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We consider a wavefront sensor combining scattering pupil with a plenoptic imager. Such a sensor utilizes the same reconstruction principle as the Hartmann-Shack sensor, however it is free from the ambiguity of the spot location caused by the periodic structure of the sensor matrix, and allows for wider range of measured aberrations. In our study, sensor with scattering pupil has demonstrated a good match between the introduced and reconstructed aberrations, both in the simulation and experiment. The concept is expected to be applicable to optical metrology of strongly distorted wavefronts, especially for measurements through dirty, distorted, or scattering windows and pupils, such as cataract eyes.
© 2014 Optical Society of America
1. Introduction
Wavefront (WF) sensing [1] represents an important branch of optical metrology. Methods of wavefront sensing can be classified to three large groups: direct phase measurements (interferometry), WF sensors based on the measurement of local WF tilts - shearing interferometers and Hartmann-Shack (HS) sensors, and sensors based on the sensing of the curvature of the WF, either by phase diversity principle, or based on the equation of irradiance transport [2].
Reliability, low sensitivity to noise and misalignments, general simplicity of implementation and broad range of data processing methods [3–6], gained the HS sensor the widest applications in the lab, industry and medicine. The popularity of the HS sensors has been further strengthened by the recent availability of affordable CMOS and CCD image sensors.
Sensing of large aberrations is also important because in many commercial sensors, the microlens array is mounted directly over the image sensor. Then, the physical size of the pupil is limited by the sensor size, which is usually of the order of several mm, while the real diameter of the aberrated pupil can vary from microns to meters. To match the pupils, an afocal (de)magnifying optical system is used. Since the WF curvature in the conjugated pupils scales as the square of magnification, and the spot density in the HS sensor is proportional to the WF curvature, small WF deviations within large apertures result in large changes of the pattern density in the conjugated HS sensor. However, the periodic structure of the microlens raster in the HS sensor, limits both the lateral resolution in the pupil and the maximum range of measurable WF tilts. Strong input aberrations can cause very strong distortions and mergers of the spot pattern, rendering impossible the correct WF reconstruction.
To find the applicability limits of a classic HS sensor, consider the simplest case of two adjacent lenslets with focal length f and pitch d, forming spot images in the focal plane as shown in Fig 1. The size of focal spots and the distance between the spots are influenced by the lenslet pitch d, the WF tilt φ and the radius of curvature of the WF R. To secure unambiguous spot indexing and WF reconstruction, the position of the focal spot should stay under the lenslet aperture. Then the maximum measurable WF tilt is limited to:
To find the minimum measurable radius of the WF curvature, we assume that two spots will merge and become undistinguishable, when the sum of spot shifts and cross sections becomes larger than the pitch d. Simple analysis leads to inequality:
where the terms under the root sign correspond to the geometrical and diffraction widening of the spot, and the third term describes the spot shift. Solution of this inequality is given by:Equation (3) defines the minimum WF curvature that can be unambiguously measured by a HS sensor. The root of the denominator defines the diffraction limits to the HS sensor pitch:
This inequality has a very simple physical meaning: the Fresnel number of the microlens raster should be greater than 2. Smaller Fresnel numbers result in merged spots even with flat input WF.
For the practically important case of , the Eq. (3) can be simplified:
The limits of lateral resolution and maximum measurable WF tilt and curvature in the classic HS scheme are explicitly defined by eq. (4), (1), and (5). Similar analysis, conducted in [5], resulted in a less explicit differential inequality, establishing the relation between the WF curvature and the local tilt. Figure 2 illustrates the degree of distortion observable in strongly aberrated HS patterns. All images are obtained with Zemax model of a classic HS sensor at the wavelength of λ = 633nm, with pupil diameter of 5 mm, sampling pitch of d = 0.5 mm and the focal length of f = 4.1 mm. The geometro-optical model is justified because the condition (4) is satisfied, and the diffraction effects can be neglected. The left image is obtained with undistorted plane wave and can be easily reconstructed. The central image corresponds to 50 waves of trefoil. The condition (1) is not satisfied and some degree of ambiguity is introduced in the definition of spot shifts. With some effort, a human would be able to establish unambiguous correspondence between the spots in the distorted pattern (central) and the initial image (left), however we do not know about any computer algorithm that would perform this task with zero ambiguity for an arbitrary aberration that does not satisfy to (1), but still satisfies to (5). The right image with strongly distorted pattern and merged spots was obtained with 100 waves of trefoil. It does not satisfy to both (1) and (5), and reconstruction of this pattern represents an extreme ill-posed problem. Thus, there is a great practical interest in the sensor configuration which is free from the limitations (1) and (5), and allows to correctly sample and measure strongly distorted wavefronts.
