## Abstract

We describe the design of a microscope combining rotational shear interferometer (RSI)-based coherence imaging with an objective lens to simultaneously obtain high numerical aperture and high depth of field imaging. We present experimental results showing the operation of this instrument.

©2001 Optical Society of America

## 1 Introduction

In conventional imaging systems there is a trade-off between numerical aperture and depth of field. We have previously discussed the elimination of this trade-off in coherence imaging systems[1, 2]. Coherence systems have an infinite depth of field, but the reconstructed image SNR degrades as a function of the number of independent source channels. One can reduce the effective number of source channels by decreasing the numerical aperture.

Microscopic applications are particularly attractive for coherence imaging because the spatial complexity of microscopic sources is often relatively low while simultaneous demand for high numerical aperture, high resolution and high depth of field is strong. Tomographic reconstruction of microscopic samples in particular requires high depth of field imaging [1].

Researchers have demonstrated various techniques to increase the depth of field of a microscope. One approach is to reduce the aperture of the exit pupil slightly, increasing the depth of field at the expense of low throughput and low resolution.

A more interesting technique is to use a cubic-phase plate, which uses an encoding mask to modulate the *k _{x}* and

*k*components of the field. Then, with appropriate decoding algorithms, an image at the focal plane is recovered with an improved depth of field. However, cubic-phase plate system still has a limited depth of field. The depth of field can be increased by making the cubic-phase plate constant larger. Again, the increased depth of field comes at the price of lower resolution [3, 4].

_{y}This paper describes a microscope based on coherence imaging systems, hereafter termed the mRSI, formed by replacing the eyepiece of a light microscope with an RSI (rotational shear interferometer). Just as the purpose of the objective/eyepiece combination in a conventional microscope is to match the numerical aperture of the source to the numerical aperture of the eye, the purpose of the objective/RSI combination in the mRSI is to match the numerical aperture of the source to the numerical aperture of the RSI. The advantage of the RSI over the eye or a camera is that one obtains infinite depth of field in the reconstructed microscopic image without sacrificing numerical aperture.

## 2 mRSI Theory and Design

For a quasi-monochromatic and spatially incoherent source, described by the radiant density *I _{s}*(

*r*), the mutual intensity [5] between two sampling points at $(\frac{\Delta x}{2},\frac{\Delta y}{2},0)$ and $(\frac{-\Delta x}{2},\frac{-\Delta y}{2},0)$ under the paraxial approximation is given by [1],

_{s}Taking an inverse Fourier Transform of Eq. 1 with respect to Δ*x* and Δ*y* we obtain

$$=\int \frac{{I}_{s}\left({r}_{s}\right)}{{z}_{s}^{2}}\delta (u-\frac{{x}_{s}}{\lambda {z}_{s}},\upsilon -\frac{{y}_{s}}{\lambda {z}_{s}}){d}^{3}{r}_{s}$$

It can be seen that *S*(*u, v*) is a line integral of $\frac{{I}_{s}\left({r}_{s}\right)}{{\mathit{z}}_{s}^{2}}$ along a ray passing through the points (*x _{s}*=λ

*z*=λ

_{s}u, ys*z*). If a system can measure the mutual intensity, then Eq. 2 can be used to obtain the source intensity.

_{s}vWe use an RSI [6, 7, 8, 9] to measure the mutual intensity function *J*. The RSI, shown schematically in Fig.1, is a Michelson-type interferometer incorporating folding mirrors(prisms) instead of flat reflectors [2]. The primary components are a beam splitter and two folding mirrors. One of the mirrors is mounted on a translation stage to vary the relative optical path Δ*z*. The fold axis of one mirror makes an angle of *θ*/2 with respect to x-axis. The fold axis of the other mirror makes an angle of -*θ*/2 with respect to x-axis.

