Table Of ContentField of view advantage of conjugate adaptive optics in
microscopy applications
Jerome Mertz,1,∗ Hari Paudel,2 and Thomas G. Bifano3
1Dept. of Biomedical Engineering, Boston University,
44 Cummington Mall, Boston, MA 02215, USA
2Dept. of Electrical Engineering, Boston University,
8 Saint Mary’s St., Boston, MA 02215, USA
5 3Photonics Center, Boston University, 8 Saint Mary’s St., Boston, MA 02215, USA
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compiled: January 13, 2015
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The imaging performance of an optical microscope can be degraded by sample-induced aberrations.
2 A general strategy to undo the effect of these aberrations is to apply wavefront correction with a
1
deformable mirror (DM). In most cases, the DM is placed conjugate to the microscope pupil, called
pupil adaptive optics (AO). When the aberrations are spatially variant, an alternative configuration
]
s involvesplacingtheDMconjugatetothemainsourceofaberrations,calledconjugateAO.Weprovide
c theoreticalandexperimentalcomparisonofbothconfigurationsforthesimplifiedcasewherespatially
i
t variant aberrations are produced by a well defined phase screen. We pay particular attention to
p
the resulting correction field of view (FOV). Conjugate AO is found to provide a significant FOV
o
advantage. While this result is well known in the astronomy community, our goal here is to recast it
.
s specifically for the optical microscopy community.
c
i OCIScodes: (110.0113)Imagingthroughturbidmedia;(110.1080)Activeoradaptiveoptics;
s
y (110.0180) Microscopy
h http://dx.doi.org/10.1364/XX.99.099999
p
[
1
v
1. Introduction rations to be compensated are spatially invariant,
5
8 Objects become blurred when they are imaged as in the case when they are produced by an index
6 through scattering media. This has been a long- of refraction mismatch at a flat interface. But an-
2 standingsourceoffrustrationinopticalmicroscopy, other reason for placing the DM in the pupil plane
0
particularly in biomedical imaging applications seemsmorehistoricalinnature,andislargelyacar-
.
1 where objects of interest are routinely embedded ryover from astronomical AO. However, one should
0
within scattering media or behind aberrating sur- bear in mind that the requirements for astronomi-
5
1 faces. A well-known strategy to counteract aber- cal compared to microscopy AO are very different.
: rations makes use of adaptive optics (AO), as bor- InastronomicalAOoneisusuallyinterestedinonly
v
rowedfromastronomicalimaging[1,2]. Theideaof very small (angular) fields of view and it is impor-
i
X adaptive optics is to insert an element in the imag- tant to bring all the DM actuators to bear on sin-
r ingoptics,typicallyadeformablemirror(DM),that gle, localized objects at a time. In microscopy AO,
a
imparts inverse aberrations to the imaged light, particularly involving widefield (i.e. non-scanning)
thus compensating for the aberrations induced by configurations, the opposite is usually true and it is
the sample or by the microscope system itself. In desirable perform AO over as large a field of view
mostcases,theDMisinsertedinthepupilplane(or (FOV) as possible. In the more general case when
conjugate plane thereof) of the microscope optics. sample-induced aberrations are spatially variant as
SuchaDMplacementisappropriatewhentheaber- opposed to invariant, a placement of the DM in the
pupil plane turns out to be a very poor choice as it
imposes a severe limitation on FOV [3]. A better
choiceistoplacetheDMinaplaneconjugatetothe
∗ Corresponding author: [email protected]
2
planewheretheaberrationsarethemostdominant, 2. Effect of a single aberrating layer
and hence deleterious. This is called conjugate AO. Weconsideratelecentricmicroscopesystem,which,
The purpose of this manuscript is to highlight the for simplicity, we take to have unit magnification
differencesbetweenconjugateandpupilAO,partic- (see Fig. 1). The complex object field located at
ularly where FOV is concerned, both theoretically the focal plane is given by E (ρ), where ρ is a
0
and experimentally. 2D lateral coordinate. This field is assumed to
be quasi-monochromatic of average wavelength λ.
