Field of view advantage of conjugate adaptive optics inmicroscopy applications

Jerome Mertz,1, ∗ Hari Paudel,2 and Thomas G. Bifano3

1Dept. of Biomedical Engineering, Boston University,

44 Cummington Mall, Boston, MA 02215, USA2Dept. of Electrical Engineering, Boston University,

8 Saint Mary’s St., Boston, MA 02215, USA3Photonics Center, Boston University, 8 Saint Mary’s St., Boston, MA 02215, USA

compiled: May 5, 2021

The imaging performance of an optical microscope can be degraded by sample-induced aberrations.A general strategy to undo the effect of these aberrations is to apply wavefront correction with adeformable mirror (DM). In most cases, the DM is placed conjugate to the microscope pupil, calledpupil adaptive optics (AO). When the aberrations are spatially variant, an alternative configurationinvolves placing the DM conjugate to the main source of aberrations, called conjugate AO. We providetheoretical and experimental comparison of both configurations for the simplified case where spatiallyvariant aberrations are produced by a well defined phase screen. We pay particular attention tothe resulting correction field of view (FOV). Conjugate AO is found to provide a significant FOVadvantage. While this result is well known in the astronomy community, our goal here is to recast itspecifically for the optical microscopy community.

OCIS codes: (110.0113) Imaging through turbid media; (110.1080) Active or adaptive optics;(110.0180) Microscopy

http://dx.doi.org/10.1364/XX.99.099999

1. IntroductionObjects become blurred when they are imagedthrough scattering media. This has been a long-standing source of frustration in optical microscopy,particularly in biomedical imaging applicationswhere objects of interest are routinely embeddedwithin scattering media or behind aberrating sur-faces. A well-known strategy to counteract aber-rations makes use of adaptive optics (AO), as bor-rowed from astronomical imaging [1, 2]. The idea ofadaptive optics is to insert an element in the imag-ing optics, typically a deformable mirror (DM), thatimparts inverse aberrations to the imaged light,thus compensating for the aberrations induced bythe sample or by the microscope system itself. Inmost cases, the DM is inserted in the pupil plane (orconjugate plane thereof) of the microscope optics.Such a DM placement is appropriate when the aber-

∗ Corresponding author: jmertz@bu.edu

rations to be compensated are spatially invariant,as in the case when they are produced by an indexof refraction mismatch at a flat interface. But an-other reason for placing the DM in the pupil planeseems more historical in nature, and is largely a car-ryover from astronomical AO. However, one shouldbear in mind that the requirements for astronomi-cal compared to microscopy AO are very different.In astronomical AO one is usually interested in onlyvery small (angular) fields of view and it is impor-tant to bring all the DM actuators to bear on sin-gle, localized objects at a time. In microscopy AO,particularly involving widefield (i.e. non-scanning)configurations, the opposite is usually true and it isdesirable perform AO over as large a field of view(FOV) as possible. In the more general case whensample-induced aberrations are spatially variant asopposed to invariant, a placement of the DM in thepupil plane turns out to be a very poor choice as itimposes a severe limitation on FOV [3]. A betterchoice is to place the DM in a plane conjugate to the

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plane where the aberrations are the most dominant,and hence deleterious. This is called conjugate AO.The purpose of this manuscript is to highlight thedifferences between conjugate and pupil AO, partic-ularly where FOV is concerned, both theoreticallyand experimentally.

To begin, we emphasize that what will be saidhere is not new. It is well known [4–7] in the as-tronomical imaging community that the FOV (or“seeing”) can be extended with the use of conju-gate AO. In this case, it is assumed that the mostimportant aberrations are produced by turbulencefrom a well defined layer in the atmosphere, and theDM is placed conjugate to this layer. This strategycan be generalized to multi-conjugate adaptive op-tics (or MCAO) where multiple DMs are conjugatedto multiple atmospheric layers.

The principle of MCAO, though well understoodin the astronomy community, seems to be less ap-preciated in the microscopy community [8]. A fewreports have discussed various benefits of MCAOin the context of microscopy [9, 10], though thesehave relied on numerical simulation only. MCAOhas also been used in retinal imaging applications[11] and in benchtop experiments designed to sim-ulate astronomical imaging [12]. Our goal here isto build on these results by providing a theoreticalframework specifically tailored to the microscopycommunity. We limit our considerations to the sim-plified case where spatially variant aberrations areassumed to arise from a single layer only. Whilesuch a case may seem overly idealized, it servesto highlight the salient features of conjugate AOregarding FOV, which is our goal here. It also be-comes relevant, for example, in sub-surface imagingapplications where the dominant aberrations arisefrom irregularities at the sample surface, as is com-mon in practice.

