Real-time Adaptive Optics with pyramid wavefront sensors: A theoretical analysis of the pyramid sensor model
RReal-time Adaptive Optics with pyramid wavefront sensors:A theoretical analysis of the pyramid sensor model
Victoria Hutterer , Ronny Ramlau and Iuliia Shatokhina October 2, 2018
Abstract
We consider the mathematical background of the wavefront sensor type that is widely used inAdaptive Optics systems for astronomy, microscopy, and ophthalmology. The theoretical analysis ofthe pyramid sensor forward operators presented in this paper is aimed at a subsequent developmentof fast and stable algorithms for wavefront reconstruction from data of this sensor type. In ouranalysis we allow the sensor to be utilized in both the modulated and non-modulated fashion. Wederive detailed mathematical models for the pyramid sensor and the physically simpler roof wavefrontsensor as well as their various approximations. Additionally, we calculate adjoint operators whichbuild preliminaries for the application of several iterative mathematical approaches for solving inverseproblems such as gradient based algorithms, Landweber iteration or Kaczmarz methods.
Ground-based telescope facilities suffer from degraded image quality caused by atmospheric turbulence.When light from a distant star passes the Earth’s atmosphere, initially planar wavefronts get distorteddue to turbulent air motions causing fluctuations of the index of refraction. Therefore, advanced AdaptiveOptics (AO) systems [35, 64] are incorporated in innovative telescope systems to mechanically correctin real-time for the distortions with deformable mirrors. The shape of the deformable mirrors is deter-mined by measuring wavefronts coming from either bright astronomical stars or artificially produced laserbeacons. The basic idea is to reflect the distorted wavefronts on a mirror that is shaped appropriatelysuch that the corrected wavefronts allow for high image quality when observed by the science camera(see Figure 1). The according positioning of the mirror actuators implies the knowledge of the incomingwavefronts. Thus, in Adaptive Optics one is interested in the reconstruction of the unknown incomingwavefront Φ from available data in order to calculate the optimal shape of the deformable mirror. Un-fortunately, there exists no optical device which is able to measure the wavefront directly. Instead, awavefront sensor (WFS) measures the time-averaged characteristic of the captured light that is related tothe incoming phase. The wavefront sensor splits the telescope aperture into many small, equally spacedsubapertures and detects the intensity of the incoming light in each of the subapertures. The numberof subapertures defines the spatial resolution of the AO system. The detector of the sensor providesthe intensity data by integrating the light from the celestial object. From the detector’s analog signal,digital sensor measurements are sampled with a time period T which is usually extremely short, i.e.,0.3 - 2 milliseconds.This paper is focused on the pyramid wavefront sensor (PWFS), which has been invented in the 1990s[60]. It is gradually gaining more and more attention from the scientific community, especially in astro-nomical AO, due to its increased sensitivity, improved signal-to-noise ratio, robustness to spatial aliasingand adjustable spatial sampling compared to the other popular wavefront sensor choice — the Shack-Hartmann (SH) sensor. Several theoretical studies [8, 21, 22, 48, 61, 62, 70, 80, 82, 81, 85] includingnumerical simulations and laboratory investigations with optical test benches [4, 33, 52, 59, 78, 84] have Industrial Mathematics Institute, Johannes Kepler University, Linz, Austria. [email protected] Industrial Mathematics Institute, Johannes Kepler University, Linz, and Johann Radon Institute for Computationaland Applied Mathematics, Linz, Austria. Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria. a r X i v : . [ a s t r o - ph . I M ] S e p igure 1: Principle of wavefront correction [2]. A deformable mirror reflects a perturbed wavefront andpropagates the corrected, planar wavefront to the science camera yielding improved image quality.confirmed the advantages of pyramid wavefront sensors while additionally promising surveys were oper-ated on sky [24, 23, 25, 27, 32, 56, 57, 58].The current development of a new era of Extremely Large Telescopes (ELTs) with primary mirrors of25 −
40 m in diameter brings new challenges to the field of Adaptive Optics. For ELTs, pyramid sensorsshow enhanced performance in real life settings, e.g., they provide the ability to sense differential pistonmodes induced by diffraction effects of realistic telescope spiders that support secondary mirrors andperform even under significant levels of non-common path aberrations [79, 20].Pyramid wavefront sensors are going to be included in many ELT instruments [5, 12, 79, 20, 31, 30,45, 51, 53, 54, 80]. We would like to mention that, apart from astronomical applications, the pyramidwavefront sensor is also applied in adaptive loops in ophthalmology [10, 13, 15, 43] and microscopy [42, 44].Therefore, wavefront reconstruction algorithms for pyramid wavefront sensors are in high demand. Themain goal of this paper is to provide an extensive mathematical analysis of the PWFS operators in orderto develop suitable wavefront reconstruction methods. So far, the existing algorithms are (with a smallnumber of exceptions as, e.g., [11, 14, 28, 29, 37, 46, 47, 48, 83]) based on a linear assumption of thepyramid sensor model [36, 41]. Nevertheless, the simplifications of the non-linear pyramid operator allowfor acceptable wavefront reconstruction quality.Basically, wavefront reconstruction from pyramid sensor data consists in solving two non-linear integralequations P Φ = [ s x , s y ]with respect to the unknown wavefront Φ, where P = [ P x , P y ] is a singular Volterra integral operatorof the first kind.In the following, we derive and analyze the mathematical model for both the non- and modulated pyramidsensor.The paper is organized as follows: In Section 2, we give a brief introduction into the physical backgroundof pyramid and roof wavefront sensors. Afterwards, we derive the singular and non-linear forward PWFSoperator P and roof sensor operators R . In order to simplify the problem of solving the WFS equation,we calculate linearizations R lin for roof wavefront sensors in Section 3. Furthermore, in Section 4, weevaluate the corresponding adjoint operators which are necessary for the application of linear iterativemethods such as gradient based algorithms or Landweber iteration for wavefront reconstruction addressedin an upcoming second part of the paper. As it can be seen in Figure 2, the main component of the pyramid sensor is a four-sided glass pyramidalprism placed in the focal plane of the telescope pupil. The incoming light is focused by the telescopeonto the prism apex. The four facets of the pyramid split the incoming light in four beams, propagatedin slightly different directions. A relay lens, placed behind the prism, re-images the four beams, allowingadjustable sampling of the four different images I ij , i, j = { , } of the aperture on the CCD camera.The two measurement sets s x , s y are obtained from the four intensity patterns as2igure 2: Scheme of the optical setup of a pyramid WFS. The circular modulation path is shown in thedashed line. s x ( x, y ) = [ I ( x, y ) + I ( x, y )] − [ I ( x, y ) + I ( x, y )] I ,s y ( x, y ) = [ I ( x, y ) + I ( x, y )] − [ I ( x, y ) + I ( x, y )] I , (1)where I is the average intensity per subaperture.For the sake of simplicity, we focus on the transmission mask modeling approach, which ignores thephase shifts introduced by the pyramidal prism and models the prism facets as transmitting only. Theinterference effects are neglected as well.A dynamic circular modulation of the incoming beam allows to increase the linear range of the pyramidsensor [81] and is also used to adjust its sensitivity. The modulation can be accomplished in several ways:either by oscillating the pyramid itself [60], with a steering mirror [8, 49], or by using a static diffusiveoptical element [49]. The circular modulation path of the focused beam on the pyramid apex is shownwith a dashed circle in Figure 2.We consider the roof wavefront sensor as a stand-alone WFS and as a simplification of the pyramidsensor. For this type of sensor, two orthogonally placed two-sided roof prisms instead of the pyramidalone are used. The orthogonality of the roof leads to a decoupling of the dependence of the sensor dataon the incoming phase in x - and y -direction. For roof wavefront sensors, a linear modulation path isadditionally considered in the literature as an approximation to the circular one.The understanding of the physics behind the pyramid sensor has changed over time. In the beginning, thePWFS was introduced with dynamic modulation of the incoming beam and described within the geometricoptics framework as a slope sensor similar to SH sensors, but having a higher sensitivity [19, 60, 62, 82].Later, the role of the beam modulation was questioned and the pyramid sensor without modulation wasstudied as well [47, 50, 61]. According to the Fourier optics based analytical model derived in [48], thenon-modulated PWFS measures a non-linear combination of one and two dimensional Hilbert transformsof the sine and cosine of the incoming distorted wavefront.Then, it was recognized that the dynamic modulation of the beam allows to strengthen the linearityof the sensor and to increase its dynamic range. Taking modulation into account within the Fourieroptics framework complicates the non-linear forward model even more. However, a linearization of bothmodels, with and without modulation, is possible under certain simplifying assumptions [8, 81]. Within3he linearized model, it was shown that the modulated pyramid sensor measures both the slope and theHilbert transform of the wavefront, depending on the frequency range and the amount of modulation [81].The full continuous measurements s x ( x, y ) and s y ( x, y ) of the pyramid wavefront sensor are not availablein practice. For the description of the discrete pyramid sensor we perform a division of the continuoustwo dimensional process into finitely many equispaced regions called subapertures. The data are thenassumed to be averaged over every subaperture which corresponds to the finite sampling of the pyramidsensor. Hence, the data grid is predetermined by a subaperture size of d · d with d = Dn , where D represents the telescope diameter, i.e., the primary mirror size, and n the number of subapertures in onedirection.Figure 3: The figures show the borders of the annular aperture mask for fixed x and y . The domainΩ x changes with x and Ω y with y respectively. In some cases the intervals are split due to the centralobstruction of the telescope.Additionally, we restrict the availability of measurements to the size of the region captured by the sensor.For several telescope systems the pupil is annular instead of circular since a secondary mirror shadesthe primary mirror, making the area of central obstruction hardly attainable for photons. Thus, theremaining light in the area of the central obstruction does not produce reliable measurements. Moreover,the incoming phases are defined on R but for the control of the deformable mirror, we are only interestedin the reconstructed wavefront shape on a restricted domain (bounded by the size of the telescope pupil).In the following, we describe the annular telescope aperture mask by Ω = Ω y × Ω x ⊆ [ − D/ , D/ asshown in Figure 3. Single lines of the annular aperture are represented by Ω x = [ a x , b x ] and Ω y = [ a y , b y ],with a x < b x , a y < b y being the borders of the pupil for fixed x and y correspondingly. The sensor providesmeasurement on the region of the CCD-detector D . Throughout the paper, we do not distinguish betweenpupil and CCD-detector and assume D = Ω. Further, the limitation onto the CCD detector (indicated asmultiplication with a characteristic function of D ) is not marked explicitly for the underlying operatorsfor simplicity of notation. However, please keep in mind the compact support of the considered functionsbecause of the restricted size of the aperture and the CCD-detector. For instance, we consider the normsin L (cid:0) R (cid:1) but since the operators map from and to functions with compact support on Ω it is equivalentto considering the norms in L (Ω). The pyramid WFS sensor model is non-linear and extensive [48, 69]. The measurements ( s nx , s ny ) of thenon-modulated sensor are connected to the wavefront Φ via a combination of 1d and 2d Hilbert transformsof sin Φ , cos Φ and their multiplications. Circular modulation complicates the model even more which4akes the task of inverting the operator P , i.e., the full Fourier optics based model of the sensor ratherdifficult.In the following P { n,c } = [ P { n,c } x , P { n,c } y ] denote the operators, that describe the pyramid sensor forwardmodel, R { n,c,l } = [ R { n,c,l } x , R { n,c,l } y ] the roof wavefront sensor operators. The linearized roof sensoroperators are indicated by R { n,c,l } ,lin = [ R { n,c,l } ,linx , R { n,c,l } ,liny ]. The superscripts { n, c, l } represent theregime in which the sensor is operated – no modulation, circular or linear modulation applied.Atmospheric turbulence is usually introduced by either using the von Karman or Kolmogorov turbulencemodel which describe the energy distribution of turbulent motions [64, 65]. Following the Kolmogorovmodel, considerations in [16] initiate to expect smoother functions for representing atmospheric turbulencethan just L with a high probability. This leads to the assumption of Φ ∈ H / (cid:0) R (cid:1) as already suggestedin [18].We start with the derivation of the analytic pyramid wavefront sensor model without taking interferenceeffects between the four pupils on the CCD-detector into account. This pyramid sensor model is knownas the transmission mask model. For the modulated sensor, we define a modulation parameter α λ = 2 παλ , (2)with α = rλ/D for a positive integer r representing the modulation radius and λ the sensing wavelength. For the pyramid wavefront sensor, we only consider the non-modulated and the sensor with circularmodulation since they make sense from the physical point of view.
