Autoconfocal microscopy with nonlinear transmitted light detection

Transcription

1 1486 J. Opt. Soc. Am. B/ Vol. 21, No. 8/ August 2004 Thomas Pons and Jerome Mertz Autoconfocal microscopy with nonlinear transmitted light detection Thomas Pons and Jerome Mertz Laboratoire de Neurophysiologie et Nouvelles Microscopies, Ecole Supérieure de Physique et de Chimie Industrielles, Institut National de la Santé et de la Recherche Medicale EPI00-02, Centre National de la Recherche Scientifique FRE2500, Paris, France Received September 23, 2003; revised manuscript received March 11, 2004; accepted March 31, 2004 We describe a simple and robust technique for transmission confocal laser scanning microscopy wherein the detection pinhole is replaced by a thin second-harmonic generation crystal. The advantage of this technique is that self-aligned confocality is achieved without the need for signal descanning. We derive the point-spread function of our instrument and quantify both signal degradation and background rejection when imaging deep within a turbid slab. As an example, we consider a slab whose index of refraction fluctuations exhibit Gaussian statistics. Our model is corroborated by experiment Optical Society of America OCIS codes: , , , , INTRODUCTION A novel technique was recently reported for transmission confocal laser scanning microscopy based on the replacement of a detection pinhole by a thin frequency-doubling crystal. 1 A pulsed infrared laser beam is focused through a sample and then imaged (refocused) onto the crystal. A short-pass filter is placed immediately behind the crystal such that only second-harmonic generation (SHG) is detected. Because the SHG power is inversely proportional to the effective area of the laser spot incident on the crystal, the crystal acts as a virtual pinhole, producing a large signal only when the laser spot is tightly focused, similarly to a physical pinhole. The notable advantage of this technique is that virtual confocality is ensured regardless of where the laser spot is focused onto the crystal, meaning that fast beam scanning is allowed without any need for elaborate descanning. We call such an instrument an autoconfocal microscope (ACM). A cursory description of an ACM based on quadratic detection and valid for thin samples only has been presented. 1 Our goal here is to characterize the imaging properties of such an ACM for samples that are both thin and thick, as defined relative to the scattering mean-free path (MFP) (l s ). In Section 2 we consider a semitransparent (or thin) sample and derive the ACM point-spread function (PSF) for both absorbing and phase-shifting point objects. We show that the PSFs of an ACM and a transmission confocal laser scanning microscope (TCLSM) are effectively the same. We qualitatively argue that optical sectioning is obtained only to the extent that scattered background is incoherent. In Section 3 we extend our discussion to thick samples and explicitly quantify the degree to which the ACM rejects scattered background a fundamental property of confocal microscopy. For simplicity, we consider only nonabsorbing media, which we characterize by a (real) refractive-index autocorrelation function. Finally, we theoretically evaluate the capacity of an ACM to distinguish a localized object of interest embedded within a turbid slab, assuming that the refractive-index fluctuations in the slab obey Gaussian statistics. Our theory is largely inspired by the seminal work of Tatarski 2 and Rytov et al. 3 used to describe imaging through the turbulent atmosphere, along with the indispensable references by Goodman 4 and Ishimaru. 5 Finally, we corroborate our theoretical analysis with simple demonstration experiments. 2. SEMITRANSPARENT SAMPLE (THICKNESS l s ) The basic layout of our ACM is shown in Fig. 1. A laser beam, depicted here as a point source, is focused into a sample by a lens of numerical aperture sin. A second lens refocuses this focal spot onto a thin nonlinear crystal (assumed thin enough that phase matching is always ensured). We consider the case of a parfocal geometry wherein the lenses are identical and of unit magnification. Generalizations to nonidentical lenses or nonunity magnifications are straightforward and will not be considered here. Our goal in this section is to derive the PSF of an ACM and discuss its capacity for axial sectioning. We consider a semitransparent sample and begin by deriving the intensity distribution incident on the image plane (i.e., on the nonlinear crystal). For ease of notation, we drop all scaling constants throughout this paper. Following the usual notational convention, 6 we write the PSFs of the lenses as h v 1, u 1 P exp iv 1 exp iu 1 1/4 sin 2 /2 2 /2 d, (1) where we adopt the axial and radial optical units u 1 4kz sin 2 ( /2) and v 1 k sin, respectively, and k is the wave vector in the sample medium. We assume that the lenses are ideal and possess no aberrations. That is, /2004/ $ Optical Society of America

