 and 2†

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Fig. 5
figure 5

Interferogram and intensity profile of reading 3 and 3†

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Fig. 6
figure 6

Interferogram and intensity profile of reading 4 and 4†

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Fig. 7
figure 7

Interferogram and intensity profile of reading 5 and 5†

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At 31.0 °C (Fig. 3), there were minimal differences in the Gaussian shape of the intensity profiles of readings 1 and 1†. However, when comparing the upper halves of each interferogram or the left-hand side portion of the intensity profiles, the distance between the first anti-node and the fifteenth anti-node was found to be approximately 240 for both readings. From Fig. 4 (37.0 °C), the increase in temperature resulted in an upwards stretching effect – though a minute value of 10 – to the upper halves of the beam with minor energy redistribution between 400 and 700 in reading 2. Consequently, the distance between first to fifteenth anti-nodes was roughly 240 and 250 for readings 2 and 2†, respectively.

In Fig. 5, due to the weaker atmospheric turbulence (i.e. smaller structure parameter \(C_{n}^{2}\)), than that in Figs. 3 and 4, there were signs of a homocentric slightly compressed dark beam centre in reading 3 with slight energy redistribution between 0 and 150, as well as between 400 and 570. The intensity profile still assumes a Gaussian profile, but with an increased stretching effect of 20 in reading 3† compared to reading 3.

The continual increase in temperature to 47.0 °C (Fig. 6) further leads to an increase in the extent of fluctuations and broadening effect in the upper region of the beam profiles with a distance of approximately 25 in the first and fifteenth anti-nodes to the right of the profile plot. No energy redistribution occurred in Reading 4† as opposed to regions 0 to 220 and 400 to 650 in Reading 4. The interferograms maintain their Gaussian shapes though signs of image blurring and beam distortion were evident in Reading 4.

The rise in heated plate temperature (Fig. 7) resulted in further stretching of the upper halves of the beam profiles with a substantial increment of approximately 40 pixels in the distance between the first and fifteenth anti-nodes of readings 5 and 5†. Moreover, there were signs of beam distortion especially within the vicinity of the centroid as well as significant and slight amounts of energy redistribution in regions 0 to 260 and 410 to 700 for reading 5. An explanation for this could be the stronger turbulence strength \(C_{n}^{2}\) from an extreme temperature above room temperature which changes the index of refraction causing random phase perturbations and therefore a defocus or distortion of the laser beam not evident in previous data.

Surprisingly, redistribution of beam energy and beam wander doesn’t significantly impact the beam centroid fluctuations except at the highest temperature (53.0 °C). However, the relative importance of these effects depends on the path length, the strength of turbulence, and the wavelength of the laser radiation [26, 27].

A comparison between ranges of refractive-index structures constant \(C_{n}^{2 }\) in literature and those of the present study are presented in Table 3. The \(C_{n}^{2 }\) in this work is within the reported range of values (2.5 × \(10^{ - 19} \;{\text{m}}^{{{\raise0.7ex\hbox{${ - 2}$} \!\mathord{\left/ {\vphantom {{ - 2} 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}\) to 1.8 × \(10^{ - 7} \;{\text{m}}^{{{\raise0.7ex\hbox{${ - 2}$} \!\mathord{\left/ {\vphantom {{ - 2} 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}\)), with discrepancies due to the different environmental situations such as differing wind speeds and temperatures at which the experiment was conducted.

Table 3 Comparison of results for \(C_{n}^{2 }\) in this work with other studies
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The refractive-index structure constant values indicate strong turbulence (\(C_{n}^{2}\) ≥ 10−13m−2/3) with the n† readings being an order of magnitude lower depicting weak perturbations. This might be attributed to the horizontal flow of heat to the laser beam following a limitation in the natural upwards flow of heat flow by convection. The results showed that an increase in the turbulence strength \(C_{n}^{2}\) is related to a decrease in the atmospheric coherence diameter (ro), and vice versa. Therefore, a stronger turbulent atmosphere further destroys the coherence properties of the output beam leading to a greater broadening effect of the beam profiles [28]. An accurate characterization of the atmospheric coherence diameter ro is important to describe the effects of atmospheric turbulence on an optical imaging system at a given location [29].

Results for the coherence diameters in this study were lower than the acceptable range (5 cm ≤ r0 ≤ 20 cm) which may be credited to the weaker optical seeing conditions from the ground level positioning of the laboratory since the coherence diameter depends on seeing conditions which increase in real atmospheric observatories [20].

Furthermore, comparing the Rytov variances, we observed that the Rytov variances were also an order of one lower in magnitude for the n† readings indicating lower scintillations as evident by the weak fluctuations depicted by the interferograms. However, the Rytov variances for all readings were in the weak scintillation range \((\sigma_{R}^{2} \ll 1)\). The observed discrepancies between the refractive-index structure constant and the Rytov variance may be attributed to the relatively short propagation paths of the laser beam [11].

As in any measurement techniques, there are some probable errors. Nevertheless, the maximum experimental error in this work is estimated to be 7%. Although heat loss to the surroundings is also a factor, it could not be numerically accounted for and as such, not included in the error estimate. Furthermore, to ensure that the optical components were free from dust particles, a commercial dust cleaner (Dust-Off) and methanol were periodically used to clean the surfaces of the optical components. Minimal levels of dust are extremely detrimental to the quality of the interferograms produced.

5 Conclusion

As a means of characterizing the positional impacts of a turbulence source on a propagating laser beam in air, two positions for a thermal turbulence source are studied. In a series of air temperatures, it was found that an increase in temperature resulted in great broadening effect of the upper region of the beam profiles, and the beam centroid was observed to drift slightly off the centre of the detector when the heated plate was positioned above (upside-down) the beam even in the short propagating distance. The observed central dip in intensity profiles is probably due to the attenuation of the laser beam at the point of discontinuity of the PDI.

However, for experimental runs where the turbulence source was below the beam, the laser beam only experienced directional fluctuations as well as image blur. Subsequent data at higher thermal turbulence showed that the severity of image blurriness and directional fluctuations increased with temperature. The refractive index structure function \(C_{n}^{2}\) values determined to understand the strength of the turbulent regions showed the values were within the strong turbulence class for both positions of the thermal source. It was concluded that the turbulence effects on a propagating laser beam were not only due to variation in turbulence strength but also a function of the turbulence source location.

Future work may include further alterations in the turbulent model such that a cylindrical shaped heating system is introduced to minimise heat loss to the environment, thus allowing for a more homogeneous heat distribution system. Another turbulent model may involve a heat plate greater in dimensions than the propagation path length to cancel out the vertical motion of heat. The experimental results herein are comparable with previous works from different researchers, which are of great importance in the missile guiding laser system, remote sensing, and communications.