1. Introduction
A wave propagating through a turbulent atmosphere is subject to perturbations of its phase and amplitude due to the fluctuations of the refractive index along the whole propagation path. These perturbations, known as wavefront distortions, severely influence the performance of optical systems that operate in or through the atmosphere, such as adaptive optics systems, interferometers, optical wireless communication systems, and laser radar systems [
1,
2,
3]. Therefore, it is important to understand how the effects of atmospheric turbulence on the propagation of a light wave can be quantified.
Clearly, the turbulence strength at each position of the propagation path contributes to the wavefront distortion. Therefore, the optical turbulence parameters that describe the optical effects of turbulence are often related to turbulence integral parameters, i.e., integrals of the refractive index structure constant Cn
2(
z) (
z denotes the position along the propagation path with the receiver plane at
z = 0) over a whole path with different path-weighting functions (PWFs). For example, the atmospheric coherence length for plane waves
r0,p has a PWF of
z0, the atmospheric coherence length for spherical waves
r0,s has a PWF of (L-
z)
5/3, and the isoplanatic angle
θ0 has a PWF of
z5/3 [
3]. In a practical situation, these optical turbulence parameters are often measured indirectly using only one measurable turbulence integral parameter, such as in the case of the differential image motion monitor (DIMM) [
4], where angle-of-arrival fluctuations are measured, and in the case of the stellar scintillation isoplanometer [
5,
6], where intensity fluctuations (scintillation) are measured. However, in many applications, some of the desired optical turbulence parameters cannot be estimated using only one measurable quantity, especially due to the anisoplanatic problems in adaptive optics [
7,
8], where the expression of related PWFs is more complex and can only be evaluated numerically. For example, in the angular or focal anisoplanatic problems in adaptive optics, the PWF of the effective phase variance (the total phase variance with the piston removed or both the piston and the tilt removed) is neither
z0 nor
z5/3, so it cannot be estimated properly by
r0 or θ
0 [
7,
8,
9].
In this paper, we relate the optical effects of turbulence to the path-weighting functions (PWFs) and describe a method (referred to as the linear combination method) that uses more than one measurable turbulence integral quantity to estimate the turbulence integral parameters that cannot be measured directly. We generalize the PWFs of measurable integral parameters and investigate their characteristics in both a plane wave model and a spherical wave model for the common case where one source is used and the receive apertures are unobscured circles. Some interesting and meaningful results have been obtained. These results may enable us to measure r0 not only at near-field but also at far-field using the covariance of tilt, and to measure θTA using a small-aperture telescope instead of a large-aperture telescope using the covariance of intensity rather than scintillation.
2. Materials and Methods
The turbulence integral parameter noted as
P is written as follows:
where
C is a constant related to
P,
L is the propagation path length through turbulence,
Cn2 is the turbulence strength,
W(
u) is the PWF, and
u =
z/
L denotes the normalized position along the path with propagation from
u = 1 to
u = 0 at the receiver. The principle of the linear combination method is that any linear combination of turbulence integral parameters corresponds to the linear combination of their PWFs. The desired integral parameter
Pd can be estimated from measurable integral parameters
Pi if only the PWF of
Pd, noted as
Wd(
u), can be approximated by linear combination of the PWFs of
Pi, noted as
Wi(
u), i.e.,
where
ai are the coefficients of
Wi(
u), and
N is the number of measurable integral parameters used in this method. In practical applications, one should first select a suitable
Wi(
u) based on the shape of
Wd(
u) and then fit
Wi(
u) to
Wd(
u) using the least square fitting method. Equation (2) shows that the more measurable integral parameters and their PWFs we know, the more desired PWFs we can approximate using this method and, thus, the more integral parameters we can estimate. Therefore, it is important to investigate more measurable quantities and their PWFs.