Definition 1.
We introduce the operators P { n,c } x in x -direction given by (cid:16) P { n,c } x Φ (cid:17) ( x, y ) := 1 π X Ω ( x, y ) p.v. Z Ω y sin [Φ( x , y ) − Φ( x, y )] · k { n,c } ( x − x ) x − x dx (3)+ 1 π X Ω y ( x ) p.v. Z Ω y Z Ω x Z Ω x sin [Φ( x , y ) − Φ( x, y )] · l { n,c } ( x − x, y − y )( x − x )( y − y )( y − y ) dy dy dx and P { n,c } y in y -direction given by (cid:16) P { n,c } y Φ (cid:17) ( x, y ) := 1 π X Ω ( x, y ) p.v. Z Ω x sin [Φ( x, y ) − Φ( x, y )] · k { n,c } ( y − y ) y − y dy (4)+ 1 π X Ω x ( y ) p.v. Z Ω y Z Ω x Z Ω y sin [Φ( x , y ) − Φ( x , y )] · l { n,c } ( x − x , y − y )( x − x )( y − y )( x − x ) dx dy dx . The functions k { n,c } are defined by k n ( x ) := 1 , k c ( x ) := J ( α λ x ) , and the functions l { n,c } by l n ( x, y ) := 1 and l c ( x, y ) := 1 T T/ Z − T/ cos[ α λ x sin(2 πt/T )] cos[ α λ y cos(2 πt/T )] dt. The function J denotes the zero-order Bessel function of the first kind given by J ( x ) = 1 π π Z cos( x sin t ) dt = 1 π π Z cos( x cos t ) dt with the modulation parameter α λ defined in (2) . p.v. Z Ω y Φ( x , y ) x − x dx = lim δ → + Z Ω y \ [ x − δ,x + δ ] Φ( x , y ) x − x dx for any wavefront Φ. In the following, p.v. R Ω y R Ω x R Ω y is always meant as abbreviation of p.v. R Ω y p.v. R Ω x p.v. R Ω y in the context of the pyramid sensor operator. Theorem 1.
The relation between the pyramid wavefront sensor data and the incoming wavefront isgiven by s { n,c } x ( x, y ) = − (cid:16) P { n,c } x Φ (cid:17) ( x, y ) ,s { n,c } y ( x, y ) = (cid:16) P { n,c } y Φ (cid:17) ( x, y ) , (5) where P { n,c } denote the pyramid sensor operators defined in (3) - (4) .Proof. The proof is given in the Appendix.
The roof WFS constitutes a part of the pyramid WFS. In the roof sensor, the pyramidal prism is replacedby two orthogonally placed two-sided roof prisms, resulting in a decoupling of x - and y -direction. There-fore, the roof WFS operators R { n,c } = [ R { n,c } x , R { n,c } y ] are much simpler, as it contains only (variationsof) 1d Hilbert transforms in one particular direction [48, 68, 81].Due to the physical setup of the roof WFS, linear modulation induces characteristics which are of inter-est especially for roof wavefront sensors. Hence, we additionally investigate the linear modulated roofwavefront sensor model. In case of circular modulation the amplitude of modulation is assumed to be α λ , already introduced in (2). For linear modulation, 2 α denotes the angle of ray displacement along thedesired direction, which is equivalent to the assumed circular modulation. We substitute the zero-orderBessel function by a sinc-term in order to describe a linear modulation instead of a circular one.For further investigations, we again do not take interference between the four beams into account andconsider the roof WFS operator R { n,c,l } on the one hand as a standalone wavefront sensor and on theother hand as an approximation to the pyramid WFS P { n,c } . Definition 2.
We introduce the operators R { n,c,l } x and R { n,c,l } y by (cid:16) R { n,c,l } x Φ (cid:17) ( x, y ) := X Ω ( x, y ) 1 π p.v. Z Ω y sin[Φ( x , y ) − Φ( x, y )] · k { n,c,l } ( x − x ) x − x dx , (6) (cid:16) R { n,c,l } y Φ (cid:17) ( x, y ) := X Ω ( x, y ) 1 π p.v. Z Ω x sin[Φ( x, y ) − Φ( x, y )] · k { n,c,l } ( y − y ) y − y dy (7) with modulation functions k n ( x ) := 1 , k c ( x ) := J ( α λ x ) , and k l ( x ) := sinc( α λ x ) . Theorem 2.
Using the operators defined in (6) - (7) the measurements of the roof WFS in the trans-mission mask model are written as s { n,c,l } x ( x, y ) = − (cid:16) R { n,c,l } x Φ (cid:17) ( x, y ) ,s { n,c,l } y ( x, y ) = (cid:16) R { n,c,l } y Φ (cid:17) ( x, y ) . Proof.
See [8, 70, 81].The operators P { n,c } x and P { n,c } y as well as R { n,c,l } x and R { n,c,l } y are constructed in the same way, oneonly has to interchange the roles of x and y in the model. In the following, we will concentrate on theoperators P { n,c } x and R { n,c,l } x since the obtained results can easily be transferred to the operators P { n,c } y and R { n,c,l } y as well. Let us now analyze the pyramid and roof sensor operators in more detail.6 roposition 1. The non-linear operators R { n,c,l } : H / (cid:0) R (cid:1) → L (cid:0) R (cid:1) , representing roof wavefrontsensors, are well-defined operators between the above given spaces.Proof. From the proof of Theorem 1 it follows that the pyramid sensor operators and (further) the roofsensor operators are non-linear and well-defined for any wavefront Φ ∈ H / (cid:0) R (cid:1) . It remains to show (cid:16) R { n,c,l } Φ (cid:17) ∈ L (cid:0) R (cid:1) . The proof uses the boundedness (1) | sin (Φ) | ≤ | Φ | and the Hölder continuity (2) with α = 5 / C > ∈ H / (cid:0) R (cid:1) (cf Sobolevembedding theorem, e.g., [1, Theorem 5.4]). We start with showing that the integrand of (6) in fact isintegrable for D (cid:16) R { n,c,l } (cid:17) ⊆ H / (cid:0) R (cid:1) . Together with (cid:12)(cid:12) k { n,c,l } (cid:12)(cid:12) ≤
1, this infers from Z Ω y (cid:12)(cid:12)(cid:12)(cid:12) sin[Φ( x , y ) − Φ( x, y )] · k { n,c,l } ( x − x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) dx (1) ≤ Z Ω y | Φ( x , y ) − Φ( x, y ) || x − x | dx (2) ≤ C Z Ω y | x − x | / | x − x | dx = C Z Ω y | x − x | / dx = 6 C (cid:16) ( b y − x ) / + ( x + a y ) / (cid:17) < ∞ for x ∈ Ω y , y ∈ Ω x . As the proper integral exists, the Cauchy principal value exists as well. It follows thatthe p.v. meaning is negligible in (6) - (7) for D (cid:16) R { n,c,l } (cid:17) ⊆ H / (cid:0) R (cid:1) . By usage of the Cauchy-Schwarzinequality (C-S), we obtain that the L -norm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R { n,c,l } x Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) = Z R (cid:12)(cid:12)(cid:12) R { n,c,l } x Φ ( x, y ) (cid:12)(cid:12)(cid:12) d ( x, y )= Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Ω ( x, y ) 1 π Z Ω y sin[Φ( x , y ) − Φ( x, y )] · k { n,c,l } ( x − x ) x − x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ( x, y ) C − S ≤ π Z Ω Z Ω y (cid:12)(cid:12)(cid:12)(cid:12) sin[Φ( x , y ) − Φ( x, y )] · k { n,c,l } ( x − x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) dx · Z Ω y dx d ( x, y ) ≤ | Ω y | π Z Ω Z Ω y (cid:18) | sin[Φ( x , y ) − Φ( x, y )] || x − x | (cid:19) dx d ( x, y ) (1) ≤ | Ω y | π Z Ω Z Ω y (cid:18) | Φ( x , y ) − Φ( x, y ) || x − x | (cid:19) dx d ( x, y ) (2) ≤ C | Ω y | π Z Ω Z Ω y | x − x | / | x − x | ! dx d ( x, y )= C | Ω y | π Z Ω Z Ω y | x − x | / dx d ( x, y ) ≤ sup x ∈ Ω y | M ( x ) | C | Ω | | Ω y | π < ∞ is finite as for Ω y ⊆ [ − D/ , D/
2] holds M ( x, y ) := Z Ω y | x − x | / dx ≤ D/ Z − D/ | x − x | / dx = 32 "(cid:18) D − x (cid:19) / + (cid:18) x + D (cid:19) / =: M ( x ) (8)and further sup x ∈ Ω y | M ( x ) | < ∞ . (9)7ence, the roof sensor operators maps incoming wavefronts in H / (cid:0) R (cid:1) to measurements in L (cid:0) R (cid:1) with compact support on the telescope pupil/detector.From the considerations in the above proof it follows the following statement: Remark 1. If Φ ∈ H / (cid:0) R (cid:1) , the principal value integrals of the operators (6) - (7) describing the roofwavefront sensor model coincide with the standard definition of Lebesgue integrals. Proposition 2.