To begin, we emphasize that what will be said
Field propagation through the microscope is taken
here is not new. It is well known [4–7] in the as-
to be through free space, except for the presence of
tronomical imaging community that the FOV (or
athinaberratinglayerlocatedadistancez fromthe
“seeing”) can be extended with the use of conju-
object, modeled as a thin phase screen of transmis-
gate AO. In this case, it is assumed that the most
siont(ρ) = eiφ(ρ), whereφ(ρ)is alocal(real)phase
important aberrations are produced by turbulence
shift. Throughout this discussion we will adopt
fromawelldefinedlayerintheatmosphere,andthe
the paraxial, or Fresnel, approximation, meaning
DM is placed conjugate to this layer. This strategy
we will consider only propagation angles that are
can be generalized to multi-conjugate adaptive op-
small. Implicitly, this means we assume our phase
tics(orMCAO)wheremultipleDMsareconjugated
screen is forward scattering only, meaning the lat-
to multiple atmospheric layers.
eral extent of its phase features is typically larger
The principle of MCAO, though well understood than λ (more on this later).
in the astronomy community, seems to be less ap-
preciated in the microscopy community [8]. A few
reports have discussed various benefits of MCAO
in the context of microscopy [9, 10], though these
have relied on numerical simulation only. MCAO
has also been used in retinal imaging applications
[11] and in benchtop experiments designed to sim-
ulate astronomical imaging [12]. Our goal here is
Fig. 1. Basic microscope layout.
to build on these results by providing a theoretical
framework specifically tailored to the microscopy As far as our imaging device is concerned, the
community. Welimitourconsiderationstothesim- object field E (ρ) propagating through the phase
0
plified case where spatially variant aberrations are screen t(ρ) is equivalent to an effective, albeit
assumed to arise from a single layer only. While scrambled, field E(ρ) propagating through free
such a case may seem overly idealized, it serves space. WecanderiveE(ρ)bypropagatingE (ρ)to
0
to highlight the salient features of conjugate AO the phase screen, multiplying it by t(ρ), and prop-
regarding FOV, which is our goal here. It also be- agating it back to the focal plane. Using standard
comesrelevant, forexample, insub-surfaceimaging Fresnel propagation integrals, we find
applications where the dominant aberrations arise
from irregularities at the sample surface, as is com-
(cid:90)(cid:90)
mon in practice. E(ρ) = ei2πκ⊥·(ρ−ρ(cid:48))E (ρ+zλκ )t(ρ(cid:48))d2ρ(cid:48)d2κ ,
0 ⊥ ⊥
Our manuscript is organized as follows. We be- (1)
gin by describing the effect on imaging of an aber- where κ is a 2D transverse spatial frequency and
⊥
rating layer an arbitrary distance from the focal λκ maybeinterpretedasapropagationangle(we
⊥
plane. We then consider the effects of compensat- have adopted similar notation as in [13]). Again, in
ing these aberrations by placing a DM first in the keeping with the paraxial approximation, the inte-
pupil plane and then in the plane conjugate to the gral over κ is assumed to span a range such that
⊥
aberrating layer. We concentrate our discussion on λκ (cid:28) 1. In reality, the range of κ will become
⊥ ⊥
theimplicationsforFOV.Finally,inthesecondhalf even more restricted by our microscope pupil, but
of this manuscript, we support our theoretical re- we do not consider this yet. Equation 1 is taken to
sults with proof of principle experiments involving bevalidindependentofourimagingdeviceandwill
image-based AO with a calibrated object and a bi- be the starting point of our discussion.
ological sample, for demonstration purposes. Our Ultimately, we will use a camera to form an im-
goal is to lay the groundwork for future bona-fide age, and thus we are interested in recording inten-
microscopy applications. sities rather than fields. With this in mind, we
3
evaluate the mutual intensity of E(ρ), defined as In effect, we assume that the phase screen is situ-
J(ρ ,ρ ) = (cid:104)E(ρ +1ρ )E∗(ρ −1ρ )(cid:105), where (cid:104)...(cid:105) ated far enough from the focal plane that light aris-
c d c 2 d c 2 d
corresponds to a time average. A tedious calcula- ing from any object point traverses a phase-screen
tion yields arealargeenoughtoencompassmanyuncorrelated,
statisticallyhomogeneousphasefeatures. Withthis
(cid:90)(cid:90)(cid:90)(cid:90) approximation, E(ρ) also becomes spatially inco-
J(ρc,ρd) = ei2πκ⊥d·(ρc−ρc(cid:48))ei2πκ⊥c·(ρd−ρd(cid:48)) herent, and Eq. 3 reduces to
J (ρ −zλκ ,ρ −zλκ )Γ (ρ (cid:48),ρ (cid:48))
0 c ⊥c d ⊥d 0 c d
(cid:90)(cid:90)
d2ρc(cid:48)d2ρd(cid:48)d2κ⊥cd2κ⊥d, I(ρ) ≈ e−i2πκ⊥·ρ(cid:48)I0(ρ+zλκ⊥)Γ¯0(ρ(cid:48))d2ρ(cid:48)d2κ⊥.