Our manuscript is organized as follows. We be-gin by describing the effect on imaging of an aber-rating layer an arbitrary distance from the focalplane. We then consider the effects of compensat-ing these aberrations by placing a DM first in thepupil plane and then in the plane conjugate to theaberrating layer. We concentrate our discussion onthe implications for FOV. Finally, in the second halfof this manuscript, we support our theoretical re-sults with proof of principle experiments involvingimage-based AO with a calibrated object and a bi-ological sample, for demonstration purposes. Ourgoal is to lay the groundwork for future bona-fidemicroscopy applications.

2. Effect of a single aberrating layerWe consider a telecentric microscope system, which,for simplicity, we take to have unit magnification(see Fig. 1). The complex object field located atthe focal plane is given by E0(ρ), where ρ is a2D lateral coordinate. This field is assumed tobe quasi-monochromatic of average wavelength λ.Field propagation through the microscope is takento be through free space, except for the presence ofa thin aberrating layer located a distance z from theobject, modeled as a thin phase screen of transmis-sion t(ρ) = eiφ(ρ), where φ(ρ) is a local (real) phaseshift. Throughout this discussion we will adoptthe paraxial, or Fresnel, approximation, meaningwe will consider only propagation angles that aresmall. Implicitly, this means we assume our phasescreen is forward scattering only, meaning the lat-eral extent of its phase features is typically largerthan λ (more on this later).

Fig. 1. Basic microscope layout.

As far as our imaging device is concerned, theobject field E0(ρ) propagating through the phasescreen t(ρ) is equivalent to an effective, albeitscrambled, field E(ρ) propagating through freespace. We can derive E(ρ) by propagating E0(ρ) tothe phase screen, multiplying it by t(ρ), and prop-agating it back to the focal plane. Using standardFresnel propagation integrals, we find

E(ρ) =

∫∫ei2πκ⊥·(ρ−ρ′)E0(ρ + zλκ⊥)t(ρ′)d2ρ′d2κ⊥,

(1)where κ⊥ is a 2D transverse spatial frequency andλκ⊥ may be interpreted as a propagation angle (wehave adopted similar notation as in [13]). Again, inkeeping with the paraxial approximation, the inte-gral over κ⊥ is assumed to span a range such thatλκ⊥ � 1. In reality, the range of κ⊥ will becomeeven more restricted by our microscope pupil, butwe do not consider this yet. Equation 1 is taken tobe valid independent of our imaging device and willbe the starting point of our discussion.

Ultimately, we will use a camera to form an im-age, and thus we are interested in recording inten-sities rather than fields. With this in mind, we

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evaluate the mutual intensity of E(ρ), defined asJ(ρc,ρd) = 〈E(ρc + 1

2ρd)E∗(ρc− 1

2ρd)〉, where 〈...〉corresponds to a time average. A tedious calcula-tion yields

J(ρc,ρd) =

∫∫∫∫ei2πκ⊥d·(ρc−ρc′)ei2πκ⊥c·(ρd−ρd′)

J0(ρc − zλκ⊥c,ρd − zλκ⊥d)Γ0(ρc′,ρd

′)

d2ρc′d2ρd

′d2κ⊥cd2κ⊥d,

(2)

where we have introduced the function Γ0(ρc,ρd) =t(ρc + 1

2ρd)t∗(ρc − 1

2ρd).So far, we have made no assumptions regarding

the object field E0(ρ). We now make the assump-tion that it is spatially incoherent, as is the case, forexample, when imaging fluorescence. That is, weformally write J0(ρc,ρd) ≈ λ2I0(ρc)δ

2(ρd), wherethe prefactor λ2 is introduced for dimensional con-sistency, but also because the coherence area ofradiating spatially incoherent fields is roughly λ2.Equation 2 then simplifies to

J(ρc,ρd) = 1z2

∫∫∫ei2πρd·(ρc−ρc

′)/zλei2πκ⊥c·(ρd−ρd′)

I0(ρc − zλκ⊥c)Γ0(ρc′,ρd

′)d2ρc′d2ρd

′d2κ⊥c.(3)

In other words, even though the object fieldE0(ρ) is taken to be spatially incoherent, the ap-parent field E(ρ) may develop spatial coherencesowing to the presence of the phase screen, a resultthat stems, in part, from the Van Cittert-Zerniketheorem (e.g. see [13]). Let us examine these spa-tial coherences more closely. To do this, we makesome assumptions regarding the phase screen.