The non-linear operators P { n,c } : H / (cid:0) R (cid:1) → L (cid:0) R (cid:1) , representing pyramid wave-front sensors, are well-defined operators between the above given spaces.Proof. As already shown in the proof of Theorem 1, the pyramid sensor operators are non-linear, well-defined operators for any wavefront Φ ∈ H / (cid:0) R (cid:1) . In order to verify that (cid:16) P { n,c } Φ (cid:17) ∈ L we split thecorresponding operators into two parts: (cid:16) P { n,c } x Φ (cid:17) ( x, y ) = (cid:16) R { n,c } x Φ (cid:17) ( x, y ) + (cid:16) S { n,c } x Φ (cid:17) ( x, y )with the roof sensor operators R { n,c } x defined in (6) and the second term (cid:16) S { n,c } x Φ (cid:17) ( x, y ) := 1 π X Ω y ( x ) p.v. Z Ω y Z Ω x Z Ω x sin [Φ( x , y ) − Φ( x, y )] · l { n,c } ( x − x, y − y )( x − x )( y − y )( y − y ) dy dy dx . With Proposition 1 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P { n,c } x Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R { n,c } x Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S { n,c } x Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) , it remains to show (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S { n,c } x Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) < ∞ .First, we focus on the non-modulated sensor and consider the operator S nx .The proof uses the L p -boundedness of the classical Hilbert transform for 1 < p < ∞ as found in, e.g., [9].For our purposes, we define the Hilbert transforms H x in x - direction and H y in y -direction by H x Φ( x, y ) := 1 π p.v. Z ∞−∞ Φ( x , y ) x − x dx , H y Φ( x, y ) := 1 π p.v. Z ∞−∞ Φ( x, y ) y − y dy . (10) Theorem 3 (Theorem 8.1.12, [9]) . For Φ ∈ L p (cid:0) R (cid:1) , < p < ∞ , the Hilbert transform defined in (10) exists almost everywhere, belongs to L p (cid:0) R (cid:1) and satisfies || H x Φ || L p ≤ c p || Φ || L p and || H y Φ || L p ≤ d p || Φ || L p (11) with constants c p , d p > . (11) is often referred to as the Marcel Riesz inequality for Hilbert transforms. The 2d Hilbert transform H xy is given as an operator H xy : L p (cid:0) R (cid:1) → L p (cid:0) R (cid:1) for 1 < p < ∞ as wellsince it can be considered as the composition H xy = H x ◦ H y .Using trigonometric formulas, we rewrite S nx into ( S nx Φ) ( x, y ) = 1 π X Ω x ( y ) p.v. Z Ω y Z Ω x Z Ω y sin [Φ( x , y ) − Φ( x, y )]( x − x )( y − y )( y − y ) dy dy dx = 1 π X Ω x ( y ) p.v. Z Ω y Z Ω x Z Ω y sin [Φ( x , y )] cos [Φ( x, y )] − cos [Φ( x , y )] sin [Φ( x, y )]( x − x )( y − y )( y − y ) dy dy dx = ( H xy (sin Φ)) ( x, y ) · ( H y (cos Φ)) ( x, y ) − ( H xy (cos Φ)) ( x, y ) · ( H y (sin Φ)) ( x, y ) and obtain for 2 ≤ p < ∞ π || S nx Φ || L p/ = || ( H xy (sin Φ)) · ( H y (cos Φ)) − ( H xy (cos Φ)) · ( H y (sin Φ)) || L p/ ≤ || ( H xy (sin Φ)) · ( H y (cos Φ)) || L p/ + || ( H xy (cos Φ)) · ( H y (sin Φ)) || L p/ ≤ || ( H xy (sin Φ)) || L p || ( H y (cos Φ)) || L p + || ( H xy (cos Φ)) || L p || ( H y (sin Φ)) || L p < ∞
8y using the generalized Hölder inequality. The multiplication with the characteristic functions is omittedabove due to simplicity of notation.It follows that ( P n Φ) ∈ L p (cid:0) R (cid:1) for 1 ≤ p < ∞ and Φ ∈ H / (cid:0) R (cid:1) .In order to prove ( P c Φ) ∈ L we use relation (29) of the proof of Theorem 1 between non-modulatedand modulated pyramid data, i.e., s cx ( x, y ) = 1 T T/ Z − T/ s nx ( x, y, t ) dt. (12)Let T denote one full time period and t ∈ [ − T / , T / M modt given by (cid:16) M modt Φ (cid:17) ( x, y ) := Φ ( x, y ) + Φ mod ( x, y, t )for the periodic tilt Φ mod inducing modulation. As in (28), this tilt is represented byΦ mod ( x, y, t ) = α λ ( x sin(2 πt/T ) + y cos(2 πt/T )) . Due to the structure of Φ mod and its compact support on the aperture, it holds Φ mod ( · , · , t ) ∈ H / (cid:0) R (cid:1) for all t ∈ [ − T / , T / M modt : H / (cid:0) R (cid:1) → H / (cid:0) R (cid:1) and further (cid:16) P n M modt Φ (cid:17) ∈ L p (cid:0) R (cid:1) ∀ t ∈ [ − T / , T / , ≤ p < ∞ . (13)Using s nx ( · , · , t ) = (cid:16) P n M modt Φ (cid:17) , (12), and the generalized Minkowski’s integral inequality (1) (cf, e.g., [34,Theorem 202],[76]), we obtain || s cx || L p = Z R | s cx ( x, y ) | p d ( x, y ) /p = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T T/ Z − T/ (cid:16) P n M modt Φ (cid:17) ( x, y ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p d ( x, y ) /p (1) ≤ T T/ Z − T/ Z R (cid:12)(cid:12)(cid:12)(cid:16) P n M modt Φ (cid:17) ( x, y ) (cid:12)(cid:12)(cid:12) p d ( x, y ) /p dt = 1 T T/ Z − T/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P n M modt Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p dt (13) < ∞ , which shows P c Φ ∈ L p (cid:0) R (cid:1) , ≤ p < ∞ for the modulated pyramid sensor operators. Note that weomitted the factor (cid:0) − (cid:1) from (5).Merely the light which is captured on the telescope pupil Ω influences the pyramid sensor response. Weconsider only the light falling on the aperture ( X Ω · Φ) but use the notation Φ ∈ H / (cid:0) R (cid:1) for bothconsidered variants of wavefronts with and without compact support on the telescope pupil.Alternatively, one can also define the pyramid sensor operators as P { n,c } x : H / (Ω) → L (Ω), i.e., on thetelescope aperture, and P { n,c } y , R { n,c,l } x , R { n,c,l } y respectively. However, since we apply, e.g., Plancherel’stheorem we define the operators on the whole R but keep in mind that one can always restrict to theregion of the telescope aperture and CCD-detector. More precisely, the pyramid sensor operators areapplied to functions with compact support on the pupil Ω and map to functions with compact supporton the detector Ω. 9 .3 Linearization of the roof sensor operators Linear approximations of WFS operators around the zero phase are sufficient in closed loop AO in whichthe wavefront sensor measures already corrected and very small incoming wavefronts. The linearizationof the operators can be obtained by different ways, e.g., by replacingsin[Φ( x , y ) − Φ( x, y )] ≈ Φ( x , y ) − Φ( x, y )which is valid in case of | Φ( x , y ) − Φ( x, y ) | << . Linearizations R { n,c,l } ,lin based on these approximations were already considered in [8, 48, 73, 81].We concentrate on linear approximations for the roof wavefront sensor operators by means of the Fréchetderivative. We start by calculating the Gâteaux derivatives. Then, we show that the Gâteaux derivativescoincide with the Fréchet derivatives and finally, we evaluate the corresponding linearizations. Theorem 4.
