1. Introduction

The pyramid wavefront sensor was first introduced by Ragazzoni [1] for astronomical applications. It is a modified version of the Foucault knife-edge test [2, 3], where a refractive element (the pyramid) is used to produce four images of the entrance pupil, and spatial modulation is introduced to control the gain of the phase slope measurements. One advantage of the pyramid wavefront sensor over the widely used Shack-Hartmannwavefront sensor seems to be increased sensitivity in closed-loop applications [4, 5], and the adjustable sensitivity and pupil sampling. The main disadvantage is the dynamic modulation, which requires a moving component inside the instrument. There are suggestions on how to remove the demand for modulation, including an extended source [6] or use of uncorrected higher-order or very fast aberrations [7, 8]. The pyramid wavefront sensor has also been implemented for ophthalmic applications [9, 10].

Adjustable modulation is important to the sensitivity and linearity of the wavefront sensor. The effects of circular modulation have been investigated by approximate geometrical theory [11, 12], and the effects of linear modulation have been partly clarified from diffraction theory [12, 13]. In this paper we investigate the response of a circularly modulated wavefront sensor, deriving and comparing results from geometrical theory, diffraction theory, and experiments. The geometrical theory is outlined in Section 1, and the diffraction theory in Section 2. Numerical and experimental results are presented in sections 3 and 4 respectively, and compared to the geometrical predictions.

2. Geometrical theory

A pyramid wavefront sensor uses a prism in the shape of a pyramid to split the field, at the image plane (or the Fourier plane of the pupil), in four distinct parts as illustrated in Fig. 1(a). Four images of the pupil are created, with intensities here labeled as I_a (x,y) (imaged through the first quadrant of the prism), I_b (x,y) (second quadrant), I_c (x,y) (third quadrant), and I_d (x,y) (fourth quadrant), where (x,y) are coordinates in the pupil plane. Without dynamic modulation the system is equivalent to a Foucault knife-edge test [2, 3] and can be used to predict the sign of the phase derivative, but not its amplitude. If modulation is introduced, e.g. as illustrated in Fig. 1(b), the situation changes. The modulation is accomplished either by moving the prism in the transverse direction on a vibrating mount, or by rotating the beam using a steering mirror at the aperture plane. The system used in this paper employs a steering mirror, but the theory applies to both cases. The movement can also follow different schemes. The most common are circular modulation, as shown in Fig. 1(b), or linear modulation, where the focused beam traces a diamond-shape around the centre of the pyramid [1]. From geometrical theory it follows that the gradient of the wavefront aberration W(x,y) is [11, 12]

 

Fig. 1. (a) Principle of the pyramid wavefront sensor. Light from the pupil is split at the Fourier plane and re-imaged as four separate pupils (only two are shown in the image). (b) The beam is modulated, i.e., rotated around the prism center.

Download Full Size | PDF

for circular modulation of radius ξ0, where the distance between aperture and Fourier plane is f and

The derivation only holds if the displacement in the Fourier plane due to the aberration, f∂W/∂x, is smaller than the modulation amplitude ξ0. For larger aberrations, S_x will saturate at ±1. For non-circular modulation, relations (1) and (2) will change [11]. However, for most modulations, for small wavefront aberrations we can conclude that

i.e., that the response functions Sx(x, y) and Sy(x, y) are proportional to the phase derivatives. The proportionality constant depends on the modulation amplitude ξ ₀: increased modulation amplitude gives a decreased sensitivity. Eq. (5) is valid for

and a similar condition holds for Eq. (6).

3. Diffraction theory

As this derivation is more complicated than the geometrical one, it is generally done in one dimension rather than two, using a roof prism in place of a pyramid. If the roof prism splits the Fourier plane along the y axis, two pupil images of intensities I ₁ (corresponding to I_a +I_d ) and I ₂ (corresponding to I_b +I_c ) are produced. All quantities can still be functions of both x and y, and in the following are taken along the x axis, so that e.g. I ₁(x) represents I ₁(x, 0). If we imagine a fixed prism, i.e., a system without modulation, the field in the images of the pupil is given by [14]

where U ₀(x)=A ₀ exp(iϕ(x)) is the optical field in the aperture and Ui(x) is the optical field in the image of the pupil. The incident intensity is assumed to be constant, I ₀=$A_0^2$, to simplify the derivation. We note that the relation between the phase ϕ(x) and the wavefront aberration W(x) used in the geometrical derivation is ϕ(x)=kW(x), where k=2π/λ is the wave number and λ the wavelength of the light. The function h_i (x) is the Fourier transform of the obscuration function H_i (ξ) applied in the Fourier plane, where ξ is the coordinate in the x direction. For the right half of the roof prism, H ₁(ξ)=1 for ξ>0 and zero otherwise, while for the left half, H ₂(ξ)=1 for ξ<0 and zero otherwise.

