3.1. Effect of Wavefront Distortion on Mixing Efficiency
The defocus-spherical distortion class of wavefront distortions includes defocus (
Z4), spherical distortion (
Z11), and higher-order spherical distortions. This type of wavefront distortion distribution, spatially uncorrelated with the polar angle
θ, forms a circularly symmetric feature centered at the origin and is classified as such. Combined with Equation (15) when subjected to defocus-spherical distortion of wavefront class only, the simulation conditions are set to photodetector quantum efficiency
η = 0.8, Planck’s constant of
h = 6.63 × 10
−34 J·s, speed of light of
c = 2.99 × 10
8 m/s, a wavelength of
λ = 1550 nm, effective noise bandwidth of Δ
f = 200 MHz, the signal optical power of
Ps = 0.36 μW, local oscillation optical power of
Pl = 0.36 mW conditions, distortion amplitude in the range of 0–10
λ, and heterodyne detection mixing efficiency with the variation of defocus-spherical distortion amplitude of class. The numerical simulation results are shown in
Figure 2.
Figure 2 shows simulations to obtain such wavefront distortion, the heterodyne detection mixing efficiency curves show the same overall change trend.
Figure 2 shows that when the distortion amplitude was less than 1
λ, the mixing efficiency was the same. With an increase in the distortion amplitude, the mixing efficiency showed different degrees of oscillation decrease. Moreover, because the proportion of the low-order distortion component in the entire wavefront distortion is much larger than that of the high-order distortion component, the wavefront distortion caused by the defocus component (
Z4) has the greatest impact on the mixing efficiency, and the impact of the high-order distortion component on the mixing efficiency gradually decreases.
The tilt-coma distortion class of wavefront distortion includes tilt (
Z2), coma (
Z8), and its higher-order coma. The high-frequency portion of such distortion completely contains the wavefront polar axis
r and polar angle
θ components with lower spatial frequencies. Only the coefficients of each component are different, making such wavefront distortion superimposed on different levels of coma components based on the tilt term. By selecting the same simulation parameters as shown in
Figure 2 and combining them with Equation (15) only subject to tilt-coma distortion class wavefront distortion, the distortion amplitude in the range of 0–10
λ, the variation curve of the heterodyne detection mixing efficiency with distortion amplitude are shown in
Figure 3.
Figure 3a shows the variation in the mixing efficiency with the distortion amplitude in
x-direction heterodyne detection, and
Figure 3b shows the variation in the mixing efficiency with the distortion amplitude in
y-direction heterodyne detection. Because the tilt-coma distortion in the
x-direction and
y-direction only has directional differences, only
Figure 3a is analyzed here. In
Figure 3a, we observe that the mixing efficiency decays rapidly to 0.4 with increasing distortion amplitude up to 1
λ, and is almost zero when the distortion amplitude is higher than 2
λ. Moreover, because the high-frequency portion of such wavefront distortion completely contains distortion components of low spatial order, however, the coefficients of each component are different, therefore the impact of such wavefront distortion on the mixing efficiency is extremely comparable. In addition, the
x-direction tilt distortion (
Z2) and
y-direction tilt distortion (
Z3) had the highest impact on the mixing efficiency.
The astigmatism class wavefront distortion increases with the Zernike order; the polar axis
r increases by a power exponent, and the polar angle
θ increases by a factor. By selecting the same simulation parameters as those presented in
Figure 2, the variation curves of the heterodyne detection mixing efficiency with the amplitude of the astigmatism class distortion in Equation (15) when subjected to only the astigmatism class wavefront distortion in the range of 0–10
λ are illustrated in
Figure 4.
Figure 4a shows the variation in the heterodyne detection mixing efficiency with the amplitude of the astigmatism class distortion in the
x-direction.
Figure 4b shows the variation in the heterodyne detection mixing efficiency with the amplitude of the astigmatism class distortion in the
y-direction. Owing to the same influence on the trend, only
Figure 4a is analyzed here. As shown in
Figure 4a, an
x-directional astigmatism amplitude of less than 1.3
λ has a greater effect on the decay of the mixing efficiency, and then the decay of the mixing efficiency gradually slows with the increase in the distortion amplitude. In addition, when the distortion amplitude was the same, the higher the order of the astigmatism wavefront distortion, the smaller the effect on the mixing efficiency. Among them, there is also a secondary (
Z12,
Z13) and high-order astigmatism class in the astigmatism class distortion, which similarly affects the mixing efficiency.
It is concluded that compared with other wavefront distortions, the defocus-spherical distortion class of wavefront distortions has different degrees of oscillation decrease in the mixing efficiency of heterodyne detection with the increase of the distortion amplitude, which is due to this type of wavefront distortion having circular symmetry. However, under the same amount of distortion amplitude, the attenuation of the mixing efficiency by tilt-coma distortion class of wavefront distortion is more obvious than that of other wavefront distortions. Among them, the tilt distortion has the most serious impact on the mixing efficiency, so it is necessary for the tilt distortion of the wavefront distortion to be corrected separately to improve mixing efficiency.