Up to this point, our calculations have been sim-ilar to those encountered in astronomical imagingthrough a turbulent atmosphere [2]. In the lattercase, the phase fluctuations imparted by t(ρ) areassumed to be random in time and long image ex-posures are taken such that Γ0(ρc,ρd) can be re-duced to its wide-sense stationary representationΓ0(ρd), which is independent of ρc. We cannot per-form such time averaging here because, in contrastto atmospheric turbulence, our phase screen is as-sumed to be static. Nevertheless, Γ0(ρc,ρd) is notisolated in Eq. 3, but rather occurs inside an inte-gral. We make an approximation of ergodicity bywriting∫

e−i2πκ⊥·ρcΓ0(ρc,ρd)d2ρc ≈ δ2(κ⊥)Γ0(ρd). (4)

In effect, we assume that the phase screen is situ-ated far enough from the focal plane that light aris-ing from any object point traverses a phase-screenarea large enough to encompass many uncorrelated,statistically homogeneous phase features. With thisapproximation, E(ρ) also becomes spatially inco-herent, and Eq. 3 reduces to

I(ρ) ≈∫∫

e−i2πκ⊥·ρ′I0(ρ + zλκ⊥)Γ0(ρ′)d2ρ′d2κ⊥.

(5)We recall that I(ρ) is the apparent object inten-

sity at the focal plane resulting from the propaga-tion of the actual object intensity I0(ρ) through thephase screen. This equation bears resemblance toand is essentially the intensity equivalent of Eq. 1,valid for spatially incoherent object fields. We notethat we still have not considered the imaging devicein our calculations. However, when we do, becausethe apparent object field remains spatially incoher-ent, we need only invoke the intensity point spreadfunction (PSF), as opposed to the amplitude PSF,to evaluate the resultant image.

To make further progress, we must make someassumptions regarding Γ0(ρ). Conventionally [14],this is done by writing Γ0(ρd) = 〈exp(iφ(ρc+

12ρd)−

iφ(ρc− 12ρd))〉, where 〈...〉 now refers to an average

over ρc. Assuming φ(ρ) are Gaussian random pro-cesses of zero mean, we find Γ0(ρd) =exp(−1

2D(ρd))

where D(ρd) = 〈(φ(ρc + 12ρd) − φ(ρc − 1

2ρd))2〉 isthe structure function of the phase variations. Tak-ing these variations to be statistically homogeneous,and characterized by a variance σφ

2 and normalizedspatial autocorrelation function γ0(ρd), we obtain

Γ0(ρd) = e−σφ2(1−γ0(ρd)), (6)

or, equivalently,

Γ0(ρd) = e−σφ2

+ e−σφ2(eσφ

2γ0(ρd) − 1). (7)

The advantage of Eq. 7 is that it identifies theeffects of the phase screen on ballistic (first term)and scattered (second term) light propagation. Weadopt the model here that the phase variations arecorrelated in a Gaussian manner over a characteris-tic length scale lφ, such that γ0(ρd) = e−ρ

2d/l

2φ . Plots

of γ0(ρd) are illustrated in Fig. 2 for various valuesof σφ

2.To gain further insight into the problem, we take

Eq. 7 one step further by recasting it yet again inthe form

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Fig. 2. Plots of Γ0(ρ) for different values of σφ2.

Γ0(ρd) ≈ e−σφ2

+ (1− e−σφ2)γ0(ρd

√1 + σφ2). (8)

Equation 8 is found to be an excellent approxi-mation to Eq. 7 for values of σφ

2 much smaller andmuch larger than one, and only deviates slightlyfrom Eq, 7 when σφ

2 is close to one. We thus adoptit as a general expression for Γ0(ρd) since it providesmuch clearer insight into its physical meaning.

As an example, let us consider the object to be apoint source located at the origin of the focal plane.That is, we write I0(ρ) = λ2I0δ

2(ρ). Plugging thisinto Eq. 5 , we find the effective object intensityto be

I(ρ) = 1z2I0

∫e−i2πρ·ρ

′/zλΓ0(ρ′)d2ρ′. (9)

Propagating this intensity through our imagingdevice and making use of Eq. 9, we derive an effec-tive PSF given by

PSF(ρ) = e−σφ2PSF0(ρ) + (1− e−σφ2)

1

πζ2e−ρ

2/ζ2 ,

(10)where PSF0(ρ) is the initial device PSF absent thephase screen, and we introduce the length scaleζ = zλ

√1 + σφ2/πlφ. A plot of this effective PSF

is illustrated in Fig. 3. We observe that it featurestwo components. The first is an attenuated versionof the initial PSF0, leading to a sharp, diffraction-limited peak that is attenuated because of loss ofballistic light due to scattering. The second is a

broad pedestal resulting from scattering due to thephase screen. An increase in ζ caused, for exam-ple, by an increase in z leads to an increase in thepedestal width and a concomitant decrease in itsheight. The proportion of the effective PSF that re-

mains ballistic is e−σφ2, while the rest is scattered.

