## A phase space model of Fourier ptychographic microscopy |

Optics Express, Vol. 22, Issue 1, pp. 338-358 (2014)

http://dx.doi.org/10.1364/OE.22.000338

Acrobat PDF (7943 KB)

### Abstract

A new computational imaging technique, termed Fourier ptychographic microscopy (FPM), uses a sequence of low-resolution images captured under varied illumination to iteratively converge upon a high-resolution complex sample estimate. Here, we propose a mathematical model of FPM that explicitly connects its operation to conventional ptychography, a common procedure applied to electron and X-ray diffractive imaging. Our mathematical framework demonstrates that under ideal illumination conditions, conventional ptychography and FPM both produce datasets that are mathematically linked by a linear transformation. We hope this finding encourages the future cross-pollination of ideas between two otherwise unconnected experimental imaging procedures. In addition, the coherence state of the illumination source used by each imaging platform is critical to successful operation, yet currently not well understood. We apply our mathematical framework to demonstrate that partial coherence uniquely alters both conventional ptychography’s and FPM’s captured data, but up to a certain threshold can still lead to accurate resolution-enhanced imaging through appropriate computational post-processing. We verify this theoretical finding through simulation and experiment.

© 2014 Optical Society of America

## 1. Introduction

1. P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature **374**, 630–632 (1995). [CrossRef]

5. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature **467**, 437–439 (2010). [CrossRef]

6. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. **35**(15), 2585–2587 (2010). [CrossRef] [PubMed]

7. A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. **28**(4), 604–612 (2011). [CrossRef]

8. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. **7**, 739–745 (2013). [CrossRef]

9. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A **339**, 521–553 (1992). [CrossRef]

11. J. N. Clark, X. Huang, R. Harder, and I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. **3**, 993 (2012). [CrossRef] [PubMed]

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature **494**, 68–71 (2013). [CrossRef] [PubMed]

## 2. Mathematically connecting conventional and Fourier ptychography

### 2.1. The conventional ptychography (CP) setup

9. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A **339**, 521–553 (1992). [CrossRef]

10. H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy **66**, 153 (1996). [CrossRef]

*a*(

*r′*) is

*ã*(

*r*)). Note that here both

*r*and

*r′*will have units of meters, since they represent the spatial axis of an imaging system’s two Fourier conjugate planes. A schematic diagram of a scanning CP setup containing two sets of such planes is in Fig. 1. While deviations exist, most recent ptychographic experiments generally follow Fig. 1’s optical outline. The following analysis considers a two-dimensional imaging geometry, for simplicity. Extension to three dimensions is direct.

*I*(

*r*) onto a shifting sample and records a series of far-field diffraction patterns. We assume

*I*(

*r*) contains an ideal point light source that produces a quasi-monochromatic plane wave (wavelength

*λ*) propagating parallel to the optical axis at a large distance

*ℓ*. The case of a non-ideal point source will be considered in Section 4. At distance

*ℓ*is an aperture plane

*A*(

*r′*) containing a lens of focal length

*f*. Directly past this plane, the optical field may be described across all space simply as

*a*(

*r′*), the aperture transmission function.

*ã*(

*r*) then interacts with a shifted sample

*ψ*to produce an exiting optical field,

*S*(

*r*). We assume the effect of sample thickness upon diffraction is negligible, allowing us to define the optical field

*S*(

*r*) directly past the sample as a multiplication of

*ã*(

*r*) and the sample transmission function

*ψ*: Here,

*x*is the sample’s shift distance perpendicular to the optical axis. The thin object approximation holds if the maximum sample thickness

*t*obeys

*δ*is the sampling resolution [14

_{res}14. K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. **59**(1), 1–99 (2010). [CrossRef]

*S*(

*r*) then propagates a large distance

*d*to far-field detector plane

*D*, where (as a first approximation) the intensity of the Fourier transform of

*S*is measured: Here,

*m*(

*x*,

*r′*) is a two dimensional function of probe shift distance (

*x*) and space (

*r′*), and comprises our data matrix. In experiment,

*m*(

*x*,

*r′*) is filled up, column-by-column, with discretized diffraction images captured at the detector for many shift distances

*x*(see example in Fig. 2). For two-dimensional images,

*m*(

*x*,

*r′*) is a four-dimensional function.