The Gâteaux derivatives (cid:16) R { n,c,l } x (cid:17) (Φ) ∈ L (cid:0) H / , L (cid:1) at Φ ∈ D (cid:16) R { n,c,l } x (cid:17) of the non-linear roof sensor operators R { n,c,l } x : D (cid:16) R { n,c,l } x (cid:17) ⊆ H / (cid:0) R (cid:1) → L (cid:0) R (cid:1) defined in (6) are givenby (cid:16)(cid:0) R { n,c,l } x (cid:1) (Φ) ψ (cid:17) ( x, y ) = X Ω ( x, y ) 1 π Z Ω y cos [Φ( x , y ) − Φ( x, y )] [ ψ ( x , y ) − ψ ( x, y )] · k { n,c,l } ( x − x )( x − x ) dx (14) and (cid:16) R { n,c,l } y (cid:17) (Φ) ∈ L (cid:0) H / , L (cid:1) respectively.Proof. We utilize the representation (cf Taylor’s theorem with Lagrange form of the remainder)sin (Φ + ψ ) = sin (Φ) + sin (Φ) ψ + 12 sin (Φ + θψ ) ψ (15)for a θ = θ (Φ , ψ ) ∈ (0 , k { n,c,l } as these functions are independent of the phase Φ anyway.The Gâteaux derivatives (cid:16) R { n,c,l } x (cid:17) (Φ) are computed as (cid:16)(cid:0) R { n,c,l } x (cid:1) (Φ) ψ (cid:17) ( x, y ) = lim t → (cid:16) R { n,c,l } x (Φ + tψ ) (cid:17) ( x, y ) − (cid:16) R { n,c,l } x Φ (cid:17) ( x, y ) t = lim t → π Z Ω y sin [Φ( x , y ) + tψ ( x , y ) − Φ( x, y ) − tψ ( x, y )] − sin [Φ( x , y ) − Φ( x, y )] t ( x − x ) dx (15) = lim t → π Z Ω y sin [Φ( x , y ) − Φ( x, y )] [ tψ ( x , y ) − tψ ( x, y )] t ( x − x )+ 12 sin [Φ( x , y ) − Φ( x, y ) + θt [ ψ ( x , y ) − ψ ( x, y )]] t [ ψ ( x , y ) − ψ ( x, y )] t ( x − x ) dx = 1 π Z Ω y cos [Φ( x , y ) − Φ( x, y )] [ ψ ( x , y ) − ψ ( x, y )]( x − x ) dx . Obviously, the Gâteaux derivatives are linear in ψ .For any Φ ∈ H / (cid:0) R (cid:1) we deduce that the Gâteaux derivatives are bounded, and further continuous inthe direction ψ , i.e., (cid:16) R { n,c,l } x (cid:17) (Φ) ∈ L (cid:0) H / , L (cid:1) , by showing (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) R { n,c,l } x (cid:17) (Φ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( H / , L ) = sup || ψ || H / =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)(cid:16) R { n,c,l } x (cid:17) (Φ) ψ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) < ∞ .
10y application of the Cauchy-Schwarz inequality, (cid:12)(cid:12) k { n,c,l } (cid:12)(cid:12) ≤
1, as well as the Hölder continuity (1) with α = 5 / C > H / (cid:0) R (cid:1) , the statement results from (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)(cid:16) R { n,c,l } x (cid:17) (Φ) ψ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R ) = Z R (cid:12)(cid:12)(cid:12)(cid:16)(cid:16) R { n,c,l } x (cid:17) (Φ) ψ (cid:17) ( x, y ) (cid:12)(cid:12)(cid:12) d ( x, y )= Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Z Ω y cos [Φ ( x , y ) − Φ ( x, y )] [ ψ ( x , y ) − ψ ( x, y )] · k { n,c,l } ( x − x ) x − x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ( x, y ) C − S ≤ π Z Ω Z Ω y (cid:12)(cid:12) cos (cid:2) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) (cid:3) · k { n,c,l } (cid:0) x − x (cid:1)(cid:12)(cid:12) dx · Z Ω y (cid:12)(cid:12)(cid:12) ψ ( x , y ) − ψ ( x, y ) x − x (cid:12)(cid:12)(cid:12) dx d ( x, y ) (1) ≤ π Z Ω Z Ω y dx · C Z Ω y (cid:18) | x − x | / | x − x | (cid:19) dx d ( x, y )= C | Ω y | π Z Ω Z Ω y | x − x | / dx d ( x, y ) (8) − (9) < ∞ . Theorem 5.
The Gâteaux derivatives (14) coincide with the Fréchet derivatives.Proof.
For the proof we use the following assertion stated in, e.g., [3, 86].
Proposition 3 (Theorem III.5.4, [86]; p.10, [3]) . If the Gâteaux derivatives (cid:16) R { n,c,l } x (cid:17) (Φ) exist for all Φ from a neighborhood of Φ ∈ D (cid:16) R { n,c,l } x (cid:17) and the mappings Φ → (cid:16) R { n,c,l } x (cid:17) (Φ) are continuous from H / (cid:0) R (cid:1) into L (cid:0) H / , L (cid:1) at Φ = Φ , then R { n,c,l } x are Fréchet differentiable at Φ . Hence, it suffices to show that for any Φ , Φ ∈ H / (cid:0) R (cid:1) holds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) R { n,c,l } x (cid:17) (Φ ) − (cid:16) R { n,c,l } x (cid:17) (Φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( H / , L ) ≤ ˜ C || Φ − Φ || H / , i.e., sup || ψ || H / =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18)(cid:16) R { n,c,l } x (cid:17) (Φ ) ψ (cid:19) − (cid:18)(cid:16) R { n,c,l } x (cid:17) (Φ ) ψ (cid:19)| {z } := U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ≤ ˜ C || Φ − Φ || H / with ˜ C < ∞ .Under the Lipschitz continuity (2) of the cosine function with the Lipschitz constant L > || U || L . = Z R (cid:12)(cid:12)(cid:12)(cid:16)(cid:0) R { n,c,l } x (cid:1) (Φ ) ψ (cid:17) ( x, y ) − (cid:16)(cid:0) R { n,c,l } x (cid:1) (Φ ) ψ (cid:17) ( x, y ) (cid:12)(cid:12)(cid:12) d ( x, y ) . = Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Z Ω y h cos (cid:2) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) (cid:3) − cos (cid:2) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) (cid:3) i · [ ψ ( x , y ) − ψ ( x, y )] · k { n,c,l } ( x − x ) x − x dx (cid:12)(cid:12)(cid:12)(cid:12) d ( x, y ) i C − S31 ≤ π Z Ω Z Ω y (cid:12)(cid:12)(cid:12) cos (cid:2) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) (cid:3) − cos (cid:2) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) (cid:3) (cid:12)(cid:12)(cid:12) dx · Z Ω y (cid:12)(cid:12)(cid:12)(cid:12) [ ψ ( x , y ) − ψ ( x, y )] · k { n,c,l } ( x − x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) dx d ( x, y ) (1) , (2) ≤ L C π Z Ω Z Ω y (cid:12)(cid:12) Φ (cid:0) x , y (cid:1) − Φ ( x, y ) − Φ (cid:0) x , y (cid:1) + Φ ( x, y ) (cid:12)(cid:12) dx · Z Ω y (cid:18) | x − x | / | x − x | (cid:19) dx d ( x, y ) . = L C π Z Ω Z Ω y (cid:12)(cid:12)(cid:2) Φ (cid:0) x , y (cid:1) − Φ (cid:0) x , y (cid:1)(cid:3) − [Φ ( x, y ) − Φ ( x, y )] (cid:12)(cid:12) dx · Z Ω y | x − x | / dx d ( x, y ) (8) ≤ L C π Z Ω Z Ω y (cid:12)(cid:12) Φ (cid:0) x , y (cid:1) − Φ (cid:0) x , y (cid:1)(cid:12)(cid:12) + | Φ ( x, y ) − Φ ( x, y ) | dx · M ( x, y ) d ( x, y ) . ≤ sup x ∈ Ω y | M ( x ) | L C π Z Ω y Z Ω x Z Ω y (cid:12)(cid:12) Φ (cid:0) x , y (cid:1) − Φ (cid:0) x , y (cid:1)(cid:12)(cid:12) dx dy dx + Z Ω y Z Ω x Z Ω y | Φ ( x, y ) − Φ ( x, y ) | dx dy dx . = sup x ∈ Ω y | M ( x ) | L C | Ω y | π (cid:0) || Φ − Φ || L + || Φ − Φ || L (cid:1) . = sup x ∈ Ω y | M ( x ) | L C | Ω y | π || Φ − Φ || L . ≤ sup x ∈ Ω y | M ( x ) | L C | Ω y | π || Φ − Φ || H / (9) ≤ ˜ C || Φ − Φ || H / , i.e., the mapping Φ → (cid:16) R { n,c,l } x (cid:17) (Φ) is continuous. Thus, the operators R { n,c,l } x representing the roofsensor are Fréchet differentiable. Theorem 6.
The linearizations R { n,c,l } ,linx : H / (cid:0) R (cid:1) → L (cid:0) R (cid:1) by means of the Fréchet derivativeof the operators R { n,c,l } introduced in (6) - (7) are given by (cid:0) R { n,c,l } ,linx Φ (cid:1) ( x, y ) := (cid:16)(cid:0) R { n,c,l } x (cid:1) (0) Φ (cid:17) ( x, y ) = X Ω ( x, y ) 1 π Z Ω y [Φ( x , y ) − Φ( x, y )] · k { n,c,l } ( x − x ) x − x dx (16) for x -direction and R { n,c,l } ,liny accordingly.Proof. The claim immediately follows from (cid:16) R { n,c,l } ,linx Φ (cid:17) := (cid:18)(cid:16) R { n,c,l } x (cid:17) (0) Φ (cid:19) , i.e., considering theFréchet derivatives (14) at Φ = 0 and in direction Φ.12 .4 Simplified linearized operators L { n,c,l } x Variations of the finite Hilbert transform operator allow to simplify the linear approximations of the roofsensor model. Let us, thus, consider the following operators.
Definition 3.
