Comparison between non-modulation four-sided and two-sided pyramid wavefront sensor

Jianxin Wang; Fuzhong Bai; Yu Ning; Linhai Huang; Shengqian Wang

doi:10.1364/OE.18.027534

1. Introduction

Wavefront sensor is one of main component of the adaptive optics, which is used to detect the wavefront aberration. Pyramid wavefront sensor (PWFS) is a novel slope sensor with dynamic modulation and is proposed by Ragazzoni [1] in 1996. Moreover, it has been successfully used to the Adaptive Optics System of the Galileo Telescope (AdOpt©TNG) [2] and Pyramir of 3.5m telescope at the Calar Alto Observatory [3]. It is also used in Multi-conjugate Adaptive optics Demonstrator systems (MAD) for Very large Telescope (VLT) [4], and Large Binocular Telescope Project (LBT) [5]. Theoretically, the performance of PWFS will excel that of the Shack-Hartmann sensor (HSWFS), and it has been reported that the performance of PWFS on the sky objects is equal or better than in that of HSWFS [6–9]. Many large telescopes, e.g., European Extremely Large Telescope (EELT), are planning to use PWFS [10]. Furthermore, they can be used to measure differential piston (DP) of the mirror segments, segment tip and tilt without changing the optical configuration [11–13].

The PWFS proposed by Ragazzoni is composed of a four-sided pyramid prism to divide the focus spot into four beams. Another type of PWFS is two-sided pyramid wavefront sensor (TSPWFS) which is composed of two 2-sided prisms and detects signals in two optical beams [14]. The working program of PWFS includes three periods since proposed by Ragazzoni: dynamic modulation with a tip-tilt mirror conjugate to the input plane to cyclically oscillate the distorted aberration [1,15–17]; static modulation with a static diffusing element [18,19] and no modulation [20].

It is shown that a PWFS without any modulation holds many advantages. First of all, its sensitivity is significantly larger than ones with modulation and SHWFS [6]. Secondly, the optical configuration will be largely simplified. Finally, the gain in magnitude is larger than that with modulation [21]. So the non-modulation PWFS and TSPWFS are concerned in this paper.

The diffraction theory of PWFS is imperfect [1,19,20,22–24], and in many papers it is identical to the theory of Foucault knife test [25], this will overestimate the performance of PWFS. Although C.Vérinaud ever doubted this equivalence, he hasn’t analyzed the characterization of non-modulation PWFS [21]. In this paper, based on wave optics the diffraction theories of PWFS and TSPWFS are presented. It is shown that that both sensors hold the same properties of slope and phase sensor. As a wavefront sensor the non-modulation PWFS and TSPWFS don’t detect the local slope of the distorted wavefront. By numerical simulation we compare the performance of TSPWFS and PWFS in slope and phase sensing and the interference between pupil images. The diffraction theory and numerical simulation results show that TSPWFS works better in precisely reconstructed wavefront and closed-loop correction.

2. Diffraction theory

2.1 PWFS

PWFS comprises two Fourier lens L₁, L₂, pyramid-shaped refractor and CCD, and its optical sketch diagram is shown in Fig. 1 .

Fig. 1 Optical sketch diagram of PWFS.

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The distorted wavefront is focused on the vertex of pyramid by L₁ and split into four beams by four facets of pyramid, and then four conjugated images of the pupil are produced on CCD by L₂. I ₁, I ₂, I ₃ and I ₄ denote their respective intensity. From the intensity distribution in the four pupils we can calculate the signals.

The complex amplitude in the pupil plane is defined as

E_{1} (x, y) = u_{0} \exp [i \frac{2 π}{λ} φ (x, y)] P,

Where

u_{0}

is amplitude,

φ (x, y)

is the phase, Pis the aperture function andλis the wavelength. The complex amplitude in the focal plane of L₁ is the Fourier transform of

E_{1} (x, y)

given by

E_{2} (u, v) = \frac{1}{λ f_{1}} F T {(E_{1} (x, y))}_{(\frac{u}{λ f_{1}}, \frac{v}{λ f_{1}}) .}

Phase mask of pyramid can be expressed by the following function

Φ_{P} = \exp [- i 2 π α_{0} (| u | + | v |)]

= \frac{1}{4} [1 + {(- 1)}^{j} sgn (u)] [1 + {(- 1)}^{k} sgn (v)] \exp (i 2 π α_{0} [{(- 1)}^{j + 1} u + {(- 1)}^{k + 1} v]) .

Where,

sgn (u) = {\begin{cases} 1 u > 0 \\ - 1 u < 0 \end{cases}

;

α_{0} = \frac{\sqrt{2}}{2} \frac{(n - 1)}{λ} α

, αis the angle between the side face and base face, n is the refractive index of the pyramid glass; j, k = 0,1 define the facet of pyramid, its distribution is shown in Fig. 2(a) . The complex amplitude refracted by pyramid is expressed as

Fig. 2 Sketch diagram of light field layout.