### 2.2. Phase space representation of CP

17. R. Horstmeyer, S. B. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express **18**(21), 22545–22555 (2010). [CrossRef] [PubMed]

*r*into a new space. Unlike the Fourier transform, which offers a one-to-one mapping between the primal variable

*r*and its conjugate

*u*(here a mapping between space and spatial frequency), this new space is two-dimensional. The WDF is a joint function of both the primal spatial variable

*r*and the conjugate spatial frequency variable

*u*. Although defined in a higher-dimensional space,

*W*maintains a one-to-one relationship with the complex function

_{ψ}*ψ*(apart from a constant phase shift). While not always exact, it is convenient to connect the value of

*W*(

*r*

_{0},

*u*

_{0}) to the amount of optical power at point

*r*

_{0}propagating in direction

*u*

_{0}. However, while the WDF is real-valued it is not necessarily non-negative, which requires this interpretation to be taken loosely.

### 2.3. Mathematical representation of Fourier ptychographic microscopy (FPM)

*m*) but does so using the unique optical setup in Fig. 3. Two primary experimental differences set FPM apart from the CP setup outlined above: an array of

_{F}*n*LEDs now occupy the illumination plane

*I*(

*r*), and the locations of the sample and aperture planes are effectively switched. Instead of recording the diffraction pattern from a small illuminated sample region, FPM images the entire sample under illumination from different directions.

*I*(

*r*) emits a quasi-monochromatic and spatially coherent field at wavelength

*λ*(partially coherent illumination is included in Section 4). Each LED sequentially illuminates the entire sample plane

*S*(

*r′*) a distance

*ℓ*away with an angled plane wave. Next, the illuminated sample is imaged by a lens of focal length

*f*located at aperture plane

*A*(

*r*). In practice, the employed lens is a microscope objective (MO), although in principle, any style of image-forming optic will result in a similar analysis. At detector plane

*D*(

*r′*), a pixel array samples the image intensity at spacing

*δ*=

_{x}*λw*/2

*f*(to avoid aliasing issues). From Eq. (1), we note that the optical field at

*A*(

*r*) is proportional to the Fourier transform of the field both at the sample plane

*S*(

*r′*) and the image plane

*D*(

*r′*), a feature that distinguishes FPM from CP and lends to its name.

*S*(

*r′*) directly past the sample plane may be written as a multiplication between the incident field and the sample transmission function

*ψ*as, Here,

*x*represents the sine of the angle at which the plane wave generated by the

*i*LED, located a distance

^{th}*h*away from the optical axis, travels:

_{i}*ℓ*the distance between the LED array and the sample. As with CP,

*x*is again connected to an illumination shift distance. Since we here define this shift distance at the illumination plane instead of the sample plane,

*x*now becomes a variable modifying the sample’s spatial frequency. The optical field

*S*(

*r′*) continues to propagate to aperture plane

*A*(

*r*), mathematically represented through the scaled Fourier transform in Eq. (1). The field is attenuated at

*A*(

*r*) by the aperture transmittance function

*a*(

*r*) (i.e., the shape of the MO pupil plane), creating the optical field, Again, we’ve neglected coordinate scaling factors for clarity (see Appendix A). Finally, this attenuated field propagates to image plane

*D*(

*r′*), represented through a scaled Fourier transform. At

*D*(

*r′*), the digital pixel array detects the field’s intensity

*m*(

_{F}*x*,

*r′*):

## 3. Visualizing connections between both ptychographic domains

**Scaling along the optical axis:**For both ptychographic procedures, distances between the optical source, sample, detector, and the lens focal length will lead to constant scaling variations along*r′*and*x*in their respective data matrices. Details of these scaling relationships are presented in Appendix A.**Sampling along***r′*: The digital detector’s sampling conditions for CP and FPM both manifest themselves along their corresponding data matrices’*r′*axis (Fig. 5, green text). For CP, the detector width must match the aperture’s maximum transmitted spatial frequency. This width defines the resolution limit of a final reconstructed image. The detector size and distance together define a geometric NA, which much match the detector pixel size to avoid aliasing [10]. For FPM, sampling along the10. H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy

**66**, 153 (1996). [CrossRef]*r′*axis follows a typical imaging setup - the detector width is paired to the imaging lens FOV, and the detector pixel size matches the imaging optics’ point-spread function (PSF) width to avoid aliasing.**Scanning along***x*: Sampling along the data matrix*x*-dimension is tied to the operation of each setup’s illumination (Fig. 5, blue text). In CP, the probe beam’s total scanning distance sets the maximum extent along*x*, which also defines the final reconstructed image’s FOV. In FPM, however, the maximum extent along*x*is set by the maximum LED-sample illumination angle. This in turn defines the final reconstructed image’s maximum resolution, as opposed to FOV. This outstanding feature of FPM allows for the extension of a lens’s typical resolution cutoff by simply illuminating the sample from large off-axis angles. Experimental uncertainty in*x*-scanning is also an important consideration. The limited accuracy of CP’s mechanical stage, caused by inter-experimental variations in movement, restricts CP resolution to approximately 1*μm*in optical arrangements [6]. The unknown angular position of FPM’s LEDs may likewise impact experimental precision, but not accuracy. Unlike CP, a single pre-calibration procedure can estimate any deviations from known LED array parameters, which can help correct precision errors in all future measurements of the same fixed FPM setup. This type of pre-calibration may also be used to remove the effect of aberrations induced by FPM’s imaging lens, which become especially prominent at high illumination angles [236. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett.