Clearly, as σφ2 increases the effective PSF becomes

progressively more diffuse.

Fig. 3. Example of an effective PSF(ρ) after degrada-tion by a phase screen. The initial diffraction-limitedPSF (PSF0) is reduced by a factor e−σφ

2

and rides on a

broader blurred background of power 1− e−σφ2

.

Some comments on the validity of Eq. 10. Tobegin, we recall that we have limited ourselvesto small propagation angles. ζ must thereforebe smaller than z, meaning we implicitly assumeλ√

1 + σφ2/lφ � 1. Moreover, in assuming thatthe light from the point object samples many phase-screen correlation areas, we implicitly restrict themicroscope NA to values much larger than lφ/z. Inturn, this means that ζ is implicitly assumed to belarger than the diffraction limited spot size λ/2NA.Bearing these assumptions in mind, we may eval-uate the Strehl ratio of our effective PSF , definedby S =PSF(0)/PSF0(0) ≈ λ2PSF(0)/πNA2. Forphase variances not so large that the Strehl ra-tio is defined primarily by ballistic light we obtain

S = e−σφ2. This result is the same as obtained

for astronomical imaging through a turbulent at-mospheric layer [1].

How do we recover from the degradation inPSF(ρ) caused by the phase screen? We turn nowto two possible strategies involving different AOconfigurations.

3. Pupil AO

The most common AO configuration involves plac-ing a DM in the pupil plane of the imaging de-vice. Wavefront correction is then based either on

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knowledge gained from a direct measurement ofthe aberrated wavefront at this plane (wavefront-sensing-based AO), or by iterative trial and error tooptimize a user-defined image metric (image-basedAO). In either case, wavefront correction is designedto undo the effects of aberrations for a particularspot in the object, typically the location of a “guidestar” [1], with the hope that the range of the correc-tion about this spot is large enough to encompassneighboring objects of interest. In astronomy par-lance, this range is called “seeing” or the “isopla-natic patch”. Here, we call it the corrected FOV ofour microscope. Our goal in this section is to derivean expression for this FOV.

To begin, we assume that the object spot whosewavefront we would like to correct is located atthe origin. This could be the location of a fluo-rescent beacon (serving as a guide star), or of anobject point we have arbitrarily selected for imageoptimization. To evaluate the aberrated wavefrontproduced by this spot, we write the object field atthis spot as E0(ρ) = λ2E0δ

2(ρ). Inserting this intoEq. 1, and performing a scaled Fourier transformto calculate the resultant field at the pupil plane,we find

Ep(ξ) =λ

fE0

∫∫e−i2πκ⊥·(ρ+

zf ξ)eiπzλκ

2⊥t(ρ)d2ρd2κ⊥,

(11)where ξ is a 2D transverse coordinate in the pupilplane and f is the focal length of the microscope ob-jective (see Fig. 1). We observe that in the absenceof a phase screen (t(ρ) = 1), Ep(ξ) becomes a planewave of constant amplitude λE0/f that is uniformlyspread across the pupil plane, as expected. In con-trast, when the phase screen is present Ep(ξ) be-comes structured both in amplitude and in phase.To correct for this effect of the phase screen, wecan act on Ep(ξ) with a wavefront corrector. Ide-ally, Ep(ξ) should be made into a plane wave again,meaning we should flatten it both in amplitude andphase. However, wavefront correctors such as DMsare never ideal and generally act on phase only.The best we can do is to flatten the phase of Ep(ξ)and leave its amplitude unchanged. Nevertheless,as we will argue below, the consequences of apply-ing phase-only wavefront correction are not all thatsevere.