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E_{3} (u, v) = E_{2} (u, v) Φ_{P} = \frac{1}{λ f_{1}} F T {(E_{1} (x, y))}_{(\frac{u}{λ f_{1}}, \frac{v}{λ f_{1}})} Φ_{P} .

The complex amplitude in CCD detector plane is written as

E_{4} (ξ, η) = \frac{1}{λ f_{2}} F T {(E_{3})}_{(\frac{ξ}{λ f_{2}}, \frac{η}{λ f_{2}})} = \frac{f_{1}}{f_{2}} [E_{1} (\frac{- f_{1}}{f_{2}} ξ, \frac{- f_{1}}{f_{2}} η) * H_{1} (ξ, η) * H_{2} (ξ, η)] .

Where,

H_{1} (ξ, η) = \frac{1}{4} [δ (ξ, η) + \frac{{(- 1)}^{j} δ (η)}{i π ξ} + \frac{{(- 1)}^{k} δ (ξ)}{i π η} - \frac{{(- 1)}^{j + k}}{π^{2} ξ η}]

,

δ ()

is Dirac function, ‘*’ denotes convolution,

H_{2} (ξ, η) = δ [ξ + λ f_{2} α_{0} {(- 1)}^{j}, η + λ f_{2} α_{0} {(- 1)}^{k}]

, its function is equal to translation motion in Eq. (5), and can be negligible in the case of calculating intensity. The location of the four pupil images are shown in Fig. 2(b).

Then Eq. (5) can be simplified:

K (ξ, η) = \frac{f_{1}}{f_{2}} E_{1} (\frac{- f_{1}}{f_{2}} ξ, \frac{- f_{1}}{f_{2}} η) * H_{1} (ξ, η) .

By substituting $\frac{- f_{1}}{f_{2}} ξ = x$ and $\frac{- f_{1}}{f_{2}} η = y$ into Eq. (6), the following expression is obtained:

\begin{matrix} K = \frac{f_{1}}{4 f_{2}} {E_{1} (x, y) * [δ (x, y) - \frac{{(- 1)}^{j} δ (y)}{i π x} - \frac{{(- 1)}^{k} δ (x)}{i π y} - \frac{{(- 1)}^{j + k}}{π^{2} x y}]} \\ = \frac{f_{1} u 0}{4 f_{2}} {B_{1} + i {(- 1)}^{j} B_{2} + i {(- 1)}^{k} B_{3} - {(- 1)}^{j + k} B_{4}} . \end{matrix}

where

\begin{matrix} B_{1} = \exp [i \frac{2 π}{λ} φ (x, y)] P \\ B_{2} = B_{1} * [\frac{δ (y)}{π x}] = \int \int_{- \infty}^{\infty} P \exp [i \frac{2 π}{λ} φ (x^{'}, y^{'})] \frac{δ (y - y^{'})}{π (x - x^{'})} d x^{'} d y^{'} \\ B_{3} = B_{1} * [\frac{δ (x)}{π y}] = \int \int_{- \infty}^{\infty} P \exp [i \frac{2 π}{λ} φ (x^{'}, y^{'})] \frac{δ (x - x^{'})}{π (y - y^{'})} d x^{'} d y^{'} \\ B_{4} = B_{1} * (\frac{1}{π^{2} x y}) = \int \int_{- \infty}^{\infty} P \exp [i \frac{2 π}{λ} φ (x^{'}, y^{'})] \frac{1}{π^{2} (x - x^{'}) (y - y^{'})} d x^{'} d y^{'} \end{matrix}

The intensity distributions of the four pupil images in CCD plane are given by

\begin{matrix} I_{1} = {| E_{41} |}^{2} = {| K |}^{2}, (j = 0, k = 0); I_{2} = {| E_{42} |}^{2} = {| K |}^{2}, (j = 1, k = 0); \\ I_{3} = {| E_{43} |}^{2} = {| K |}^{2}, (j = 1, k = 1); I_{4} = {| E_{44} |}^{2} = {| K |}^{2}, (j = 0, k = 1). \end{matrix}

The signals $S_{x}$ and $S_{y}$ are determined by calculating the intensity distributions of the four image pupils according to

S_{x} = \frac{(I_{1} + I_{4}) - (I_{2} + I_{3})}{{| u_{0} |}^{2} {(\frac{f_{1}}{f_{2}})}^{2}} = \frac{1}{2} [R_{e} (i B_{1}^{*} B_{2}) - R_{e} (i B_{4}^{*} B_{3})],

S_{y} = \frac{(I_{1} + I_{2}) - (I_{3} + I_{4})}{{| u_{0} |}^{2} {(\frac{f_{1}}{f_{2}})}^{2}} = \frac{1}{2} [R_{e} (i B_{1}^{*} B_{3}) - R_{e} (i B_{4}^{*} B_{2})] .

