1. Introduction

The distribution as a function of height h of the refractive-index structure constant (also known as optical-turbulence), C ² _N (h), is a key parameter in the field of high angular-resolution imaging through the atmosphere. For example, in optical astronomy, measurements of C ² _N (h) are required to develop adaptive optical systems for ground-based telescopes and to evaluate existing or potential astronomical sites.

The Scintillation Detection and Ranging (SCIDAR) technique, proposed by Vernin and Roddier in 1973 [2], is aimed at the measurement of the optical-turbulence profile. The method and the physics involved have thoroughly been treated by a number of authors [3, 4, 5, 6, 7]. Here we only recall the guidelines of the principle.

The spatial autocovariance of the single-star scintillation produced by a turbulent layer at an altitude h, strength C ² _N and thickness δh is given by

where r stands for the modulus of the position vector r and K(r,h) is given by the Fourier transform of the power spectrum W_I of the irradiance fluctuations. An expression of W_I can be found in Reference [8]. For completeness, here we give an expression for K(r,h) [7] in the case of Kolmogorov turbulence and weak perturbation approximation:

where λ is the wavelength, k=2π/λ and u is the spatial frequency. It is worth noting that the scintillation variance, 𝓒(0), is proportional to h ^5/6, as can be easily deduced from Eqs. 1 and 2 by changing the integration variable to ξ=h ^1/2 f.

The SCIDAR method consists of the following: Light coming from two stars separated by an angle θ and crossing a turbulent layer at an altitude h casts on the ground two identical scintillation patterns shifted from one another by a distance θh. The spatial autocovariance of the compound scintillation exhibits peaks at positions r=±θh with an amplitude proportional to the C ² _N value associated to that layer. The determination of the position and amplitude of those peaks leads to C ² _N (h). This is the principle of the so-called Classical SCIDAR (CS), in which the scintillation is recorded at ground level by taking images of the telescope pupil while pointing a double star. As the scintillation variance produced by a turbulent layer at an altitude h is proportional to h ^5/6, the CS is blind to turbulence close to the ground, which constitutes a major disadvantage because the most intense turbulence is often located at ground level [9, 10].

To circumvent this limitation, Fuchs et al. in 1994 [11] proposed to optically shift the measurement plane a distance d below the pupil. For the scintillation variance to be significant, d must be of the order of 1 km or larger. This is the principle of the Generalized SCIDAR (GS) which was first implemented by Avila et al. in 1997 [12]. In the GS, a turbulent layer at an altitude h produces autocovariance peaks at positions r=±θ (h+d), with an amplitude proportional to (h+d)^5/6. The cut of the peak centered at r=θ (h+d), along the direction of the double-star separation is given by

In the realistic case of multiple layers, the autocovariance corresponding to each layer add up because of the statistical independence of the scintillation produced in each layer. Hence, Eq. 3 becomes:

For h between −d and 0, C ² _N(h)=0 because that space is virtual. To invert Eq. 4 and determine C ² _N(h), a number of methods have been used like Maximum Entropy [12], Maximum likelihood [6, 1] or CLEAN [7, 13].

Two quantities related to the GS technique that are helpful in this paper are the full width at half maximum of the scintillation autocovariance [7]:

where λ is the wavelength, and the maximum altitude for which C ² _N values can be measured [13]:

where D is the telescope pupil diameter.

The procedure to estimate the scintillation autocovariance 𝓒 is to compute the mean auto-correlation of double-star scintillation images, normalized by the autocorrelation of the mean image. In the CS - where images are taken at the telescope pupil - this computation leads analytically to 𝓒 [3]. However for the GS, Johnston et al. pointed out in 2002 [1] that the result of this procedure is not equal to 𝓒. The discrepancy is due to the shift of the out-of-focus pupil images produced by each star on the detector. Those authors analyzed this effect only for turbulence at ground level (h=0). Here we generalize the analysis to turbulence at any height. In §2 we present the detailed analytical development of the mean autocorrelation of double-star scintillation images and its normalization by the autocorrelation of the mean image. Both configurations, CS and GS are treated. §3 is devoted to the estimate of the error produced by this normalization in GS measurements. Finally the conclusions are given in §4. Our results can be applied to correct all the GS measurements that have been affected by the erroneous normalization [12, 14, 6, 15, 16, 17, 7, 18, 9, 19, 20, 21, 22, 23, 13, 24].

