1. Introduction

Widefield phase imaging is of general utility in the biological sciences. The most popular microscopy techniques make use of Zernike phase contrast (PC) [1, 2] or Nomarski differential interference contrast (DIC) [3, 4]. In both techniques, uniform white light transilluminates a sample. Phase variations provoked by the sample are converted to intensity variations by specialized optics in the light path and are then detected by a widefield camera. DIC differs from PC in that it is sensitive to phase gradients along a particular lateral direction and hence confers an apparent 3D relief to the sample, producing shadow effects as if the illumination were incident from an angle.

Alternative phase imaging techniques also produce a similar 3D relief appearance. These techniques undoubtedly originated from the observation familiar to most microscopists that DIC-like shadow effects can be produced simply by partially blocking the light path before or after the sample. When this partial block is placed near a Fourier plane relative to the sample plane, then the sample illumination appears uniform and the resulting image bears striking resemblance to a DIC image, despite the fact that no specialized optics are required whatsoever. Variations of this technique involve placing a partial block in the illumination path (sometimes referred to as oblique[5, 6] or anaxial[7] illumination, or Dodt contrast[8]), or in the detection path (often referred to as Schlieren imaging[9, 10, 11, 12, 13]). In its usual implementation, Schlieren imaging makes use of slit (i.e. coherent) illumination, in which case a half-block in the detection path produces phase contrast by a Hilbert transform[14]. A more elaborate variation of Schlieren imaging, known as Hoffman contrast, also involves slit illumination but uses a specialized partial beam “modulator” with three levels of opacity[15]. Generalizations of Schlieren imaging to incoherent illumination have also been studied[11].

We present a rigorous description that encompasses all of the above techniques by including the possibility of partial-blocks in both the illumination and detection apertures. While such a possibility was previously considered for fixed partial aperture blocks [6, 13], its analysis remained largely qualitative. Our description is general in that we allow the possibilities of partial-blocks of arbitrary offsets and fill-factors. Different offsets and fill-factors are found to lead to different degrees of contrast that range from fully brightfield to fully darkfield. As such, we refer to our general technique as “graded-field” contrast. For simplicity, we confine our analysis to square apertures and consider white light illumination of arbitrary spatial coherence. Finally, we present examples of graded-field imaging of neurons in unlabeled rat brain slices.

2. Formalism

The basic microscope layout is shown in Fig. (1). This is a conventional transillumination widefield microscope which comprises Köhler illumination and afocal imaging. The illumination source, typically a lamp, is imaged onto the back focal plane of the condenser lens fa (equivalently, this source can be located exactly at the back focal plane rather than imaged onto it, as depicted in Fig. 1). We refer to the condenser back focal plane as the illumination aperture. The transilluminated light emanating from the sample, located at plane 1, is then imaged onto a CCD camera, located at plane 3. For simplicity, we consider unit magnification only, though our formalism may easily be generalized to arbitrary magnification. The back aperture of the objective lens fb is located at plane 2. We refer to the objective back aperture as the detection aperture. In both the illumination and detection apertures we introduce translatable light blocks that partially block these apertures to arbitrary fill factors. In Fig. (1) these blocks are introduced from the same side, along the transverse x-axis. Again for simplicity, we consider square apertures which allows us to treat the x and y axes independently, Henceforth, we will consider only the x axis, with the understanding that conventional brightfield imaging occurs along the y axis.

Our strategy will be to derive the mutual coherence incident on (superscript in) and emanating from (superscript out) the sample, followed by the intensity distribution incident on the CCD camera. We begin by noting that the field amplitudes in planes separated by a single lens are related by simple Fourier transforms. Planes 0 and 2 are conjugate to one another and will be referred to as Fourier planes relative to the sample (1) and image (3) planes (also conjugate to one another).

 

Fig. 1. Experimental setup: incoherent white light transilluminates a sample in a 6 f (unit-magnification) imaging line. Partial beam blocks are introduced in the illumination (top) and/or detection (bottom) apertures.

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If the illumination field amplitude at plane 0 (just before the partial block) is given by E ₀(x0), then the field amplitude incident on the sample is

where we have used the Fourier coordinate ξ₀=kx ₀/f_a (k=2π/λ and λ is the average illumination wavelength) and α_1,2=ka _1,2/f_a , where a ₁ and a ₂ represent the physical limits of the illumination aperture as prescribed by the position of its partial-beam block (i.e. a _1,2 are spatial coordinates whereas α_1,2 are Fourier coordinates). If the block is removed then a ₁→-a and a ₂→a, meaning that the fully opened aperture is a square of spatial dimension 2a. We omitted the prefactor in front of the integral in Eq. (1) since this plays no role in our discussion.