The sign $R_{e} ()$ denotes real part; Eq. (8) and Eq. (9) can be expressed as

\begin{matrix} S_{x} = \frac{1}{2} \int_{- P (y)}^{P (y)} \frac{\sin {\frac{2 π}{λ} [φ (x, y) - φ (x^{'}, y)]}}{π (x - x^{'})} d x^{'} \\ + \frac{1}{2} \int_{- P (x)}^{P (x)} d y_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{\sin {\frac{2 π}{λ} [φ (x, y_{2}) - φ (x_{1}, y_{1})]}}{π^{3} (x - x_{1}) (y - y_{1}) (y - y_{2})} d x_{1}, \end{matrix}

\begin{matrix} S_{y} = \frac{1}{2} \int_{- P (x)}^{P (x)} \frac{\sin {\frac{2 π}{λ} [φ (x, y) - φ (x, y^{'})]}}{π (y - y^{'})} d y^{'} \\ + \frac{1}{2} \int_{- P (y)}^{P (y)} d x_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{\sin [\frac{2 π}{λ} [φ (x_{2}, y) - φ (x_{1}, y_{1})]]}{π^{3} (x - x_{1}) (y - y_{1}) (x - x_{2})} d x_{1} . \end{matrix}

The integration regions $[- P (x), P (x)]$ and $[- P (y), P (y)]$ in Eqs. (10) and (11) are the crossing points when x and y axes passing through the considered point $(x, y)$ interact with boundary of pupil function respectively.

2.2 TSPWFS

The configuration of TSPWFS is shown in Fig. 3 , in which the signals in x and y directions are measured by two optical beams. Each beam passes through a 4f magnification imaging system and a prism. The prism is placed at the back focal plane of the first lens in the 4f system, and magnification ratio is determined by the ratio of f ₄ to f₃, where f ₃ is focal length of L₃ or L₄, and f ₄ is focal length of L₅ or L₆. Edges of the two prisms are perpendicular to each other, one is along horizontal direction, and another is along vertical direction. Each optical beam produces two pupil images, $I_{x +}$ , $I_{x -}$ , $I_{y +}$ and $I_{y -}$ denote the intensity of pupil images split along the positive and negative x and y axis respectively. By analyzing the intensity difference between two images along the x and y direction we can obtain the signals G_x and G_y respectively. In this paper two couples of pupils (x and y) are imaged on the same detector simultaneously while it works almost equally as imaged on two detectors.

Fig. 3 Schematic diagram of TSPWFS.

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The complex amplitude $U_{1} (x, y)$ in the pupil plane defined by Eq. (1) is divided into two beams by beam splitting mirror (BS), and the transmission and reflection ratio of BS is a/b, the complex amplitude at the back focal plane of L₃ and L₅ are shown:

U_{2 x} (u, v) = \frac{\sqrt{a}}{λ f_{3}} F T {(U_{1} (x, y))}_{(\frac{u}{λ f_{3}}, \frac{v}{λ f_{3}})},

U_{2 y} (u, v) = \frac{\sqrt{b}}{λ f_{3}} F T {(U_{1} (x, y))}_{(\frac{u}{λ f_{3}}, \frac{v}{λ f_{3}})} .

According to edge direction parallel to vertical and horizontal line, the phase masks of prism are defined as:

Φ_{x} = \exp [- i 2 π α^{'} (| u |)] = \frac{1}{2} [1 + {(- 1)}^{t} sgn (u)] \exp [i {(- 1)}^{t + 1} 2 π α^{'} u],

Φ_{y} = \exp [- i 2 π α^{'} (| v |)] = \frac{1}{2} [1 + {(- 1)}^{t} sgn (v)] \exp [i {(- 1)}^{t + 1} 2 π α^{'} v] .

where t = 0,1 denote the light fall on facet at the positive and negative half-plane respectively;

α^{'} = \frac{(n - 1) α}{λ}

, the complex amplitude refracted by prisms are expressed as

U_{3 x} (u, v) = U_{2 x} (u, v) Φ_{x} = \frac{\sqrt{a}}{λ f_{3}} F T {(U_{1} (x, y))}_{(\frac{u}{λ f_{3}}, \frac{v}{λ f_{3}})} Φ_{x},

U_{3 y} (u, v) = U_{2 y} (u, v) Φ_{y} = \frac{\sqrt{b}}{λ f_{3}} F T {(U_{1} (x, y))}_{(\frac{u}{λ f_{3}}, \frac{v}{λ f_{3}})} Φ_{y} .