2 Thomas Pons and Jerome Mertz Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. B 1487 Fig. 1. Parfocal unit magnification configuration of a quadratic detection ACM. A scanning laser beam is focused onto a sample and refocused onto a thin SHG crystal. The resulting SHG is isolated (filter) and detected by a photomultiplier tube (PMT). the coordinates of the lens pupil functions P( ) are normalized such that P( 1) 1 and P( 1) 0. To determine the SHG power produced by the crystal, we evaluate the electric field at the image plane given by U v h v 1, u 1 t v 1, u 1 h v v 1, u 1 dv 1 du 1, (2) where t(v 1, u 1 ) is the three-dimensional object transmission function, and we neglect multiple scattering since here we consider only semitransparent samples. A. Transparent Sample We begin by treating the simplest case of a completely transparent sample that produces no scattering. In this case t(v 1, u 1 ) (u 1 ) and the field distribution at the image plane becomes U 0 h v 1,0 h v v 1,0 dv 1 h, 0. (3) Accordingly, the intensity distribution at this plane becomes I 0 U 0 2 P 1 P 2 exp iv 1 2 d 1 d 2. Equation (4) represents a ballistic light distribution, since it arises from unscattered transmitted laser light only. Making use of the variable changes c ( 1 2 )/2 and d 1 2, we find that I 0 ( ) is the Fourier transform of the function (4) H 0 d P c d /2 P c d /2 d c. (5) Equation (5) is the diffraction-limited optical transfer function of a simple lens. This is expected since our parfocal two-lens system is equivalent to a single lens when the sample is transparent. The functions I 0 ( ) and H 0 ( d ) play important roles below. B. Point-Spread Function To derive the PSF in our microscope configuration, we suppose that our sample now contains a single point perturbation located at the position (v, u ). That is, we write 7 t v 1, u 1 ; v, u u 1 v 1 v u 1 u, (6) where is the modulus of the transmission perturbation, assumed small. The real part of corresponds to an absorption perturbation whereas the imaginary part corresponds to a phase perturbation. For simplicity we assume that the sample is scanned in three-dimensions, with the understanding that formally equivalent results are obtained if the beam is scanned instead of the sample. The perturbed intensity distribution at the image plane is where I v; v, u U 0 v U v; v, u 2, (7) U v; v, u h v, u h v v, u, (8) and, accordingly, the resultant SHG power produced by the crystal is SHG v, u I 2 v;v, u dv S 0 4Re S 1 v, u..., (9) expanded only to the first-order perturbation in. zeroth-order ballistic component is defined by S 0 U 0 4 dv I 0 2 dv H 0 2 d d d, The (10) where d is interpreted as a normalized spatial frequency, 8 and the last equality is an expression of Parseval s theorem. The first-order term, corresponding to the product of a scattered field and three ballistic fields, is defined by S 1 v, u I 0 U 0 U * v; v, u dv. (11) As is apparent from Eq. (11), the function I 0 ( ) plays an identical role here as a pinhole transmission function in standard confocal microscopy hence the appellation autoconfocal microscopy for our technique. Figure 2 depicts various PSFs obtained for purely absorbing or phase-shifting perturbations. We recall that the function I 0 ( ) represents the distribution of ballistic light at the crystal plane. As defined by Eq. (4), I 0 ( ) is the Airy function J 1 ( )/ 2 whose effect, as observed from Fig. 2, is essentially identical to that of a standard TCLSM. For an amplitude perturbation, we find the contrast S 1 /S 0 of our ACM to be the same as that of a standard TCLSM whose pinhole radius is 1.65 optical units. This comparison provides a convenient estimate for the effective pinhole size of our ACM. The theoretical results shown in Fig. 2 can be compared with the experimental results of Ref. 1, bearing in mind that the point perturbation in this reference (a latex bead) provoked both a phase shift and an effective absorption, since the light scattered by the bead was partially clipped by the lens pupil. We also note that, even though a pure phase-shifting perturbation does not change the total power incident on the image plane, it can, according to Fig. 2(c), lead to an increase in power transmitted through a finite (but nonzero) size pinhole.