We define the mutual coherence[16] by

where the overbar represents a time average. From this, we obtain

This is the mutual coherence function incident on the sample for an arbitrary illumination mutual coherence. For the particular case where the illumination is uniform in intensity but spatially incoherent we write[16]

We then obtain

where we have defined x _1d=x′ ₁-x ₁ and

Throughout this paper, we will use the subscripts d to denote the difference between two variables and c to denote their average (or center). In other words, α_d denotes the width of the illumination aperture and α_c its offset (in Fourier coordinates).

Three comments can be made here. On examining Eq. (5) we note that the mutual coherence incident on the sample is independent of x _1c, meaning that the intensity incident on the sample is spatially uniform independently of the aperture block position. This follows from the general principle of Köhler illumination. Moreover, because the aperture size is finite, the incident illumination is now partially coherent over an approximate range 2π/α_d , as prescribed by the sinc function in Eq. (5). Finally, the effect of an offset in the aperture is to introduce a complex phase in the (otherwise real) incident mutual coherence. This phase will play a critical role in producing a gradient-field contrast.

We now derive the intensity distribution I ₃(x ₃) at the CCD camera given an arbitrary mutual coherence $J_1^{\mathit{\text{out}}}$(x ₁,x′ ₁) emanating from the sample. Following the same line of reasoning as above, we obtain

where we have used the Fourier coordinates ξ₂=kx ₂/f_b and β _1,2=kb _1,2/f_b , where b ₁ and b ₂ represent the physical (spatial) limits of the detection aperture as prescribed by the position of its partial-beam block. As above, if the beam block is removed, then the detection aperture is a square of spatial dimension 2b.

Similarly

where the integration limits can be extented to infinity. Combining Eqs. (7) and (8) and integrating over the detection aperture coordinates, we find

Finally, we obtain

where we have introduced the transfer function

and made use of the definition I ₃(x ₃)=J ₃(x ₃,x ₃). We have also recast mutual coherence in the form J ₁(x _1c,x _1d)= $\overline{E_1(x_{1c}-x_{1d}/2)E_1^{*}(x_{1c}+x_{1d}/2)}$ .

The function G ₁₃ in Eq. (10) can be thought of as a kind of point spread function (PSF), however instead operating on an incoherent intensity, as does a normal PSF, it operates on a partially coherent intensity. As in the illumination case, an offset in the detection aperture introduces a complex phase in G ₁₃. As will be seen below, the interplay between this phase and the phase in Eq. (5) will ultimately determine the contrast level in our graded-field microscope.

So far we have derived the mutual coherence incident on the sample, and also the intensity distribution incident on the camera given an arbitrary output mutual coherence emanating from the sample. The remaining step is to establish the link between the mutual coherences incident and emanating fromthe sample. This is done by assuming that the sample is thin and of arbitrary complex amplitude transmitivity t(x ₁) such that

The aim of our microscope is to reveal phase variations in t(x ₁). From the definition of mutual coherence, we obtain the relation

where we have introduced the mutual coherence transmitivity

The link between $J_1^{\mathit{\text{in}}}$ and $J_1^{\mathit{\text{out}}}$ has therefore been established, at least formally. Equations (5), (10) and (13) are the main results of this section. These define the relation between the observed intensity I ₃ at the camera and the phase variations in the sample manifest in T. This relationship will be explicitly derived below.

3. Symmetry properties

In our model the sample is entirely characterized by its complex amplitude transmittivity t(x ₁), which we assume to be arbitrary and hence impose no conditions on its profile. By construction, however, the associated mutual coherence transmittivity obeys the following condition

We can therefore decompose this complex transmitivity into its real and imaginary components such that T(x _1c,x _1d)=T_r (x _1c,x _1d)+iT_i (x _1c,x _1d), and note that the real functions T_r (x _1c,x _1d) and T_i (x _1c,x _1d) are respectively even and odd in x _1d.

Similarly, we decompose J ₁(x _1d) into its real and imaginary components such that J ₁(x _1d)=J_r (x _1d)+iJ_i (x _1d), where

and note that J_r (x _1d) and J_i (x _1d) are respectively even and odd in x _1d.

Finally, we decompose G ₁₃(x ₃+x _1c,x _1d) into its real and imaginary components

respectively even and odd in x _1d.