Two relay lenses L₄ and L₆ altogether form four images on the CCD detector, the complex amplitude are shown as

U_{4 x} (ξ, η) = \frac{1}{λ f_{4}} F T {(U_{3 x})}_{(\frac{ξ}{λ f_{4}}, \frac{η}{λ f_{4}})}

U_{4 y} (ξ, η) = \frac{1}{λ f_{4}} F T {(U_{3 y})}_{(\frac{ξ}{λ f_{4}}, \frac{η}{λ f_{4}})}

The signals are determined according to:

G_{x} = \frac{I_{x -} - I_{x +}}{{(\frac{f_{3}}{f_{4}})}^{2} {| u_{0} |}^{2} a},

G_{y} = \frac{I_{y -} - I_{y +}}{{(\frac{f_{3}}{f_{4}})}^{2} {| u_{0} |}^{2} b} .

The relationship between signals and aberration are expressed as

G_{x} = \int_{- P (y)}^{P (y)} \frac{\sin {\frac{2 π}{λ} [φ (x, y) - φ (x^{'}, y)]}}{π (x - x^{'})} d x^{'},

G_{y} = \int_{- P (x)}^{P (x)} \frac{\sin {\frac{2 π}{λ} [φ (x, y) - φ (x, y^{'})]}}{π (y - y^{'})} d y^{'} .

2.3 Discussion

For PWFS and TSPWFS, both of them divide the far-field spot at the back focal plane and detect the signals on the conjugate pupil plane. By comparing Eqs. (10) and (11) with Eqs. (22) and (23) we can find that the principle of TSPWFS is equivalent to Foucault knife theory and the signals measured by PWFS contain a cross term including signals in x and y direction. However the cross term don’t affect the sign of signals but its amplitude.

The distorted wavefront can be expanded on a set of Zernike modes [26], it can be written as $φ (x, y) = \sum_{m = 1}^{N} a_{m} Z_{m} (x, y)$ , N stands for the total number of Zernike mode, $a_{m}$ stands for the Zernike coefficients of the phase expansion, and $Z_{m} (x, y)$ is the Zernike polynomials.

In the case of small aberration, the sine term of Eqs. (10) and (11) with Eqs. (22) and (23) can be substituted by its first term of Taylor series expansion, and they can be expressed as

\begin{array}{l} S_{x} = \sum_{m = 1}^{N} a_{m} {\int_{- P (y)}^{P (y)} \frac{Z_{m} (x, y) - Z_{m} (x^{'}, y)}{λ (x - x^{'})} d x^{'} + \\ \int_{- P (x)}^{P (x)} d y_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{Z_{m} (x, y_{2}) - Z_{m} (x_{1}, y_{1})}{λ π^{2} (x - x_{1}) (y - y_{1}) (y - y_{2})} d x_{1}}, \end{array}

\begin{array}{l} S_{y} = \sum_{m = 1}^{N} a_{m} {\int_{- P (x)}^{P (x)} \frac{Z_{m} (x, y) - Z_{m} (x, y^{'})}{λ (y - y^{'})} d y^{'} + \\ \int_{- P (y)}^{P (y)} d x_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{Z_{m} (x_{2}, y) - Z_{m} (x_{1}, y_{1})}{λ π^{2} (x - x_{1}) (y - y_{1}) (x - x_{2})} d x_{1}} . \end{array}

G_{x} = \sum_{m = 1}^{N} a_{m} \int_{- P (y)}^{P (y)} \frac{2}{λ} \frac{Z_{m} (x, y) - Z_{m} (x^{'}, y)}{(x - x^{'})} d x^{'},

G_{y} = \sum_{m = 1}^{N} a_{m} \int_{- P (x)}^{P (x)} \frac{2}{λ} \frac{Z_{m} (x, y) - Z_{m} (x, y^{'})}{(y - y^{'})} d y^{'} .

The integral terms in Eqs. (24) and (25) and Eqs. (26) and (27) are linear with $S_{x}$ $S_{y}$ and $G_{x}$ $G_{y}$ respectively.

In the case of large aberration, the relationship between the detected signals and measured wavefront is nonlinear, but the polarities keep unchanged so it can work in a closed-loop correction.

The diffraction theory of non-modulation PWFS and TSPWFS shows that the linear dependence of the PWFS and TSPWFS signals on wavefront aberration is only valid for small aberrations. In the linear regime, they hold the same property as HSWFS and can reconstruct wavefront with modal method, while their linear construction bases are not exactly the gradient of Zernike modes.

3. Wavefront reconstruction

3.1 Slope sensing

Equations (24) and (25) and Eqs. (26) and (27) are valid when the peak-to- valley (PV) value of distorted wavefront is less than 0.5λ, so non-modulation PWFS and TSPWFS can work like HSWFS.