In effect, the DM is itself a phase screen whosetransmission function can be written as tdm(ξ). Tocorrect for the phase variations in Ep(ξ), the DMshould impart its own phase variations that are pre-cisely the negative of those of Ep(ξ). That is, the

DM should effectively phase conjugate Ep(ξ). Thisoccurs when

tdm(ξ) =E∗p(ξ)

|Ep(ξ)|. (12)

At this point, the math becomes difficult andwe need to introduce a simplification to proceed.Specifically, we replace the denominator in Eq. 12with the (constant) field amplitude that would bepresent without the phase screen, and write

tdm(ξ) =

∫∫ei2πκ⊥·(ρ+

zf ξ)e−iπzλκ

2⊥t∗(ρ)d2ρd2κ⊥.

(13)This simplification has no effect on the average in-

tensity at the pupil plane, and we may justify it onthe grounds that, with the aid of the approximationin Eq. 4, we maintain |tdm(ξ)| ≈ 1, in agreementwith the phase-only nature of our DM. Neverthe-less, this simplification does somewhat modify thefield statistics at the pupil plane, the ramificationsof which will be discussed below.

We recall that tdm(ξ) here is the shape appliedto the DM that performs optimal AO correctionexactly at the origin of the focal plane. We cannow evaluate the spatial range, or FOV, over whichthis correction remains effective. To do this, we be-gin now with an arbitrary (albeit spatially incoher-ent) object field. The effective field at the objectplane that takes into account the aberrations im-parted by phase screen is given by E(ρ) in Eq. 1.To take into account the additional, hopefully cor-rective, effect of the DM, we propagate this field tothe pupil (a scaled Fourier transform), multiply itby tdm(ξ), and then propagate it back to the focalplane (a scaled inverse Fourier transform). Usingthe same math and associated approximations asin the previous section, and keeping only dominantphase correlations, we arrive at

Iao(ρ) ≈ I0(ρ)Γ2t (ρ). (14)

Again, this result is subject to the same condi-tions of validity as before. Moreover, we have as-sumed that the DM spans the entire pupil (not toosmall), and its actuators are sufficiently dense toaccurately represent tdm(ξ) (more on this below).

Equation 14 warrants scrutiny. I0(ρ) is the ac-tual object intensity; Iao(ρ) is the effective objectintensity as observed through the phase screen andcorrected by the pupil DM. The window of AO cor-rection is thus characterized by Γ2

t (ρ). From Eq. 6,

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we note that this window is equivalent to Γ0(ρ),but with a phase variance σφ

2 that is effectivelydoubled. We can thus recast it using the approx-imation provided by Eq. 8. However, it is at thispoint that we introduce a small correction to ourresults. Namely, we write

Γ2t (ρd) ≈ e−2σφ

2+ (1− e−2σφ2)

π

4γ0(ρd

√1 + 2σφ2).

(15)The reader can observe that we have introduced

an extra factor of π/4 in the second term. Thejustification for this term is as follows. We recallthat the reason for separating Γ0(ρd) into two termswas to better identify the ballistic and scatteringpropagation components. The second term arisesfrom light that is scattered by the phase screen.For light that originates from a point at the focalplane, the scattered component of this light thenimpinges the pupil plane with spatially varying am-plitude and phase. Indeed this scattered compo-nent takes on the characteristics of fully developedspeckle [14] (fully developed because we have re-moved from it all ballistic contribution). Upon cor-recting the wavefront of this speckle field, the bestthe DM can do is flatten its phase. The amplitudedistribution of the speckle field remains unchanged,meaning it continues to obey Rayleigh statistics. Inthe typical case that the microscope pupil is physi-cally smaller than the field distribution at the pupilplane (i.e. there is some energy loss), then the bestthe DM can do is yield a corrected image intensityfrom this scattered light component that is π/4 re-duced compared to the same component with nophase screen [16]. We need to include this correc-tion factor in Eq. 15 because our derivation madeuse of Eq. 13 for the optimal DM profile rather thanthe more correct Eq. 12.

As an aside, we note that the spatial extent ofthe speckle grains at the pupil plane is inverselyrelated to the spatial extent of the aberrated inten-sity pattern at the object plane, and is thus givenby, roughly, λf/2ζ (akin to the Fried parameterin astronomical imaging [15]). The DM actuatorsmust be smaller than this to avoid undersamplingof tp(ξ).