**35**(15), 2585–2587 (2010). [CrossRef] [PubMed]].23. G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express

**21**(13), 15131–15143 (2013). [CrossRef] [PubMed]The sampling rate along the*x*-dimension of both data matrices is set by the number of captured images. Our above model assumes the WDF is ideally discretized, requiring the number of detector pixels along*r′*to match the number of collected images along*x*. In practice, accurate high-resolution sample reconstruction does not require full population of*m*(*x*,*r′*) or*m*(_{F}*x*,*r′*) along*x*[21]. Under-sampling along21. M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy

**108**, 481–487 (2008). [CrossRef]*x*remains an effective strategy because the WDF is a redundant 2D representation of a complex 1D signal. Phase retrieval algorithms, such as those used in [1–81. P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature

**374**, 630–632 (1995). [CrossRef]], exploit this redundancy to faithfully reconstruct sample and probe functions from under-sampled data matrices, as also explored in [228. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon.

**7**, 739–745 (2013). [CrossRef]]. Strictly speaking, such under-sampling along22. C. Teale, D. Adams, M. Murnane, H. Kapteyn, and D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express

**21**(6), 6783–6793 (2012). [CrossRef]*x*invalidates Eq. (12)’s exact equality. However, the high-resolution solutions for samples, probes and apertures demonstrated in [1**374**, 630–632 (1995). [CrossRef]**7**, 739–745 (2013). [CrossRef]**Data matrix blur kernel:**CP’s finite probe width causes blurring between images, and the finite extent of its aperture will typically define the maximum spatial frequency cutoff for each image. These limiting effects respectfully manifest themselves along the*r*and*u*dimensions of CP’s aperture WDF,*W*(_{ã}*r*,*u*), shown in the bottom of Fig. 5. Convolution with*W*(_{ã}*r*,*u*) in Eq. (5) describes how sample information is blurred during the detection process. Since it is zero beyond a certain cutoff value along*u*,*W*(_{ã}*r*,*u*) removes from the data matrix any sample information above this associated spatial frequency range. FPM’s rotated blur kernel*W*(_{ã}*u*,*r*) is defined by its imaging aperture. It also blurs and cutoffs sample information from the data matrix in a similar manner as CP’s blur kernel, and may additionally contain the effects of optical aberrations from the imaging lens, as previously noted.

*w*= 37.5 mm, focal length

*f*= 105 mm) creates a sinc of estimated width 18

*μ*m (peak-to-zero) at the sample from an LED located

*ℓ*= 300 mm away. The sample plane contains a grating with 4

*μ*m minimum feature size that is shifted in 4

*μ*m steps. In Fig. 2 and Fig. 4, the simulated grating is 0.4 mm wide, while in Section 5 it is 1.33 mm wide. We assume a 4 mm-wide detector containing 4

*μ*m pixels with full factor captures its diffraction pattern, which approximately requires

*d*= 30 mm, assuming free space propagation. For FPM, we assume a similar lens (with parameters

*w*= 37.5 mm,

*d*= 300 mm and

_{o}*d*= 105 mm) images the sample onto an identical detector. FPM’s LED array is fixed at a distance

_{i}*l*= 100 mm and illuminates the same sample. The array extends across a total distance

*h*= 24 mm perpendicular to the optical axis, yielding a 240

*μ*m pitch for Fig. 2 and Fig. 4. One important parameter still missing from the above analysis is the light’s coherence state, connected to the active area of each optical source. We will now extend our phase space model to account for this critical effect.

## 4. A complete statistical model with partially coherent light

11. J. N. Clark, X. Huang, R. Harder, and I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. **3**, 993 (2012). [CrossRef] [PubMed]

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature **494**, 68–71 (2013). [CrossRef] [PubMed]

### 4.1. Partially coherent source description

*U*(

*r*,

*t*) as a temporally stationary stochastic process and examining its correlation across space and time: 〈

*U*(

*r*

_{1},

*t*

_{1})

*U*(

^{*}*r*

_{2},

*t*

_{2})〉 = Γ̃(

*r*

_{1},

*r*

_{2},

*τ*). Here, Γ̃ is the light’s mutual coherence,

*τ*=

*t*

_{2}−

*t*

_{1}is a constant time difference, and the expectation value is performed over time. From the Weiner-Khinchine theorem, the cross-spectral density (CSD) of this stochastic process is defined as Γ(