3 1488 J. Opt. Soc. Am. B/ Vol. 21, No. 8/ August 2004 Thomas Pons and Jerome Mertz C. Optical Sectioning It is well known that the main advantage of confocal fluorescence microscopy is its capacity for out-of-focus fluorescence background rejection. In particular, a uniformly fluorescent transverse slice produces a signal that scales as u 2 s, where u s is its axial distance from the focal plane. Such a scaling law, which is necessary for optical sectioning, applies even in a transmission geometry because of the incoherent (random phase) nature of fluorescence emission. However, there is a fundamental difference between TCLSMs that are based on fluorescence and on transmission. Whereas a fluorescence microscope exhibits a dark background in the absence of a sample, an ACM, in contrast, exhibits a bright background, stemming from the term S 0 in Eq. (9). This background cannot be easily eliminated. Moreover, the capacity of an ACM for optical sectioning is sample dependent. This problem is readily apparent if one considers simple samples such as a uniformly phase-shifting or absorbing transverse slice. The ACM signals produced by either of these samples is independent of u s and no optical sectioning is possible (this inability to reject a uniform background is sometimes referred to as the missing-cone problem). 9 However, samples of interest are rarely so simple. If one considers a transverse slice made of point-size perturbations that produce random phase shifts, the signal produced by an ACM then crucially depends on u s. The transmittance of such a slice can be modeled as t v 1, u 1 ; u s u 1 n n v 1 v n u 1 u s, (12) where n is randomly distributed in phase. If a large number of these perturbations is illuminated by the laser beam, insertion of Eq. (12) into Eq. (9) leads to a cancellation of the S 1 term, leaving the second-order term as a sample-dependent response. Such a response exhibits optical sectioning since it scales with u s in the same way as a fluorescence confocal response. In effect, by imposing random phases on n we have mimicked the incoherence of a fluorescence signal. This argument also applies to finite-size perturbations. The only requirement here for an optical sectioning response is that the number of illuminated perturbations must be large ( 1). This condition is always met, in theory, since the laser beam area expands with increasing u s. The tightness of the optical sectioning depends on how small u s is when this condition is met. Fig. 2. Quadratic detection ACM PSFs. First-order signal responses produced by a point object that is purely absorbing, (a) axial and (b) radial, or purely phase shifting, (c) axial with null radial response; o.u., optical units. 3. THICK SAMPLE (THICKNESS l s ) In practice, samples of interest are often highly scattering, leading to severe limitations on imaging depth. Our goal in this section is to quantify these limitations by extending our above analysis to thick samples. We consider an intermediate regime often encountered in biological imaging wherein light propagating through a sample is neither wholly ballistic nor wholly diffusive. 5 In particular, we consider scattering that is dominantly forward directed. Such scattering arises from samples that provoke local phase variations that do not significantly deflect the light field but nonetheless highly degrade image quality. We adopt the geometry shown in Fig. 3. The sample consists of a slab of thickness L, into which a small object of interest is embedded. As in Subsection 2.B, we suppose the object provokes a localized amplitude or phase perturbation whose signal we wish to evaluate. We derive both signal and background as a function of slab position (or, equivalently, object depth). For simplicity, we assume that the object is situated exactly at the focal point and that the slab medium is nonabsorbing, homogeneous, and isotropic. These assumptions allow us to emphasize the main features of our results, although they are not fundamental to our analysis.