We define the integral operators L { n,c,l } = h L { n,c,l } x , L { n,c,l } y i by ( L { n,c,l } x Φ)( x, y ) := 1 π p.v. Z Ω y Φ( x , y ) k { n,c,l } ( x − x ) x − x dx (17) and L { n,c,l } y accordingly. As derived in [26, 76], L { n,c,l } are bounded operators on L p (cid:0) R (cid:1) for 1 < p < ∞ due to the structureof the functions k { n,c,l } introducing modulation. According to the pyramid and roof sensor model, weconsider the operators as L { n,c,l } : H / (cid:0) R (cid:1) → L (cid:0) R (cid:1) . In case of no modulation the operator L nx coincides with the finite Hilbert transform – a singular Cauchy integral operator. Using the above definedoperators, the linearized roof sensor measurements read as s { n,c,l } ,linx ( x, y ) = − (cid:16) R { n,c,l } ,linx Φ (cid:17) ( x, y )= − X Ω ( x, y ) h(cid:16) L { n,c,l } x Φ (cid:17) ( x, y ) − Φ( x, y ) · (cid:16) L { n,c,l } x (cid:17) ( x, y ) i . (18)Equation (18) provides two possibilities for wavefront reconstruction. As it is shown in [68, 69], whenan AO system enters the closed loop, the first term in the forward model gains in importance and anassumption of neglecting the second term in the reconstruction procedure is justifiable. Therefore, onecould either use the full linearized roof sensor model or ignore the second term (cid:16) L { n,c,l } x (cid:17) as it is done inseveral existing algorithms for pyramid wavefront sensor [41], e.g., the Preprocessed Cumulative Recon-structor with Domain decomposition (P-CuReD) [72, 73], the Pyramid Fourier Transform Reconstructor[71, 74], the Finite Hilbert Transform Reconstructor [70], or the Singular Value Type Reconstructor [38]. Several iterative algorithms for solving inverse problems (e.g., the Landweber iteration, steepest descentmethod or conjugate gradient method for the normal equation) include the application of adjoint opera-tors. In order to make these methods suitable for wavefront reconstruction, we derive the adjoints of theunderlying operators.First, we will evaluate the Fourier transforms of the one-term assumptions L { n,c,l } defined in (17) andafterwards use Plancherel’s theorem to calculate the corresponding adjoints.The underlying operators L { n,c,l } are defined from H / (cid:0) R (cid:1) into L (cid:0) R (cid:1) . In order to calculate thecorresponding adjoint operators from L (cid:0) R (cid:1) into H / (cid:0) R (cid:1) , we introduce the embedding operator i s : H / → L (19)and derive the adjoints according to [63] as, e.g., (cid:16) L { n,c,l } ,linx (cid:17) ∗ : L → H / with (cid:16) L { n,c,l } ,linx (cid:17) ∗ = i ∗ s (cid:16) ˜ L { n,c,l } ,linx (cid:17) ∗ for (cid:16) ˜ L { n,c,l } ,linx (cid:17) ∗ : L → L . For simplicity, we use the notation (cid:16) L { n,c,l } ,linx (cid:17) ∗ instead of (cid:16) ˜ L { n,c,l } ,linx (cid:17) ∗ and omit the multiplication with the aperture mask in the following. The adjoints of the roof sensoroperators are considered accordingly. Proposition 4.
The 1d Fourier transforms in x -direction of the operators L { n,c,l } x defined in (17) aregiven by (cid:16) L { n,c,l } x Φ (cid:17)b ( ξ, y ) = c { n,c,l } ( ξ ) · b Φ ( ξ, y ) (20)13 ith c n ( ξ ) = i sgn ( ξ ) (21) for the non-modulated sensor, c c ( ξ ) = i ( sgn ( ξ ) , for | ξ | > αλ , π arcsin (cid:0) ξ λα (cid:1) , for | ξ | ≤ αλ (22) for the circularly modulated sensor, and c l ( ξ ) = i ( sgn ( ξ ) , for | ξ | > αλ ,ξ λα , for | ξ | ≤ αλ (23) for the linearly modulated sensor. The Fourier transforms of L { n,c,l } y are represented accordingly.Proof. Similar considerations are contained in [73, 81]. Since we examine the 1d Hilbert transform L ( R ) → L ( R ) we fix y ∈ Ω x and investigate the operators L { n,c,l } x : H / ( R ) ⊆ L ( R ) → L ( R )defined according to (17) without indicating the fixed y specifically. For fixed x ∈ Ω y the operators L { n,c,l } y : H / ( R ) ⊆ L ( R ) → L ( R ) are analyzed respectively. We introduce the even kernel functions v { n,c,l } by v { n,c,l } ( x ) := p.v. k { n,c,l } ( x ) x and obtain (cid:16) L { n,c,l } x Φ (cid:17) ( x, y ) = − π (cid:16) Φ ( · , y ) ∗ v { n,c,l } (cid:17) ( x )= − π ∞ Z −∞ Φ ( x , y ) v { n,c,l } ( x − x ) dx = lim δ → + π x − δ Z −∞ Φ ( x , y ) k { n,c,l } ( x − x ) x − x dx + ∞ Z x + δ Φ ( x , y ) k { n,c,l } ( x − x ) x − x dx . (24)By the convolution theorem, the 1d convolution in (24) is a multiplication in the Fourier domain, i.e., (cid:16) L { n,c,l } x Φ (cid:17)b ( ξ, y ) = − r π b Φ ( ξ, y ) · (cid:92) v { n,c,l } ( ξ ) . (25)As already used in [73, 81], the Fourier transforms of the kernel functions are calculated as c v n ( ξ ) = − i q π sgn ( ξ )for the non-modulated sensor, b v c ( ξ ) = − i (p π sgn ( ξ ) , for | ξ | > αλ , q π arcsin (cid:0) ξ λα (cid:1) , for | ξ | ≤ αλ for the circularly modulated sensor, and b v l ( ξ ) = − i (p π sgn ( ξ ) , for | ξ | > αλ , p π ξ λα , for | ξ | ≤ αλ for the linearly modulated sensor.The claim of the Proposition follows by (25) and c { n,c,l } = − q π (cid:92) v { n,c,l } .14ote that by the isometry of the Fourier transform we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L { n,c,l } x Φ ( · , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) L { n,c,l } x Φ (cid:17)b ( · , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c { n,c,l } · b Φ ( · , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = ˜ c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b Φ ( · , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L , i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L { n,c,l } x Φ ( · , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L = ˜ c || Φ ( · , y ) || L for a constant 0 < ˜ c < ∞ . (cid:16) L { n,c,l } x (cid:17) ∗ in L ( R ) As previously mentioned, it is sufficient to derive the adjoints as operators from L into itself and usethe embedding operator (19) in order to obtain adjoint operators from L into H / [63]. Proposition 5.
The adjoints (cid:16) L { n,c,l } x (cid:17) ∗ : L (cid:0) R (cid:1) → L (cid:0) R (cid:1) of the operators L { n,c,l } x defined in (17) are given by (cid:16)(cid:16) L { n,c,l } x (cid:17) ∗ Ψ (cid:17) ( x, y ) = − π p.v. Z Ω y Ψ( x , y ) · k { n,c,l } ( x − x ) x − x dx , and (cid:16) L { n,c,l } y (cid:17) ∗ accordingly, i.e., L { n,c,l } are skew-adjoint in L (cid:0) R (cid:1) .Proof. The proof is performed in the Fourier domain. We have to consider the L ( C )-inner product dueto c { n,c,l } ∈ C defined in (21) - (23). We use Plancherel’s theorem and the equality c { n,c,l } = − c { n,c,l } for the complex conjugate (1) .For any Φ , Ψ ∈ L (cid:0) R (cid:1) with support on Ω and y ∈ Ω x holds h (cid:16) L { n,c,l } x Φ (cid:17) ( · , y ) , Ψ ( · , y ) i L ( C ) = h (cid:16) L { n,c,l } x Φ (cid:17)b ( · , y ) , b Ψ ( · , y ) i L ( C ) = Z R (cid:16) L { n,c,l } x Φ (cid:17)b ( ξ, y ) b Ψ( ξ, y ) dξ (20) = Z R (cid:16) c { n,c,l } ( ξ ) · b Φ( ξ, y ) (cid:17) b Ψ( ξ, y ) dξ (1) = − Z R b Φ( ξ, y ) (cid:16) c { n,c,l } ( ξ ) · b Ψ( ξ, y ) (cid:17) dξ = h b Φ ( · , y ) , (cid:16)(cid:16) L { n,c,l } x (cid:17) ∗ Ψ (cid:17)b ( · , y ) i L ( C ) = h Φ ( · , y ) , (cid:16)(cid:16) L { n,c,l } x (cid:17) ∗ Ψ (cid:17) ( · , y ) i L ( C ) with (cid:16)(cid:16) L { n,c,l } x (cid:17) ∗ Ψ (cid:17)b ( ξ, y ) = − c { n,c,l } ( ξ ) b Ψ ( ξ, y ), i.e., (cid:16) L { n,c,l } x (cid:17) ∗ = − L { n,c,l } x . (cid:16) R { n,c,l } ,linx (cid:17) ∗ in L ( R ) Proposition 6.
The adjoints (cid:16) R { n,c,l } ,linx (cid:17) ∗ : L (cid:0) R (cid:1) → L (cid:0) R (cid:1) of the linearized roof sensor operators R { n,c,l } ,linx defined in (16) are given by (cid:16)(cid:16) R { n,c,l } ,linx (cid:17) ∗ Ψ (cid:17) ( x, y ) = (cid:16)(cid:16) L { n,c,l } x (cid:17) ∗ Ψ (cid:17) ( x, y ) − Ψ( x, y ) (cid:16) L { n,c,l } x (cid:17) ( x, y )= − π p.v. Z Ω y [Ψ( x , y ) + Ψ( x, y )] · k { n,c,l } ( x − x ) x − x dx and (cid:16) R { n,c,l } ,liny (cid:17) ∗ respectively. roof. We choose any Φ , Ψ ∈ L (cid:0) R (cid:1) with support on the telescope pupil Ω. Due to the linearity of theinner product and with representation (18), it holds that h (cid:16) R { n,c,l } ,linx Φ (cid:17) , Ψ i = hX Ω (cid:16) L { n,c,l } x Φ − Φ L { n,c,l } x (cid:17) , Ψ i = h L { n,c,l } x Φ − Φ L { n,c,l } x , Ψ i = h L { n,c,l } x Φ , Ψ i − h Φ L { n,c,l } x , Ψ i = h Φ , (cid:16) L { n,c,l } x (cid:17) ∗ Ψ i − h Φ , Ψ L { n,c,l } x i = h Φ , (cid:16) L { n,c,l } x (cid:17) ∗ Ψ − Ψ L { n,c,l } x i = h Φ , (cid:16)(cid:16) R { n,c,l } ,linx (cid:17) ∗ Ψ (cid:17) i , where we consider the inner product with respect to L (cid:0) R (cid:1) or L (Ω) respectively. Conclusion
In this paper, we have considered the mathematical background for the pyramid wavefront sensor whichis widely used in Adaptive Optics systems in areas such as astronomy, microscopy, and ophthalmology.The theoretical analysis of the forward operators of pyramid and roof wavefront sensors presented in thispaper was aimed at the subsequent development of fast and stable algorithms for wavefront reconstructionfrom pyramid sensor data. The interference effects as well as the phase shift introduced by the pyramidalprism were neglected for simplicity. The analysis allows any kind of modulation (no, circular, linear) tobe applied to the sensors. Due to the closed loop operation assumption, we can linearize the initiallynon-linear forward operators. Additionally, the closed loop setting allows to simplify the linearizedoperators further, ending up with measurements described by one term comparable to a finite Hilberttransform in case of the non-modulated sensor. Moreover, for the considered operators we derived thecorresponding adjoint operators. These are used in the upcoming second part of the paper [39] whichwill be devoted to the application of iterative algorithms to the problem of wavefront reconstruction frompyramid wavefront sensor data. An extension of the analysis (linearization and calculation of adjoints)to the full pyramid sensor operator as well as the adaption of the aperture mask to segmented pupils onELTs [17, 40, 55, 66, 67] is dedicated to future work.