(2)
(5)
wherewehaveintroducedthefunctionΓ (ρ ,ρ ) = We recall that I(ρ) is the apparent object inten-
0 c d
t(ρ + 1ρ )t∗(ρ − 1ρ ). sity at the focal plane resulting from the propaga-
c 2 d c 2 d
So far, we have made no assumptions regarding tionoftheactualobjectintensityI (ρ)throughthe
0
the object field E (ρ). We now make the assump- phase screen. This equation bears resemblance to
0
tionthatitisspatiallyincoherent,asisthecase,for and is essentially the intensity equivalent of Eq. 1,
example, when imaging fluorescence. That is, we valid for spatially incoherent object fields. We note
formally write J (ρ ,ρ ) ≈ λ2I (ρ )δ2(ρ ), where thatwestillhavenotconsideredtheimagingdevice
0 c d 0 c d
the prefactor λ2 is introduced for dimensional con- in our calculations. However, when we do, because
sistency, but also because the coherence area of the apparent object field remains spatially incoher-
radiating spatially incoherent fields is roughly λ2. ent, we need only invoke the intensity point spread
Equation 2 then simplifies to function (PSF), as opposed to the amplitude PSF,
to evaluate the resultant image.
(cid:90)(cid:90)(cid:90) To make further progress, we must make some
J(ρc,ρd) = z12 ei2πρd·(ρc−ρc(cid:48))/zλei2πκ⊥c·(ρd−ρd(cid:48)) assumptions regarding Γ¯0(ρ). Conventionally [14],
thisisdonebywritingΓ¯ (ρ ) = (cid:104)exp(iφ(ρ +1ρ )−
I (ρ −zλκ )Γ (ρ (cid:48),ρ (cid:48))d2ρ (cid:48)d2ρ (cid:48)d2κ . 0 d c 2 d
0 c ⊥c 0 c d c d ⊥c iφ(ρ −1ρ ))(cid:105), where (cid:104)...(cid:105) now refers to an average
(3) c 2 d
over ρ . Assuming φ(ρ) are Gaussian random pro-
c
In other words, even though the object field cessesofzeromean,wefindΓ¯ (ρ ) =exp(−1D(ρ ))
0 d 2 d
E (ρ) is taken to be spatially incoherent, the ap- where D(ρ ) = (cid:104)(φ(ρ + 1ρ )−φ(ρ − 1ρ ))2(cid:105) is
0 d c 2 d c 2 d
parent field E(ρ) may develop spatial coherences the structure function of the phase variations. Tak-
owing to the presence of the phase screen, a result ingthesevariationstobestatisticallyhomogeneous,
that stems, in part, from the Van Cittert-Zernike andcharacterizedbyavarianceσ 2 andnormalized
φ
theorem (e.g. see [13]). Let us examine these spa- spatial autocorrelation function γ (ρ ), we obtain
0 d
tial coherences more closely. To do this, we make
some assumptions regarding the phase screen.
Up to this point, our calculations have been sim- Γ¯0(ρd) = e−σφ2(1−γ0(ρd)), (6)
ilar to those encountered in astronomical imaging
or, equivalently,
through a turbulent atmosphere [2]. In the latter
case, the phase fluctuations imparted by t(ρ) are
assumed to be random in time and long image ex- Γ¯ (ρ ) = e−σφ2 +e−σφ2(eσφ2γ0(ρd)−1). (7)
0 d
posures are taken such that Γ (ρ ,ρ ) can be re-
0 c d
duced to its wide-sense stationary representation The advantage of Eq. 7 is that it identifies the
Γ¯0(ρd), which is independent of ρc. We cannot per- effects of the phase screen on ballistic (first term)
form such time averaging here because, in contrast and scattered (second term) light propagation. We
to atmospheric turbulence, our phase screen is as- adopt the model here that the phase variations are
sumed to be static. Nevertheless, Γ0(ρc,ρd) is not correlated in a Gaussian manner over a characteris-
isolated in Eq. 3, but rather occurs inside an inte- ticlengthscalelφ,suchthatγ0(ρd) = e−ρ2d/lφ2. Plots
gral. We make an approximation of ergodicity by of γ (ρ ) are illustrated in Fig. 2 for various values
0 d
writing of σ 2.
φ
To gain further insight into the problem, we take
(cid:90)
Eq. 7 one step further by recasting it yet again in
e−i2πκ⊥·ρcΓ (ρ ,ρ )d2ρ ≈ δ2(κ )Γ¯ (ρ ). (4)
0 c d c ⊥ 0 d the form
4
broad pedestal resulting from scattering due to the
phase screen. An increase in ζ caused, for exam-
ple, by an increase in z leads to an increase in the
pedestal width and a concomitant decrease in its
height. TheproportionoftheeffectivePSFthatre-
mains ballistic is e−σφ2, while the rest is scattered.
Clearly, as σ 2 increases the effective PSF becomes
φ
progressively more diffuse.
Fig. 2. Plots of Γ (ρ) for different values of σ 2.