Our strategy will be to isolate only the even components in the integrand of Eq. (10). To this end, we further define the sample independent transfer functions

which are again respectively even and odd in x _1d. Through simple trigonometric relations we find

where

Finally, keeping only the even terms in Eq. (10), we obtain

This equation along with the Eqs. (22), (23) and (24) constitute the main results of this section. In effect, Eq. (25) explicitly relates the sample transmittivity, encoded in T(x _1c,x _1d), to the final intensity detected by the CCD camera. The functions K_e (x ₃+x _1c,x _1d) and K_o (x ₃+x _1c,x _1d) play dual roles. They serve as transfer functions for the coordinate x _1c whereas they serve as window functions for the coordinate x _1d. As transfer functions, they essentially image a sample point x _1c to a detector point x ₃.with an attendant loss of resolution characterized by the width of K(x ₃+x _1c,x _1d). As window functions their role is more subtle in that they reveal non-local features in the sample transmittivity. The roles of K_e and K_o as window functions are shown in Fig. (2) for various states of aperture blocking.

A benefit of the coordinate transformation in Eqs (6) is that it effectively isolates the roles of the aperture widths and aperture offsets in Eqs. (22) and (23). The aperture widths appear only in K and govern image resolution. The aperture offsets, on the other hand, govern the relative weights of K_e and K_o . If neither aperture is offset (α _c =β _c =0) then K_e →K and K_o →0. This corresponds to a standard brightfield configuration and the K_e component in Eq. (25) can be thought of as performing brightfield imaging. When aperture offsets are introduced, then K_o begins to play a role at the expense of K_e . Because of the asymmetry in x _1d, this K_o component can be thought of as performing gradient imaging. As we will see below, K_e produces amplitude contrast whereas K_o produces phase contrast.

 

Fig. 2. Plots of the window functions Ke(0, x1d) and K_o (0, x _1d) for different aperture block configurations (shown schematically on left). K_e reveals amplitude fluctuations whereas K_o reveals phase-gradient fluctuations in the sample. B characterizes the net brightfield contribution, or background level, as defined by the integral of K_e (Eq. (26)). The argument x _1d is normalized to be unitless.

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4. Graded field tuning

A better appreciation of the effects of K_e and K_o is gained by considering their net contributions. We quantify these contributions by evaluating the integrals of K_e and K_o over x _1c and x 1_d . It is clear that the integral of K_o vanishes because K_o is odd in x _1d. In general, however, the integral of K_e does not vanish and we define the net brightfield contribution by

where K_total corresponds to the total brightfield contribution when the aperture blocks are removed. In other words, the parameter B characterizes the extent to which the microscope contrast is graded. If B=0 then the contrast is fully darkfield; if B=1 then the contrast is fully brightfield. B can be tuned within this range simply by adjusting the aperture offsets. We note that the integrals over x _1c in Eq. (26) are straightforward from:

A pictoral representation of B is shown in Fig. (2). B can be thought of as the relative amount of ballistic (i.e. unscattered) illumination light that arrives at the detector. When the illumination and detection apertures are fully open then all the ballistic illumination arrives at the detector and B=1. When beam blocks are introduced, then B is reduced in proportion to the to the intersection of the aperture areas. Finally, when the beam blocks cover exactly half of their respective apertures, then B=0. In this case, there is no brightfield background and the contrast is purely darkfield. The illumination light that directly traverses the sample (i.e. ballistic light) is completely blocked and the only light impinging on the detector arises from sample scattering.

5. Phase versus amplitude contrast

We distinguish phase versus amplitude contrast by explicitly separating the sample complex transmittivity t(x ₁) into real and imaginary components

where the real functions q(x ₁) and p(x ₁) respectively impart amplitude and phase variations to the transmitted light. We assume that the sample is thin, meaning both q(x ₁) and p(x ₁) are much smaller than 1. To first order, we then write

thereby segregating amplitude and phase variations into the real and imaginary parts of T(x _1c,x _1d) respectively.

The above relations suggest that the brightfield component K_e in Eq. (10) produces amplitude contrast while the gradient component K_o produces phase contrast. More precisely, K_e reveals q(x _1c) only when B>0 (if B=0 then K_e reveals the local curvature in q(x _1c), or the second derivative q″(x _1c)). In contrast, K_o reveals the local slope in p(x _1c), or the first derivative p′(x _1c), thereby further refining its role as performing “phase gradient” imaging (the possibility of a partial aperture block to reveal phase gradients has been previously recognized[11]). Our phase gradient imaging is sensitive to the sign of p′(x _1c). Positive phase gradients lead to an increase in detected intensity whereas negative phase gradients lead to a decrease in detected intensity. We note that in the case of darkfield imaging (B=0) then the detected intensity cannot be less than zero and our phase gradient contrast becomes one-sided. That is, positive phase gradients are detected whereas negative phase gradients are not. In particular, one can verify that when B=0 (i.e. α_c =α_d /2 and β_c =β_d /2) and when the sample complex transmittivity is a pure phase (i.e. t(x ₁)=exp[i ϕ(x ₁)]) then the K_e and K_o integrals in Eq. (25) exactly cancel for negative phase gradients. However once a small brightfield background is introduced by slightly opening one or both aperture blocks (B≲0), then negative phase gradients become observable and the image takes on a 3D relief appearance, much like a DIC image. Experimental examples of graded-field imaging are shown in Fig. (3).