To summarize, Γ2t (ρd) represents the window over

which pupil-based AO is effective. For phase varia-tions of standard deviation σφ one radian or larger,this window is predominantly defined by the sec-ond term and yields a FOV of diameter a0 ≈2lφ/

√1 + 2σφ2. This FOV becomes narrower with

increasing phase variations, approaching the limitof√

2lφ/σφ. In other words, in the regime where

Fig. 4. Interpretation of what defines FOV for pupilAO. As an object point ρ1 becomes displaced from theguide-star point ρ0, its wavefront correction becomes un-correlated with the fixed phase screen aberrations, andAO fails.

the phase variations are large (i.e. the regime whereAO is potentially most useful!) the FOV dependsnot on the characteristic scale lφ of the aberratingfeatures, but rather on their (inverse) characteris-tic slope. This result is similar to a general re-sult for imaging through phase screens discussed in[14]. Moreover, the FOV is independent of the loca-tion z of the phase screen. These observations canreadily be understood from Fig. 4. We recall thatany wavefront correction provided by a pupil-planeDM must be spatially invariant in the sense that itmust be imparted equally to all object points. Thewavefront correction can be thought of, therefore,as figuratively tracking each object point. However,for object points displaced from the guide star thewavefront correction becomes rapidly uncorrelatedwith the aberrations it is intended to correct. Inthis case, the correction actually produces worseimaging than if there were no DM at all, becauseit leads to the presence of two uncorrelated phasescreens in the imaging optics rather than one (hencethe effective doubling of σφ

2). Because the micro-scope system is telecentric, this FOV is independentof z.

We close this section with a reminder that ourderivation of a0 presumed that pupil AO was askedto optimize the image at a single point only, namelyat the origin. One might wonder what would hap-pen if it were asked to optimize over a larger region,say of size A. This problem is tantamount to op-timizing multiple guide stars simultaneously. Suchsimultaneous optimization is possible; however, itis known to lead to reduced contrast enhancementat each guide star by a factor of N , the number of

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guide stars [16]. That is, while pupil AO can, inprinciple , correct over a range A larger than a0,the quality of this correction, as measured by con-trast enhancement, is expected to rapidly degradeas (a0/A)2.

4. Conjugate AO

In the previous section we characterized a prob-lem (often debilitating) of pupil-plane AO that itcannot provide extended FOVs. This disadvan-tage was interpreted as arising from the propertythat the pupil-plane AO wavefront correction effec-tively tracks the different points in the object planewhile the aberrating phase screen remains fixed. Asolution to this problem is clear. The wavefrontcorrection should instead be locked to the phasescreen rather than to the object points. This canbe achieved by placing the DM conjugate to thephase screen itself, called conjugate AO.

Fig. 5. Interpretation of what defines FOV for conjugateAO. The fixed wavefront correction cancels the aberra-tions caused by the fixed phase screen. The FOV isthen limited by size of the projected DM itself, albeitwith blurred edges dependent on distance z of the phasescreen and the microscope NA.

What is the FOV associated with conjugate AO?Based on the above interpretation, it is clear thatthe FOV must be the size of the DM projected ontothe phase screen. For example, if the DM is con-jugate to the phase screen with unit magnification,then the FOV is the same size as the DM itself.Again, since our microscope system is telecentric,this FOV is independent of the z location of thephase screen, provided the DM is maintained con-jugate. However, one must be careful with this laststatement since there may be edge effects that aredependent on z. Such effects are best appreciatedwith a simple ray optic picture as shown in Fig. 5.

The microscope pupil defines a characteristic max-imum cone size (i.e. NA) of light that can be col-lected from any object point. If this cone size islarge enough that it spans the projected DM for allobject points within the FOV, then conjugate AOcorrects equally well throughout this FOV. On theother hand, if the cone size is smaller (as shown),then the DM causes vignetting. This vignettingbegins to occur at distances roughly zNA from theFOV edges (note: the DM here is assumed to be ob-structing beyond its edges). Nevertheless, despitethis potential issue of edge effects, it is clear (andwell known [4]) that the FOV provided by conjugateAO can be significantly larger than that providedby pupil AO.

5. Experimental demonstrationThe FOV advantage of conjugate AO can bedemonstrated experimentally. In most cases, AOis applied in laser scanning microscope configura-tions [17–27]. We apply it here instead in a bright-field configuration. Our setup is illustrated in Fig. 6and is basically a 1.33× magnification microscopemodified to accommodate a pupil or conjugate DM(Boston Micromachine Corp. MultiDM, 140 actu-ators in a square 12 × 12 array without the corneractuators, 400µm actuator pitch). A thin transmis-sion object is trans-illuminated by a LED (660 nm).A thick diffuser is inserted in the LED path, justbefore the object, to ensure object spatial incoher-ence. The NA of the microscope is about 0.04, asdefined by a ≈8 mm diameter iris pupil. We notethat there is a factor of about two demagnificationin the imaging optics from this pupil to the pupilDM to properly match their respective sizes. A mir-ror blank is inserted in the place of the conjugateDM when the system is in a pupil DM configura-tion, and vice versa. The camera is a ThorlabsDCC1545M CMOS (pixel size 5.2µm square).

