1. Introduction

When designing a wavefront curvature sensor [1], the propagation distance (the separation between measurement planes and the plane of the lens, inversely proportional to the defocus distance) dominates some measurement performances such as sensitivity and accuracy. Some methods based on geometrical optics are presented to determine the propagation distance. Roddier et al. [1] put forward the lower bound of the defocus distance, which is the separation between the defocus plane and the focal plane. Soto et al. [2] determined the propagation distance by optimizing the variance in experimental measurements. The analysis of curvature sensing for extreme adaptive optics (XAO, designed to reach very high Strehl ratios and therefore a high contrast) performed by Guyon [3] refers to the influence of the Talbot effect. Recently, Guyon et al. [4] presented the effect of the propagation distance on wavefront curvature sensing sensitivity because of the Talbot effect, in which the upper bound of the optimal propagation distance is determined by the requirement of wavefront measurement linearity and the residual aberrations in low-order modes. In [4], dual-stroke wavefront curvature sensing with two defocus distances is introduced, which offers a significant performance gain. Nevertheless, to our best knowledge, a theoretical analysis on the choice of the optimal propagation distance with diffraction optics has not yet been found because of the complexity of the diffraction formulations [5].

This paper presents the frequency analysis of wavefront curvature sensing from the viewpoint of diffraction optics. The wavefront aberration is decomposed into a series of Fourier harmonics at various frequencies. The fact that the Fourier harmonics are the eigenmodes of the curvature operator allows the deduction of the Talbot effect in wavefront curvature sensing. Usually the Talbot effect is analyzed with the wave propagation transfer function approach [6]. Therefore, the curvature signal produced by the sine wave phase distribution is also calculated with the transfer function approach, which is called the frequency analysis of wavefront curvature sensing. The analysis shows that the intensity of the curvature signal is a sine function of the product of the squared frequency and the propagation distance. By maximizing the curvature signal (or the sensitivity of wavefront curvature sensing), the optimum propagation distance is equal to one quarter of the Talbot length at the critical frequency. Following the determination of the propagation distance, the intensity of the curvature signal sinusoidally varies with the squared frequencies, vanishing at some higher frequency bands just like a comb filter. To cover these insensitive bands, wavefront curvature sensing with dual propagation distances or with multi-propagation distances is proposed, which is named dual-distance (dual-z) or multi-distance (multi-z) wavefront curvature sensing. The concept of dual-z wavefront curvature sensing has been employed virtually in the adaptive optics system for astronomy in order to adapt to different operation stages [7]. Only if the comb filter is established will dual-z or multi-z wavefront curvature sensing efficiently cover the higher frequency. The analysis in this paper is also applicable for similar problems such as the phase diversity technique, electron microscopy, x-ray crystallography, etc. [8].

The outline of the paper is as follows. In Section 2, we analyze the curvature signal of the sine wave phase distortion with the transfer function approach. Section 3 introduces how to determine the optimum propagation distance. In Section 4, we derive the comb filter of wavefront curvature sensing. In Section 5, we propose the dual-z or the multi-z wavefront curvature sensor, which can cover higher frequencies. Finally, the conclusions are presented in Section 6.

2. Analysis of wavefront curvature sensing with the transfer function approach

2.1 Talbot effect in wavefront curvature sensing

The wavefront curvature sensing technique developed by Roddier [1] is schematically described in Fig. 1. The distorted wavefront φ is focused by lens L with focal length f_L. The normalized difference of intensities on two defocus planes M ₁ and M ₂ with defocus distance l is proportional to the local wavefront curvature. Based on the imaging principle, two defocus planes M ₁ and M ₂ correspond to two measurement planes P ₁ and P ₂ symmetrically located before and after the lens with a propagation distance of $z=\frac{f_L^2}{l}$ [1]. Consequently, two irradiance distributions on the defocus planes are miniatures of two corresponding measurement planes.

 

Fig. 1. Schematic diagram of curvature sensing; two defocus planes M ₁ and M ₂ correspond to measurement planes P ₁ and P ₂, respectively.

Download Full Size | PDF

According to the intensity transport equation (ITE) [9], the intensity distributions [10] on planes P ₁ and P ₂ are linear functions of the local wavefront curvature ∇² φ on the entrance pupil plane (i.e. the plane of the lens),

where the intensity on the entrance pupil plane A is unity, $k=\frac{2\pi }{\lambda }$ is the wavenumber, λ is the wavelength, and ∇² is the Laplacian or curvature operator.