4 Thomas Pons and Jerome Mertz Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. B 1489 A. Background Before deriving the signal produced by an isolated perturbation of interest, we derive the associated background in the absence of any specific perturbation. As in Subsection 2.A, we must calculate field U 0 at the image plane. This time, however, we take into account the phase shifts incurred by the light upon propagation through the entire slab thickness. These are L ( ) A ( ) B ( ). By correspondence with Eqs. (1) and (3), we write U 0 v P 1 exp iv 1 exp i L 1 d 1, (15) leading to S 0 P 1 P 2 P 3 P 4 exp iv K 1,2,3,4 L d 1 d 2 d 3 d 4 dv, (16) where we have defined Fig. 3. (a) Configuration in which a point object is embedded at a depth L A within a turbid slab of thickness L. (b) Equivalent configuration in which phase variations provoked by the slab are projected into appropriate lens pupils. As is apparent from Fig. 3(a), the sample can be thought of as two adjacent semislabs of thicknesses L A and L B, situated, respectively, before and after the object plane. By assumption, backscattered light is neglected and we assume that the light that traverses these semislabs travels from left to right only. The semislabs provoke random phase variations in the light field whose effects can be examined separately: semislab A defocuses the light as it propagates to the object plane, and semislab B further defocuses the light as it continues to propagate to the image plane. Bearing this picture in mind, we develop a formalism based on the alternative equivalent geometry shown in Fig. 3(b), where we project the phase variations provoked, respectively, by semislabs A and B into the pupil functions of the corresponding lenses. In other words, we mimic the defocusing effects of the semislabs by introducing lens aberrations and write P A,B P exp i A,B. (13) The statistics of these aberrations must be correctly defined so as to match those of the semislabs. We discuss how to define these statistics in Subsection 3.C. For now we characterize the lens aberrations by their autocorrelation function, which, by assumption of transverse homogeneity and isotropy, is a function only of the distance between the aberration coordinates. For lens A we write A d A 1 A 2, (14) where d 1 2 and the brackets correspond to an ensemble average, and we assume that (A) ( d ) 0 for sufficiently large d. A similar equation applies to lens B. Also, since the phase variations provoked by the semislabs are assumed to be uncorrelated, A ( 1 ) B ( 2 ) 0. K L 1,2,3,4 exp i L 1 L 2 L 3 L 4. (17) Since we are concerned here with a typical background, we perform an ensemble average of K 1,2,3,4 L. By assumption, the slab is thick enough that L represents a sum of many independent phase variations, and we write K 1,2,3,4 L exp 1 1 i j L i L j 2, i, j (18) where we have invoked the central limit theorem and made use of the relation exp(i ) exp( 1/2 2 ), which is valid for Gaussian variables. An integration of Eq. (16) over variable v imposes the constraint , leading to the simplification K L 1,2,3,4 H L 2 d exp 1 2, (19) where L 1 L 3 L 1 L 4 L 2 L 4 L 2 L 3, and we have introduced the transfer function (20) H L d exp L 0 L d. (21) The exponent in Eq. (21) is often referred to as (twice) the structure function of the phase variation. 2 The physical meaning of H L is elaborated on below. We note here that, if the slab is transparent (or nonexistent), H L ( d ) 1 for all d. If, instead, the slab is thick enough to provoke significant phase variations, H L ( d ) rapidly decays from unity at d 0 to a small baseline value of exp (L) (0) (see Fig. 4). We define below what we mean by significant and assume for now that H L is sufficiently peaked around the origin that H 2 L takes on nonnegligible

5 1490 J. Opt. Soc. Am. B/ Vol. 21, No. 8/ August 2004 Thomas Pons and Jerome Mertz values in Eq. (19) only when d 0. As a result, the main contribution in the integration in Eq. (16) comes from the region where K 1,2,3,4 L H 2 L ( d ). We then obtain S 0 P c d /2 P c d /2 P c d /2 P c d /2 H 2 L d d c d c d d, (22) which, with Eq. (5), simplifies to S 0 H 0 2 d H L 2 d d d, (23) where S 0 is the average background SHG power obtained when only the slab is taken into account and nothing more (i.e., no object of interest lies at the focal center). A comparison of approximation (23) with Eq. (10) suggests that H L ( d ) can be interpreted as a filter function similar to H 0 ( d ). 