From Eq. (8), the relation [7, 12]

where R is the radius of the aperture, can be derived through evaluation of the Fourier transforms h ₁(x) and h ₂(x) [3, 14]. If the phase is not taken along the x axis, the equation still holds but the integration limits must be adjusted.

The modulation can be introduced as part of the phase function ϕ(x, t)=ϕ _mod(x, t)+φ(x), where t represents time, ϕ _mod(x, t) is the modulation and φ(x) contains the wavefront to be measured. Assuming a linear modulation ϕ _mod(x, t)=a ₀ tx/(T/2), where a ₀=kξ ₀/f, and integrating over one full time period T, gives the result [8, 11, 12, 15]

Just as for the geometrical derivation, however, the type of modulation will affect the result. If we assume instead a sinusoidal modulation in time, ϕ _mod(x, t)=a ₀ xsin(2πt/T), corresponding to circular modulation in the two-dimensional case, we find (see Appendix A)

where J ₀ is the zero-order Bessel function of the first kind.

As a ₀ will take on large values even for small modulation amplitudes ξ ₀, both the sinc function in Eq. (10) and the Bessel function in Eq. (11) can be considered as narrow. The following derivation can be used for both integrals, but will be completed only for the sinusoidal modulation. First we must consider the value taken on by sin [φ(x′)-φ(x)]/(x′-x) when x′→x. Both the nominator f(x′)=sin [φ(x′)-φ(x)]→0 and denominator g(x′)=x′-x→0 as x′→x. Furthermore, g(x′)≠0 for x′≠x. Then l’Hôpital’s rule can be applied, stating that

Since f′(x′)=φ′(x′)cos[φ(x′)-φ(x)]→φ′(x) as x′→x, and g′(x)=1, Eq. (11) now approximates as

where the Bessel function has been assumed narrow enough that the rest of the integral can be considered as slowly varying. Evaluating the integral yields a constant inversely proportional to a ₀ [16, Eq. 6.511.1], and consequently

Comparison to Eqs. (5) and (6) shows that the same linear relationship has been derived in both cases. Since we assumed constant incident intensity, $A_0^2$ is identical to the geometrical approximation of 〈I ₁+I ₂〉, and consequently 〈I ₁-I ₂〉 is approximately proportional to S_x .

This result depends on the amplitude of modulation in relation to the severity of the phase errors. The condition that J ₀ [a ₀(x′-x)] is much narrower than sin [φ(x′)-φ(x)]/(x′-x) can be approximated by demanding that the first zero of the Bessel function occurs much closer to x′=x than that of the sinc function, i.e., that

Apart from a small change in the constants, and taking the fact that φ(x)=kW(x) into account, this is the same condition as for the geometrical derivation (Eq. (7)).

The linear relationship in Eq. (14) is derived from Eq. (11) using the approximation of small wavefront inclinations compared to the modulation introduced. However, using another less drastic set of approximations, it is possible to derive a result very similar to Eqs. (1) and (2). Two approximations are required. The first is to extend the integration range from [-R, R] to (-∞,∞), which is valid if the phase is sought at a point well inside the aperture. It can also be regarded at part of the geometrical approximation – diffraction at the edges of the aperture is disregarded. The second is to locally approximate the aberration by tilt. Since only points x′ close to x will contribute to the integration, we replace φ(x)-φ(x′) by c(x-x′), where c=∂φ/∂x is the local phase derivative. A change of variables z=a ₀(x-x′) then allows us to write Eq. (11) as

which can be evaluated using the identity [16, Eq. 6.693.7]

and the fact that the integrand is even. The result is that

provided |∂φ/∂x| < kξ ₀/f. For larger values of the derivative, the expression will saturate at plus/minus a constant. This is the same relation as the geometrical result in Eqs. (1) and (2).