Analogous to the Zernike decomposition, wavefront φ(ξ,η) may be expressed as a summation of sine wave functions at various spatial frequencies where the (ξ,η) plane is the entrance pupil plane of lens L. The general component is a cos (k_ξξ + k_ηη) + b sin (k_ξξ + k_ηη), which allows for any orientation on the (ξ,η) plane and any amplitude where k_ξ or k_η are the angular spatial frequency in radians per unit of ξ, or η.

By rotating and shifting the (ξ,η) axes, we can return to sin(k_ξξ) with any amplitude, and this expression will suffice for the following discussion:

where f ₀ is the fundamental frequency; B_n and θ_n are the coefficient and the initial phase of the n^th order harmonic, respectively; n is an integer; and the piston term (n = 0) of phase is omitted. The Fourier harmonics or the sine wave functions are the eigenmodes of the Laplacian or the curvature operator [11]. The curvature of an arbitrary sine wave phase B = B sin (2πfξ + θ) is given by

Substituting Eq. (3) into Eq. (1), the intensity patterns on planes P ₁ and P ₂ conform to the wavefront map

The accurate reappearance in intensity of the periodic phase structure is essentially the Talbot effect [12]. Accordingly, the Talbot effect will be instrumental in analyzing curvature sensing from the viewpoint of diffraction optics. On the one hand, diffraction transforms the invisible phase information into a measurable intensity distribution. Therefore, the information transformation should be studied with the diffraction optics approach, which reflects the intrinsic properties of curvature sensing. On the other hand, the Talbot effect is the straightforward representation of Fresnel diffraction in the aspect of the frequency spectrum. Then, the transfer function approach used in the Talbot effect will considerably simplify the analysis of curvature sensing, benefiting from the powerful capabilities of frequency analysis.

2.2 Curvature signal calculated with transfer function approach

We revisit the Talbot effect [12,13] of a sine wave phase structure in curvature sensing. In any diffraction problem involving a periodic structure, the transfer function approach will yield the simplest calculation. On the entrance pupil plane, the complex amplitude of an arbitrary sine wave phase is described by

where $f=\frac{1}{L}$ is the spatial frequency, L is the period, 2B is the modulated amplitude, J_q is the q^th order Bessel function of the first kind, q is an integer, and j is √-1 . With the transfer function approach, we obtain the complex amplitude on plane P ₁ to be given by

where x is the axis on plane P ₁ parallel to the ξ axis.

It can be seen from Eq. (6) that if the distance $z=n\frac{2}{\lambda f^2}=n\frac{2L^2}{\lambda }$, the Talbot effect of the periodic phase structure is similar to that of amplitude grating, where n is an integer. We denote the Talbot length as $Z_r=\frac{2}{\lambda f^2}=\frac{2L^2}{\lambda }$, which is equal to that of amplitude grating. Suppose that distance z satisfies $z=\frac{n{Z}_T}{2}$; the intensity on plane P ₁ is equal to unity, which suggests the sinusoidal phase component remains invisible at multiples of one half of the Talbot length. No intensity information converted from the phase distortion exists at those positions.

Calculated with the transfer function approach, the intensity pattern on plane P ₁ can be given as

where the term else contains higher order and more complex harmonic patterns. The weighting factor of each harmonic component in Eq. (7) is determined by the product of J_q(B). In the range of β≪1, the magnitudes of J ₀(B) and J ₀(B) are much larger than other higher-order magnitudes. A comparison of the coefficients of the first 5 harmonic components is shown in Fig. 2. Accordingly, the else term in Eq. (7) can be ignored in the small perturbation approximation (i.e. B ≪ 1).

 

Fig. 2. Comparison of the coef icients of the first 5 harmonic components.

Download Full Size | PDF

As demonstrated in [12], the intensity distribution on the (2n-1)Z_T/4 planes can be expressed only for the sinusoidal phase distribution as

After combining Eq. (8), Eq. (7) can be simplified as

In a similar way, the intensity pattern on plane P ₂ with distance -z can be written as

Following the curvature sensing principle, curvature signal S is given as

Equation (11) shows that the curvature signal is a representation of the sine wave phase distribution, except that the amplitude is modulated sinusoidally by the product of the propagation distance z and the squared frequency f. The sinusoidal modulation predicted by the transfer function approach cannot be obtained by geometrical optics. Equation (11) is the core of this paper, from which we deduce the optimum propagation distance and the frequency filter of curvature sensing.