To introduce spatially variant aberrations in ourmicroscope, we inserted a phase screen a distancez (here 35 mm) from the object. The phase screenconsisted of a blank microscope slide onto whichwas spray-painted a thin layer of clear acrylic. Theprofile of this phase screen was measured by a ZygoNT6000 white-light interferometer and exhibitedrms phase variations on the order of σφ ≈ 9.5 ra-dians (see discussion below), corresponding to justabout the limit of what can be corrected by our DMbased on its 3.5µm peak-to-valley stroke. A char-acteristic spatial scale of the phase variations wasvery roughly estimated to be of order lφ ≈ 1 mm,leading to an anticipated pupil AO correction rangeof about a0 ≈ 150µm.

8

Fig. 6. Experimental setup. Lens focal lengths are f1 = 100 mm, f2 = 100 mm, f3 = 100 mm, f4 = 200 mm,f5 = 75 mm, and f6 = 50 mm. Solid vertical lines correspond to object planes; dashed vertical line correspond topupil planes. Note that in reality the DMs are reflective and the layout was doubly folded.

Admittedly, a scenario where a large empty spaceseparates an object from an aberrating layer is un-likely to be found in actual microscopy applications.The purpose of this experiment is only to highlightsome salient features underlying conjugate versuspupil AO.

As noted previously, when AO is used in mi-croscopy it is typically applied to laser scanning mi-croscopes where different wavefront corrections aretargeted sequentially to different spots distributedthroughout the sample. In our case, we wish to ap-ply AO in a non-scanning bright-field configuration.That is, we wish to apply a single wavefront correc-tion to the entire image. To determine an optimizedDM shape, we used an image-based stochastic par-allel gradient descent (SPGD) procedure that op-timizes a particular image metric [28]. Here, themetric was chosen to be the image contrast mea-sured within a user-selected square zone centeredabout the image origin (contrast being defined asthe square root of the image variance divided bythe mean). The application of contrast-based wave-front optimization works well when the object is alocalized guide star; however, it is known to failwhen the object is arbitrary and extended, sinceit tends to re-distribute light into mottled patternsrather than improve image sharpness. To circum-vent this problem we proceeded in two steps. First,we inserted as an object an array of apertures atthe focal plane (see Fig. 7). This calibration ob-ject served as a well-defined homogeneous array ofguide stars, enabling contrast-based DM optimiza-tion to perform adequately. Second, following DMoptimization (pupil or conjugate), we replaced theaperture array with an object of interest, namely athin section of H&E-stained mammal muscle ten-don mounted on a microscope slide (Carolina Bio-logical Supply Co.).

The results for pupil AO are shown in Fig. 7.

Fig. 7. Experimental results for pupil AO. Uncorrected(DM flat) images of an aperture array of period 200µm(a) without and (b) with the presence of an aberratingphase screen. (c) Same image after AO correction thatoptimized contrast in a zone about the origin of size250 × 250µm2 (single guide star). Uncorrected imagesof tissue section (d) without and (e) with phase-screen,and (f) after AO correction using the same DM patternas established for (c). Image sizes are 4 × 4mm2 at thefocal plane.

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Fig. 8. Experimental results for conjugate AO. Images ofaperture array after AO correction that optimized con-trast in a zone about the origin of size (a) 250×250µm2

(single guide star), and (b) 4×4mm2 (entire image). (c)and (d) are corresponding images of tissue section.

Fig. 9. Recapitulation of results. 600× 600µm2 blowupof zones about the origin taken from images of tissuesection (a) uncorrected without aberrations, (b) uncor-rected with aberrations, (c) pupil-AO corrected, and (d)conjugate-AO corrected (taken from Fig. 8d).

In this case, the correction zone was chosen to besmall, such that it spanned only a single guide star(correction zones larger than this led to DM itera-tions that failed to converge or increase contrast).

Clearly, pupil AO was effective at improving imagequality near the origin. However, just as clearly,it was ineffective at improving quality even a smalldistance from the origin. The same was true whenwe inserted the tissue sample, as can be seen inFig. 7. Based on the properties of the phase screen,the FOV of the wavefront correction was expectedto be about a0 ≈ 150µm, which is in rough agree-ment with experiment .