4. Numerical evaluation

Equation (11) is readily evaluated numerically, as in Fig. 2 for λ=632.8nm, ξ 0=1.281mm, f=163mm, and R=1.64mm. The parameters are the same as for the experimental results in Section 4, for easier comparison. The aberration is assumed to be defocus, given by

 

Fig. 2. Values of 〈I ₁-I ₂〉 in the aperture, numerically obtained from the integral in Eq. (11). System parameters are λ=632.8nm, ξ ₀=1.281mm, f=163mm, and R=1.64mm. The aberration is defocus as given by Eq. (19), where n ranges from -40 to 40 in steps of 5.

Download Full Size | PDF

where n=-40,-35, …, 40. Defocus was used, rather than any other aberration, for two particular reasons. First, the phase derivative is proportional to x, so Fig. 2 in itself becomes an assessment of the linearity. Second, it is easy to introduce defocus in an experimental situation. The curves are nearly linear until x reaches a certain value, where 〈I ₁-I ₂〉 saturates. There are two kinds of nonlinearity: one that starts when the phase derivative becomes too big, showing as a deviation from linear increase in Fig. 2. The second kind occurs very close to the aperture edge, where the response suddenly drops very low. Of those two phenomena, the first is of particular interest since it reflects the linearity of 〈I ₁-I ₂〉 versus dφ/dx. Since the phase derivatives can be found explicitly from the phase in Eq. (19), it is a simple procedure to rearrange the result as in Fig. 3. In doing this, the points at the edge of the aperture are removed to better show the linearity without the influences of the edge effect. It shows the geometrical (solid lines) and diffraction (dots) predictions for ξ ₀=1.281mm, corresponding to a steering mirror voltage of 3 V (see section 4), and for ξ ₀=0.427mm, corresponding to a voltage of 1 V. The diffraction results have been normalized.

 

Fig. 3. Pyramid response 〈I ₁-I ₂〉 as a function of the incident-phase derivative, according to the geometrical prediction (solid lines) and diffraction (dots) and assuming a sinusoidal modulation. System parameters are the same as in Fig. 2, except that the modulation amplitude is ξ ₀=1.281mm (3V) and ξ ₀=0.427mm (1V).

Download Full Size | PDF

The smallest modulation corresponds to the highest sensitivity, i.e., the steepest graph, and vice versa. Geometrical and diffraction predictions agree very closely, and the relationship between the phase derivative and the wavefront sensor response is indeed linear for relatively small values. The numerical values will be slightly different if another kind of aberration is used, but the general result remains the same.

According to the analytical result in Eq. (15), <I ₁ - I ₂> is supposed to be linear if |∂φ/∂x|≪1.0×10⁵m^-1 for 3 V modulation and 3.4×10⁴ m ^-1 for 1 V modulation. From Fig. 3 we can see that they remain nearly linear up to roughly |∂φ/∂x|=0.5×10⁵m^-1 and 2×10⁴ m ^-1 respectively. Considering that Eq. (15) is approximate, the agreement is good.

The diffraction prediction depends closely on the modulation model used. Figure 4 shows the same results, but with the diffraction result calculated using linear modulation according to Eq. (10).

 

Fig. 4. Same as Fig. 3, except Eq. (10) was used to calculate the diffraction result. For linear modulation the diffraction result and the geometrical approximation (not shown) are indistinguishable when placed on the same graph. The geometrical approximation with circular modulation (solid line) is therefore plotted here to allow easier comparison with Figs. 3 and 6.

Download Full Size | PDF

5. Experiment

A pyramid wavefront sensor, constructed as part of an ophthalmic closed-loop adaptive optics system [10], was used to obtain experimental results. This reference contains a detailed schematic of the experimental setup. For these measurements, the deformable mirror was left in its bias position and the sensor used to measure the wavefront gradients of a collimated laser beam (radius 3 mm), which entered the system at the normal location of the ocular pupil. This ocular pupil is conjugated to a steering mirror where a sinusoidal modulation of the beam about the pyramid tip is introduced. The apparatus includes a Badal stage consisting of an optical trombone formed by two mirrors mounted on a manual translation stage. The stage is normally used to compensate for the subject’s natural defocus (approximately ± 2 diopters over a 6 mm pupil), but was employed in this work to introduce specific, known amounts of defocus into the originally collimated laser beam. An objective characterization of the Badal stage was carried out using a commercial interferometer to probe the introduced defocus at each stage position, b=0, 1, …, 27 mm. The optical path probed by the interferometer included the Badal stage and the deformable mirror, which was fixed in the bias position for this and all subsequent measurements. The results fitted well to the function

where $C_2^0$ is the Zernike coefficient in µm.