4 By limiting the extent of the spatial frequencies that are transferred to the image plane, H L ( d ) provokes a blurring of the focal spot incident on the SHG crystal. Hence, although the presence of the slab does not alter the total power incident on the crystal, it does lead to a reduction in the resultant SHG that the crystal produces. The intrinsic sensitivity of nonlinear detection to defocusing is the basis of ACM background rejection. B. Signal We now derive the signal produced by the point object located at the focal center. We use the same formalism developed above to derive background, but this time we treat the semislabs individually. Referring to Eq. (8) and explicitly identifying the respective phase aberrations in lenses A and B, we write U v P 1 P 2 exp iv 2 exp i A 1 B 2 d 1 d 2. (24) We restrict our analysis here to the first-order perturbation in both absorption and phase contrasts. This firstorder signal [Eq. (11)] becomes S 1 P 1 P 2 P 3 P 4 P 6 exp iv K A 1,2,3,4 K B 1,2,3,6 d 1 d 2 d 3 d 4 d 6 dv, (25) where we have used definitions for K A and K B similar to those in Eq. (17) and adjusted our indices in accord with Eq. (18). Integration over variable v leads to the constraint d , and, following the same reasoning as in Subsection 3.A, we obtain K B 1,2,3,6 H B 2 d, (26) K A 1,2,3,4 H A d H A d, (27) where we have defined d 3 4 and have assumed that H 2 B ( d ) is nonnegligible only for small d, as before, leading to the restriction 6 3. The signal produced by a localized amplitude perturbation is then given by S 1 H 0 d H A d d d H 0 d H B 2 d H A d d d. (28) We note that approximation (28) resembles approximation (23) except that a component of the light transmitted through semislab A prior to its interaction with the object has been isolated (first integral). We also remind the reader that S 1 reveals a phase gradient rather than a phase exactly at the focal center [see Fig. 2(c)]. Approximations (23) and (28) are the main results of this subsection and represent formal expressions for the background and highest-order signal obtained when we used a quadratic detection ACM to image inside a thick slab. C. Gaussian Refractive-Index Fluctuations So far we have made no assumptions about the detailed nature of the phase fluctuations introduced by the slab. We consider here the specific example for which these are produced by refractive-index fluctuations that obey locally Gaussian statistics. Such statistics are routinely used to describe scattering media and are particularly convenient because of their tractability. To this end, we define a transverse autocorrelation function for the refractiveindex fluctuations: n 1 n 2 n 2 exp d 2 /l n 2, (29) Fig. 4. Filter function H L as a function of normalized spatial frequency d. Ballistic light is attenuated by a factor ( exp ) L (0) (dotted line). Nonballistic light is transmitted only below the cutoff frequency 3dB. where we have reverted to the lab-frame coordinate system (, z) relative to the focal center, and l n is a characteristic fluctuation scale, assumed to be the same in all three dimensions. If light propagates an axial distance z l n, it incurs a phase shift k z. On the other hand, for longer axial distances z l n, the phase shift is no longer proportional to the propagation distance but instead performs a random walk with step size kl n. In the latter case the variance of the phase fluctuations, as opposed to their amplitude, scales linearly with axial propagation distance, and we write z d zl n k 2 n 1 n 2, (30)

6 Thomas Pons and Jerome Mertz Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. B 1491 where d 1 2, and z is assumed to be small enough that we can neglect beam convergence or divergence (the derivation of this expression is presented in more detail in Ref. 5). As described above, we use the technique of projecting the slab fluctuations into the lens pupils, which requires the coordinate transformation d d z sin. 8 Referring to Eq. (21), we then obtain H z d exp z 2 1 z d z sin, (31) where 2 k 2 l n n 2 is the variance of the phase fluctuations per unit propagation distance, and we define ( z) ( d ) ( z) ( d )/ ( z) (0). We note that ( z) ( d ) is always equal to one at the origin but becomes more and more narrowly peaked as the propagation distance through the slab increases. To derive the filter function through a thick slab, not just a thin slice, we must take beam convergence or divergence into account. Since the filter functions for sequential slices of thickness z are assumed to operate independently, we make the approximation H L d L H z d. (32) This last step represents one of the main advantages of our having projected the phase fluctuations from the slab (spatial coordinates) to the lens pupils (frequency coordinates), where the filter functions operate multiplicatively. 