 

Fig. 3. Experimental images of pyramidal neurons in an acute rat hippocampus slice (depth ~50µm). Images were taken with no aperture blocks (fully brightfield configuration - panel A), with a partial beam block in the illumination (panel B) or (panel C) detection aperture, and with partial beam blocks in both apertures (panel D). In the experimental setup, the illumination aperture was larger than the detection aperture. The slices were 400mm thick; the rat was 15 days old.

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6. Image resolution

We recall that K_e and K_o serve as transfer functions for the coordinate x _1c and as window functions for the coordinate x _1d. Image resolution therefore depends on the ranges allowed for these coordinates. In general, these ranges are determined by the widths and offsets of the illumination and detection apertures, whose roles, according to Eqs. (22) and (23), can be analyzed separately.

We begin by analyzing the role of the aperture widths. Since these widths appear only in K (Eq. (24)), their role is the same for both K_e and K_o . We consider two limits that characterize qualitatively different states of illumination, which we refer to as incoherent (a_d ≫b_d ) and coherent (a_d ≪b_d ).

In the case a_d ≫b_d (incoherent illumination) we can make the approximation

The roles of K as transfer and window functions have become separated here, and the ranges of x _1d and x _1c are roughly defined by |x _1d≳ 2π/α_d and |x ₃+x _1c|≳π/β_d . Because K is localized to small x _1d’s, it effectively reduces to an incoherent PSF where the observed light intensity at the detector position x ₃ depends only on the transmitted light intensity in the vicinity of the sample position x _1c.

In the opposite case a_d ≪b_d (coherent illumination) we can make the approximation

The roles of K as a window and transfer function remain inseparable. The ranges of x _1d and x _1c are now roughly defined by |x _1d|≳2π/β_d and |x ₃+x _1c|≳π/β_d , which are no longer independent. The observed light intensity at the detector position x ₃ depends on the distribution of transmitted light fields (as opposed to intensities) in the vicinity of the sample position x _1c. We note that the limit a_d ≪b_d effectively corresponds to slit illumination, as is the case for both Schlieren and Hoffman contrast techniques[10, 12, 15].

In both cases of incoherent and coherent illumination, transfer function resolution from sample position x _1c to detector position x ₃ is determined by the detection aperture width b_d alone. When bd becomes large, then K is non-zero only when x ₃≈-x _1c and the transfer function resolution becomes high (note: the image is inverted relative to the object).

Final image resolution also depends on the resolution of the window function for coordinate x _1d, since ultimately this window function is responsible for revealing non-local transmission variations in the sample, such as phase gradients. The range allowed for |x _1d| depends not only on the aperture widths, as discussed above, but also on the aperture offsets. If the aperture offsets are too large, then the cosine and sine terms in Eqs. (22) and (23) oscillate too rapidly, and both K_e and K_o are ineffective at producing contrast. If instead the aperture offsets are too small, then the sine term in Eq. (23) oscillates too slowly, and K_o is ineffective at producing phase gradient contrast. As a rule of thumb, graded-field microscopy is effective at revealing phase gradients when α_c +β_c ≈ηα_d for incoherent illumination, or α_c +β_c ≈ηβ_d for coherent illumination, where η ranges between $\frac{1}{2}$ and 1 (a larger η leads to greater suppression of brightfield background). A simple conclusion may be drawn from this rule of thumb: If only a single aperture block is used to reveal phase gradients, then for incoherent illumination it is more effective to partially block the illumination aperture (as corroborated by Fig. (3)), whereas for coherent illumination it is more effective to partially block the detection aperture. In both cases, the effect of a second aperture block is then to suppress brightfield background. An advantage of incoherent over coherent illumination is that it allows higher phase gradient resolution since it makes use of a larger illumination aperture. Correspondingly, this is an advantage of graded-field microscopy, which allows the use of a large illumination aperture, over Schlieren or Hoffman contrast microscopies, both of which use slit illumination.

7. Conclusion

In summary, we have presented the formalism for a general graded-field microscopy technique involving the partial blocking of illumination and detection apertures. The resultant aperture offsets lead to phase gradient imaging as well as adjustable brightfield background suppresion. The main advantage of graded-field microscopy is its extreme simplicity and versatility. Graded-field microscopy requires no specialized optics (as opposed to DIC or Zernike phase contrast), and can be implemented with any widefield microscope. Moreover, because it makes use of adjustable aperture blocks, graded-field micirscopy can readily be tuned over different imaging configurations ranging from fully brightfield to fully darkfield. The adavntages of graded-field microscopy should make it useful for general phase-contrast imaging applications.