The integral term in Eqs. (12) and (13) can be written as

T_{x m} = \int_{- P (y)}^{P (y)} \frac{Z_{m} (x, y) - Z_{m} (x^{'}, y)}{λ (x - x^{'})} d x^{'} + \int_{- P (x)}^{P (x)} d y_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{Z_{m} (x, y_{2}) - Z_{m} (x_{1}, y_{1})}{λ π^{2} (x - x_{1}) (y - y_{1}) (y - y_{2})} d x_{1},

T_{y m} = \int_{- P (x)}^{P (x)} \frac{Z_{m} (x, y) - Z_{m} (x, y^{'})}{λ (y - y^{'})} d y^{'} + \int_{- P (y)}^{P (y)} d x_{2} \int_{- y_{0}}^{y_{0}} d y_{1} \int_{- P (y_{1})}^{P (y_{1})} \frac{Z_{m} (x_{2}, y) - Z_{m} (x_{1}, y_{1})}{λ π^{2} (x - x_{1}) (y - y_{1}) (x - x_{2})} d x_{1} .

The signals $S_{x}$ and $S_{y}$ are the linear summation of the $T_{x m}$ and $T_{y m}$ , in matrix form $S = T A$ where, $S = {[S_{x}, S_{y}]}^{'}$ , response matrix Tis composed of $T_{x m}$ and $T_{y m}$ , Ais Zernike coefficient vector.

The integral term in Eqs. (28) and (29) can be expressed as

Q_{x m} = \int_{- P (y)}^{P (y)} \frac{2}{λ} \frac{Z_{m} (x, y) - Z_{m} (x^{'}, y)}{(x - x^{'})} d x^{'},

Q_{y m} = \int_{- P (x)}^{P (x)} \frac{2}{λ} \frac{Z_{m} (x, y) - Z_{m} (x, y^{'})}{(y - y^{'})} d y^{'} .

The TSPWFS signals $G_{x}$ and $G_{y}$ are the linear summation of $Q_{x m}$ and $Q_{y m}$ , in matrix form Eqs. (26) and (27) can be written as $G = Q A$ , where $G = {[G_{x}, G_{y}]}^{'}$ , response matrixQis composed of $Q_{x m}$ and $Q_{y m}$ . If we obtain the matrixT and G, by using SVD standard algorithm, then the pseudo-inverse $T^{+}$ and $G^{+}$ of the PWFS and TSPWFS response matrix respectively, the so called reconstruction matrix are determined. The Zernike coefficient Adetected with PWFS and TSPWF are determined by the equations $A = T^{+} S$ and $A = G^{+} Q$ respectively. Equations (28)–(31) demonstrate that the response matrix of PWFS has no analytic solution while TSPWFS has.

The linearity is small for PWFS and TSPWFS. We can use dynamic modulation to increase linearity and dynamic range at the expense of sensitivity.

3.2 Phase sensing

The Eqs. (10) and (11) with Eqs. (22) and (23) show that the measured signals are the distance weighted sum of the sinus of the phase difference between each point on the pupil function. More distant points, with a higher probability of differing more in phase, will be weighted less. In the case of large aberration, it makes sure Eqs. (10) and (11) with Eqs. (22) and (23) can still provide the correct sign of the aberration when the sine terms make the approximation of sin(x) ~ x. With linear wavefront reconstruction, we can obtain an estimated wavefront, and then the deformable mirror is employed to correct the aberration until the residual error is acceptable. Even though the detected signal is not exactly equivalent with the wavefront, the closed-loop can be converged anyway.

The reconstruction matrix is defined as slope sensing, in each closed-loop operation, for PWFS and TSPWFS we obtain the Zernike modes vectorAby $A = T^{+} S$ and $A = G^{+} Q$ respectively.

4. Numerical simulation

Both PWFS and TSPWFS use refract prism (four facets or two facets) to divide the focal plane. Through comparing with Foucault knife-edge test this kind of wavefront sensor can detect simultaneously signals in x and y direction. The key component of this kind wavefront sensor is a four-sided pyramid (seen Fig. 1) or two-sided pyramid (seen Fig. 3), and the base angle is usually less than 5°. There are two models to describe phase mask of the prism: one is amplitude mask algorithm (AMA) [23], another is phase mask algorithm (PMA) [27]. In the former method the images are infinitely distant from each other. While in the latter method, the interference between the pupil images is taken into account, and it is a real simulation of the optical setup. Here we choose the phase mask model.

Magnification of 4f imaging system is unity; the size of all matrices used is 511 × 511, where the imbedded circle pupil has a diameter of 63 points. The point spread function (PSF) sampling pixel is 21 and can meet the high accuracy requirement [27]. The phase screen mentioned in the numerical simulation consists of 3 to 65 Zernike modes, in which the coefficients follow the Kolmogorov power spectrum [28]. The separation between adjacent pupils must choose properly to decrease the effect of interference between the pupil images as well as make the best of the detector pixels. The parameter M is the ratio of center distance between adjacent pupils to the diameter of one pupil image.