5 Approximation (32) is a product over the entire slab thickness and can be evaluated by integrating the exponent in approximation (31). We obtain the approximate expression H L d exp L 2 1 exp L 2 exp 2 d 2 V/l 2 n, (33) where we have defined V L (z sin ) 2 z, which approximately corresponds to the volume of the laser beam inside the slab [shaded region in Fig. 3(a)]. As described in Subsection 3.A and is explicit in approximation (33), H L ( d ) consists of a spatial frequency-independent baseline (first term) onto which a narrow peak around the origin (second term) is superposed (see Fig. 4). The physical meaning of these terms is as follows: H L ( d ) represents the effect of the slab on the transmitted beam. This effect is twofold. The baseline term in approximation (33) is an expression of Lambert s law and describes the frequency-independent attenuation of the ballistic (nonscattered) light transmitted through the slab. With this interpretation, the scattering MFP of the slab is defined as l s 2. The peak term in approximation (33) represents the effect of H L ( d ) on the rest of the light transmitted through the slab that has been scattered. Whereas low spatial frequencies are efficiently transmitted, frequencies higher than cutoff 3dB l n l s /V are severely attenuated. We remind the reader that a diffraction-limited focus requires a transmission of frequencies up to d 1. Hence, inasmuch as 3dB 1 (we quantify this below), the second term in H L ( d ) leads to a significant blurring of the nonballistic light at the image plane. We now directly evaluate the background produced by the SHG crystal. For convenience, we make two approximations. First, even though H 0 ( d ) as defined by Eq. (5) can be expressed analytically, 8,10 we adopt the much simpler Gaussian beam approximation H 0 ( d ) exp( 2 d ), which is valid to within the paraxial limit. Second, we relate l n to the more experimentally accessible transport MFP, defined by l s * l s /(1 cos s ), where s is the deflection angle occasioned by a single scattering event. 5 For Gaussian refractive-index fluctuations [Eq. (29)], these are approximately related by k 2 l 2 n l s */l s. 5 As an example, l n is of the order of 1 m for most biological tissues of interest, meaning that the scattering is highly forward directed at optical wavelengths and l s * is typically times longer than l s. 11 Using approximation (33) and performing the integral in approximation (23), we obtain S 0 SHG 0 exp 2L/l s 1 exp 2L/l s R l s *, V, (34) where SHG 0 corresponds to the SHG power obtained if there were no slab (L 0). As discussed above, the effect of the slab is to convert nonscattered ballistic light into scattered light. The thicker the slab, the more this conversion is complete, and the first and second terms in approximation (34) correspond to these ballistic and nonballistic components, respectively. However, the nonballistic component is significantly rejected here by the factor l R l s *, V s * l s * k V 2. (35) This rejection factor is a fundamental consequence of the fact that defocused nonballistic light is ineffective in producing SHG. The greater the defocusing, the greater the 2 rejection, as indicated by the relation R(l s *, V) 3dB. Moreover, the rejection depends only on the intrinsic slab parameter l s * and on extrinsic parameters such as slab thickness and position along the optical axis, both of which govern the interaction volume through the geometric relation V V A V B sin2 L 3 A L 3 B. (36) 3 An illustration of R for different V values is shown in Fig. 5. In this experimental example, the slab is thick enough that the ballistic component can be neglected, meaning that S 0 R(l s *, V). We emphasize that, even when the interaction volume is at a minimum here (slab is centered on the focal plane), the rejection factor still remains considerably smaller than one, indicating that nonballistic light is highly defocused and justifying a posteriori the assumptions that led to approximation (22). The theoretical fit shown in Fig. 5 contains no free parameters and is remarkably accurate despite the simplicity of Eq. (35). We note that, for this example and those pre-

7 1492 J. Opt. Soc. Am. B/ Vol. 21, No. 8/August 2004 Thomas Pons and Jerome Mertz Fig. 5. Background ACM signal (filled circles) obtained experimentally when translating a scattering slab in the z direction (slab is centered when z slab 0). The slab is composed of 1- m latex beads in agarose ( 870 nm, l s 39 m, l s * 490 m, L 340 m). The theoretical trace (solid curve) from Eq. (35) is shown for comparison. The leftmost bracketed terms in approximation (37) represent the laser intensity incident exactly at the point object, consisting of ballistic and nonballistic components. The latter component is diminished by the factor R(l s, V A ) because of spreading the nonballistic light. Using the same apparatus as described in Ref. 1, we experimentally corroborate the validity of our model with test slabs consisting of 1- m latex beads embedded in scattering media (themselves composed of latex beads, some of which are fluorescent, in agarose gels). The parameters l s and l s * can be prescribed for each slab based on the sizes and concentrations of the beads. Moreover, parameter l s can easily be verified by monitoring the average two-photon excited fluorescence (TPEF) signal produced by the fluorescent beads, which is known to decay as exp( 2L A /l s ) to moderate depths. 