Acknowledgements
This work has been partly supported by the Austrian Federal Ministry of Science and Research (HRSM)and the Austrian Science Fund (F68-N36, project 5). The authors thank the Austrian Adaptive Opticsteam for fruitful discussions and especially Simon Hubmer for his support.
Appendix
Proof of Theorem 1.
First, we consider the non-modulated case.Let Φ ∈ H / (cid:0) R (cid:1) denote the phase screen in radians coming into the telescope. Using wave opticsbased models and assuming constant amplitude over the full telescope pupil Ω = Ω y × Ω x , the complexamplitude (wave) ψ aper corresponding to this incoming phase reads as ψ aper ( x, y ) = X Ω ( x, y ) · exp ( − i Φ( x, y )) . Note that ψ aper ∈ L p (cid:0) R (cid:1) for all 1 ≤ p ≤ ∞ due to the compact support of ψ aper , the continuity of Φ,and further the continuity of ψ aper on Ω y × Ω x . In order to assume the continuity of ψ aper on R weslightly modify the telescope aperture mask X Ω and approximate it by X (cid:15) Ω ∈ C ∞ (cid:0) R (cid:1) fulfilling X Ω = X (cid:15) Ω on Ω.The idea of the extended mask is to smoothen the sharp edges of the telescope pupil in order to guarantee ψ (cid:15)aper ∈ H / (cid:0) R (cid:1) using ψ (cid:15)aper ( x, y ) := X (cid:15) Ω ( x, y ) · exp ( − i Φ( x, y )) . X (cid:15) Ω ∈ C ∞ (cid:0) R (cid:1) .One possible representation of the modification X (cid:15) Ω can be constructed utilizing the following Lemma forthe sets Ω = Ω y × Ω x = [ a y , b y ] × [ a x , b x ] and Ω (cid:15) = ( a y − (cid:15), b y + (cid:15) ) × ( a x − (cid:15), b x + (cid:15) ) for a small (cid:15) > ⊂ Ω (cid:15) . Lemma 1 (Lemma 4.2, [77]) . Let Ω ⊂ R n and Ω (cid:15) ⊂ R n be bounded with Ω ⊂ Ω (cid:15) and Ω (cid:15) open. Then,there exists a real-valued function X (cid:15) Ω ∈ C ∞ (Ω (cid:15) ) with ≤ X (cid:15) Ω ( z ) ≤ for z ∈ Ω (cid:15) and X (cid:15) Ω ( z ) = 1 for z ∈ Ω . Outside of Ω (cid:15) we extend X (cid:15) Ω with zeros and obtain X (cid:15) Ω ∈ C ∞ (cid:0) R (cid:1) .We investigate the construction of the new smooth aperture mask X (cid:15) Ω in more detail. As in [77], weconsider the smooth function f ∈ C ∞ (cid:0) R (cid:1) , f ( z ) := c exp − − | z | ! , for | z | < , , for | z | ≥ , having compact support on [ − , . The constant c ∈ R is chosen such that Z R f ( z ) dz = Z | z |≤ f ( z ) dz = 1 . Additionally, we introduce f (cid:15) ( z ) := 1 (cid:15) f (cid:16) z(cid:15) (cid:17) for (cid:15) >
0. Using this function, S. L. Sobolev established a method [75] which is utilized for the smoothingof the characteristic function describing the telescope pupil. With the coordinate transformation z − (cid:15)z = z , (26)for the function X Ω ∈ L p (cid:0) R (cid:1) , 1 ≤ p ≤ ∞ we build the average function X (cid:15) Ω ∈ C ∞ (cid:0) R (cid:1) by X (cid:15) Ω ( z ) = Z R X Ω ( z − (cid:15)z ) f ( z ) dz (26) = Z R (cid:15) X Ω ( z ) f (cid:18) z − z (cid:15) (cid:19) dz = Z | z − z |≤ (cid:15) f (cid:15) ( z − z ) X Ω ( z ) dz . Altogether, ψ (cid:15)aper is an element of H / (cid:0) R (cid:1) using that from Φ ∈ H / it follows e − i Φ ∈ H / asstated in [6, 7].The adaption of the aperture mask is necessary to guarantee a well-defined mathematical derivation ofthe pyramid wavefront sensor model. Please note that ||X Ω − X (cid:15) Ω || L ( R ) = O (cid:0) (cid:15) (cid:1) , and therefore X (cid:15) Ω is an arbitrarily good approximation of the aperture mask X Ω .The following physical argument supports the usage of a smoothed aperture mask Ω (cid:15) instead of the onewith the sharp edges. Both masks have a compact support. Therefore, on the Fourier domain they areboth represented with infinite spectra. Since (cid:15) < d (with d denoting the telescope subaperture size) issmall, the difference between the two masks, when looking in the Fourier domain, appears only in the veryhigh frequency components. Because the pyramidal prism is a physical device of finite size, it anywaycuts off high frequency components of the input. Additionally, the sensor brings spatial discretization inthe model due to subaperture averaging. As a result, the spectra of the resulting sensor data containfrequencies only up to a given cut-off frequency defined via the subaperture size d as f cut = 1 / (2 d ).17herefore, in practice there is no difference between the smoothed and the sharp aperture mask. In thefollowing and throughout the paper we write X Ω but keep in mind that we always mean X (cid:15) Ω for a small (cid:15) > P SF pyr = F − d { OT F pyr } . (27)Within the transmission mask approach [48], the OTF only takes splitting of the light into account andignores the phase shifts introduced by the pyramid facets. It is represented as a sum of 2d Heavisidefunctions OT F pyr ( ξ, η ) = X m =0 1 X n =0 T mn ( ξ, η )with T mn ( ξ, η ) = H d [( − m ξ, ( − n η ] := ( , if ( − m ξ ≥ , ( − n η ≥ , , otherwise.The 2d Heaviside function is the product of two 1d Heaviside functions H d ( ξ, η ) = H d ( ξ ) · H d ( η )and the 1d Heaviside function can be represented as H d ( ξ ) = 12 + 12 · sgn ( ξ ) . Therefore H d ( ξ, η ) = 14 [1 + sgn ( ξ ) + sgn ( η ) + sgn ( ξ ) · sgn ( η )] , which for g m ( ξ ) := sgn (( − m ξ ) results in T mn ( ξ, η ) = 14 [1 + g m ( ξ ) + g n ( η ) + g m ( ξ ) · g n ( η )] . Please note that in the above notation, g m is always meant as function of the first variable ξ and g n as function of the second variable η for 2d considerations. Furthermore, F x will denote the 1d Fouriertransform in the first variable and F y the 1d Fourier transform in the second variable. With (27) and dueto the linearity of the Fourier transform, the PSF of the pyramid is represented as a sum of four PSFs P SF pyr = F − d n X m =0 1 X n =0 T mn o = X m =0 1 X n =0 F − d { T mn } = X m =0 1 X n =0 F − d (cid:26)
14 [1 + g m + g n + g m · g n ] (cid:27)| {z } =: P SF mnpyr . The OTF is a sum of products of functions depending either on ξ or η . Therefore, the inverse 2d Fouriertransform reduces to products of 1d inverse Fourier transforms. For P SF mnpyr , we obtain
P SF mnpyr ( x, y ) = 14 (cid:2) F − x { } ( x ) · F − y { } ( y )+ F − x { g m } ( x ) · F − y { } ( y )+ F − x { } ( x ) · F − y { g n } ( y )+ F − x { g m } ( x ) · F − y { g n } ( y ) (cid:3) . ϕ , we introduce the delta distribution δ defined as h δ, ϕ i = ϕ (0). This applicationis well-defined for continuous functions, e.g., ϕ ∈ H / (cid:15) ( R ), and, on account of this, δ ∈ H − / − (cid:15) ( R )for (cid:15) >