0 φ
(cid:113)
Γ¯ (ρ ) ≈ e−σφ2 +(1−e−σφ2)γ (ρ 1+σ 2). (8)
0 d 0 d φ
Fig. 3. Example of an effective PSF(ρ) after degrada-
tion by a phase screen. The initial diffraction-limited
Equation 8 is found to be an excellent approxi-
mation to Eq. 7 for values of σ 2 much smaller and PSF (PSF0) is reduced by a factor e−σφ2 and rides on a
much larger than one, and onφly deviates slightly broader blurred background of power 1−e−σφ2.
from Eq, 7 when σ 2 is close to one. We thus adopt
φ
itasageneralexpressionforΓ (ρ )sinceitprovides Some comments on the validity of Eq. 10. To
0 d
much clearer insight into its physical meaning. begin, we recall that we have limited ourselves
As an example, let us consider the object to be a to small propagation angles. ζ must therefore
pointsourcelocatedattheoriginofthefocalplane. be smaller than z, meaning we implicitly assume
(cid:112)
That is, we write I0(ρ) = λ2I0δ2(ρ). Plugging this λ 1+σφ2/lφ (cid:28) 1. Moreover, in assuming that
into Eq. 5 , we find the effective object intensity thelightfromthepointobjectsamplesmanyphase-
to be screen correlation areas, we implicitly restrict the
microscope NA to values much larger than l /z. In
φ
turn, this means that ζ is implicitly assumed to be
(cid:90)
I(ρ) = 1 I e−i2πρ·ρ(cid:48)/zλΓ¯ (ρ(cid:48))d2ρ(cid:48). (9) larger than the diffraction limited spot size λ/2NA.
z2 0 0 Bearing these assumptions in mind, we may eval-
uate the Strehl ratio of our effective PSF , defined
Propagating this intensity through our imaging
by S =PSF(0)/PSF (0) ≈ λ2PSF(0)/πNA2. For
device and making use of Eq. 9, we derive an effec- 0
phase variances not so large that the Strehl ra-
tive PSF given by
tio is defined primarily by ballistic light we obtain
S = e−σφ2. This result is the same as obtained
PSF(ρ) = e−σφ2PSF (ρ)+(1−e−σφ2) 1 e−ρ2/ζ2, for astronomical imaging through a turbulent at-
0 πζ2 mospheric layer [1].
(10) How do we recover from the degradation in
where PSF0(ρ) is the initial device PSF absent the PSF(ρ) caused by the phase screen? We turn now
phase screen, and we introduce the length scale to two possible strategies involving different AO
(cid:112)
ζ = zλ 1+σφ2/πlφ. A plot of this effective PSF configurations.
is illustrated in Fig. 3. We observe that it features
two components. The first is an attenuated version 3. Pupil AO
of the initial PSF , leading to a sharp, diffraction- The most common AO configuration involves plac-
0
limited peak that is attenuated because of loss of ing a DM in the pupil plane of the imaging de-
ballistic light due to scattering. The second is a vice. Wavefront correction is then based either on
5
knowledge gained from a direct measurement of DM should effectively phase conjugate E (ξ). This
p
the aberrated wavefront at this plane (wavefront- occurs when
sensing-basedAO),orbyiterativetrialanderrorto
optimize a user-defined image metric (image-based
E∗(ξ)
AO).Ineithercase,wavefrontcorrectionisdesigned t (ξ) = p . (12)
dm
|E (ξ)|
to undo the effects of aberrations for a particular p
spotintheobject, typicallythelocationofa“guide
At this point, the math becomes difficult and
star”[1],withthehopethattherangeofthecorrec-
we need to introduce a simplification to proceed.
tion about this spot is large enough to encompass
Specifically, we replace the denominator in Eq. 12
neighboring objects of interest. In astronomy par-
with the (constant) field amplitude that would be
lance, this range is called “seeing” or the “isopla-
present without the phase screen, and write
natic patch”. Here, we call it the corrected FOV of
ourmicroscope. Ourgoalinthissectionistoderive
(cid:90)(cid:90) z
an expression for this FOV. tdm(ξ) = ei2πκ⊥·(ρ+fξ)e−iπzλκ2⊥t∗(ρ)d2ρd2κ⊥.