Atmospheric turbulence causes phase and amplitude perturbations, and at present, only the tilt phase and piston amplitude, corresponding to the angle of arrival and the intensity of the light collected with an aperture, can be measured directly and relatively easily. In practice, the variance of the angle of arrival and the variance of intensity are used to obtain turbulence information through their PWFs, which is exactly the case in the DIMM and the stellar scintillation isoplanometer, respectively. However, the covariance functions for the angle of arrival or intensity provide more turbulence information due to their potentially large number of PWFs. The covariance of the tilt phase can be measured from the covariance of the angle of arrival [
3,
4], and the covariance of the piston amplitude can be measured from the covariance of intensity [
6,
10].
Assuming the Rytov approximation and Kolmogorov turbulence, using the analytic approach developed by Sasiela [
3], the covariance functions for the phase and the log-amplitude related quantities can be derived as follows:
where
φ and α denote the phase and the log-amplitude related quantities, respectively;
d is the center-to-center distance of two receive apertures; the optical wavenumber
k = 2
π/
λ, where
λ is the wavelength of the observed beacon; and
WC,φ(
u) and
WC,χ(
u) are path-weighting functions which are given by the following:
where
κ is the spatial wavenumber transverse to the
z direction; γ is the propagation parameter, which depends on the geometric divergence of the light wave and has the simple value of
γ = 1 for plane waves and
γ = 1−
z/
L for spherical waves;
Jn is the
nth order Bessel function of the first kind; and
F(γκ) is the filter function, which depends on propagation geometry, such as the size of the receive aperture and the distributed source, and measured related quantities, such as the tilt phase or piston amplitude. For circular apertures with diameter
D, the filter function for Zernike tilt and piston are as follows [
3]:
For a uniformly illuminated circular source with diameter
Ds located at a distance
LS, considering that the uniform circular source consists of incoherent point beacons, the source filter function is as follows [
3]:
The final filter function is the product of any produced filter functions. Make a change to the variables by replacing
κD with
x and use the replacement
FN =
D2/
λL, where
FN is the Fresnel number, to simplify the final expression of the PWFs for the tilt phase (i.e., substituting
F(γκ) =
Ft(γκ) *
Fs(γκ) into Equation (4)) and piston amplitude (i.e., substituting
F(γκ) =
Fp(γκ) *
Fs(γκ) into Equation (4)), noted as
WC,T(
u),
WC,P(
u), to the following:
From Equation (8), it is clear that the PWFs for the tilt phase and piston amplitude depend on four variables:
γ,
d/
D,
FN, and
DSL/DLS (i.e., the ratio of the angular diameter of the source to that of the receive aperture observed at plane
z =
L). In the following sections
DSL/DLS will be noted as
R for simplicity. Any change in these four variables will result in a new PWF. Obviously, we can obtain more PWFs by using the covariance function instead of the variance function, which is the special case of the covariance function for
d = 0. The geometry and variables of the covariance functions are shown in
Figure 1 for reference.
For convenience later, denote WT(u) as the normalized PWF for the covariance of the tilt phase, and WP(u) as the normalized PWF for the covariance of piston amplitude. To obtain qualitative knowledge of the effects of these four variables on the PWFs, next we will give the PWFs in a range of typical conditions by numerical evaluation.
4. Discussion
From the above numerical results, we can see how each of the four variables (γ, d/D, FN, and R) influence the curve shapes of PWFs, which is helpful in the creation of a desired PWF using the linear combination method (Equation (2)). When we focused on the PWFs for different FN values without changing the other variables, we found that the effect of diffraction on the covariance of the tilt phase or piston amplitude is far smaller than that on the variance.
Note that the PWFs investigated here only considered the common case where one source is used and the receive apertures are unobscured circles. When the shape of the receive aperture is changed or two sources are used, the curve shapes of the related PWFs will also change, as in the case of MASS [
13] (Multi-Aperture Scintillation Sensor), where the receive aperture is annulus, or SloDAR [
14] (Slope Detection And Ranging) and SciDAR [
15,
16] (Scintillation Detection And Ranging), where two sources are used. Thus, there are potentially a large number of various
Wi(
u) values that we can obtain, which would make the linear combination method more powerful.