*r*

_{1},

*r*

_{2},

*ω*) = ∫ Γ̃(

*r*

_{1},

*r*

_{2},

*τ*)

*e*

^{−jωτ}

*dτ*. The spectral density

*C*(

*r*,

*ω*) = Γ(

*r*,

*r*,

*ω*) represents the intensity of light at location

*r*at a certain frequency

*ω*. We will assume our illumination sources are fully spatially incoherent within their photon-generating area, leading to a CSD function at source plane

*I*, where

*C*represents the geometric shape of the source intensity for each frequency

*ω*(typically a circ-function in two dimensions),

*γ*is its spatial coherence cross section and

*δ*is a Dirac delta function. For the remainder of this section, we will drop spectral dependance on

*ω*for simplicity, assuming a notch filter is used in experiment to effectively isolate a narrow spectrum from the source. Although not detailed here, effects of a spectrally broad (i.e., temporally incoherent) source are an important consideration and may be included through incoherent superposition of the following equations. The Van Cittert-Zernike theorem relates Eq. (13)’s CSD of the source Γ

*in to the CSD a distance*

_{I}*z*away, Γ

*: where a constant multiplier is neglected for simplicity, Δ*

_{z}*r′*=

*r′*

_{1}−

*r′*

_{2}and

*C*, and the CSD function Γ

*at any subsequent plane a large distance*

_{z}*z*from this source.

### 4.2. CP with partially coherent light

*A*(

*r′*). Here, the light’s CSD function Γ

*(*

_{ℓ}*r′*

_{1}−

*r′*

_{2}) is given by Eq. (14), with

*z*=

*ℓ*. The aperture

*a*(

*r′*) then modulates Γ

*(*

_{ℓ}*r′*

_{1}−

*r′*

_{2}) before the light is focused by the lens to the sample plane, mathematically expressed by applying a Fourier transform kernel to each spatial coordinate

*r′*

_{1}and

*r′*

_{2}. Multiplying Γ

*(*

_{ℓ}*r′*

_{1}−

*r′*

_{2}) in Eq. (14) with aperture function

*a*and Fourier transforming the result leads to an input-output (i.e., source-to-sample plane) CSD relationship defined by a convolution [24

24. D. Brady, *Optical Imaging and Spectroscopy* (John Wiley & Sons, 2009). [CrossRef]

*S*and we have used the coordinate variable replacement

*p*=

*r′*for notational clarity. We have omitted a constant scaling of

*p*by 1

*/λℓ*and

*r*

_{1}and

*r*

_{2}by 1

*/λf*, for simplicity. With Eq. (15), we now have a full statistical description of CP’s focused probe beam illuminating the sample. Our previous representation of the focused probe beam as a fully coherent field, simply described by

*ã*(

*r*), is no longer valid now that the source has finite spatial extent. We can update our original expression for the intensity at the detector

*m*(

*x*,

*r′*) in Eq. (4) to reflect our new partially coherent probe beam with a simple replacement. Instead of multiplying the sample

*ψ*with coherent probe wave

*ã*, we multiply

*ψ*with the probe wave CSD in Eq. (15): Plugging Eq. (15) into Eq. (16) and performing several straightforward manipulations (outlined in Appendix C) produces the following mathematical description of the CP data matrix

*m*(

*r′*,

*x*) in terms of the aperture’s WDF, the sample’s WDF, and the illumination source’s geometric shape

*C*: Partially coherent light alters CP’s data matrix with an additional convolution along the scan variable

*x*(Fig. 6(a)). The goal of ptychographic data post-processing under partially coherent illumination is to recover a complex description of the sample

*W*from data matrix

_{ψ}*m*(

*x*,

*r′*) by deconvolving the effects of both

*W*and

_{ã}*C*. This is identical to the coherent case, but with an additional (yet still separable) blurring term.

### 4.3. FPM with partially coherent light

*array*of spatially offset and partially coherent LEDs at its illumination plane. Using

*x*to represent the distance from a given LED to the optical axis, the CSD of one LED may be expressed by modifying Eq. (13) to incorporate a spatial offset by

*x*: Γ

*(*

_{I}*r*

_{1},

*r*

_{2}) =

*γ*

^{2}

*C*(

*r*

_{1}−

*x*)

*δ*(

*r*

_{1}−

*r*

_{2}). This LED’s shifted source light first illuminates the sample at plane

*S*(

*r′*). Again neglecting its quadratic phase and constant scaling terms for simplicity, Eq. (14) can propagate Γ