0. The distribution ( p.v. x ) is defined via the Cauchy principal value by D (cid:18) p.v. x (cid:19) , ϕ E = lim (cid:15) → + Z | x | >(cid:15) ϕ ( x ) x dx = π ( H x ϕ ) (0)for the 1d Hilbert transform defined according to (10). As the Hilbert transform H x : L ( R ) → L ( R )is a well-defined operator (see, e.g., [9, Theorem 8.1.12]), the evaluation of h ( p.v. x ) , ψ i is well-definedfor ψ ∈ H / (cid:15) ( R ) ⊂ L ( R ), and thus ( p.v. x ) ∈ H − / − (cid:15) ( R ).Specifically, F − x { } ( x ) = √ π · δ ( x ) ∈ H − / − (cid:15) ( R ) , F − x { sgn ( · ) } ( x ) = i · r π · p.v. x ∈ H − / − (cid:15) ( R ) , and in the same way F − x { g m } ( x ) = i · r π · p.v. − m x = i · ( − m · r π · p.v. x ∈ H − / − (cid:15) ( R ) . Using the notations δ x in case the delta distribution is only applied in x -direction and δ y accordingly, v x := ( p.v. x ) and v y := ( p.v. y ) as well as v xy := ( p.v. xy ), the 2d P SF mnpyr ∈ H − − (cid:15) (cid:0) R (cid:1) is given by P SF mnpyr = π · δ x δ y + 12 · i · ( − m · v x δ y + 12 · i · ( − n · δ x v y + 12 π · ( − m + n +1 · v xy . According to the standard description of optical systems, the complex amplitude ψ det coming to the detec-tor plane is the inverse 2d Fourier transform of the complex amplitude after the pyramid (cid:0) ψ (cid:15)aper · OT F pyr (cid:1) which results (by application of the convolution theorem) in a convolution of the incoming complex am-plitude ψ (cid:15)aper with the point spread function of the glass pyramid as described, e.g., in [48]. This stepcan now mathematically be written in the sense of distributions as a shifted PSF distribution applied tothe complex amplitude of the incoming phase ψ det ( x, y ) = 12 π h P SF pyr (( x, y ) − ( · , · )) , ψ (cid:15)aper i . Then, by linearity, the four independent beams ψ mndet , m, n ∈ { , } , falling onto the detector are given by ψ mndet ( x, y ) = 12 π h P SF mnpyr (( x, y ) − ( · , · )) , ψ (cid:15)aper i = 12 π h (cid:16) π · δ x δ y + 12 · i · ( − m · v x δ y + 12 · i · ( − n · δ x v y + 12 π · ( − m + n +1 · v xy (cid:17) (( x, y ) − ( · , · )) , ψ (cid:15)aper i = 14 h ( δ x δ y ) (( x, y ) − ( · , · )) , ψ (cid:15)aper i + ( − m i π h ( v x δ y ) (( x, y ) − ( · , · )) , ψ (cid:15)aper i + ( − n i π h ( δ x v y ) (( x, y ) − ( · , · )) , ψ (cid:15)aper i + ( − m + n +1 π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i = 14 ψ (cid:15)aper ( x, y ) + ( − m i π h v x ( x − · ) , ψ (cid:15)aper ( · , y ) i + ( − n i π h v y ( y − · ) , ψ (cid:15)aper ( x, · ) i + ( − m + n +1 π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i . The four complex amplitudes are explicitly formulated as ψ det ( x, y ) = 14 ψ (cid:15)aper ( x, y ) + i π h v x ( x − · ) , ψ (cid:15)aper ( · , y ) i + i π h v y ( y − · ) , ψ (cid:15)aper ( x, · ) i − π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i ,ψ det ( x, y ) = 14 ψ (cid:15)aper ( x, y ) + i π h v x ( x − · ) , ψ (cid:15)aper ( · , y ) i i π h v y ( y − · ) , ψ (cid:15)aper ( x, · ) i + 14 π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i ,ψ det ( x, y ) = 14 ψ (cid:15)aper ( x, y ) − i π h v x ( x − · ) , ψ (cid:15)aper ( · , y ) i + i π h v y ( y − · ) , ψ (cid:15)aper ( x, · ) i + 14 π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i ,ψ det ( x, y ) = 14 ψ (cid:15)aper ( x, y ) − i π h v x ( x − · ) , ψ (cid:15)aper ( · , y ) i− i π h v y ( y − · ) , ψ (cid:15)aper ( x, · ) i − π h v xy (( x, y ) − ( · , · )) , ψ (cid:15)aper i . Now, the intensities on the detector are computed as I mn ( x, y ) = ψ mndet ( x, y ) ψ mndet ( x, y ) , m, n ∈ { , } . If we abbreviate ψ (cid:15)aper by ψ and omit the arguments for simplicity of notation the four intensities areevaluated as I ( x, y ) = ψ det ( x, y ) ψ det ( x, y )= (cid:20) ψ + i π h v x , ψ i + i π h v y , ψ i − π h v xy , ψ i (cid:21) · (cid:20) ψ − i π h v x , ψ i − i π h v y , ψ i − π h v xy , ψ i (cid:21) = 116 ψψ − i π ψ h v x , ψ i − i π ψ h v y , ψ i − π ψ h v xy , ψ i + i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i − i π h v x , ψ ih v xy , ψ i + i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ + i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i ,I ( x, y ) = ψ det ( x, y ) ψ det ( x, y )= (cid:20) ψ + i π h v x , ψ i − i π h v y , ψ i + 14 π h v xy , ψ i (cid:21) · (cid:20) ψ − i π h v x , ψ i + i π h v y , ψ i + 14 π h v xy , ψ i (cid:21) = 116 ψψ − i π ψ h v x , ψ i + i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i + i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i ,I ( x, y ) = ψ det ( x, y ) ψ det ( x, y )= (cid:20) ψ − i π h v x , ψ i + i π h v y , ψ i + 14 π h v xy , ψ i (cid:21) · (cid:20) ψ + i π h v x , ψ i − i π h v y , ψ i + 14 π h v xy , ψ i (cid:21) = 116 ψψ + i π ψ h v x , ψ i − i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i − i π h v x , ψ ih v xy , ψ i + i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i + i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ + i π h v xy , ψ ih v x , ψ i − i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i ,I ( x, y ) = ψ det ( x, y ) ψ det ( x, y )= (cid:20) ψ − i π h v x , ψ i − i π h v y , ψ i − π h v xy , ψ i (cid:21) · (cid:20) ψ + i π h v x , ψ i + i π h v y , ψ i − π h v xy , ψ i (cid:21) = 116 ψψ + i π ψ h v x , ψ i + i π ψ h v y , ψ i − π ψ h v xy , ψ i− i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i + i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i − i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i . Taking the sums according to (1), we obtain the non-modulated (indicated by the superscript n ) pyramidsensor data s nx in x -direction as I · s nx ( x, y ) = [ I ( x, y ) + I ( x, y )] − [ I ( x, y ) + I ( x, y )]= 116 ψψ − i π ψ h v x , ψ i + i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i + i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i + 116 ψψ − i π ψ h v x , ψ i − i π ψ h v y , ψ i − π ψ h v xy , ψ i + i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i − i π h v x , ψ ih v xy , ψ i + i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ + i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i− ψψ − i π ψ h v x , ψ i − i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i + i π h v x , ψ i ψ − π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i − i π h v x , ψ ih v xy , ψ i + i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i − π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ + i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i − π h v xy , ψ ih v xy , ψ i− ψψ − i π ψ h v x , ψ i + i π ψ h v y , ψ i − π ψ h v xy , ψ i + i π h v x , ψ i ψ − π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i − π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i − π h v xy , ψ ih v xy , ψ i , I · s nx ( x, y ) = − i π ψ h v x , ψ i + i π h v x , ψ i ψ − i π h v y , ψ ih v xy , ψ i + i π h v xy , ψ ih v y , ψ i = − i π (cid:2) ψ h v x , ψ i − h v x , ψ i ψ (cid:3) − i π (cid:2) h v y , ψ ih v xy , ψ i − h v xy , ψ ih v y , ψ i (cid:3) = − i π h ψ ( x, y ) h v x ( x − · ) , ψ ( · , y ) i − h v x ( x − · ) , ψ ( · , y ) i ψ ( x, y ) i − i π h h v y ( y − · ) , ψ ( x, · ) ih v xy ( x − · , y − · ) , ψ i − h v xy ( x − · , y − · ) , ψ ih v y ( y − · ) , ψ ( x, · ) i i . This can further be formulated as I · s nx ( x, y ) = − i π X Ω ( x, y ) · exp ( − i Φ ( x, y )) p.v. Z R X Ω ( x , y ) · exp ( i Φ ( x , y )) 1 x − x dx − X Ω ( x, y ) · exp ( i Φ ( x, y )) p.v. Z R X Ω ( x , y ) · exp ( − i Φ ( x , y )) 1 x − x dx − i π p.v. Z R X Ω ( x, y ) · exp ( − i Φ ( x, y )) 1 y − y dy · p.v. Z R p.v. Z R X Ω ( x , y ) · exp ( i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx − p.v. Z R X Ω ( x, y ) · exp ( i Φ ( x, y )) 1 y − y dy · p.v. Z R p.v. Z R X Ω ( x , y ) · exp ( − i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx = − i π X Ω ( x, y ) · exp ( − i Φ ( x, y )) p.v. Z Ω y exp ( i Φ ( x , y )) 1 x − x dx − X Ω ( x, y ) · exp ( i Φ ( x, y )) p.v. Z Ω y exp ( − i Φ ( x , y )) 1 x − x dx − i π X Ω y ( x ) p.v. Z Ω x exp ( − i Φ ( x, y )) 1 y − y dy · p.v. Z Ω y p.v. Z Ω x exp ( i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx − X Ω y ( x ) p.v. Z Ω x exp ( i Φ ( x, y )) 1 y − y dy · p.v. Z Ω y p.v. Z Ω x exp ( − i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx . With Euler’s and trigonometric formulas we obtain I · s nx ( x, y ) = − i π X Ω ( x, y ) i p.v. Z Ω y sin [Φ ( x , y ) − Φ ( x, y )] x − x dx i π X Ω y ( x ) i p.v. Z Ω x p.v. Z Ω y p.v. Z Ω x sin [Φ ( x , y ) − Φ ( x, y )]( x − x ) ( y − y ) ( y − y ) dy dx dy = X Ω ( x, y ) 12 π p.v. Z Ω y