The intensity distributions of astigmation aberration on the detector with RMS = 0.3λ obtained by PWFS and TSPWFS respectively are shown in Fig. 4 .

Fig. 4 Pupil images in the CCD plane when an astigmation aberration incidences respectively two kinds of wavefront sensors.

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The size of detector is same for PWFS and TSPWFS, the parameter M for PWFS and TSPWFS with two CCD detectors is equal to 1.5 and 1.05 for TSPWFS with one CCD detector. In Fig. 4, the light between the pupil images is resulted by interference between pupil images.

4.1The effect of images interference

To make best use of detector pixels as well as weaken the interference between images, images on the detector should be properly arranged. It is shown in Fig. 5 .

Fig. 5 The arrangement of images on detector. (a) images by PWFS (b) images detected simultaneously by TSPWFS with one detector (c) images split along x directional by TSPWFS detect by one detector.

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To compare the TSPWFS performance with one or two detectors we define the parameter M′.

M' = {\begin{cases} M PWFS; TSPWFS with 2 CCD \\ \sqrt{2} M TSPWFS with 1 CCD \end{cases} .

For PWFS and TSPWFS, the parameter M is large than 1 and the minimum size of detector is ${[(M + 1) \times r]}^{2}$ , and ris the radius of pupil image.

The signal difference between PMA and AMA is due to pupils’ interference. Figure 6 is the RMS of signals difference changing with M′, and the wavefront used in Figs. 6(a) and (b) is Z₅.

Fig. 6 Signal difference is due to interference between images. (a) the difference between TSPWFS signals with 1 or 2 CCD detectors (b) difference between signals with PMA and AMA models.

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The difference between TSPWFS signals obtained by one or two detectors changing with M′ is illustrated in Fig. 6(a). It demonstrates that the larger the RMS value of wavefront, the less the difference between signals, and the signal difference with different magnitude of aberration diminishes until M′ is equal to 1.85. The signal difference obtained with PMA and AMA by PWFS or TSPWFS is shown in Fig. 6(b), where the RMS of Z₅ is 0.1λ. The numerical simulations illustrate that interference between TSPWFS images is less than that of PWFS. Moreover, for PWFS and TSPWFS when M is near 4 and 3 respectively, the interference between images can be neglected. So in the following numerical simulation M′ is 4.

Using PWFS and TSPWFS to measure a frame of $d / r_{0} = 2$ phase screen with RMS = 0.08λ and PV = 0.41λ, where d is the input aperture diameter, r ₀ is the Fried parameter. The x and y directional signals measured by both sensors are illustrated in Fig. 7 , the pictures on the left show the PWFS signals, pictures on the right show TSPWFS signals. The sign of the signals detected by PWFS and TSPWFS are identical, while PWFS signals are stronger than TSPWFS signals and the amplitude difference is greater in the edge than in the inner.

Fig. 7 The signals of a d/r₀ = 2 phase screen in x and y direction measured by PWFS and TSPWFS. (a) and (c) show PWFS signals in x-direction and y-direction respectively (b) and (d) denote TSPWFS signals in x-direction and y-direction respectively.

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4.2 Establishment of the response matrix

From the upper theory analysis, we know that the response matrix of PWFS does not have analytic solution. In the case of small aberration the relationship between the measured signals of PWFS and the respond matrix is linear, we can build the response matrix in the linear region by recording the response of PWFS to a series of given wavefronts e.g. different Zernike modes. To compare the intrinsic property of both sensors, response matrix of TSPWFS in the following simulation can be obtained according to the same procedure. We should firstly determine the magnitude of each Zernike mode for establishing the response matrix to keep wavefront sensors in linear region.

In simulation, ten groups of 65 Zernike modes with their RMS values varying from 0.01λ to 0.1λ are given. A series of response matrixes Tand Q to the given modes are obtained. Using these matrixes to reconstruct the phase screen as same as in Fig. 7, the residual errors between the reconstructed wavefront and the original wavefront are obtained. The reconstruction precision is defined as the RMS ratio of residual wavefront to original wavefront, $β = {RMS}_{e} / {RMS}_{0}$ . The results are shown in Fig. 8 . The condition number of reconstruction matrix is an estimation of the reconstruction stability [29]. Large condition number predicates less stability. The condition number of response matrix built with different amplitude is demonstrated in Fig. 8(a). It shows that for both sensors the condition number of response matrix will decrease as the RMS magnitude increases. Condition number of PWFS is larger than that of TSPWFS under all conditions. Figure 8(b) shows that to reconstruct the same aberration by both sensors the residual errors of PWFS are lager than TSPWFS with each response matrix built at same RMS amplitude. The larger the given magnitude, the smaller of condition number, however at larger given magnitude the nonlinear characteristic will decrease the reconstruction accuracy. For two sensors with different magnitudes of given Zernike modes to establishing the response matrixes, there are appropriate regions of minimum errors (0.06 and 0.08 for PWFS and TSPWFS respectively). In the later simulation we choose 0.07λ as the RMS value of both PWFS and TSPWFS.