Figure 5 shows the response of the wavefront sensor as a function of position in the pupil at each Badal stage setting, i.e., for different amounts of defocus. Measurements are plotted at the entrance pupil of the system, which has a radius of 3 mm. All other data is plotted in the re-imaged pupil coincident with the steering mirror of the pyramid wavefront sensor, which has a radius of 1.64 mm. In acquiring these data, the pupil sampling was set at the highest possible rate and care was taken to ensure no saturation of the pyramid output at any Badal setting.

 

Fig. 5. Values of S_x in the aperture, obtained experimentally at 3 V modulation (ξ0=1.281mm). Each curve corresponds to one Badal stage position and consequently to a certain amount of defocus given by Eq. (20). The stage position b = 0, 1, …, 27 mm.

Download Full Size | PDF

Since the amount of defocus is known from Eq. 20, it is relatively simple to calculate the phase derivative at each measurement point, and to re-arrange the gradient cross-sections into the graph of Fig. 6. In the process, the three pixels closest to each pupil edge were removed since they are influenced by the edge effect. This graph, which is very similar to Fig. 3, shows the experimentally obtained response function of the pyramid wavefront sensor along with the geometrically predicted response. It naturally shows a bit more variation than the numerically obtained prediction, but still confirms the earlier results. The small oscillations in the numerical values of 〈I ₁-I ₂〉 in Fig. 4 occur because they are not normalized by 〈I ₁+I ₂〉. In the experimental results in Fig. 6, proper normalization is used and no such oscillations are observed.

Although the intention is to introduce only defocus with the Badal stage, some small amounts of the other aberrations are also introduced. Tilt, in particular, has the effect of shifting the zero-crossing of the experimental gradient cross-sections to slightly below the S_x =0 line in Fig. 5. The same data for measurements taken at 1V modulation show a larger downward shift of the zero crossing, which is to be expected as at the lower modulation the pyramid wavefront sensor is more sensitive. We have added the corresponding amount of tilt to the interferometric values for ∂φ/∂x before producing Fig. 6. In addition, one variable in producing the graph of Fig. 6 is the zero position of the Badal stage, which is not necessarily at the position indicated by Eq. (20). As mentioned earlier, interferometric measurements were used to probe that part of the experiment that included the Badal stage and deformable mirror, and these results were used to produce Eq. (20). These measurements however, could not encompass the entire optical path from ocular pupil to wavefront-sensing camera. For example, if the pyramid itself is not located exactly at the Fourier plane, there can be some small extra defocus in the system. And indeed we find that a small (+1 mm) shift in the zero position of the Badal stage is required to obtain good agreement between experiment and theory in Fig. 6. Both experimental curves (1V and 3V) split into two branches, although the split is more pronounced for the higher modulation voltage. We have found that all curves starting above zero in Fig. 5 contribute to one branch in Fig. 6, while all starting below zero contribute to the second branch. We do not yet have a satisfactory explanation for why this happens. Despite these small differences, the experimental curves agree very well with the predictions from theory.

 

Fig. 6. Pyramid response S_x as a function of the incident-phase derivative, according to the geometrical prediction (solid lines) and experiment (dots). System parameters are the same as in Fig. 2, except that the modulation amplitude is ξ ₀=1.281mm (3V) and ξ ₀=0.427mm (1V). Values of defocus for the geometrical curves are obtained from Eq. (20).

Download Full Size | PDF

The focus of this paper has been on the linearity of the sensor response S_x to the wavefront derivative, but we also notice a second nonlinearity that appears close to the pupil edges. Both in diffraction theory (Fig. 2) and in experimental results (Fig. 5) the response drops at the edges of the aperture. This indicates that wavefront slopes measured at the edge of the pupil are generally underestimated. Since the sign is still correct for any significant amount of aberration, the effects in closed-loop operation should be small.

6. Conclusions

In this paper, we show that the same results can be obtained from the simple geometrical prediction of pyramid sensor linearity as from an analytical derivation from diffraction theory. To our knowledge, this extension of the theory to diffraction is done for the first time here. We also perform an experimental test of linearity using defocus as the input aberration. Considering the response of the wavefront sensor to a specified aberration, we find good agreement between the geometrical prediction, the numerically obtained diffraction prediction, and experimental results. We conclude that it is perfectly valid to use the simpler geometrical approximation rather than the more involved diffraction calculations when analyzing this kind of system, except possibly at the edges of the pupil.