sin [Φ ( x , y ) − Φ ( x, y )] x − x dx + X Ω y ( x ) 12 π p.v. Z Ω y p.v. Z Ω x p.v. Z Ω x sin [Φ ( x , y ) − Φ ( x, y )]( x − x ) ( y − y ) ( y − y ) dy dy dx . Taking the sums according to (1), the non-modulated pyramid sensor data s ny in y -direction are writtenas I · s ny ( x, y ) = [ I ( x, y ) + I ( x, y )] − [ I ( x, y ) + I ( x, y )]= 116 ψψ − i π ψ h v x , ψ i + i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i + i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i + 116 ψψ + i π ψ h v x , ψ i + i π ψ h v y , ψ i − π ψ h v xy , ψ i− i π h v x , ψ i ψ + 116 π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i + 116 π h v y , ψ ih v y , ψ i + i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i − i π h v xy , ψ ih v y , ψ i + 116 π h v xy , ψ ih v xy , ψ i− ψψ + i π ψ h v x , ψ i + i π ψ h v y , ψ i + 116 π ψ h v xy , ψ i− i π h v x , ψ i ψ − π h v x , ψ ih v x , ψ i − π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ − π h v y , ψ ih v x , ψ i − π h v y , ψ ih v y , ψ i + i π h v y , ψ ih v xy , ψ i + 116 π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i − i π h v xy , ψ ih v y , ψ i − π h v xy , ψ ih v xy , ψ i− ψψ − i π ψ h v x , ψ i + i π ψ h v y , ψ i − π ψ h v xy , ψ i + i π h v x , ψ i ψ − π h v x , ψ ih v x , ψ i + 116 π h v x , ψ ih v y , ψ i + i π h v x , ψ ih v xy , ψ i− i π h v y , ψ i ψ + 116 π h v y , ψ ih v x , ψ i − π h v y , ψ ih v y , ψ i − i π h v y , ψ ih v xy , ψ i− π h v xy , ψ i ψ − i π h v xy , ψ ih v x , ψ i + i π h v xy , ψ ih v y , ψ i − π h v xy , ψ ih v xy , ψ i , which is equivalent to I · s ny ( x, y ) = i π ψ h v y , ψ i + i π h v x , ψ ih v xy , ψ i − i π h v y , ψ i ψ − i π h v xy , ψ ih v x , ψ i = i π (cid:2) ψ h v y , ψ i − h v y , ψ i ψ (cid:3) + i π (cid:2) h v x , ψ ih v xy , ψ i − h v xy , ψ ih v x , ψ i (cid:3) = i π h ψ ( x, y ) h v y ( y − · ) , ψ ( x, · ) i − h v y ( y − · ) , ψ ( x, · ) i ψ ( x, y ) i + i π h h v x ( x − · ) , ψ ( · , y ) ih v xy ( x − · , y − · ) , ψ i − h v xy ( x − · , y − · ) , ψ ih v x ( x − · ) , ψ ( · , y ) i i I · s ny ( x, y ) = i π X Ω ( x, y ) · exp ( − i Φ ( x, y )) p.v. Z R X Ω ( x, y ) · exp ( i Φ ( x, y )) 1 y − y dy − X Ω ( x, y ) · exp ( i Φ ( x, y )) p.v. Z R X Ω ( x, y ) · exp ( − i Φ ( x, y )) 1 y − y dy + i π p.v. Z R X Ω ( x , y ) · exp ( − i Φ ( x , y )) 1 x − x dx · p.v. Z R p.v. Z R X Ω ( x , y ) · exp ( i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx − p.v. Z R X Ω ( x , y ) · exp ( i Φ ( x , y )) 1 x − x dx · p.v. Z R p.v. Z R X Ω ( x , y ) · exp ( − i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx = i π X Ω ( x, y ) · exp ( − i Φ ( x, y )) p.v. Z Ω x exp ( i Φ ( x, y )) 1 y − y dy − X Ω ( x, y ) · exp ( i Φ ( x, y )) p.v. Z Ω x exp ( − i Φ ( x, y )) 1 y − y dy + i π X Ω x ( y ) p.v. Z Ω y exp ( − i Φ ( x , y )) 1 x − x dx · p.v. Z Ω y p.v. Z Ω x exp ( i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx − X Ω x ( y ) p.v. Z Ω y exp ( i Φ ( x , y )) 1 x − x dx · p.v. Z Ω y p.v. Z Ω x exp ( − i Φ ( x , y )) 1( x − x ) ( y − y ) dy dx . Using Euler’s and trigonometric formulas we get I · s ny ( x, y ) = i π X Ω ( x, y ) i p.v. Z Ω x sin [Φ ( x, y ) − Φ ( x, y )] y − y dy + i π X Ω x ( y ) i p.v. Z Ω y p.v. Z Ω y p.v. Z Ω x sin [Φ ( x , y ) − Φ ( x , y )]( x − x ) ( y − y ) ( x − x ) dy dx dx = − X Ω ( x, y ) 12 π p.v. Z Ω x sin [Φ ( x, y ) − Φ ( x, y )] y − y dy − X Ω x ( y ) 12 π p.v. Z Ω y p.v. Z Ω x p.v. Z Ω y sin [Φ ( x , y ) − Φ ( x , y )]( x − x ) ( y − y ) ( x − x ) dx dy dx . mod ( x, y, t ) = α λ ( x sin(2 πt/T ) + y cos(2 πt/T )) (28)introducing the circular modulation path to the incoming screen Φ. The constant α λ denotes the modu-lation parameter defined in (2). Clearly, by using the non-modulated model from above, one obtains foreach time step t the non-modulated measurements s nx ( x, y, t ) , s ny ( x, y, t ) corresponding to the tilted phaseΦ( x, y ) + Φ mod ( x, y, t ).As the second step, one has to integrate these time-dependent non-modulated pyramid measurements s nx ( x, y, t ) , s ny ( x, y, t ) over one full time period T , which gives the measurements of the circularly modulatedpyramid wavefront sensor as s cx ( x, y ) = 1 T T/ Z − T/ s nx ( x, y, t ) dt, (29) s cy ( x, y ) = 1 T T/ Z − T/ s ny ( x, y, t ) dt. Thus, the modulated sensor measurements are described by s cx ( x, y ) = 1 T T/ Z − T/ π X Ω ( x, y ) p.v. Z Ω y sin[Φ( x , y ) + Φ mod ( x , y, t ) − Φ( x, y ) − Φ mod ( x, y, t )] x − x dx dt + 1 T T/ Z − T/ π X Ω y ( x ) p.v. Z Ω y Z Ω x Z Ω x sin[Φ( x , y ) + Φ mod ( x , y , t ) − Φ( x, y ) − Φ mod ( x, y , t )]( x − x )( y − y )( y − y ) dy dy dx dt. First, we want to separate the parts which depend on time to be able to integrate them, s cx ( x, y ) = 1 T T/ Z − T/ π X Ω ( x, y ) p.v. Z Ω y sin (cid:2) (Φ( x , y ) − Φ( x, y )) + (cid:0) Φ mod ( x , y, t ) − Φ mod ( x, y, t ) (cid:1)(cid:3) x − x dx dt + 1 T T/ Z − T/ π X Ω y ( x ) p.v. Z Ω y Z Ω x Z Ω x sin[(Φ( x , y ) − Φ( x, y )) + (Φ mod ( x , y , t ) − Φ mod ( x, y , t ))]( x − x )( y − y )( y − y ) dy dy dx dt. Note that the modulation function Φ mod is linear in the first two arguments, i.e., Φ mod ( x , y, t ) − Φ mod ( x, y, t ) = α λ x sin(2 πt/T ) + α λ y cos(2 πt/T ) − α λ x sin(2 πt/T ) − α λ y cos(2 πt/T )= α λ ( x − x ) sin(2 πt/T )= Φ mod ( x − x, , t ) (30) and Φ mod ( x , y , t ) − Φ mod ( x, y , t ) = α λ x sin(2 πt/T ) + α λ y cos(2 πt/T ) − α λ x sin(2 πt/T ) − α λ y cos(2 πt/T )= α λ ( x − x ) sin(2 πt/T ) + α λ ( y − y ) cos(2 πt/T )= Φ mod ( x − x, y − y , t ) . (31) s cx ( x, y ) = 1 T T/ Z − T/ π X Ω ( x, y ) p.v. Z Ω y sin[(Φ( x , y ) − Φ( x, y )) + Φ mod ( x − x, , t )] x − x dx dt + 1 T T/ Z − T/ π X Ω y ( x ) p.v. Z Ω y Z Ω x Z Ω x sin[(Φ( x , y ) − Φ( x, y )) + Φ mod ( x − x, y − y , t )]( x − x )( y − y )( y − y ) dy dy dx dt. Using trigonometric formulas, we separate the time-dependent parts s cx ( x, y ) = X Ω ( x, y ) π p.v. Z Ω y sin[Φ( x , y ) − Φ( x, y )] x − x T T/ Z − T/ cos (cid:2) Φ mod ( x − x, , t ) (cid:3) dt dx + 12 π p.v. Z Ω y cos[Φ( x , y ) − Φ( x, y )] x − x T T/ Z − T/ sin (cid:2) Φ mod ( x − x, , t ) (cid:3) dt dx + X Ω y ( x ) π p.v. Z Ω y Z Ω x Z Ω x sin[Φ( x , y ) − Φ( x, y )]( x − x )( y − y )( y − y ) T T/ Z − T/ cos (cid:2) Φ mod ( x − x, y − y , t ) (cid:3) dt dy dy dx + 12 π p.v. Z Ω y Z Ω x Z Ω x cos[Φ( x , y ) − Φ( x, y )]( x − x )( y − y )( y − y ) T T/ Z − T/ sin (cid:2) Φ mod ( x − x, y − y , t ) (cid:3) dt dy dy dx . The second and the fourth terms equal zero, since the integrands are odd functions. After substitutionof the explicit expressions (30)-(31) for Φ mod , the remaining time integrals simplify to1 T T/ Z − T/ cos (cid:2) Φ mod ( x − x, y − y , t ) (cid:3) dt = 1 T T/ Z − T/ cos [ α λ ( x − x ) sin(2 πt/T ) + α λ ( y − y ) cos(2 πt/T )] dt = 1 T T/ Z − T/ cos [ α λ ( x − x ) sin(2 πt/T )] cos [ α λ ( y − y ) cos(2 πt/T )] dt − T T/ Z − T/ sin [ α λ ( x − x ) sin(2 πt/T )] sin [ α λ ( y − y ) cos(2 πt/T )] dt = 1 T T/ Z − T/ cos [ α λ ( x − x ) sin(2 πt/T )] cos [ α λ ( y − y ) cos(2 πt/T )] dt − . and 1 T T/ Z − T/ cos (cid:2) Φ mod ( x − x, , t ) (cid:3) dt = 1 T T/ Z − T/ cos [ α λ ( x − x ) sin(2 πt/T )] dt = 12 π π Z − π cos [ α λ ( x − x ) sin( t )] dt = J [ α λ ( x − x )] , t = 2 πt/T and the definition of the zero-order Bessel function J ( x ) = 1 π π Z cos( x sin t ) dt = 12 π π Z − π cos( x sin t ) dt. All steps of the proof can by performed for the data s { c } y analogously. References [1] R. A. Adams.
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