If the incident field is *V*(*x _{f}, y_{f}, z_{f}*), then the field produced on the output arm by reflection from the mirrors is

*V*(

*x*cos

_{f}*θ*+

*yf*sin

*θ*,

*xf*sin

*θ*-

*yf*cos

*θ*,

*zf*+Δ

*z*) and

*V*(

*xf*cos

*θ*-

*yf*sin

*θ*,-

*xf*sin

*θ*-

*yf*cos

*θ*,

*zf*). Thus, the intensity detected at point (

*xf, yf*) on the output arm of the RSI is given by,

where

$$\Delta y({x}_{f},{y}_{f})=2{x}_{f}\mathrm{sin}2\theta \phantom{\rule{.5em}{0ex}}\mathrm{and}$$

$$q({x}_{f},{y}_{f},{x}_{g},{y}_{g})=({y}_{f}{x}_{g}-{x}_{f}{y}_{g})\mathrm{sin}\left(2\theta \right)$$

Here, (*x _{g}, y_{g}*) is the focal plane center position in global coordinates [2]. Eq. 3 relates the signal measured by the RSI as a function of lateral interferometer position (

*x*) and the longitudinal scan position to the mutual intensity

_{g}, y_{g}*J*(Δ

*x*,Δ

*y*,

*q*,Δ

*z*). Once the mutual intensity is known, Eq. 2 can be applied to obtain the object intensity.

Since image reconstruction involves a digital Fourier transformation, the RSI is fundamentally a sampling instrument. The numerical aperture of the RSI is determined by the sampling rate in *xf, yf* plane and the shear angle *θ*. Assuming a pixel size of Δ in the output plane, the numerical aperture of the RSI is

A typical value of the numerical aperture of the RSI is 0.14 at a shear angle of 30o. Fig. 2 shows a schematic diagram of our optical system. The object is placed under a Zeiss Axioplan transmission-light microscope [10]. As discussed in the previous section, the design goal of the mRSI is to match the numerical aperture of an RSI to the radiant (spatial) bandwidth of the microscopic source. This match is achieved using a high numerical aperture microscope objective. Since the microscope objectives are also designed under paraxial approximations, the approximations used in Eq. 1 are still valid for the mRSI.

Typically microscope objectives are designed to image the source 0.15 m from the objective. However, we are physically constrained as we cannot place the RSI directly at the output of the microscope. In our system configuration, the RSI input aperture is over 0.75m away from the microscope output aperture. Thus, there is a simultaneous requirement for the optical system to match the numerical aperture of the objective with that of the RSI and to redirect the optical field into the RSI input aperture. These requirements can be met by using two intermediate relay lenses of large focal lengths.

The Zeiss objective [10] used in the microscope has a magnification of 10 and a numerical aperture of 0.30. The lens *L*
_{2} (focal length 0.25*m*) and lens *L*
_{3} (*f*=0.30*m*) matche the numerical aperture of the RSI to that of the objective.

Since mRSI is a combination of the RSI and the microscope objective, the angular resolution of the mRSI will depend on the angular resolution of the RSI and that of the microscope objective. The angular resolution of the microscope objective is given by

The aperture diameter of the objective is 5.7*mm*. So the angular resolution of the objective is 0.038 radians. The angular resolution of the RSI is given by

where *N* is the number of pixels per axis in the RSI sensor plane. We use a red interference filter of bandwidth 3nm centered at λ=0.633*µm* to ensure that the light is quasi-monochromatic. The pixel size of the CCD camera is Δ=9*µm* and there are 512 pixels per axis on the RSI sensor plane. Note that the angular resolution of the RSI can be varied by adjusting the shear angle *θ*.

Thus, the angular resolution of the mRSI will be limited by the angular resolution of the microscope. The shear angle of the RSI is set such that the angular resolution of the RSI is smaller than that of the objective. For this mRSI, the shear angle is set at *θ*=30°. At this shear angle, the angular resolution of the RSI is 2.7×10^{-4}(< 0.038)radians.

## 3 Results and Analysis

To demonstrate the high depth of field imaging with the mRSI, we have used a diffraction mask with a spatial frequency of 20 lines/mm. The object has been placed at various distances of (a) 4 mm, (b) 6 mm, (c) 13 mm and (d) 30 mm from the microscope objective.