To begin, we assume that the object spot whose
(13)
wavefront we would like to correct is located at
Thissimplificationhasnoeffectontheaveragein-
the origin. This could be the location of a fluo-
tensity at the pupil plane, and we may justify it on
rescent beacon (serving as a guide star), or of an
thegroundsthat,withtheaidoftheapproximation
object point we have arbitrarily selected for image
in Eq. 4, we maintain |t (ξ)| ≈ 1, in agreement
optimization. To evaluate the aberrated wavefront dm
with the phase-only nature of our DM. Neverthe-
produced by this spot, we write the object field at
less, this simplification does somewhat modify the
this spot as E (ρ) = λ2E δ2(ρ). Inserting this into
0 0
field statistics at the pupil plane, the ramifications
Eq. 1, and performing a scaled Fourier transform
of which will be discussed below.
to calculate the resultant field at the pupil plane,
We recall that t (ξ) here is the shape applied
we find dm
to the DM that performs optimal AO correction
exactly at the origin of the focal plane. We can
λ (cid:90)(cid:90) z now evaluate the spatial range, or FOV, over which
Ep(ξ) = E0 e−i2πκ⊥·(ρ+fξ)eiπzλκ2⊥t(ρ)d2ρd2κ⊥, this correction remains effective. To do this, we be-
f
(11) gin now with an arbitrary (albeit spatially incoher-
where ξ is a 2D transverse coordinate in the pupil ent) object field. The effective field at the object
planeandf isthefocallengthofthemicroscopeob- plane that takes into account the aberrations im-
jective (see Fig. 1). We observe that in the absence parted by phase screen is given by E(ρ) in Eq. 1.
ofaphasescreen(t(ρ) = 1), E (ξ)becomesaplane To take into account the additional, hopefully cor-
p
waveofconstantamplitudeλE /f thatisuniformly rective, effect of the DM, we propagate this field to
0
spread across the pupil plane, as expected. In con- the pupil (a scaled Fourier transform), multiply it
trast, when the phase screen is present Ep(ξ) be- by tdm(ξ), and then propagate it back to the focal
comes structured both in amplitude and in phase. plane (a scaled inverse Fourier transform). Using
To correct for this effect of the phase screen, we the same math and associated approximations as
can act on E (ξ) with a wavefront corrector. Ide- in the previous section, and keeping only dominant
p
ally, E (ξ)shouldbemadeintoaplanewaveagain, phase correlations, we arrive at
p
meaningweshouldflattenitbothinamplitudeand
phase. However, wavefront correctors such as DMs I (ρ) ≈ I (ρ)Γ¯2(ρ). (14)
are never ideal and generally act on phase only. ao 0 t
The best we can do is to flatten the phase of E (ξ) Again, this result is subject to the same condi-
p
and leave its amplitude unchanged. Nevertheless, tions of validity as before. Moreover, we have as-
as we will argue below, the consequences of apply- sumed that the DM spans the entire pupil (not too
ing phase-only wavefront correction are not all that small), and its actuators are sufficiently dense to
severe. accurately represent t (ξ) (more on this below).
dm
In effect, the DM is itself a phase screen whose Equation 14 warrants scrutiny. I (ρ) is the ac-
0
transmission function can be written as t (ξ). To tual object intensity; I (ρ) is the effective object
dm ao
correct for the phase variations in E (ξ), the DM intensity as observed through the phase screen and
p
shouldimpartitsownphasevariationsthatarepre- corrected by the pupil DM. The window of AO cor-
cisely the negative of those of E (ξ). That is, the rection is thus characterized by Γ¯2(ρ). From Eq. 6,
p t
6
we note that this window is equivalent to Γ¯ (ρ),
0
but with a phase variance σ 2 that is effectively
φ
doubled. We can thus recast it using the approx-
imation provided by Eq. 8. However, it is at this
point that we introduce a small correction to our
results. Namely, we write
π (cid:113)
Γ¯2(ρ ) ≈ e−2σφ2 +(1−e−2σφ2) γ (ρ 1+2σ 2).
t d 4 0 d φ
(15)
The reader can observe that we have introduced
an extra factor of π/4 in the second term. The
justification for this term is as follows. We recall
thatthereasonforseparatingΓ¯ (ρ )intotwoterms Fig. 4. Interpretation of what defines FOV for pupil
0 d
was to better identify the ballistic and scattering AO. As an object point ρ1 becomes displaced from the
propagation components. The second term arises guide-starpointρ0,itswavefrontcorrectionbecomesun-
correlated with the fixed phase screen aberrations, and
from light that is scattered by the phase screen.
AO fails.