The analysis presented in this paper is conducted considering stellar sources at the zenith. In the case of sources located at an angle z from the zenith, the results remain valid if one replaces h by hsec(z).

2. Analytical development

2.1. Autocovariance of double-star scintillation images

Let us assume a single turbulent layer at an altitude h above the telescope pupil plane. The detector plane is made the conjugate of a plane at a distance d below the pupil plane. The instantaneous image produced by a single star on the detector can be written as:

where A represents the mean irradiance, f (r) is the random spatial modulation factor produced by the scintillation, r is the position vector on the detection plane and P(r) stands for the pupil irradiance transmittance. The diffraction of the pupil due to the propagation distance d is neglected, to disentangle this effect from the problem investigated here. A brief discussion of the effect of pupil diffraction in GS-kind measurements can be found in reference [13]. We make the following hypothesis on the terms of Eq. 7: A is considered to be constant, f is a dimensionless, real, stationary, homogeneous, isotropic and ergodic random function with mean value equal to 1, and P is equal to 1 inside the pupil and 0 outside.

For a double star, the image captured on the detector is the sum of the contribution coming from each star. Assuming that star 1 is aligned along the optical axis and star 2 is separated by an angle θ and the individual stellar irradiances are A ₁ and A ₂, the image can be written as:

Note that on the off-axis-star term the scintillation pattern f is shifted due to the image plane displacement (d) and the layer altitude (h), whereas the pupil P is shifted only due to d. This feature is illustrated in Fig. 1. The particular case of h = 0 – which leads to identical shifts for the off-axis terms f and P – was analyzed by Johnston et al. [1]. Equation 8 is the starting point for the generalization of that work.

The mean autocorrelation of images described by Eq. 8 is:

Symbols 〈〉 and ⋆ denote ensemble average and correlation, respectively. Writing explicitly the correlation in the first term of the right-hand-side of Eq. 9, this term reads:

Rearranging terms, and remembering that A ₁ and P are constants, we have:

Because of the hypothesis made on f, 〈f(ρ) f(ρ+r)〉 is only a function of r and can therefore be out of the integral over ρ. Defining

the first term, T ₁ of Eq. 9 becomes:

C(r) is related to the scintillation autocovariance (Eq. 1) simply by

because 〈f〉=1.

 

Fig. 1. Optical beams from the on-axis (solid lines) and off-axis (dashed lines) stars, reaching the virtual analysis plane. Two triangles are emphasized in thick lines. The bases of the taller and smaller triangles indicate the separation of the scintillation patterns (drawn as two identical speckles) and the pupil projections respectively.

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Performing an analogous treatment on the second, third and fourth terms of Eq. 9, we have:

Note that the width ofC(r) is given by Eq. 5 and the footprint S(r) is equal to twice the telescope pupil diameter 2D. Thus C(r) is much narrower than S(r).

2.2. Autocorrelation of the average of the double-star scintillation images

The mean image, calculated from Eq. 8, is simply

because 〈f〉=1 and A ₁, A ₂ and P are constant. Applying similar considerations as those who led to Eq. 16, the autocorrelation of the mean image is written as

In Fig. 2, the black line represents a cut of Γ_〈I〉(r) along the double star separation, for three ranges of values of dθ that will be of interest below.

2.3. Normalization

The quantity we want to isolate is either of the lateral autocovariance peaks,C(r−θ(d+h))-1 or C(r+θ(d+h))-1 (cf. Eq. 15), from which C ² _N (h) can be retrieved. For that purpose, the procedure generally used in scidar-like experiments consists of the following computation:

Three cases can be distinguished, depending on the value of dθ. In the first two, the operation performed with Eq. 19 indeed cancels out the terms corresponding to the pupil autocorrelation, leaving the scintillation autocovariance term isolated:

• Case 1. If d=0, like in the CS case, Eq. 19 becomes:

where

and

Therefore, knowing the stellar irradiances and measuring Λ(r), one can determine either of the lateral autocovariance peaks, as long as they are not superimposed, i.e. as long as θh is larger than the peaks width L (see Eq. 5). This case is represented in Fig. 2(a). It is useful to express a and b in terms of the difference of the magnitudes of the two stars Δm≡|m ₁−m ₂|. Knowing that m _1,2≡-2.5log₁₀(A _1,2) and defining α≡A ₁/A ₂=10^−0.4Δm, one can show that

and

• Case 2. For dθ≥D, like in the Low Layer SCIDAR [13], the lateral peaks, C(r+θ(d+h)) and C(r−θ(d+h)) lie further apart from the origin than the region where S(r) is non-zero. This can be seen in Fig. 2(b) where S(r) is represented by the magenta dashed line and the scintillation covariance peaks are the red curves. The lateral peaks lie in zones where the black line is overlapped with the green and yellow lines, i.e., where only S(r+θd) or S(r−θd) contribute to Γ_〈I〉(r). In this case the three additive terms of Eqs. 16 and 18 can be treated separately in Eq. 19, yielding:

Note that the stellar irradiances get eliminated in the expression of the central, left and right-hand-side autocovariance peaks, Λ_c(r), Λ_l(r) and Λ_r(r), whereas if d=0 they do appear in the expression of Λ(r).

• Case 3. For 0<dθ<D, like in most of the GS measurements, the lateral scintillation covariance peaks lie in zones where the central pupil autocorrelation S(r) overlap the lateral pupil autocorrelations S(r+θd) and S(r−θd) (see Fig. 2c). Within those zones of overlapping, Γ_〈I〉(r) (black line) is larger than S(r+θd) and S(r−θd). Thereby, the normalization performed using Eq. 19 no longer results in the cancellation of those terms. To prove that analytically, without loss of generality one can focus on the zone of influence of the right-hand-side covariance peak, namely, r >θ d. Assuming that θ d is much larger than the width L of the covariance peaks (Eq. 5), in this zone, the other two covariance peaks C(r) and C(r+θ(d+h)) are equal to 1 (see Eq. 12 and remember that 〈f〉=1). Hence, when substituting Eqs. 16 and 18 into the numerator of Eq. 19, the terms containing S(r) and S(r+θd) cancel out. Equation 19 can then be written as:

or, introducing factors a and b defined in Eqs. 21 and 22:

This expression shows that because of the troublesome influence of the central and left-hand-side pupil autocorrelation terms, S(r) and S(r+θd), the term S(r−θd) in the numerator cannot be canceled. The effect of this inexact normalization is calculated in § 3.

In reference [1] similar results are obtained, except that the analysis there is limited to ground turbulence (h=0) and that the condition they impose for the case corresponding to our case 2 is stronger: dθ>2D.

 

Fig. 2. Schematic representation of a cut of Γ_〈I〉(r) (black line) along the double-star separation, for (a) d=0, (b) dθ≥D and (c) 0<dθ<D. The green, magenta and yellow dashed lines in (a) and (b) represent A ₁ A ₂ S(r+θd), (A ² ₁+A ² ₂) S(r) and A ₁ A ₂ S(r−θd), respectively. Red curves represent the scintillation autocovariance terms.

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Sometimes the function that is considered for retrieving the scintillation autocovariance in GS is the difference of the cross-sections of Λ parallel and perpendicular to the star separation: Ψ(x)=Λ_‖(x)-Λ_‖(x), where x is the abscissa of the axis aligned along the direction of θ, Λ_‖(x)=Λ(x,0) and Λ_⊥(x)=Λ(0,x). This procedure was introduced for the CS [3] in order to eliminate the central autocovariance peak which can be superimposed on the wanted lateral peaks. For the GS, if the three autocovariance peaks are well separated (i.e. θ(d+h)>2L), which is usually the case, there is no need of performing that operation. Nevertheless, it is often done. It can easily be shown that in that case, Λ_⊥(x)=0 and Λ_‖(x) is the one-dimensional expression of Eq. 29. So, our analysis is valid for the GS data reductions based on the function Ψ(x).

Note that for the CS (case 1 above), Ψ(x) can also be obtained by calculating the difference of the parallel and perpendicular cross-sections of Γ_I and divide the result by the parallel cross-section of Γ_〈I〉. However, if this procedure is used in the GS, the result cannot be reduced to Eq. 29. Therefore, our analysis does not account for that data reduction procedure.

3. Error induced by the inexact normalization in GS

In most GS measurements that fall in Case 3 described above, Λ(r), defined in Eq. 19, has been thought to be equal to

In other words, it was thought that the normalization defined in Eq. 19 would provide the same result as in the case of the CS (Eq. 20), with the only difference that the lateral covariance peaks would be shifted by ±θd.