The object is illuminated with an incoherent white light source, and an interference pattern is formed on the CCD detector array. The dynamic range of the detector is 10 bits. A spectral filter enforces the quasimonochromatic assumption, and the bandwidth of the filter determines the fringe modulation. Large bandwidth filter would decrease the interference fringe modulation leading to a bad SNR, whereas a very narrow filter will reduce the throughput of the system.

For a quasimonochromatic field, the intensity measured at a point on the RSI sensor plane is

The amplitude and phase of the mutual intensity can be isolated by sampling the output plane intensity as a function of optical path difference. For example, if one measures *I*(*xf, yf*) for *l*=0, *l*=λ/4 and *l*=λ/2, then one can separate *J*(Δ*x*,Δ*y*) from the background intensity.

In practice more than three values of *l* are used to reduce the effects of the errors in the translation stage. We choose 16 values of *l* (*l _{n}*=

*n*λ/16,

*n*=0, 1.., 15), and an image is captured corresponding to each delay.

*J*(Δ

*x*,Δ

*y*) can be found using these images and Eq. 8. A discrete Fourier transform of J will then give the source intensity.

The corresponding images obtained after the digital analysis are shown in Fig. 3. The digital analysis took around 30s for each reconstructed image. Since the detector measures the intensity of the light, which will necessarily be positive, the Fourier transform will always result in a large DC peak at the point (0,0). This point contains no information and hence has been blocked in these images. The angular scale of the image is around 10o and the spatial scale is 1.5*mm* for Fig. 3(a). Note that the spatial scales of the images reconstructed depend on the position of the object from the objective.

The working distance of the objective used was 5.7*mm*. The depth of field of a conventional microscope is given by [11] $D=\frac{\lambda}{{(N.A)}^{2}}$ which is around 9*µm*. However, it can be clearly seen from the images taken from the mRSI that even if the object is placed around 3cm away the lens, the image is still in focus. We note that in a conventional microscope, the position of the object from the objective determines the focus of the image and the angular magnification. An advantage of the mRSI is that this distance determines only the angular magnification of the system and not the focus of the final reconstruction.

To further illustrate high depth of field imaging with the mRSI, we have taken two diffraction masks (one containing horizontal lines and the other containing vertical lines) and placed one on top of the other. Fig. 4 shows the images obtained from a camera (320×240 pixels) placed at the output of the microscope. We could focus to either the horizontal lines or to the vertical lines but both lines could not be focussed at a time.

Fig. 5 shows the images of the same masks obtained by the mRSI. It is evident that both the lines are in focus even if an air gap of 10*mm* is introduced between them. The spatial scales of the two masks in Fig. 5(b) are different as they are kept at different distances from the objective.

Thus, high depth of field microscopic images can be obtained with an mRSI. Unfortunately, this very desirable quality comes at a price. In lens-based systems, the signal to noise ratio is proportional to √*N*, where *N* is the number of photons collected. However, in RSI imaging, the signal-to-noise ratio is proportional to $\sqrt{N/P}$ where *P* is the number of pixels in the RSI image plane [2]. So, the SNR is worse than the standard system by a factor of 1/√*P*.

## 4 Conclusion

The mRSI is capable of imaging objects through a microscopic objective with a high depth of field. This should have applications to areas such as biology, fluid mechanics, and material science. In this work, we have shown the design and analysis of the mRSI, along with experimental results. To design the mRSI, it was necessary to calculate the numerical aperture of the RSI and match this with the numerical aperture of the objective lens. Also, we used relay lenses to map the field into the RSI aperture.

For the experimental results in this paper, we studied diffraction masks at various distances from the objective. Comparing this with the images from a camera, we could see that the mRSI results in a high depth of field.

## 5 Acknowledgements

The authors are grateful to Daniel Marks, Jason Gallicchio, Evan Cull, Ronald Stack and Remy Tumbar for comments and assistance with this project. This work was supported by the Defense Advanced Research Projects Agency through Army Research Office grant DAAG55-98-1-0039.

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