For light that originates from a point at the focal
plane, the scattered component of this light then
impingesthepupilplanewithspatiallyvaryingam-
thephasevariationsarelarge(i.e. theregimewhere
plitude and phase. Indeed this scattered compo-
AO is potentially most useful!) the FOV depends
nent takes on the characteristics of fully developed
not on the characteristic scale l of the aberrating
speckle [14] (fully developed because we have re- φ
features, but rather on their (inverse) characteris-
moved from it all ballistic contribution). Upon cor-
tic slope. This result is similar to a general re-
recting the wavefront of this speckle field, the best
sult for imaging through phase screens discussed in
the DM can do is flatten its phase. The amplitude
[14]. Moreover,theFOVisindependentoftheloca-
distributionofthespecklefieldremainsunchanged,
tion z of the phase screen. These observations can
meaningitcontinuestoobeyRayleighstatistics. In
readily be understood from Fig. 4. We recall that
the typical case that the microscope pupil is physi-
any wavefront correction provided by a pupil-plane
cally smaller than the field distribution at the pupil
DM must be spatially invariant in the sense that it
plane (i.e. there is some energy loss), then the best
must be imparted equally to all object points. The
the DM can do is yield a corrected image intensity
wavefront correction can be thought of, therefore,
from this scattered light component that is π/4 re-
asfigurativelytrackingeachobjectpoint. However,
duced compared to the same component with no
for object points displaced from the guide star the
phase screen [16]. We need to include this correc-
wavefront correction becomes rapidly uncorrelated
tion factor in Eq. 15 because our derivation made
with the aberrations it is intended to correct. In
useofEq.13fortheoptimalDMprofileratherthan
this case, the correction actually produces worse
the more correct Eq. 12.
imaging than if there were no DM at all, because
As an aside, we note that the spatial extent of
it leads to the presence of two uncorrelated phase
the speckle grains at the pupil plane is inversely
screensintheimagingopticsratherthanone(hence
related to the spatial extent of the aberrated inten-
the effective doubling of σ 2). Because the micro-
sity pattern at the object plane, and is thus given φ
scopesystemistelecentric,thisFOVisindependent
by, roughly, λf/2ζ (akin to the Fried parameter
of z.
in astronomical imaging [15]). The DM actuators
must be smaller than this to avoid undersampling We close this section with a reminder that our
of t (ξ). derivation of a presumed that pupil AO was asked
p 0
Tosummarize,Γ¯2(ρ )representsthewindowover tooptimizetheimageatasinglepointonly,namely
t d
which pupil-based AO is effective. For phase varia- at the origin. One might wonder what would hap-
tions of standard deviation σ one radian or larger, penifitwereaskedtooptimizeoveralargerregion,
φ
this window is predominantly defined by the sec- say of size A. This problem is tantamount to op-
ond term and yields a FOV of diameter a ≈ timizing multiple guide stars simultaneously. Such
0
(cid:112)
2l / 1+2σ 2. This FOV becomes narrower with simultaneous optimization is possible; however, it
φ φ
increasing phase variations, approaching the limit is known to lead to reduced contrast enhancement
√
of 2l /σ . In other words, in the regime where at each guide star by a factor of N, the number of
φ φ
7
guide stars [16]. That is, while pupil AO can, in The microscope pupil defines a characteristic max-
principle , correct over a range A larger than a , imum cone size (i.e. NA) of light that can be col-
0
the quality of this correction, as measured by con- lected from any object point. If this cone size is
trast enhancement, is expected to rapidly degrade large enough that it spans the projected DM for all
as (a /A)2. object points within the FOV, then conjugate AO
0
corrects equally well throughout this FOV. On the
other hand, if the cone size is smaller (as shown),
4. Conjugate AO
then the DM causes vignetting. This vignetting
In the previous section we characterized a prob-
begins to occur at distances roughly zNA from the
lem (often debilitating) of pupil-plane AO that it
FOVedges(note: theDMhereisassumedtobeob-
cannot provide extended FOVs. This disadvan-
structing beyond its edges). Nevertheless, despite
tage was interpreted as arising from the property
this potential issue of edge effects, it is clear (and
that the pupil-plane AO wavefront correction effec-
wellknown[4])thattheFOVprovidedbyconjugate
tively tracks the different points in the object plane
AO can be significantly larger than that provided
while the aberrating phase screen remains fixed. A
by pupil AO.
solution to this problem is clear. The wavefront
correction should instead be locked to the phase 5. Experimental demonstration
screen rather than to the object points. This can The FOV advantage of conjugate AO can be
be achieved by placing the DM conjugate to the demonstrated experimentally. In most cases, AO
phase screen itself, called conjugate AO. is applied in laser scanning microscope configura-
tions [17–27]. We apply it here instead in a bright-
fieldconfiguration. OursetupisillustratedinFig.6
and is basically a 1.33× magnification microscope
modified to accommodate a pupil or conjugate DM
(Boston Micromachine Corp. MultiDM, 140 actu-
ators in a square 12×12 array without the corner
actuators, 400µmactuatorpitch). Athintransmis-
sionobjectistrans-illuminatedbyaLED(660nm).