Fig. 2 Simulated (Zemax) undistorted spot pattern (left), patterns distorted by a 50 and 100 waves trefoil aberration (middle and right).
2. Principle of operation
Consider an optical system formed by the input pupil and the image sensor positioned at distance F behind the pupil, as shown in Fig. 3 (left). The pupil is illuminated by a light beam, which forms illuminated spot with intensity I(x, y) = I(r) in the image sensor plane. The WF tilt t in the pupil can be calculated from the position c of the center of gravity of the illuminated spot:
where ρ is the coordinate vector in the sensor plane and c0 is the pupil center of gravity.Fig. 3 Optical setup for measurement of the WF tilt in the pupil (left), plenoptic configuration that allows for measurement of local WF tilts in virtual subapertures (center) and scheme of pixel indexing in the image plane(right).
If the input WF is aberrated, the aberration can be reconstructed from the local tilts in the subapertures defined over the pupil. The local tilts are defined a
where tS is the local tilt corresponding to some arbitrary subaperture, IS is the intensity distribution caused by the subaperture, and cS0 is the center of gravity of the subaperture. In all cases the integration should be conducted over an area large enough to accommodate all the light passed through the pupil.Traditionally, the subapertures in the WF sensor are defined physically, by covering the pupil with a screen or a lenslet array, resulting in either the Hartmann, or Hartmann-Shack configuration. However, even with clear pupil, the subapertures can be defined virtually, by using imaging system formed by a combination of a microlens raster with an image sensor. This configuration is shown in Fig. 3 (center). Each microlens produces a pupil image, according to the thin lens equation:
where F is the distance from the pupil to the microlens raster, l is the distance from the microlens raster to the image sensor, and f is the focal length of the microlens raster. The condition should be also satisfied to secure proper separation between multiple pupil images.In the paraxial approximation, each lenslet builds an image of the pupil consisting of M × N pixels, each pixel indexed as m, n. The focal raster has dimensions of I × J lenslets, with each individual lenslet indexed as i, j, with coordinates (x, y)i,j. With this, a global pixel indexing i, j, m, n can be introduced. The indexing geometry is illustrated in Fig. 3 (right), where indices i, j address the particular lenslet and the indices m, n address the particular pixel under the chosen lenslet, which also corresponds to a certain position in the pupil.
Thus, the intensity distribution in the array focal plane U can be indexed with 2 pairs of indices as Ui,j;m,n. Here i, j corresponds to a certain lenslet. The pair m, n corresponds both to a certain position in the pupil (subaperture), and to the conjugated position in the pupil image under certain lenslet. By fixing the first index pair, one can obtain the intensity distribution under a certain lenslet, and by fixing the second index pair, one can obtain the intensity distribution formed by a particular subaperture with indices m′, n′: Ui,j;m′,n′. The center of gravity of the intensity distribution related to the virtual subaperture with indices m′, n′ can be calculated, similar to (7) as
The coordinate arrays (X, Y), corresponding to the local WF tilts in the virtual subapertures with indices m = 1...M, n = 1...N, can be used for WF reconstruction, employing one of existing methods of wavefront reconstruction from a two-dimensional array of local wavefront tilts [3–6].The existence and the measurability of the WF by definition means that in the geometric approximation there is only one single ray, passing through the WF surface in each point of the pupil. This means that summations in Eq. (10) contain only one non-zero term Ui′,j′;m′,n′. Then ∑i,j Ui,j;m′,n′ = Ui′,j′;m′,n′, Eq. (10) reduces to
and the coordinate of the center of gravity of any virtual subaperture can be defined with the precision that does not exceed the size of a single lenslet. Although, some rough WF reconstruction is possible even in this degenerate case, a much more precise solution can be obtained with a special pupil function, that for each point of the WF distributes the light over a number of lenslets, preserving the position of the center of gravity, corresponding to the initial ray.This can be achieved by introducing some form of pupil scattering. The scattering function should satisfy to the following requirements:
- the center of gravity of the intensity distribution, formed by any virtual subaperture, should be preserved after scattering;
- the scattering angle ϕ should be of the order of d/f
- the size of the plenoptic sensor S = I · d, formed by the microlens raster and the light sensor, should be large enough to catch all scattered light: I · d > D + ϕF for a simple scattering pupil, or I · d > ϕF for a combination of a scattering pupil with a collective lens.