*(*

_{I}*r*

_{1},

*r*

_{2}) to the sample plane

*S*(

*r′*) to express the CSD at the sample, Γ

*: where (*

_{S}*ρ*

_{1},

*ρ*

_{2}) have replaced (

*r′*

_{1},

*r′*

_{2}) as the sample’s spatial coordinates at

*S*(

*r′*), for notational clarity. This illumination light is then modulated (i.e., multiplied) by the sample transmission function

*ψ*and subsequently imaged onto the detector plane. As in the previous subsection, the transformation of the CSD from the sample to the detector plane is given by a convolution of each spatial variable

*ρ*

_{1}and

*ρ*

_{2}with a coherent impulse response [24

24. D. Brady, *Optical Imaging and Spectroscopy* (John Wiley & Sons, 2009). [CrossRef]

*ã*: Γ

*(*

_{D}*r′*

_{1},

*r′*

_{2}) is the CSD of partially coherent light at the detector. The imaging system’s coherent impulse response

*ã*is typically a scaled sinc function. The measured intensity at the detector is given by evaluating Γ

*at one spatial location*

_{D}*r′*=

*r′*

_{1}=

*r′*

_{2}. This allows us to express FPM’s measured data as

*m*(

_{F}*x*,

*r′*) = Γ

*(*

_{D}*r′*,

*r′*), where

*m*(

_{F}*x*,

*r′*) is the same data matrix from Section 3. By substituting Eq. (18) into Eq. (19) and setting

*r′*

_{1}=

*r′*

_{2}=

*r′*we obtain the following expression for the recorded image intensity

*m*(

_{F}*x*,

*r′*) as a function of LED offset

*x*and detector position

*r′*: Equation (20) resembles our coherent FPM data matrix expression in Eq. (10), but now with an additional

*C̃*term accounting for partial coherence effects. As detailed in Appendix D, Eq. (20) may be rearranged into a final expression in terms of the aperture WDF, sample WDF, and LED source geometry: Comparing Eq. (21) to Eq. (11)’s coherent description of FPM, we see that partial coherence manifests itself as an additional convolution along the data matrix

*x*-dimension (Fig. 6(b)). Practically, this indicates each FPM image, captured from a different LED and compiled along

*x*, will begin to look increasingly similar with increasingly incoherent illumination. In the limit of a completely incoherent source, spatial shifting will leave all image features nearly unchanged. Since this blur remains a separable function, it is still possible to deconvolve the effects of both

*C*and

*W*to obtain an accurate sample estimate

_{a}*W*. Comparing Eq. (21) to Eq. (17)’s expression for partially coherent CP, we find a new primary difference between the two setups: while partial coherence alters both data matrices along the

_{ψ}*x*dimension (the scan variable), it changes the underlying structure of each data matrix differently, since each is rotated by 90° with respect to the other. Put simply, using a partially coherent source in a CP setup blurs together the sample’s spatial information within its recorded data matrix. In FPM, using an array of partially coherent sources blurs the sample’s spatial frequency content, as Fig. 6 clearly depicts.

## 5. Case study: CP and FPM under partially coherent illumination

*C*(

*p*) [12

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature **494**, 68–71 (2013). [CrossRef] [PubMed]

*C*(

*p*), offering a more robust procedure when an estimate of the illumination source’s shape is available. Third, our experiment tracks the slow degradation of phase imaging performance as a function of decreasing source coherence. To the best of our knowledge, it is still not currently well-understood why phase acquisition is possible yet noisy with low-coherence illumination, and our findings may generalize to benefit this area of investigation.

*m*and

*m*match, after a rotation. Second, the listed parameters require both setups to use the same lens numerical aperture, detector pixel size and count, and nearly the same total optical path length, offering as even a comparison as possible. Third, the parameters correspond closely with previous optical CP [6

_{F}**35**(15), 2585–2587 (2010). [CrossRef] [PubMed]

7. A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. **28**(4), 604–612 (2011). [CrossRef]

**7**, 739–745 (2013). [CrossRef]

*x*by a similar factor.

### 5.1. Simulation

^{3}pixels each, which combine to form each data matrix. Note that all figures display the central 350-pixel area of each captured image to aid in visualization. As in Fig. 2 and Fig. 4, our sample here is a chirped grating with minimum feature size of 4

*μm*. Unlike previous simulations, the grating is now 1.33 mm-wide and is of a slightly different structure to match our experimental sample (see Fig. 7(d)). We first apply a Fresnel-based propagation simulation to create this grating’s CP and FPM data matrices under partially coherent illumination, as in Fig. 7(a)–(b). We then numerically compute Eq. (17) and Eq. (21) using the same grating function

*ψ*(including all relevant scaling factors in Appendix A). In doing so, we find agreement up to an average error of

*<*1% caused by numerical approximation, which verifies our phase space formulation.