3.2. Evaluation of Mixing Efficiency and Mixing Gain
The influence of the wavefront distortion generated in the atmospheric turbulence on the coherent detection system decreases the mixing efficiency of the coherent detection system with the increase of the turbulence intensity. Therefore, we further analyzed the correction of wavefront distortion for the improvement of the mixing efficiency of the coherent detection system, and with the increase of turbulence intensity, we considered that a larger wavefront correction order is required to compensate for the phase wavefront distortion, to achieve the best compensation for the mixing efficiency and mixing gain of the coherent detection system at the receiving end.
Regarding the correction of wavefront distortion, Zernike polynomials are used to describe the initial wavefront distortion under each specific atmospheric turbulence intensity
D/
r0, and a 340-order distortion matrix is generated. Wherein, as the wavefront correction order increases, the wavefront distortion is iteratively corrected until the wavefront distortion phase is corrected to a theoretical value of 0, which is the end of the wavefront distortion correction. Then, the influence curve of the order of wavefront correction on the mixing efficiency of the coherent detection system is fitted using a polynomial curve.
Figure 5 shows the iterative curve of wavefront distortion correction under a specific turbulence intensity.
Figure 5a–d show that under a certain atmospheric turbulence intensity, as the wavefront correction order increases, the mixing efficiency is improved; and as the atmospheric turbulence intensity increases, the mixing efficiency curve jitters more obviously. This is because the coefficient of wavefront distortion is related to the turbulence intensity
D/
r0. The greater the intensity of atmospheric turbulence, the more obvious the change of the wavefront distortion coefficient during the correction process, and the more jittery the mixing efficiency presented.
Next, we further discuss the optimal correction order of wavefront distortion required when the performance of the coherent detection system at the receiving end is determined, and the improvement limit of the mixing efficiency with the correction order of wavefront distortion.
When the bit error rate (BER) of the coherent detection system at the receiving end is determined, the simulation conditions are set as follows: receiving aperture
D = 105 mm, corresponding to
D/
r0 is 0.2, 2, 5, and 8; quantum efficiency of the photodetector
η = 0.8, Planck’ constant
h = 6.63 × 10
−34 J·s, speed of light
c = 2.99 × 10
8 m/s, wavelength
λ = 1550 nm, signal optical power
Ps = 0.36 μW, local oscillation optical power
Pl = 0.36 mW, and effective noise bandwidth Δ
f = 200 MHz. For the coherent detection system with specific performance indicators, the optimal wavefront correction order is calculated under different turbulence intensities, and the simulation results are shown in
Figure 6.
When the BER of the coherent detection system at the receiving end is on the order of 10
−9, the corresponding mixing efficiency is 0.7. To meet the system performance index, the optimal correction order required is shown in
Figure 6. Among them, when observing
Figure 6a, it can be concluded that when
D/
r0 is 2 in heterodyne detection, the correction 10th order mixing efficiency reaches the system requirements; when
D/
r0 is 5, the correction 59th order mixing efficiency reaches 0.7; when
D/
r0 is 8, the correction 69th order mixing efficiency is required to reach 0.7;
Figure 6b shows the order relationship of wavefront correction to meet the requirements of the coherent detection system in homodyne detection. When
D/
r0 is 2, 5, and 8, the mixing efficiency of the wavefront correction at the 10th, 61st, and 72nd order reaches 0.7; When the minimum turbulence intensity
D/
r0 is 0.2, the mixing efficiency after wavefront distortion correction is greater than 0.7.
The simulation conditions were set as follows: receiving aperture
D = 105 mm, corresponding to
D/
r0 of 0.2, 2, 5, and 8; quantum efficiency of the photodetector
η = 0.8; Planck’s constant
h = 6.63 × 10
−34 J·s; the speed of light
c = 2.99 × 10
8 m/s; wavelength
λ = 1550 nm; signal optical power
Ps = 0.36 μW, local oscillation optical power
Pl = 0.36 mW, and effective noise bandwidth Δ
f = 200 MHz. When the wavefront correction order
N in Equation (15) changes, the extreme values of the heterodyne detection mixing efficiency calculated using Equation (10) are obtained using numerical calculations. The simulation results are shown in
Figure 7.
Figure 7 shows that the mixing efficiency is significantly improved for a certain number of wavefront correction orders. The mixing efficiency converges to a stable value by increasing the number of correction orders. Among them, the influence of low-order wavefront correction on the mixing efficiency is obvious. The corresponding mixing efficiency increases to 97%, 51%, 12%, and 4% when
D/
r0 is 0.2, 2, 5, and 8, respectively, and only the tilt term is corrected. Following the correction of the defocus term, the mixing efficiency increases to 98%, 55%, 16%, and 6%; following the correction of the astigmatism term, the mixing efficiency increases to 99%, 60%, 22%, and 8%. When
D/
r0 is 0.2 in the heterodyne detection, only the 5th-order wavefront correction is completed, and the mixing efficiency reaches the extreme value. When
D/
r0 is 2, the correction 85th order mixing efficiency reaches the extreme value. When
D/
r0 is 5, the mixing efficiency below 231 orders of wavefront correction can reach an extreme value. When
D/
r0 is 8, it is necessary to correct the mixing efficiency below 264 orders to reach the extreme value.