The relative reconstruction error for a single wavefront slope measured over an area formed by IJ pixels is proportional to [7]:
where B is the bit depth of the image sensor, and IJ is the number of pixels used to calculate the slope. The rationale for this expression is very simple: regardless of the sensor physics, the average error in a single pixel is equal or greater than the half of the last bit value. Averaging over IJ pixels reduces the error by times. It follows from (12) that for precise slope calculation, we need either a very high bit depth of the readout, or averaging over a large number of illuminated pixels. Since the majority of industrial wavefront sensors is based on 8-bit cameras (B = 8) with relatively high readout errors, one needs to maximize the number of active pixels used for slope calculation to achieve a good slope precision. In the case of plenoptic geometry, this requirement is equivalent to maximizing of the number of illuminated lenslets in the focal plane.To show that the scattering does not change the position of the center of gravity for an arbitrary WF tilt introduced to the subaperture, consider an optical system formed by a pupil function U(r)exp(iφ(r)), where U is the amplitude transmission and φ is the phase delay, r = (x′, y′) is the coordinate, and z is the positions of the intensity sensor, relative to the pupil. Assuming the pupil is illuminated by a plane wave with constant amplitude A and zero phase, the field in the image plane, in the Fresnel approximation, can be expressed as
where ρ = (x, y), λ is the wavelength, k = 2π/λ, . Integral (13) can be expressed in terms of the Fourier transform ℱ with frequency coordinates corresponding to ρ/λz: It follows directly from the properties of Fourier transform, that the introduction of WF tilt t = (tx′, ty′) in the input pupil in the form of Aeikt·r, results in the lateral translation of the field intensity I in the image plane by zt: It(ρ, z) = I(ρ − zt, z), where I = |E|2. Although the intensity I depends on the pupil function U and the phase φ, the lateral translation of the intensity distribution depends only on the WF tilt and does not depend on the pupil function. This proves our initial suggestion that introduction of a scattering function in the pupil preserves the changes in local WF tilts, if the sensor is calibrated and used with the same constant pupil function.Expressions 10 were obtained in paraxial approximation. In practice the pupil images formed by off-axis lenslets experience geometrical distortion. Sensor calibration is required to establish the proper correspondence between the pupil and the pupil images. Such a calibration can be obtained with a strong scatterer in the pupil. The scattering angle should satisfy to , to obtain complete sharp pupil images for all lenslets.
To illuminate the maximum number of pixels and secure the widest possible dynamic range, the optical system of such a sensor should satisfy to the following conditions:
- The scattering angle introduced by the pupil function should be matched to the numerical aperture of the focal microlens raster, to guarantee that each microlens produces a quasi-uniformly illuminated image of the system pupil, and the images produced by adjacent microlenses do not overlap.
- The numerical aperture of the collective lens, if such a lens is used, should be matched to the numerical aperture of the microlens raster, to secure the optimal filling of the image plane and to maximize the range of measurable tilts.
Example of a WF sensor, based on a scattering pupil is shown in Fig. 4 (center). Even wider range of aberrations can be measured with a scatterer combined with a collective lens, as shown in the same figure (right).
Fig. 4 Plenoptic WF sensor described in [8] (left), WF sensor with scattering pupil (middle), same with a collective lens (right).
3. Simulation and experiment
We have built a ray-tracing model of a plenoptic WF sensor with scattering pupil, optimized for the measurement of extended range of aberrations. The model was based on a scheme shown in the centre of Fig. 3, with D = 5 mm, F = 100 mm, l = 9 mm, f = 8.31 mm, d = 0.5 mm, M = N = 50, the sensor size S = M · d = 25 mm, and λ = 633 nm. Pupil scattering was modeled with a gaussian scattering screen with scattering diagram width of σ = 0.01 rad. Figure 5 shows the intensity patterns corresponding to the flat WF in the input of the sensor, and the calibration pattern obtained with additional strong lambertian scattering screen positioned in the pupil.
Fig. 5 Simulated response of the plenoptic sensor for a flat WF (left) and calibration pattern obtained with a strong scattering screen in the pupil (right), with a small part of magnified calibration image shown in the inset.