*C*will allow both setups to maintain high-resolution imaging performance using larger, brighter optical sources (i.e., with higher photon throughput). As a standard benchmark, we apply the well-known Wiener filter in our deconvolution attempt. Previously used to recover complex sample data in [9

9. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A **339**, 521–553 (1992). [CrossRef]

**66**, 153 (1996). [CrossRef]

**494**, 68–71 (2013). [CrossRef] [PubMed]

18. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

19. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1562 (2009). [CrossRef] [PubMed]

*μm*-diameter active area (0.11° angular extent). The associated Wiener deconvolution outputs are shown directly to the right. Gaussian noise (normalized variance of 10

^{−3}) was added to the data before deconvolution. Noise variance and source size were assumed as prior knowledge to assure optimal filter performance. Figure 7(c) plots the average root-mean-squared error (RMSE) of recovered data matrices as a function of source diameter after Wiener deconvolution. Each point in this plot is an average over 10 experiments with noise variances ranging evenly from 10

^{−2}to 10

^{−4}. The linear process of recovering a sample estimate from its coherent data matrix ensures sample reconstruction RMSE will follow a similar curve. CP and FPM setups that do not create a fully sampled data matrix (i.e., that under-sample along

*x*) still benefit from a similar deconvolution approach. While beyond the scope of this work, we have successfully applied a blind deconvolution algorithm to under-sampled CP and FPM data matrices to achieve nearly equivalent coherence removal performance.

### 5.2. Experiment

*I*to facilitate easy variation of LED coherence area. This variation was achieved by placing pinholes of different diameter (100

*μ*m–1000

*μ*m) directly in front of the active area of a 532 nm central-wavelength diode. Note that while sufficient for the current experiment, a mechanical stage setup offers resolutions that are generally inferior to LED array-based FPM, since mechanical motion introduces the same inaccuracies limiting CP’s achievable resolution. Second, an

*f*= 50 mm,

*w*= 50 mm collection lens was inserted 50 mm in front of the LED source to assure uniform illumination of the sample. We experimentally determined this lens has minimal effect on the coherence area at the detector plane. Our imaging setup used a

*f*= 105 mm,

*w*= 37.5 mm compound lens (Nikon Micro-Nikkor f/2.8G) positioned

*d*= 300 mm from the sample that imaged onto a 4.54

_{o}*μm*pixel CMOS array (Prosilica-GX 1920).

*μ*m steps across 25 mm, for a total of 100 samples along

*x*. This sampling rate is approximately 4–5 times higher than prior demonstrations of FPM [8

**7**, 739–745 (2013). [CrossRef]

26. X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. **38**(2), 4845–4848 (2013). [CrossRef] [PubMed]

*x*, we capture an image of the linear grating and select a single row of the CMOS detector array to form data matrix column

*x*. Each image’s maximum pixel value is scaled to 1 (i.e., each data matrix column in Fig. 8 is normalized to it’s maximum value), which enhances the appearance of noise in low-intensity areas but aids with visualization of coherence effects. The wiggling effect observable within the experimental data matrix (i.e., shifting of the grating image as a function of illumination angle) has two primary causes. First, shifting at the image plane may occur for samples not in ideal focus, which our detector’s slight undersampling prevents an exact verification of. Second, the grating’s finite thickness (3 mm) does not accurately match the thin object approximation from Section 2, leading to an unaccounted for phase modification that manifests itself as this irregular artifact.

*m*(

_{F}*x*,

*r′*) broadens along the

*x*-dimension when using a larger-diameter source, as the convolution relationship in Eq. (21) predicts. Conceptually, an increasingly incoherent source will extend the lens’s coherent spatial frequency cutoff at

*k·*NA to its incoherent spatial frequency cutoff at 2

*k·*NA, hence broadening what is captured along

*x*. This slight improvement in spatial resolution is also present (although difficult to discern) within each individual image along the

*r′*-dimension. Second, Eq. (21)’s convolution also predicts features along

*x*to blur with increased incoherence, which is clearly observed at the edge of the diffraction cone. As just noted, this blurring does not impact the spatial resolution of each image, but instead causes images captured by adjacent LEDs to become increasingly similar, and thus harder to accurately extract sample phase from. Finally, incoherent illumination still allows the FPM setup to acquire high-frequency sample information that otherwise would not be captured by a conventional imaging setup. This is indicated by the dark “tails” at the bottom of each data matrix, which represent high-frequency grating information that is diffracted into the imaging lens from an off-axis LED, otherwise cutoff from a single image. The density of this high-frequency information tail decreases with increasingly incoherent illumination. However, it is still clearly present with a low-coherence source, thus allowing computational improvement of a reconstructed image’s resolution beyond the conventional imaging lens NA cutoff. This information-preserving feature of ptychography in the presence of incoherent light is a very powerful tool that has yet to be studied in full, and is the main conclusion of this experiment.