Fig. 8 Comparison of response matrix obtained with different amplitude of Zernike modes. (a) the curve of the condition number of response matrix changing with different given RMS values of Zernike modes to obtain corresponding response matrix (b) the reconstruction precision changing with the Zernike mode RMS to obtain the response matrix.

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4.3 Measurements of single Zernike modes

We reconstruct two sets of the first 30 order Zernike modes with original wavefront RMS 0.02λ and 0.15λ separately by using the established reconstruction matrix $T^{+}$ and $Q^{+}$ , and the simulation results are illustrated in Fig. 9 .

Fig. 9 Reconstruction precisions of PWFS and TSPWFS for single Zernike modes. original wavefront RMS values of (a) and (b) are 0.02λ and 0.15λ respectively.

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From the results shown in Fig. 9 the following conclusion can be obtained. (1) Reconstruction precision of TSPWFS is higher than PWFS for most Zernike aberrations; (2) The reconstruction precision of higher order Zernike is worse than the lower order Zernike aberration; (3) In the case of large distortion the linear relationship between signals and phase is destroyed, and the reconstruction precision doesn’t satisfy the wavefront measurement requirement. However the sign of signals is right, they can work in closed-loop correction.

4.4 Correction of static aberration

We make a comparison between PWFS and TSPWFS on the accuracy and the speed of the closed-loop with different sampling rate. Additionally, the effect of CCD read-out noise is considered.

In the adaptive optics system, the feedback algorithm commonly used is a pure integrator [30]. The closed-loop operation can be expressed as $φ_{L} = φ_{L - 1} + g Δ φ$ , where $φ_{L}$ and $φ_{L - 1}$ are the value of the feedback wavefront by the DM at the $L^{t h}$ and $L - 1^{t h}$ iteration. Where, $Δ φ$ is the reconstruction residual wavefront andgis the loop gain.

In the numerical simulation the wavefront corrector is a liquid-crystal spatial light modulator, and the reconstruction matrix is built as before. The grey level of images on the CCD plane ranges from 0 to 255, and the CCD readout noise is Gaussian white noise with a zero mean value. In closed-loop correction, we apply a series of gto correct a frame phase screen and select the one to keep the closed-loop correction steady and fast. The evolution of SR with different loop gain is shown in Fig. 10 , and the phase screen is d/r ₀ = 30 (RMS = 1.40λ, PV = 6.19λ). According to the results from Figs. 10(a) and 10(b), we choose 1.5 and 1.6 for respectively PWFS and TSPWFS as the optimal loop gain.

Fig. 10 The optimal loop gain for PWFS and TSPWFS. (a) and (b) are the evolution of SR with different loop gain by PWFS and TSPWFS respectively.

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Three typical frames of phase screen are select to deeply investigate the properties of both sensors and listed in Tab. 1(a). The closed-loop iteration number, strehl ratio after correction as well as the optimal loop gain are listed in Tab. 1(b). Tab.1 demonstrates that for PWFS or TSPWFS, the performance of closed-loop correction with phase screen d/r ₀ = 60 is better than that with phase screen d/r ₀ = 100. The reason is that the spatial frequency of atmosphere turbulence in the latter is higher than in the former. When read-out noise is considered, the speed of loop closure with PWFS as wavefront sensor apparently decreases and the optimal loop gain for both of sensors obviously increases. Closed-loop speed of both sensors decreases largely when spatial resolution of aberration increases. Moreover, TSPWFS exceeds PWFS in loop speed and correction accuracy.

Tab. 1. Detecting condition and closed-loop performance by PWFS and TSPWFS

View Table

4.4 Choice of the parameter M

The optimal choice of the parameter M must make a tradeoff between the effect of interference on reconstruction precision and detector pixels utilization efficiency. By numerical simulation we give an instruction on the parameter selection. To compare the TSPWFS performance with one or two detectors, firstly we will use the parameter M′.

Figure 4 demonstrates that for PWFS the interference exists mainly in the center area between adjacent images and for TSPWFS with two detectors in the area between opposite images. When four images of TSPWFS are projected on one detector, we must consider interference not only between images but also on the signal of other pair images.

The intensity in the line perpendicular to center line between two images is normalized by the maximum of intensity on the detector, and is shown in Fig. 11 . The aberration used in Figs. 11(a) and (b) is Z₅. Figure 11(a) is the curve of normalized intensity of a pair of TSPWFS images with RMS 0.1λ and different M′, which demonstrates that the half width of interference pattern between images is about $0.8 \times r$ , and it is almost unchanging, when M becomes larger. To avoid interference on the signal of other pair images, for TSPWFS with one detector, M′ should be larger than 1.8, or M >1.27. Figure 11(b) is the normalized intensity changing with different RMS value and M′ = 1.47 (M = 1.04). When the aberration is larger, the effective area of interference is smaller.