It is interesting to evaluate the effect of this misinterpretation. Function F(r) is indeed what one wants to estimate experimentally in GS measurements. However, the quantity that has actually been measured is Eq. 29. The corresponding relative error is:

Replacing Eqs. 29 and 30 and using the definition of a and b (Eqs. 21 and 22), it is straight forward to show that

This equation has been calculated for the right-hand-side covariance peak (r>θd). For the left-hand-side peak, one interchanges −θ d and +θd, yielding the same result because S(r) is an even function.

Using Eq. 31, F(r) can be estimated from the measurement of Λ(r) and the calculation of ε (r), which involves either analytical or experimental determination of the pupil autocorrelation S(r) and the knowledge of the conjugation altitude d and the double star parameters θ, a and b. S(r) can be calculated for any pupil function P(r) using the Wiener-Khinchin theorem [25] which states that

where 𝓕 and 𝓕 ⁻¹ are the Fourier transform and its inverse operators.

In § 3.1 an approximate analytical expression of the relative error ε is calculated for the particular case of a circular pupil without central obscuration. The interest of having such an expression is to understand the influence of the observational parameters on ε and to be able to make a rapid estimate of its value. In § 3.2, exact values of ε are computed for pupils obscured by a secondary mirror and for different values of the observational parameters. The reader interested only in the exact values of ε can skip § 3.1.

3.1. Approximate expression for a full circular pupil

Let us start by defining a disk of diameter D as:

and set the pupil P(r) equal to 𝓓(r,D). The autocorrelation of P(r) is computed using Eq. 33. Let S _𝓓(r) be the autocorrelation for the case of P(r)=𝓓(r,D). As P(r) is centro-symmetric, so is S_𝓓(r). A diametral cut of S_𝓓(r) is shown in Fig. 3(a). To simplify Eq. 32 we can make the following approximation:

 

Fig. 3. Diametral cut of the normalized autocorrelation of a disk with diameter D without central obscuration (a) and with a central obscuration of diameter 0.3D (b).

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For r>θd, which is the case for Eq. 32, all the arguments of S in that equation are positive. We can then ignore the absolute value in Eq. 35. In the substitution of S(r) by S_𝓓(r) in Eq. 32, attention must be paid for the cases where SD vanishes. Three different situations can occur:

1. Neither S_𝓓(r+θd) nor S_𝓓(r) nor S_𝓓(r-θd) are null. This occurs for r+θd<D.

2. S_𝓓(r+θd)=0 but S_𝓓(r)≠0 and S_𝓓(r−θd)≠0, which takes place when r+θd≥0 but r<D.

3. S_𝓓(r+θd)=S_𝓓(r)=0 and S_𝓓(r−θd)≠0, which happens if r+θd≥0, r≥D but r−θd<D.

Taking into account this matter while plugging Eq. 35 into Eq.32 and noticing from Eqs. 23 and 24 that a+2b=1, simple algebraic manipulations lead to:

The right-hand-side peak of the scintillation autocovariance is centered at r=θ(d+h) (see Eq. 16 and Fig. 2). Replacing r by this expression in Eq. 36 gives

where h_max, as defined in Eq. 6, has been introduced.

 

Fig. 4. (a) Relative error as a function of the altitude h of the turbulence, for a circular pupil of diameter D=1 m and b=0.25. Each color corresponds to a different value of θ as indicated in seconds of arc by the color labels on the plot. Solid lines are obtained by the exact computation of ε following Eqs. 32 and 33. Dotted lines correspond to the approximate expression ε_𝓓 given in Eq. 37. (b) Same as (a) but for a circular pupil with a central obscuration of diameter 0.3D.

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The last equation shows that for the two first cases, if θd increases then the relative error increases. Note From Eq. 24 that the maximum value of b is 1/4, which occurs when Δm=0. Therefore 1-2b>0. Moreover, 1−b>b. Thus, in the second case, if θ h increases, the denominator decreases faster than the numerator, implying that ε_𝓓 increases. In the first case too, if θ h increases ε_𝓓 increases. Concerning the dependence on D, similar arguments lead to the conclusion that if D increases, the error decreases. In the third case, the relative error is independent of θ, d, h and D. From Eq. 36 it can be deduced that in the three cases, ε𝓓>0. This means, from Eq. 31, that Λ(r)>F(r), resulting in an overestimation of the turbulence intensity.