In some applications, the PWFs of
Pd (the desired turbulence integral parameter) have complex analytic expressions, most of which can only be evaluated numerically, and
Pd is often difficult to measure at present. For example, in the angular or focal anisoplanatic problems in adaptive optics, the effective phase variance σ
2Eff (the total phase variance σ
2φ with the piston removed or both the piston σ
2P and the tilt σ
2TA removed, i.e., σ
2Eff = σ
2φ−σ
2P (−
σ2TA)) is such a
Pd [
3,
7,
8,
9]. For natural guide star adaptive optics (NGS AO), it is well known that σ
2φ = (
θ/
θ0)
5/3, and one can easily derive σ
2P = 4.25 σ
2TA = 4.25 (
θ/
θTA)
2 for small angles, where
θTA is the tilt isoplanatic angle [
1,
3,
17]. Therefore, if
θTA can be measured along with
θ0, one can obtain the effective phase variance in NGS AO using the relationship σ
2Eff = (
θ/
θ0)
5/3 − 4.25(
θ/
θTA)
2 (−(
θ/
θTA)
2). As far as we know, the direct measurement of
θTA has not been reported. Based on the linear combination method proposed in this paper, we have proposed one direct measurement scheme of
θTA in our other accompanying manuscript [
18].
In some other applications, though expressions for the PWF of
Pd are simple, it cannot be measured under non-ideal conditions. For example,
r0,p and
θ0 in a finite distance cannot be measured using the conventional method (DIMM and stellar scintillation isoplanometer) since the light wave propagating in the turbulence path is not a plane wave. Generally, the light wave is a spherical wave for a point beacon in a finite distance. Then, the parameter measured using DIMM is actually
r0,s rather than
r0,p, and the parameter measured using the conventional isoplanometer has a PWF with a curve shape similar to the red dashed line in
Figure 5a rather than the red dashed line in
Figure 3a (i.e., the well-known
u5/3).
The linear combination method shows promise of estimating the real-time
Pd with high accuracy if the suitable
Wi(
u) can be found. As an application example of this method, one can measure the isoplanatic angle in a finite distance through the combination of one spherical wave scintillation and two covariances of intensity in three receive apertures [
10], and the validity of this method was proven by real data from the validation experiment [
19].
One can also roughly estimate
Pd through the optical turbulence profile obtained from those turbulence profilers [
12,
13,
14,
15,
16]. Nevertheless, the linear combination method offers another feasible solution.
5. Conclusions
It was found that the path-weighting functions (PWFs) for covariance depend on four variables:
γ,
d/
D,
FN, and
DSL/DLS. In other words, the optical effects of turbulence are influenced by not only the turbulence strength but also these four variables. Numerical results show that the effect of diffraction on the variance of the tilt phase or piston amplitude ismore significant than it is on the covariance, e.g., the two curves of
WT(
u) for
d = 0
D,
FN = 5 and
d = 2
D,
FN = 1 almost coincide with each other, and it is possible to measure the tilt isoplanatic angle through the covariance of intensity rather than scintillation. More specifically, the curve of the PWFs for the tilt phase shifts upwards as the value of
d/
D or
FN increases, or the value of
R (
R =
DSL/DLS) decreases (see
Figure 2 and
Figure 4). The peak position of
WP(u) shifts to the start of the path as the value of
d/
D or
FN decreases, or the value of
R increases (see
Figure 3 and
Figure 5).
In the future, work will be conducted to prove the validity of measuring the tilt isoplanatic angle through the covariance of intensity and to investigate more suitable PWFs for practical applications. The linear combination method is an effective and powerful method for measuring the turbulence integral parameters of interest as there are a potentially large number of various PWFs that we can obtain.