3. Selection of optimum propagation distance for wavefront curvature sensing

The propagation distance dominates the measurement performances of curvature sensing. From the viewpoint of frequency analysis, the optimum propagation distance is selected by maximizing the measurement sensitivity. From Eq. (11) it can be seen that for a certain frequency f, the curvature signals have the largest amplitude or the highest sensitivity at the propagation distances z = (2n - 1) Z_T/4, which can be expressed as

On the contrary, with propagation distances z = nZ_T/2, the curvature signal is null visibility for frequency f. In other words, the curvature sensor cannot detect phase distortions at the spatial frequency of f with propagation distances z = nZ_T/2.

Curvature sensing is a central difference approximation of the ITE [10]. On account of shorter propagation distance and higher measurement accuracy, we only consider one quarter of the Talbot length. The relationship between the optimum propagation distance z _opt and the frequency f is given by

The optimal propagation distances are distinct for individual spatial frequencies, even if only one quarter of the Talbot length is considered. Which spatial frequency should be considered especially? The power spectrum of phase fluctuations should be referenced as the standard to select it, such as the Wiener spectrum of phase fluctuations due to Kolmogoroff turbulence. In this paper, the critical spatial frequency f_k can be chosen as the average power point at which the power spectrum density is equal to the average power spectrum density in the measured or compensated frequency band. The wavefront curvature sensor designed with the critical frequency will cover a broader frequency spectrum and receive more power. Following the selection of critical frequency f_k, the propagation distance z_k of curvature sensing is determined by Eq. (13).

4. Filter model of wavefront curvature sensing

The filter model of curvature sensing models the measurement sensitivity at various spatial frequencies. Following the determination of propagation distance z_k, the sensitivity of curvature sensing depends on the sine function of the squared frequency. Substituting z in Eq. (11) with z_k, the curvature signal for phase distortions at various spatial frequencies yields

For propagation distance z_k, curvature sensing is quite sensitive to phase distortions near frequencies √2n-1 f_k. On the contrary, it is insensitive to phase distortions near frequencies √2n f_k. We call this sensitivity spectrum a comb filter. It implies that the curvature sensing technique does not impose the upper bound for the transferable spatial frequency. Physically, its upper bound is just limited by diffraction, i.e., f < λ^-1. In addition, considering the blurring effect induced by the wavefront aberration, diffraction effects limit spatial resolution of the measurements to √λz[14]. Practically, it is limited ultimately by the sizes of the two defocus spots and the resolutions of detectors.

We calculate the curvature signal of the phase distortion at various spatial frequencies. The comb filter of curvature sensing can be schematically shown in Fig. 3. The conditions adopted in the simulation are presented as follows: the frequency range is from 1 m^-1 to 30 m^-1, the critical frequency f_k is selected as 10 m^-1, the determined optimum propagation distance z_k is 7901 m, the modulated amplitude of phase distortion is 0.2 rad, the wavelength is 0.6328 μm , and the shade of color in the image stands for the magnitude of the curvature signals.

 

Fig. 3. Filter model of curvature sensing shown in Eq. (14).

Download Full Size | PDF

What are the impacts of the comb filter on wavefront measurement and adaptive compensation? To begin with, the comb filter dramatically increases complexities and difficulties of the reconstruction algorithms. The curvature signal within the aperture does not contain wavefront information at some frequencies. To retrieve the whole wavefront, reconstruction algorithms must recover the absent phase components with edge information. A lot of effort is dedicated to the edge condition. For instance, the Gerchberg–Saxton algorithm [15] consists of a great deal of iterative loops; the Green’s function algorithm [16] needs the huge Green’s function matrices. Moreover, the comb filter property, to a large extent, lowers the compensation efficiency of the adaptive optics system at some frequencies. Bimorph or membrane deformation mirrors cannot shape proper compensating profiles with just edge signals, so the absence of some curvature signals increases residual errors.