12 Two regimes can be distinguished based on the relative contributions of ballistic and nonballistic components in the average SHG signal [approximation (34)]. If the ballistic component is dominant [first term in approximation (34)], then S 1 is essentially independent of L A or V A, meaning that the signal produced by a point object of interest, whether absorbing or phase shifting, remains the same at all depths throughout the slab. This regime is illustrated in Fig. 6(b). If the nonballistic component is dominant instead [second term in approximation (34)], then the amount of ballistic light incident on the SHG crystal is negligible. This should not be confused, however, with the amount of ballistic light incident on the point object itself, which can be much greater and lead to contrast. This second regime is illustrated in Fig. 6(c). Hence, although only nonballistic light produces a signal in this second regime, highresolution images can nonetheless be obtained. The signal here decays with object depth. From approximation sented henceforth, V is always large enough that we can use the approximation R(l s *, V) l s */k 2 V. To evaluate the capacity of our ACM to perform deep imaging in a scattering slab, we consider, as in Subsection 2.C, the signal produced by a point perturbation of interest located at the focal center. Depth L A of this perturbation relative to the slab surface is governed by the slab position, which in turn governs L B, V A, V B, and V (only L remains unchanged). Approximating the filter functions in approximation (28), as was done above to obtain approximation (33), we arrive at S 1 exp L A /l s 1 exp L A /l s R l s *, V A exp L L B /l s 1 exp L L B /l s R l s *, V V B. (37) Fig. 6. (a) Simultaneous TPEF (left) and ACM (right) images of a 500-nm fluorescent latex bead. (b) Corresponding ACM contrast (squares) produced by nonfluorescent beads as a function of penetration depth L A in a scattering medium ( 870 nm, l s 126 m, l s * 1800 m, L 700 m) consisting of 1- mdiameter nonfluorescent beads; and sparsely distributed 0.5- mdiameter fluorescent beads. (c) ACM contrast as a function of depth for a different scattering medium ( 870 nm, l s 39 m, l s * 490 m, L 340 m). The solid line represents the theoretical traces for ACM contrast decay as a function of depth (no free parameters). For reference we illustrate decays of the TPEF signal (circles).

8 Thomas Pons and Jerome Mertz Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. B 1493 (37) we infer the rough scaling law for moderate depths S 1 exp( L A / l s ), where the slab MFP has been effectively lengthened by the factor 1/(1 3l s /L), which is valid for L l s. This apparent lengthening of the MFP stems directly from the effectiveness of nonlinear detection in rejecting the nonballistic background. 4. CONCLUSION In summary, the main advantages of the ACM are that it allows fast beam scanning, provides effective background rejection, and can be readily combined with TPEF microscopy. The performance of an ACM is essentially the same as that of a standard TCLSM. The capacity of an ACM for depth penetration depends on the net amount of ballistic light that traverses the slab. If the slab is thin, both the background and the signal are independent of depth. If the slab is thick, the background scales approximately inversely with the light slab interaction volume whereas the signal decays moderately with depth, in accord with the simple model presented above. ACKNOWLEDGMENT We thank Changhuei Yang and Laurent Moreaux for their initial contributions to this research and the Institut National de la Santé et de la Recherche Médicale (INSERM) for financial support. J. Mertz (jmertz@bu.edu) is now with the Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, Massachusetts REFERENCES 1. C. Yang and J. Mertz, Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection, Opt. Lett. 28, (2003). 2. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967). 3. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Heidelberg, 1987). 4. J. W. Goodman, Statistical Optics (Wiley, New York, 1984). 5. A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Wave Theory (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1978). 6. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984). 7. C. J. R. Sheppard and X. Q. Mao, Three-dimensional imaging in a microscope, J. Opt. Soc. Am. A 6, (1989). 8. J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill New York, 1968). 9. N. Streibl, Depth transfer by an imaging system, Opt. Acta 31, (1984). 10. M. Born and E. Wolf, Principles of Optics (Academic, New York, 1970). 11. V. Tuchin, Tissue Optics (SPIE Press, Bellingham, Wash. 2000). 12. E. Beaurepaire and J. Mertz, Epifluorescence collection in two-photon microscopy, Appl. Opt. 41, (2002).