Fig. 11 The width of interference area of TSPWFS with two detectors. (a) the normalized intensity distributes along the line perpendicular to the center line between two opposite images with different center distance M′, and the aberration is Z₅ with RMS 0.1λ (b) the distribution of normalized intensity changes with same M′ but different RMS value of aberration.

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Because the effect of interference on wavefront measurement differs from on the closed-loop correction, the measurement accuracy (Fig. 12(a) ) and the residual wavefront of closed-loop correction (Fig. 12(b)) with different M′ were simulated.

Fig. 12 The effect of interference between pupil images with different M′. (a) the measurement accuracy of d/r ₀ = 2 changing with M′ (b) the closed-loop performance of d/r ₀ = 20 changing with M′.

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The phase screens used in Figs. 12(a) and (b) are d/r ₀ = 2 (RMS = 0.09λ, PV = 0.51λ) and d/r ₀ = 20 (RMS = 0.92λ, PV = 2.88λ) respectively. Figure 12(a) shows that with the best accuracy for PWFS the parameter M′ is close to 4.5, while for TSPWFS with one or two detectors the parameter M′ is about 3. Figure 12(b) shows that for PWFS and TSPWFS with one detector the parameter M′ is 1.3 and 1.8 respectively, the accuracy can meets the accuracy requirement. Figures 12(a) and (b) illustrate that when images are tightly contacted (M = 1), the performance of PWFS and TSPWFS are worst. Accuracy with TSPWFS is higher than with PWFS in both cases of wavefront measurement and closed-loop correction. The performance of TSPWFS in closed-loop correction by one detector is equal to by two detectors when M′ is larger than 1.8. For TSPWFS with one detector, when M′ <1.67, the accuracy is no more than PWFS. From the simulation in Fig. 11(a), we know that it is due to the images locating in another interference area. From the simulation, we can conclude: in the case of wavefront measurement, the optimal M for PWFS and TSPWFS with one detector can be 3 and 1.8, and for closed-loop correction is 1.3 for both sensors.

5. Conclusion

By using optics wave theory, the diffraction theory of PWFS and TSPWFS has been deduced. The principle of TSPWFS is equivalent to the theory of Foucault knife test. Compared with TSPWFS, the signals obtained by PWFS contain a cross term from signals in x and y directions.

The magnitude of Zernike modes for establishing the response matrix for PWFS and TSPWFS was optimized. The numerical simulations show that the precision of TSPWFS is higher than that of PWFS in both cases of small and large aberration. PWFS is more influenced by the read-out noise, and the interference between images affects slightly on the closed-loop correction. Additionally, the ability of PWFS and TSPWFS is subjected to the spatial samplings of wavefront. The speed and performance of closed-loop correction depend on the wavefront sensor, the loop gain, the amplitude of distorted wavefront and the spatial sampling. For closed-loop correction, the optimum distance between adjacent images for both sensors is 1.3 times image diameter.

Both wavefront sensors can be used in the AO system, and TSPWFS will excel PWFS in performance.

Acknowledgments

The authors thank Prof. Wenhan Jiang and the reviewers for their helpful suggestions, and the work has been funded by the Yong Scientists Fund of National Natural Science Foundation of China (61008038).

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(a)	$d / r_{0}$	30		60		100
	RMS(λ)	1.40		2.51		2.15
	PV(λ)	6.19		11.04		13.8
	Variance of noise (grey level)	0	0.03 × 255	0	0.03 ×255	0	0.03 × 255
	Measurement sampling	63 × 63 (circle pupil aperture)

	After correction
(b)	g	PWFS	1.50	5.10	1.50	5.10	1.50	5.10
	g	TSPWFS	1.60	3.80	1.60	3.80	1.60	3.80
	SR	PWFS	0.995	0.995	0.993	0.993	0.664	0.664
	SR	TSPWFS	0.999	0.999	0.999	0.999	0.884	0.884
	Loop number	PWFS	18	21	34	49	63	78
	Loop number	TSPWFS	15	16	30	34	47	50

Comparison between non-modulation four-sided and two-sided pyramid wavefront sensor

Abstract

1. Introduction

2. Diffraction theory

2.1 PWFS

2.2 TSPWFS

2.3 Discussion

3. Wavefront reconstruction

3.1 Slope sensing

3.2 Phase sensing

4. Numerical simulation

4.1The effect of images interference

4.2 Establishment of the response matrix

4.3 Measurements of single Zernike modes

4.4 Correction of static aberration

4.4 Choice of the parameter M

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Tables (1)

Equations (35)

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