By selecting the same simulation parameters shown in
Figure 7, the extreme values of the homodyne detection mixing efficiency in Equation (13) can be obtained by numerical calculations when the wavefront correction order
N in Equation (15) varies. When
D/
r0 takes different values, the relationship curves of the mixing efficiency with the change in the wavefront correction order are shown in
Figure 8.
Figure 8 shows that when
D/
r0 is 0.2, the wavefront correction is the same as that under heterodyne detection, and only the correction of 5th-order mixing efficiency converges to 1. When
D/
r0 is 2, the correction of the 87th-order mixing efficiency is required to reach the extreme value. When
D/
r0 is 5, the correction of the mixing efficiency below order 234 reaches the extreme value. When
D/
r0 is 8, it needs to correct the mixing efficiency below order 269 to converge to 1. It is concluded that the effect of the low-order wavefront correction on the mixing efficiency of the system receiving end in homodyne detection is the same as that in heterodyne detection. This is because the integral cosine term of the wavefront phase difference between the signal light and local oscillator light is much larger than the integral sine term, and the influence of low-order wavefront distortion on the mixing efficiency integral sine term is negligible.
Combining
Figure 7 and
Figure 8, we find that under the influence of atmospheric turbulence, the wavefront correction order required for homodyne detection to reach the wavefront distortion correction limit is higher than that required for heterodyne detection, owing to the higher requirement for spatial phase matching.
According to the trend of wavefront distortion correction with mixing efficiency, we found that when the atmospheric coherence length
r0 is certain and the correction of wavefront distortion is of a specific order, there exists a theoretically optimal receiving aperture
D that maximizes the mixing gain. Under the conditions of a wavelength of
λ = 1550 nm, the signal optical power of
Ps = 0.36 μW, local oscillation optical power of
Pl = 0.36 mW, and atmospheric coherence length of
r0 = 0.05 m, the mixing gain in heterodyne detection and homodyne detection calculated in Equation (14) can be obtained via simulation when the wavefront correction order
N in Equation (15) varied. Among them, the simulation results of the curve mixing gain changing with the receiving aperture
D in different detection methods are shown in
Figure 9, and the optimal mixing gain results are shown in
Figure 10.
Figure 9 shows that as the order of wavefront correction increases, the mixing gain increases gradually; and as the receiving aperture
D increases, the mixing gain reaches the maximum, and then shows a downward trend. Among them, comparing
Figure 9a,b, it is found that the mixing gain obtained by homodyne detection is greater than that obtained by heterodyne detection because the detection sensitivity of homodyne detection itself is better than that of heterodyne detection.
Observing
Figure 10 shows that when the
D/
r0 ratio is 7.6, the best heterodyne detection mixing gain is 8.5 dB, and the best homodyne detection mixing gain is 11.2 dB when correcting the wavefront distortion of the first five orders; the best heterodyne detection mixing gain is 10.3 dB and the best homodyne detection mixing gain is 13.3 dB when correcting the wavefront distortion of the first 15 orders. The mixing gain increase with the wavefront correction order. When the wavefront distortion correction is below the 35th order, the maximum heterodyne detection mixing gain rises to 12.1 dB, and the maximum homodyne detection mixing gain rises to 15.1 dB. The homodyne detection mixing gain is higher than the heterodyne detection mixing gain of about 3 dB.
When considering different detection methods, the same simulation parameters as those in
Figure 9 were selected, and under the condition of atmospheric coherence length
r0 = 0.01 m, the curves of the mixing gain changing with the receiving aperture
D in heterodyne detection and homodyne detection are shown in
Figure 11, and the optimal mixing gain results are shown in
Figure 12.
Figure 11 shows that as the order of wavefront correction increases, the mixing gain increases gradually; and as the receiving aperture
D increases, the mixing gain reaches the maximum, and then the mixing gain showed an oscillation decrease. The oscillation mainly comes from the fact that with the increase of turbulence intensity, the energy of the main lobe of the wavefront distortion spreads to the side lobes, which leads to the decrease of the matching degree of the optical field at the receiving end and thus the oscillation phenomenon.
Figure 12 shows that the mixing gain increased as the wavefront correction order gradually increased. In correcting the first five orders of wavefront distortion, the best heterodyne detection mixing gain was 8.3 dB, and the best homodyne detection mixing gain was 11.1 dB. When the wavefront distortion was corrected to below 35 orders, the maximum heterodyne detection mixing gain increased to 12.5 dB, and the maximum homodyne detection mixing gain rose to 15.5 dB. The mixing gain of heterodyne and homodyne detection was optimal when the
D/
r0 ratio was 11.5. We find that the optimal mixing gain in homodyne detection was generally higher than that in heterodyne detection by approximately 3 dB.
Combining the simulation results of receiving aperture D and mixing gain under different turbulence conditions, it is concluded that as the turbulence intensity increases, the mixing gain is more sensitive to the selection of the receiving aperture. Under a specific turbulence intensity, different orders of wavefront distortion correction will have a suitable receiving aperture to maximize the mixing gain, which provides a quantitative reference for the selection of an optical system in practice.