The results of conjugate AO are shown in Fig. 8.In this case, we chose correction zones that wereprogressively increased in size (only two of whichare shown). As is apparent both with the guide-star array and tissue sample, the resultant FOV in-creases with correction zone size, approaching thesize of the full unaberrated image (Fig. 7a). Ofparticular interest are Figs. 8a,c. The zone of cor-rection here was 250 × 250µm2, meaning AO wasasked to optimize the contrast of the central guidestar only. And yet clearly the resultant correctionFOV is much larger than this. In this case, the cor-rection FOV corresponds roughly to the region ofthe phase screen illuminated by the central guidestar, as defined by the microscope pupil. That is,it is of size roughly zNA.

Upon closer inspection of the corrected imagesnear the origin (Fig. 9), we observe that conjugateAO did not provide quite as crisp a correction aspupil AO, One possible reason for this is that only afew conjugate DM actuators contributed to the cor-rection of this small region, whereas for the pupilDM all the actuators contributed. Conjugate AOmay have thus suffered from a slight problem of in-sufficient actuator sampling (a problem that can becorrected with improved sampling). Another possi-ble reason is that edge effects may have underminedthe correction efficacy.

Finally, we compare topography maps of the opti-mized DM shapes with those of the phase screen it-self. The correspondence is apparent, both in formand amplitude. Specifically, we note a factor of≈ 4 difference in topography amplitude betweenthe measured phase screen and DM shapes. Thisarises in part from the impact of the acrylic topog-raphy on its wavefront (the acrylic index of refrac-tion ≈ 1.5 leads to an optical path difference thatis about half the local topographic height). An-other factor of two arises from the fact that our DMoperates in reflection mode, leading to an effectivewavefront doubling. In our case the phase-screenaberrations were measured to be 1030 nm rms inwavefront, consistent with an acrylic topography of2060 nm, and corrected by a DM shape of 514 nm

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Fig. 10. Comparison of the wavefront shapes roughly atthe same location of (top) the phase screen and (bot-tom) the DM after conjugate AO with full-image con-trast correction (i.e. same as in Fig. 8b). In both cases,the wavefront shapes were measured with a Zygo white-light interferometer.

rms. We recall that phase is related to wavefrontby φ(ρ) = 2π

λ W (ρ).

6. DiscussionIn view of the apparent FOV advantage of con-jugate over pupil AO, one may wonder why it isnot more prevalent in the microscopy community.There may be several reasons for this. To begin,the most egregious aberrations caused by a sampleare often not those induced by laterally spatiallyvariant sample features but rather by a laterally in-variant index of refraction mismatch. In this case,conjugate AO does not help and pupil AO is pre-scribed instead. Moreover, the sample may not beexhibit aberrations in the form of a single, dominantphase screen (as in our idealized, and certainly con-trived, demonstration experiment), in which casethere may be a difficulty in determining an opti-

mal DM location. Even in the case where thereindeed exists a well-defined, dominant aberratinglayer, the placement of the DM in its conjugateplane may not be so straightforward. For exam-ple, in our experimental demonstration the DM wasplaced a distance z beyond the nominal microscopeimage plane (see Fig. 6). However, in the moreusual case of a microscope with magnification Mmuch greater than unity, the DM should be placedinstead a distance M2z from the image plane. Forlarge M this distance may be problematic and re-quire additional re-imaging optics. There is alsothe issue of DM actuator size. For pupil AO, theDM actuators should be smaller than λf/ζ; for con-jugate AO they should be smaller than lφ, which,for weak phase variance, is more restrictive by afactor ≈ z/f (magnification notwithstanding). Fi-nally, we must consider how the DM optimizationis actually performed. In our demonstration exper-iment we had the luxury of being able to swap in aguide-star array to aid iterative image-based opti-mization. This is generally not possible in prac-tice, and an alternative solution must be found.Ideally, it would be best to measure the aberra-tions directly using a wavefront sensor; however,this becomes problematic because standard wave-front sensors such as Shack-Hartmann sensors [1]only work well with quasi-collimated wavefronts.To our knowledge, extended-source wavefront sen-sors (e.g. [29–31]) have not yet been applied to AOin microscopy configurations.

Nevertheless, the above caveats are generallytechnical in nature. Considering the potentially sig-nificant FOV advantage of conjugate AO, its futureimplementation, perhaps in conjunction with pupilwavefront correction, may well prove to become anew standard in AO applied to microscopy.

Acknowledgments

T. Bifano acknowledges a financial interest inBoston Micromachines Corporation. Support forH. Paudel was provided by the Industry/UniversityCooperative Research Center for Biophotonic Sen-sors and Systems.

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