Figure 4a shows plots of ε_𝓓 and ε as a function of h, for D=1 m, d=4 km, b=1/4 and different values of θ. The altitude takes values within the height-range where the optical turbulence strength can be significant in the atmosphere. The exact values of the relative error ε are computed following Eqs. 32 and 33 and the relation r=θ(d+h). Comparing the solid and dashed lines for a given value of θ, it can be seen that the exact and approximate expressions of the relative error follow the same trend and give similar values. The mean relative difference between ε and ε_𝓓 for the values plotted in Fig. 4 is 23%. The curves saturate at (b−1)/b and are plotted only for h<h _max for each value of θ.

3.2. Exact values for a circular pupil with central obscuration

Most of the time – if not always – the GS technique has been implemented in telescopes with pupils centrally-obscured by a secondary mirror. The shape of the autocorrelation S of such a pupil is different from that of a non-obscured pupil, as shown in Fig. 3. This affects the value of ε. In Fig. 4b, solid lines represent ε for the same conditions as in Fig. 4a but with a central obscuration of diameter 0.3D, which is a common value for optical telescopes. Like in Fig. 4a, dashed lines represent ε_𝓓. The mean and maximum difference between the exact and the approximate computations – for the parameter ranges considered in Fig. 4b – are 36% and 77% respectively, which are non-negligible. Nevertheless, if one wishes a rapid estimate of ε within a factor of 1.77 with respect to the exact value, Eq. 37 is a useful expression. An interesting feature is that ε does not increase monotonically as h increases, but has a local minimum.

Figure 5 shows four panels containing plots of ε (h). In each panel all the parameters but one remain constant. One of those parameters, e, is defined as follows: eD is the diameter of the secondary mirror. By examining Figs. 4b, 5a, 5b, 5c and 5d one can understand the effect of each parameter: h, θ, D, d, e and b. There appears to be two characteristic altitudes in each plot: the error remains mostly constant up to a given altitude h ₁, then it increases rapidly as h increases up to another altitude h ₂ from which ε remains strictly equal to (1−b)/b until h reaches h _max. For example, in Fig 5b, for d=10 km (red curve), h ₁~16 km and h ₂~30 km. Those “cut-off” altitudes h ₁ and h ₂ depend strongly on D and θ, less strongly on d and seem to be independent of e and b. For a given altitude of the turbulence, closer double stars and/or larger telescopes and/or analysis plane closer to the pupil give smaller errors, unless the saturation regime is reached. Larger secondary shadows produce larger oscillations of ε (h) for h<h ₁.

There are parameter combinations for which the relative error is negligible and others for which it is dramatic. For example, for a layer at h=2 km, using a double star of separation θ=5″, equal-magnitude components (b=0.25) on a 2-m telescope with a secondary mirror of diameter 0.6 m and the analysis plane at d=2 km, the relative error is ε=0.048, whereas for h=22 km, θ=8″, stars with 1 magnitude difference (b=0.203), D=1m, e=0.3 and d=4km, the relative error is ε=3.86. In such a case, considering Eq. 31 and the proportionality between C ² _N and the scintillation autocovariance (Eq. 1), to obtain the correct C ² _N value one would have to multiply the erroneous value by 1/(1+3.86)~0.21.

4. Conclusion

The problem concerning the normalization procedure in the GS technique, which was raised and studied by Johnston et al. in 2002 [1] in the particular case of turbulence at ground level, has been analyzed in the general case of turbulence at any altitude h. We provide a procedure to calculate and correct the error induced by the inaccurate normalization for pupils of any shape and give an approximate analytical expression of that error in the case of a circular pupil with no central obscuration. The approximate expression is useful to make a rapid estimate of the error and to understand the influence of the observational parameters. For an accurate correction of C ² _N (h) measurements the exact expression of the error (Eq. 32) should be used. It has been shown that the error is more significant for higher turbulent layers, smaller telescopes, longer distances of the analysis plane from the pupil, wider double-star separations, and larger differences of stellar magnitudes.

A forthcoming article will be devoted to the application of the results presented here on the correction of some GS measurements.

 

Fig. 5. Relative error as a function of the altitude h of the turbulence, for: (a) different values of D and θ=10″, d=4 km, e=0.3 and b=0.25; (b) different values of d and D=1.5 m, e=0.3, θ=7.5″ and b=0.25; (c) different values of e and D=1.5 m, d=4 km, θ=7.5″ and b=0.25; (d)different values of b and D=1.5 m, d=4 km, θ=7.5″ and e=0.3

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