A 50 D defocus and trefoil with PV amplitude of ∼ 200λ were reconstructed from the sensor data with a good agreement between the aberration input and the reconstruction, as shown in Fig. 6. The local WF tilts were calculated according to (10), and the WF was reconstructed in the form of decomposition over Zernike polynomials. The amplitude of the aberration of the order of ∼ 200λ is far out of range of the ordinary HS WF sensor, as illustrated in Fig. 2.
Fig. 6 Simulated intensity pattern (1000 × 1000 pixels) and the WF reconstruction over 18 × 18 subapertures for a 50 D defocus (top row) and trefoil with amplitude of 200λ (bottom row).
An experimental setup was also based on a scheme shown in the center of Fig. 3, with D = 2.5 mm, F = 41.6 mm, l = 10.45 mm, f = 8.31 mm, d = 0.5 mm, M = N = 10, and λ = 633 nm. The aberration represented by a test object, was placed in a collimated beam of a He-Ne laser, immediately in front of the diffuser DF, represented by an orthogonal microlens array with 150 μm pitch and 3.5 mm focal length. The intensity pattern was registered with UI-2210M camera from IDS Imaging, with 1/2” CCD having square 10μm pixels.
Aberrations of tilt, defocus and astigmatism have been simultaneously introduced in the pupil with a tilted and decentered plano-convex lens with optical power of 10 D. Figures 7(a) and 7(b) show the experimental intensity distributions, registered for unaberrated and aberrated WF. Intensity modulation was observed in the raw images due to the regular structure of microlenses in the diffusor. During the processing, the modulation was removed using Gaussian blur filter of 8 pixels width, which is acceptable for low-order reconstruction. The local WF tilts were calculated according to expressions (10), and the WF was reconstructed in the form of decomposition over Zernike polynomials. The reconstructed WF is shown in Fig. 7(c).
Fig. 7 Experimentally registered intensity distributions for unaberrated WF (a), for decentered plano-convex lens with optical power of 10 D introduced in the pupil (b), and reconstructed aberrated WF (c).
The defocus term, reconstructed from the experiment, has an amplitude of ∼7.8 μm, in a good agreement with the theoretically expected value of 7.8125 μm. The dynamic range of the sensor was limited by the sensitive area of the available CCD, a larger sensor would allow to measure aberrations with larger dynamic range.
We obtained a good match between the introduced and reconstructed aberration, proving the feasibility of the proposed approach, although the sensitivity, precision and the resolution of the sensor are subject to further optimization.
4. Discussion
The proposed sensor, in similar conditions, is able to measure a much wider range of local tilts, compared to the traditional HS sensor. It is easy to see that since all virtual subapertures are projected onto the same sensor area, the maximum range of measurable local tilts is defined by the ratio of sensor size S, to the focal length F:
Since S is not limited by the pupil diameter D, or the subaperture size d, the measurable range of local tilts can be wider than that of the classic HS sensor. In practice the range is limited by the sensor size and the geometrical distortions of the intensity pattern at the sensor edges.
Also the limitation (5) on the local curvature is relaxed considerably: all subapertures are projected onto the same sensor area and the problem of spot overlapping and unambiguous indexing does not exist. As with the tilt, the minimum measurable curvature is limited by the sensor size S.
The proposed concept should not be confused with a WF sensor working with extended source. Strictly speaking, WF measurement in a system forming an image of an extended object represent a severely underdetermined problem, because the number of unique wavefronts, travelling through the system, is as large as the number of resolved points in the object image. In the contrary, scattering pupil preserves the uniqueness of the wavefront, as it allows to reconstruct the original ray directions with diffraction limited precision. The scattering in the pupil is employed to spread the light over a larger number of pixels, with the goal to improve the precision of the measurement of local WF tilts.
In conclusion, the plenoptic WF sensor with scattering pupil allows for optimal combination of pupil resolution and slope definition, as it is free from the limitations on the range of spot movements, typical for the classic HS sensor. Extended range makes it a good candidate for an optical shop instrument, especially for the measurement of large aberrations. It also can be (after proper calibration) used in practical situations, where strongly distorted WF should be characterized through a static scattering or distorting pupil, that does not allow for any direct placement of sampling optics. One of perspective applications would be the WF sensing in aberrated and strongly scattering cataract human eyes.
Acknowledgments
Authors are grateful to Dr. Vsevolod (Seva) Patlan for fruitful discussions.
References and links
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