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Hartmann-Shack test with random masks for modal wavefront reconstruction

Oleg Soloviev and Gleb Vdovin

Open Access Open Access

Abstract

The paper discusses the influence of the geometry of a Hartmann-(Shack) wavefront sensor on the total error of modal wavefront reconstruction. A mathematical model is proposed, which describes the modal wavefront reconstruction in terms of linear operators. The model covers the most general case and is not limited by the orthogonality of decomposition basis or by the method chosen for decomposition. The total reconstruction error is calculated for any given statistics of the wavefront to be measured. Based on this estimate, the total reconstruction error is calculated for regular and randomised Hartmann masks. The calculations demonstrate that random masks with non-regular Fourier spectra provide absolute minimum error and allow to double the number of decomposition modes.

©2005 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. Differences between ideal reconstruction matrix H (∞×∞ identity matrix) and non-ideal Hartmann matrix H and corresponding modal reconstruction errors. The series truncation error appears due to the discarding in reconstruction high-order modes (marked with blue colour in the diagram). These modes, however, can be sensed as low-order ones (nonzero elements in the “tail” of the matrix, marked with orange), the effect known as aliasing in sampling theory. If the first N columns do not form a unity matrix, low-order modes are also reconstructed with aliasing error, often called cross-coupling or cross-talk in the literature.

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Fig. 2.
Fig. 2. A schematic sketch of the sensing of high- and low-order modes for a regular and an irregular sensor geometry in the Fourier domain. Averaging of the slopes over subapertures acts as a low-pass filter (multiplication with 𝓕 χa ). During this operation high-order modes are transformed into low-order ones (aliasing). Even in the case where filtered functions are quite dissimilar, sampling with a periodic centre distribution (convolution with a periodical spectrum) could result in significant spectrum distortion and similar spectra (cross-talk). Irregular geometry, with the spectrum of the distribution of centres imitating a δ-function, should not have the cross-talk effect.

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Fig. 3.
Fig. 3. 5 masks used for calculation of H, shown in the unit circles. The subapertures are circles with radius 1 11 , except (b) with radius 1 9 . Each mask contains 61 subapertures, except (a) with 91 holes.

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Fig. 4.
Fig. 4. Fourier transforms of the s/a centres for Hex61, move61, and MC61 masks.

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Fig. 5.
Fig. 5. Matrices H, obtained for various masks for N = 40 ((a) – (e)) and N = 80 ((f) –(j)) decomposition modes. Absolute values of non-zero elements are indicated by a linear gray-scale level for values in [0,1] and a logarithmic hue scale for values greater than 1 (see Fig. (k)). This figure is advised to be viewed on-screen and with increased zoom level.

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Fig. 6.
Fig. 6. An infinite sum hl1j (c l 1 c l 2〉 hjl2 for H move61 for N = 15 decomposition modes is approximated by a finite one with l 1,l 2 =N +1,…,L and plotted against L.

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Fig. 7.
Fig. 7. Sum of aliasing and truncation errors and measurement error for the 5 masks shown in Fig. 3 and the Kolmogorov turbulence statistical model.

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Fig. 8.
Fig. 8. H for decomposition by averaged (top picture in each pair) and point gradients for various N.

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Fig. 9.
Fig. 9. Aliasing and measurement error, in (D/r 0) ( D r 0 ) 5 3 and σg2 resp., for reconstruction with averaged and point gradients.

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Tables (1)

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Table 1. Properties of the HS sensing for various mask types

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Equations (44)