A thick diffuser is inserted in the LED path, just
before the object, to ensure object spatial incoher-
ence. The NA of the microscope is about 0.04, as
defined by a ≈8 mm diameter iris pupil. We note
that there is a factor of about two demagnification
in the imaging optics from this pupil to the pupil
Fig.5. InterpretationofwhatdefinesFOVforconjugate DMtoproperlymatchtheirrespectivesizes. Amir-
AO. The fixed wavefront correction cancels the aberra- ror blank is inserted in the place of the conjugate
tions caused by the fixed phase screen. The FOV is DM when the system is in a pupil DM configura-
then limited by size of the projected DM itself, albeit
tion, and vice versa. The camera is a Thorlabs
withblurrededgesdependentondistancez ofthephase
DCC1545M CMOS (pixel size 5.2µm square).
screen and the microscope NA.
To introduce spatially variant aberrations in our
microscope, we inserted a phase screen a distance
What is the FOV associated with conjugate AO? z (here 35 mm) from the object. The phase screen
Based on the above interpretation, it is clear that consisted of a blank microscope slide onto which
theFOVmustbethesizeoftheDMprojectedonto was spray-painted a thin layer of clear acrylic. The
the phase screen. For example, if the DM is con- profile of this phase screen was measured by a Zygo
jugate to the phase screen with unit magnification, NT6000 white-light interferometer and exhibited
then the FOV is the same size as the DM itself. rms phase variations on the order of σ ≈ 9.5 ra-
φ
Again, since our microscope system is telecentric, dians (see discussion below), corresponding to just
this FOV is independent of the z location of the aboutthelimitofwhatcanbecorrectedbyourDM
phase screen, provided the DM is maintained con- based on its 3.5µm peak-to-valley stroke. A char-
jugate. However, one must be careful with this last acteristic spatial scale of the phase variations was
statement since there may be edge effects that are very roughly estimated to be of order l ≈ 1 mm,
φ
dependent on z. Such effects are best appreciated leadingtoananticipatedpupilAOcorrectionrange
with a simple ray optic picture as shown in Fig. 5. of about a ≈ 150µm.
0
8
Fig. 6. Experimental setup. Lens focal lengths are f = 100 mm, f = 100 mm, f = 100 mm, f = 200 mm,
1 2 3 4
f = 75 mm, and f = 50 mm. Solid vertical lines correspond to object planes; dashed vertical line correspond to
5 6
pupil planes. Note that in reality the DMs are reflective and the layout was doubly folded.
Admittedly, ascenariowherealargeemptyspace
separates an object from an aberrating layer is un-
likelytobefoundinactualmicroscopyapplications.
The purpose of this experiment is only to highlight
some salient features underlying conjugate versus
pupil AO.
As noted previously, when AO is used in mi-
croscopyitistypicallyappliedtolaserscanningmi-
croscopes where different wavefront corrections are
targeted sequentially to different spots distributed
throughout the sample. In our case, we wish to ap-
plyAOinanon-scanningbright-fieldconfiguration.
That is, we wish to apply a single wavefront correc-
tiontotheentireimage. Todetermineanoptimized
DM shape, we used an image-based stochastic par-
allel gradient descent (SPGD) procedure that op-
timizes a particular image metric [28]. Here, the
metric was chosen to be the image contrast mea-
sured within a user-selected square zone centered
about the image origin (contrast being defined as
the square root of the image variance divided by
themean). Theapplicationofcontrast-basedwave-
front optimization works well when the object is a
localized guide star; however, it is known to fail
when the object is arbitrary and extended, since
it tends to re-distribute light into mottled patterns
rather than improve image sharpness. To circum-
vent this problem we proceeded in two steps. First, Fig. 7. Experimental results for pupil AO. Uncorrected
we inserted as an object an array of apertures at (DM flat) images of an aperture array of period 200µm
the focal plane (see Fig. 7). This calibration ob- (a) without and (b) with the presence of an aberrating
ject served as a well-defined homogeneous array of phase screen. (c) Same image after AO correction that
guide stars, enabling contrast-based DM optimiza- optimized contrast in a zone about the origin of size
250×250µm2 (single guide star). Uncorrected images
tion to perform adequately. Second, following DM
of tissue section (d) without and (e) with phase-screen,
optimization (pupil or conjugate), we replaced the
and (f) after AO correction using the same DM pattern
aperture array with an object of interest, namely a
as established for (c). Image sizes are 4×4mm2 at the
thin section of H&E-stained mammal muscle ten-
focal plane.
don mounted on a microscope slide (Carolina Bio-
logical Supply Co.).
The results for pupil AO are shown in Fig. 7.
9
Clearly, pupil AO was effective at improving image
quality near the origin. However, just as clearly,
it was ineffective at improving quality even a small
distance from the origin. The same was true when
we inserted the tissue sample, as can be seen in
Fig. 7. Based on the properties of the phase screen,
the FOV of the wavefront correction was expected
to be about a ≈ 150µm, which is in rough agree-
0
ment with experiment .
The results of conjugate AO are shown in Fig. 8.