## 6. Conclusion and future work

*C*(

*p*), will also improve removal of partial coherence effects. Second, Eq. (12) suggests that alternative optical setups can capture the data matrix

*m*under different linear transformations (e.g., a matrix rotation that is not 90°, or another isomorphic transform besides rotation). These alternatives to CP and FPM will most likely offer application-specific advantages. For example, one could imagine both shifting the sample across a limited range and using a small number of illumination sources to increase collection efficiency. This specific joint CP-FPM setup may benefit applications only tolerating minimal movement, but many other hybrid designs may be easily imagined to fulfill niche design constraints. Finally, we minimally considered the computational post-processing aspect of ptychography in our analysis. As recently demonstrated, phase space offers a rich array of image reconstruction tools [22

22. C. Teale, D. Adams, M. Murnane, H. Kapteyn, and D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express **21**(6), 6783–6793 (2012). [CrossRef]

19. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1562 (2009). [CrossRef] [PubMed]

20. A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy **120**, 64–72 (2012). [CrossRef] [PubMed]

**494**, 68–71 (2013). [CrossRef] [PubMed]

## Appendix A: Phase space expressions with scaling factors included

*λf*scaling factor [13], where

*λ*is wavelength and

*f*the lens focal length. Second, propagation from the sample to the detector includes a similar scaling factor by

*λd*, with

*d*the detector distance. A scaled version of Eq. (3) is thus, where the subscripts indicate the original and transformed coordinates used within the Fourier transform exponent. This can be rewritten in integral form as, From here, a scaled Wigner convolution relationship is found as, where the

*ψ*subscript indicates the coordinate system of

_{λ}_{f}*W*is multiplicatively scaled by a constant

_{ψ}*λf*factor. Pre-integral multiplicative constants are omitted for clarity. Equation (24) includes three primary effects of scaling. First is the

*λf*scaling factor along

*W*’s spatial variable

_{ψ}*r*, which also necessarily requires the phase space function’s spatial frequency variable

*u*to be contracted by the same proportion before computing the convolution. Second, the resulting data matrix’s

*r′*coordinate is scaled by a

*λf/d*factor, and third its

*x*coordinate by 1

*/λf*.

*d*is the distance from the sample to the lens and

_{o}*d*is the distance from the lens to the detector (Fig. 3). Straightforward manipulations following Appendix B’s steps lead to, where

_{i}*ψ*

_{λdo}here indicates

*W*is fully scaled by a constant factor 1

_{ψ}*/λd*. Again, three main differences are apparent comparing the above to the FPM convolution expression in Eq. (11):

_{o}*r′*is scaled by

*λd*,

_{o}/d_{i}*x*is scaled by

*d*, and

_{o}*W*’s joint coordinates are scaled by 1

_{ψ}*/λd*before convolution. Similar manipulations yield scaling factors for data matrices containing the effects of partially coherent illumination.

_{o}## Appendix B: The Wigner representation of the FPM data matrix

*ψ*and

*ã*, for direct comparison with Eq. (5). This can be achieved by taking advantage of the Wigner distribution function (WDF). As noted in Section 2, the WDF is a convenient tool to achieve variable separation. The WDFs describing

*ψ*and

*a*are obtained by first transforming (

*r*

_{1},

*r*

_{2}) to center-difference coordinates (

*r*,

*y*), using

*r*

_{1}=

*r*+

*y/*2 and

*r*

_{2}=

*r*−

*y/*2: where in the exponent we use the fact that

*r*

_{1}−

*r*

_{2}=

*y*. Following the definition of the WDF in Eq. (6), we can define the WDF of

*the Fourier transform*of our sample function

*ψ̃*as, Applying an inverse Fourier transform to both sides of Eq. (28) yields, The Wigner distribution of the aperture function

*a*(

*r*),

*W*(

_{a}*r*,

*u*), will take a similar form as Eq. (28). As we will see next, it is more useful to express the WDF of the aperture with a shifted spatial frequency term,

*W*(

_{a}*r*,

*r′*−

*u*): The inverse Fourier transform of Eq. (30) yields, Inserting Eq. (29) and Eq. (31) into Eq. (27) and noting all terms in the exponent cancel produces a near-final FPM data matrix expression: where the pre-integral multiplier is omitted for clarity. To fully connect FPM’s data matrix with CP’s in Eq. (5), we can take advantage of a convenient property of the Wigner distribution. As a function of both space and spatial frequency, it is clear that

*W*(

_{ψ}*r*,

*u*) must contain the same information as when it is applied to the sample’s Fourier transform,

*W*(

_{ψ̃}*r*,

*u*). The two Wigner functions are connected by, The Wigner distribution of the sample’s Fourier transform