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f ˜ = 𝓗 f = ∑ j = 1 N λ j f j = λ j f j ,
f = c i g i + c 0 ,
𝓗 g i = ∑ j = 1 N h i j f j ,
𝓗 f = ( f 1 , f 2 , … , f N ) · H · ( c 1 c 2 … ) = h i j c i f j ,
λ j = h i j c i .
ε = ∥ 𝓗 f − f ∥
〈 ε 〉 = 〈 ∥ f − f ˜ ∥ 〉 = ∥ 〈 f 〉 − 〈 f ˜ 〉 ∥ = ∥ 〈 c i 〉 g i − 〈 λ j 〉 f j ∥ = ∥ 〈 c i 〉 g i − 〈 c i 〉 h i j f j ∥ ,
〈 ε 2 〉 = 〈 ( f − f ˜ , f − f ˜ ) 〉 = 〈 f f 〉 − 2 〈 f f ˜ 〉 + 〈 f ˜ f ˜ 〉
= 〈 c i 1 c i 2 〉 g i 1 g i 2 − 2 〈 c i λ j 〉 g i f j + 〈 λ j 1 λ j 2 〉 f j 1 f j 2
= 〈 c i 1 c i 2 〉 g i 1 g i 2 − 2 〈 c i 1 c i 2 〉 h j * i ′ g i 1 f j + h i 1 j 1 〈 c i 1 c i 2 〉 h j * i ′ ( f j 1 , f j 2 ) ,
f j = g j ,
g i g i ′ = g i i ′
〈 ε 2 〉 = 〈 c i 1 c i 2 〉 g i 1 g i 2 − 2 〈 c i λ j 〉 g i g j + 〈 λ j 1 λ j 2 〉 g j 1 g j 2 = 〈 c i c i 〉 − 2 〈 c j λ j 〉 + 〈 λ j λ j 〉 ,
〈 ε 2 〉 = 〈 c i c i 〉 − 2 〈 c j c i 〉 h j * i + h i 1 j 〈 c i 1 c i 2 〉 h j * i 2 .
h i j = δ i j ,
𝓗 = ( ∑ i = 1 ∞ c i g i ) = ∑ j = 1 N c j g j .
〈 ε 〉 = ∥ ∑ l = N + 1 ∞ c l g l ∥
〈 ε 2 〉 = 〈 c i c i 〉 − 〈 c j λ j 〉 = 〈 c l c l 〉 = def σ tr 2 .
h j j ′ = δ j j ′ .
〈 ε 2 〉 = 〈 c j c j 〉 + 〈 c l c l 〉 − 2 〈 c j c j ′ 〉 h j * j ′ − 2 〈 c j c l 〉 h j * l
+ h j 1 j 〈 c j 1 c j 2 〉 h j * j 2 + h l 1 j 〈 c l 1 c j 2 〉 h j * j 2 + h j 1 j 〈 c j 1 c l 2 〉 h j * l 2 + h l 1 j 〈 c l 1 c l 2 〉 h j * l 2
= 〈 c j c j 〉 + 〈 c l c l 〉 − 2 〈 c j c j 〉 − 2 〈 c j c l 〉 h j * l + 〈 c j c j 〉 + h l j 〈 c l c j 〉 + 〈 c j c l 〉 h j * l + h l 1 j 〈 c l 1 c l 2 〉 h j * l 2
= 〈 c l c l 〉 + h l 1 j 〈 c l 1 c l 2 〉 h j * l 2 .
σ al 2 = def h l 1 j 〈 c l 1 c l 2 〉 h j * l 2 ,
1 S a k ( ∫ a k ∂ f ∂ x d r , ∫ a k ∂ f ∂ y d r ) ,
χ a k ( r ) = χ a ( r − r k ) ,
1 S a ∫ ℝ 2 χ a ( r − r k ) ∇ f ( r ) d r = 1 S a χ a * ∇ f
∑ i = 1 m δ ( r − r k ) .
𝓖 f = def 1 S a ∑ k = 1 m δ ( r − r k ) ( χ a * ∇ f ) .
dim Im 𝓗 ≤ rank 𝓖 .
𝓕 ( 𝓖 f ) = 1 S a 𝓕 ( ∑ k = 1 m δ ( r − r k ) ) * ( 𝓕 χ a ∙ 𝓕 ∇ f ) .
f ̂ j = 𝓖 f j ,
f ̂ j = ∑ k = 1 m δ ( r − r k ) ∇ f j .
∥ 𝓖 − λ j f ̂ j ∥ = min x 1 , … x N ∥ 𝓖 − x j f ̂ j ∥
λ j = 𝓛 ( 𝓖 f ) = ( 𝓛 ◦ 𝓖 ) f = 𝓗 f ,
H = L · G .
L = G N +
λ j = 𝓛 ( 𝓖 f + e k ) = 𝓗 f + 𝓛 e k .
〈 e k 〉 = 0 , 〈 e k , e k ′ 〉 = σ g 2 δ k ′ k , 〈 c i e k 〉 = 0 ,
〈 ε 2 〉 = 〈 c i c i 〉 − 2 〈 c j c i 〉 h j i + h i 1 j 〈 c i 1 c i 2 〉 h j i 2 + l k 1 j l j k 2 σ g 2 ,
σ meas 2 = l k 1 j l j k 2 σ g 2
x i y j ↦ 1 π a 2 ∫ x 2 + y 2 ≤ a 2 x i y j dxdy = { a i + j Γ ( i + 1 2 ) Γ ( j + 1 2 ) π Γ ( i + j 2 + 2 ) , i and j are even , 0 , otherwise .
〈 c l c l 〉 ≍ 0.274 N − 0.8428 ( D r 0 ) 5 3 ,
L ˙ = G ˙ N +
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