In this case, we chose correction zones that were
progressively increased in size (only two of which
are shown). As is apparent both with the guide-
star array and tissue sample, the resultant FOV in-
creases with correction zone size, approaching the
size of the full unaberrated image (Fig. 7a). Of
particular interest are Figs. 8a,c. The zone of cor-
rection here was 250 × 250µm2, meaning AO was
Fig.8. ExperimentalresultsforconjugateAO.Imagesof asked to optimize the contrast of the central guide
aperture array after AO correction that optimized con- star only. And yet clearly the resultant correction
trastinazoneabouttheoriginofsize(a)250×250µm2 FOV is much larger than this. In this case, the cor-
(singleguidestar),and(b)4×4mm2 (entireimage). (c) rection FOV corresponds roughly to the region of
and (d) are corresponding images of tissue section. the phase screen illuminated by the central guide
star, as defined by the microscope pupil. That is,
it is of size roughly zNA.
Upon closer inspection of the corrected images
near the origin (Fig. 9), we observe that conjugate
AO did not provide quite as crisp a correction as
pupilAO,Onepossiblereasonforthisisthatonlya
fewconjugateDMactuatorscontributedtothecor-
rection of this small region, whereas for the pupil
DM all the actuators contributed. Conjugate AO
may have thus suffered from a slight problem of in-
sufficient actuator sampling (a problem that can be
corrected with improved sampling). Another possi-
blereasonisthatedgeeffectsmayhaveundermined
the correction efficacy.
Finally,wecomparetopographymapsoftheopti-
mized DM shapes with those of the phase screen it-
self. The correspondence is apparent, both in form
and amplitude. Specifically, we note a factor of
≈ 4 difference in topography amplitude between
Fig. 9. Recapitulation of results. 600×600µm2 blowup the measured phase screen and DM shapes. This
of zones about the origin taken from images of tissue arises in part from the impact of the acrylic topog-
section (a) uncorrected without aberrations, (b) uncor- raphy on its wavefront (the acrylic index of refrac-
rectedwithaberrations,(c)pupil-AOcorrected,and(d) tion ≈ 1.5 leads to an optical path difference that
conjugate-AO corrected (taken from Fig. 8d). is about half the local topographic height). An-
otherfactoroftwoarisesfromthefactthatourDM
operates in reflection mode, leading to an effective
In this case, the correction zone was chosen to be wavefront doubling. In our case the phase-screen
small, such that it spanned only a single guide star aberrations were measured to be 1030 nm rms in
(correction zones larger than this led to DM itera- wavefront, consistent with an acrylic topography of
tions that failed to converge or increase contrast). 2060 nm, and corrected by a DM shape of 514 nm
10
mal DM location. Even in the case where there
indeed exists a well-defined, dominant aberrating
layer, the placement of the DM in its conjugate
plane may not be so straightforward. For exam-
ple,inourexperimentaldemonstrationtheDMwas
placed a distance z beyond the nominal microscope
image plane (see Fig. 6). However, in the more
usual case of a microscope with magnification M
much greater than unity, the DM should be placed
instead a distance M2z from the image plane. For
large M this distance may be problematic and re-
quire additional re-imaging optics. There is also
the issue of DM actuator size. For pupil AO, the
DMactuatorsshouldbesmallerthanλf/ζ;forcon-
jugate AO they should be smaller than l , which,
φ
for weak phase variance, is more restrictive by a
factor ≈ z/f (magnification notwithstanding). Fi-
nally, we must consider how the DM optimization
is actually performed. In our demonstration exper-
iment we had the luxury of being able to swap in a
guide-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-
Fig. 10. Comparison of the wavefront shapes roughly at sors (e.g. [29–31]) have not yet been applied to AO
the same location of (top) the phase screen and (bot- in microscopy configurations.
tom) the DM after conjugate AO with full-image con-
Nevertheless, the above caveats are generally
trast correction (i.e. same as in Fig. 8b). In both cases,
technicalinnature. Consideringthepotentiallysig-
the wavefrontshapes were measured witha Zygo white-
nificant FOV advantage of conjugate AO, its future
light interferometer.
implementation, perhaps in conjunction with pupil
wavefront correction, may well prove to become a
rms. We recall that phase is related to wavefront new standard in AO applied to microscopy.
by φ(ρ) = 2πW(ρ).
λ Acknowledgments
6. Discussion T. Bifano acknowledges a financial interest in
In view of the apparent FOV advantage of con- Boston Micromachines Corporation. Support for
jugate over pupil AO, one may wonder why it is H. Paudel was provided by the Industry/University
not more prevalent in the microscopy community. Cooperative Research Center for Biophotonic Sen-
There may be several reasons for this. To begin, sors and Systems.
the most egregious aberrations caused by a sample
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