5. The design of dual-z or multi-z wavefront curvature sensing

To relieve the impact of the comb filter, greater propagation distance should be considered in one curvature sensor to fill up insensitive frequency bands, which are named dual-z or multi-z curvature sensing. Dual-z or multi-z curvature sensing can cover a wider frequency spectrum and will lighten the dependence of reconstruction algorithms on edge conditions. The dual-z curvature sensor has been equipped virtually in adaptive optics systems for astronomy [6] to adapt to different compensating stages because distinct spatial frequencies are required before and after closing the loop. Most recently, dual-z curvature sensing was referred to by Guyon [3]. Only if the comb filter model is established will dual-z or multi-z curvature sensing cover a wider spatial frequency band.

The first propagation distance z_k is the conventional optimum propagation distance, determined by f_k and Eq. (13). Following the determination of z_k, curvature sensing is not sensitive to the frequencies of √2f_k, √4f_k, √6f_k, √8f_k, √10f_k, √12f_k but to the frequencies of f_k, √3f_k, √5f_k, √7f_k, √9f_k, √11f_k, √13f_k. To measure the distortions at frequencies insensitive for z_k, a shorter propagation distance z_k2 should be included in the same curvature sensing. For example, z _k2 could be conveniently chosen as the optimum distance for the frequency of √2f_k by Eq. (13), which is given by

Then z_k2 is sensitive to the frequencies of √2f_k, √6f_k, √10f_k, √14f_k, √18f_k.

There still exist some frequencies that cannot be measured with z_k and z _k2, such as √4f_k, √8f_k, √12f_k, √16f. A shorter propagation distance z _t3 sensitive to the frequency of √4f_k can be introduced into the same curvature sensor, and therefore the insensitive frequencies just leave √8f_k, √16f, √24f_k. Similarly, z_k3 is given by

If necessary, more propagation distances may be contained to cover a wider frequency spectrum. The covering effects with dual-z or multi-z schemes are illustrated in Fig. 4. The parameters in simulations are the same as those in Fig. 3. The diagram is divided into four parts from top to bottom. The corresponding propagation distance of each part is z_k, z_k/2 , z_k/4, and z_k/8, respectively. From Fig. 4 it can be seen that the extra propagation distances can fill up the insensitive frequency bands of the first propagation distance. Due to the spreading width of the passband, the covering effect of only two propagation distances, such as z_k and z_k/8, can go beyond the prospects.

 

Fig. 4. Covering effect of dual-z or multi-z curvature sensing.

Download Full Size | PDF

The implementation of dual-z or multi-z curvature sensing may rely on diverse techniques. For instance, the vibrating membrane [3,4,17] used in the XAO is driven to vibrate at two different amplitudes to produce the two required propagation distances, which is called dual-stroke curvature wavefront sensing. Also, crossed defocus grating [18,19,20,21] used in close proximity to a lens can simultaneously produce four different propagation distances, which is employed in the laser M ² factor measurement. Phase defocus gratings can adjust the photons flowing into different defocus spots.

In connection with the multi-z scheme, the total amount of light available from the reference source has to be shared between these sensing planes within a given period of time, which could result in insufficient photon flux for wavefront measurement and increase noise. A difficult compromise therefore needs to be made between two opposite requirements: the spatial resolution of curvature sensing and the signal noise of the measurement, which is similar to the compromise between the total number of subapertures and the measurement noise of the Hartmann wavefront sensor. Fortunately, low-order aberrations correspond to a longer propagation distance or a shorter defocus distance than the high-order ones, resulting in smaller defocus images, which need less photons than those that high-order ones need. The photons should be reasonably shared at different defocus planes through modulating the vibrating periods of the membrane or the diffractive efficiencies of the crossed defocus gratings. Detailed schemes and implementation are beyond the scope of this paper.

6. Conclusion

Frequency analysis of curvature sensing is performed by use of the transfer function approach. The curvature signal of sine wave phase distortion is sinusoidally modulated by the product of the propagation distance and the squared frequency. We formulate the optimum propagation distance in terms of one quarter of the Talbot length at the critical frequency. The comb filter is estimated as a sine function of the squared frequency. Dual-z or multi-z curvature sensing is described to cover a wider spatial frequency spectrum. This paper shows the capability of the transfer function approach in sensitivity analysis of curvature sensing. In the future, analysis with the transfer function approach will involve other performances of curvature sensing, such as the influence of the edge signal, accuracy, and noise propagation, etc. The proposed dual-z or multi-z curvature sensing will be demonstrated experimentally. The new reconstruction algorithm available for du l-z or multi-z curvature sensing will be studied.