*ψ̃*is given by the Wigner distribution of the sample

*ψ*in Eq. (28) but rotated 90° [15]. Applying this property to both WDFs in Eq. (32), without swapping the dummy convolution variables, produces an expression directly comparable with CP’s Eq. (5): which is in Eq. (11). Two manipulations applied to Eq. (34) lead to the 90° rotation relationship between the CP and FPM data matrices in Eq. (12). First, both the data matrix on the left of Eq. (34) and the two Wigner functions on the right of Eq. (34) must be rotated by swapping the order of their variables (i.e.,

*m*(

_{F}*x*,

*r′*) becomes

*m*(

_{F}*r′*,

*x*)). Second, the dummy integration variables

*x*and

*r′*must switch from one Wigner function to the other (i.e.,

*x*goes to

*W*and

_{ã}*r′*goes to

*W*). Comparing the result of these operations with Eq. (5) leads to Eq. (12)’s linear transform.

_{ψ}## Appendix C: Conventional ptychography with partially coherent source, derivation

*m*(

*x*,

*r′*) in Eq. (16) to produce, Next, we perform the variable substitution

*r*

_{1}=

*r*+

*y/*2 and

*r*

_{2}=

*r*−

*y/*2 to create, Following the same steps as Eq. (28) – Eq. (29), we may replace the

*dy*and

*d*(−

*r′*−

*u*) integrals drop to leave, the final expression in Eq. (17). The finite extent of the incoherent source

*C*(

*p*) alters our original expression for CP’s data matrix through a convolution along the

*x*-dimension of the data matrix, similar to FPM as derived in Appendix D.

## Appendix D: FPM with partially coherent sources, derivation

*C̃*(

*ρ*

_{1}−

*ρ*

_{2}) = ∫

*C*(

*p*)exp(−

*jkp*(

*ρ*

_{1}−

*ρ*

_{2}))

*d p*into Eq. (20) leads to, Then, we can make a variable substitution

*t*=

*x*+

*p*to create, As above, we first make the variable substitution

*ρ*

_{1}=

*r*+

*y/*2 and

*ρ*

_{2}=

*r*−

*y/*2: Then, substituting the following Wigner distributions, into our expression for intensity at the detector and noting all exponential terms cancel (similar to what is shown in Eq. (39)) yields, To convert this into a form directly comparable to both CP and FPM under coherent illumination, we first remove

*t*from Eq. (46) using the relationship

*t*−

*x*=

*p*. Second, we note that the position of the

*x*and

*r′*variables are along opposite dimensions of

*W*and

_{ψ}*W*as compared with our previous data matrix expression in Eq. (11). Swapping the order of variables on both sides of Eq. (46) produces, which is directly comparable to Eq. (11). Now, partial coherence effects are included with a convolution with source shape

_{ã}*C*along the data matrix

*x*dimension. Comparing the above equation to Eq. (17) reveals that although

*C*blurs both data matrices along

*x*, the WDFs describing each are rotated by 90° with respect to the other, leading partial coherence to mix together the data captured by CP and FPM in a different fashion.

## Acknowledgments

## References and links

1. | P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature |

2. | F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B |

3. | J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL |

4. | P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science |

5. | M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature |

6. | A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. |

7. | A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. |

8. | G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. |

9. | J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A |

10. | H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy |

11. | J. N. Clark, X. Huang, R. Harder, and I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. |

12. | P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature |

13. | J. Goodman, |

14. | K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. |

15. | M. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, |

16. | M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A |

17. | R. Horstmeyer, S. B. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express |

18. | H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. |

19. | A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy |

20. | A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy |

21. | M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy |

22. | C. Teale, D. Adams, M. Murnane, H. Kapteyn, and D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express |

23. | G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express |

24. | D. Brady, |

25. | R. G. Brown and P. Y. C. Hwang, |

26. | X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. |

**OCIS Codes**

(110.4980) Imaging systems : Partial coherence in imaging

(110.1758) Imaging systems : Computational imaging

(080.5084) Geometric optics : Phase space methods of analysis

(070.7425) Fourier optics and signal processing : Quasi-probability distribution functions

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 22, 2013

Revised Manuscript: December 16, 2013

Manuscript Accepted: December 17, 2013

Published: January 2, 2014

**Virtual Issues**

Vol. 9, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Roarke Horstmeyer and Changhuei Yang, "A phase space model of Fourier ptychographic microscopy," Opt. Express **22**, 338-358 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-1-338

Sort: Year | Journal | Reset

### References

- P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995). [CrossRef]
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- J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007). [CrossRef]
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- G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]
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- C. Teale, D. Adams, M. Murnane, H. Kapteyn, D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012). [CrossRef]
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