## Abstract

We present a far-field imaging system with a one-dimensional form factor based on coupling light into the side of an optical fiber. The point spread function of this threadlike camera is determined analytically and confirmed experimentally. Because the system is one-dimensional, high resolution is available in one spatial dimension. An imaging device is demonstrated with an angular resolution of 100 micro-radians. Diffraction-limited imaging is achieved for aperture lengths as large as 1 cm. An image is formed from a light field produced by a Dammann grating illuminated by a laser.

© 2016 Optical Society of America

## 1. Introduction

Far-field imaging systems range in size from relatively small (e.g. cell phone cameras) to very large (e.g. astronomical telescopes). The form factor or the size and shape of these systems often is an important design constraint, especially when used in aerospace and surveillance applications where weight is a prime consideration. It is interesting to ask whether far-field resolution can be preserved when strict limits are placed on the physical size of an imaging system along any or all of its three spatial dimensions. In the current paper, we investigate the limit where two of the dimensions are reduced to very small values. Specifically, we propose an imaging system effectively constrained to a linear form factor, which maintains high angle resolution.

The resolution of any imaging system, so long as it relies on propagating waves, has a diffraction limit proportional to the numerical aperture, NA [1]. For a lens based system, *NA* = *n*sin(atan(*D*/2*d _{o}*)), where

*D*is the lens aperture,

*d*is the object distance, and

_{o}*n*is the index of refraction of the surrounding medium. Since we are interested in resolving distant objects,

*d*

_{0}≫

*D*. And, as one generally does not have control over the index of the surrounding medium, the above equation implies that a large receiving aperture size

*D*is essential to achieve high object space angular resolution.

The diffraction limit restricts the height and width of the aperture, but not the total depth of the system. Different architectures have been designed to decrease system depth while preserving resolution. For example, a Cassegrain telecope uses a secondary mirror to decrease the required tube length for the same effective focal length. The folded optic technique was taken to its limit in [3]. In that design, light entered through an annular aperture around the periphery and, through successive reflections, was channeled to a detector array in the center. Although this technique is very effective using a moderate number of folds, the depth can not be scaled down indefinitely.

The diffraction limit is not unique to lens or mirror-based systems. The input angular acceptance of a grating-coupled waveguide has also been shown to be inversely proportional to the grating width [2]. In a previous work [4], we described an imaging device based on such a slab waveguide that was capable of high object-space resolution while maintaining a *two-dimensional* form factor. A diffraction grating was used to couple far-field object points into discrete waveguide modes. Although the active part was only a few wavelengths thick, the imaging system was shown to have an angular resolution commensurate with the diffraction limit from the 6mm aperture of the device. An image was then produced by scanning over object space.

In the present work, the imaging system form factor is further restricted to be *one-dimensional*. Such shapes can satisfy demanding packaging requirements and may find utility in aerodynamic surfaces where the cross-sectional area needs to be minimized. One-dimensional systems also can be made considerably lighter than their two-dimensional counterparts. Whereas our planar device required support material to ensure optical flatness, the thread-like optical system is simply stretched between two points on an existing structure. The surface of this structure need not be optically flat. A diagram of the envisioned system architecture is shown in Fig. 1. Light from an object in the far-field is coupled into a single-mode fiber by means of a tilted fiber Bragg grating. A photo-diode is integrated onto the end of the fiber to detect the power coupled into the fiber core. Scanning is then used to interrogate various points in the object space. Because the optical system’s resolution is inextricably linked to its aperture size, this thread-like form factor can only have high far-field resolution in one dimension. Specifically, the angular width in the *x*′-direction of Fig. 1 is inversely proportional to the length of the grating. In contrast, the resolution in the *y*′-direction is inherently much lower.

There are many existing imaging systems that use optical fibers. These include conventional endoscopy, lensless systems [5, 6], and optical coherence tomography [7]. By adapting super-resolution techniques, such as STED [8] and two-photon fluorescence [9], fiber systems have been able to exceed their diffraction limit. All of these can be classified as optical probes, where the fiber is used to transmit light from a place with limited accessibility to a remote system for further processing. In the existing fiber imaging technologies, high spatial resolution is only possible if the fiber end-facet or lens assembly is in close proximity to the object. In contrast, we are interested in far-field imaging such as provided by a telephoto lens. This system is unique in that the imaging pupil is distributed across the length of the fiber, significantly increasing the far-field resolution, albeit in one dimension. Also, because the proposed architecture has only the single fiber and a detector, the system has much lower total volume and weight.

This paper explores the imaging characteristics of this thread-like optical system, where the light is coupled through the side of the fiber. First, the radiation modes of the tilted fiber Bragg grating (TFBG) are reviewed. Next, the angular resolution is described analytically and confirmed experimentally. Then, an image of a simple object is recorded. The final section contains a brief discussion of some practical limitations of the technique.

## 2. Tilted fiber Bragg grating far-field response

Tilted fiber Bragg gratings have been used for a variety of sensing applications ranging from hydrophones and accelerometers to measuring temperature, strain, pressure, refractive index, and biological/chemical concentrations [10]. In most of these applications, the coupling strength between modes is modified by various physical effects, allowing sensitive measurements of these parameters. The tilted fiber Bragg grating in the current application, however, simply serves as a convenient mechanism to couple a well defined light distribution from free space into a fiber mode through the fiber cladding. To obtain a qualitative estimate of the far-field resolution of this system, consider injecting light into the fiber core through its end facet. The far-field pattern of the out-coupled light produced by the TFBG is expected to be narrow in the long direction of the fiber, consistent with diffraction from a long, truncated wavefront. If we now view this structure in reverse as an imaging system, light in the far field should couple back into the fiber mode with an angular discrimination given by this same pattern. Assuming that the grating is high quality and does not scatter excessive amounts of light, all background light outside this angular band will be effectively rejected. More specifically, the resulting modal power excited in the fiber is proportional to the square of the overlap integral of the radiation pattern with the complex light field. By scanning over angle, one can build up a one-dimensional image where the radiation pattern acts as an analog to the point spread function, PSF, of a conventional imaging system. Since the magnitude and phase of the point spread function fully describe an imaging device, it is important to calculate these for an imaging thread.

Consider a mode traveling along the fiber with propagation constant *β _{l,m}*. The grating diffracts light into discrete orders whose longitudinal wavevectors,

*k*, are given by the phase matching condition

_{z}*is the grating period and*

_{g}*θ*is the grating tilt angle (Fig. 2). Because the magnitude of the wavevector must be conserved, the radial component is given by where

_{g}*λ*

_{0}is the free space wavelength and

*n*is the index in the region of interest.

*k*and

_{r}*k*are constants for all azimuthal angles

_{z}*ϕ*, therefore the radiation mode moves away from the fiber along a cone whose angle to the fiber axis is The tilt in the grating results in the power being non-uniformly distributed along

*ϕ*. This distribution has been calculated by the volume current method [11] as well as coupled mode theory [12]. The ray along the cone with maximum power,

*ϕ*= 0, will be considered the imaging axis for the imaging thread and is designated as

*z*′. The plane normal to the imaging axis and at a distance

*R*

_{0}from the grating is defined as the object plane. The object plane coordinates

*x*′ and

*y*′ correspond to infinitesimal changes in

*θ*and

_{c}*ϕ*respectively at the origin. The intersection of the object plane with the radiation cone can be shown, after some algebra, to be parameterized by Eqs. (4) and (5).

*θ*= π/2. To validate the shape of the radiation pattern, a 1.3

_{c}*μ*m beam was end-fired into a CorActive UVS-652 single mode fiber containing a TFBG. The grating was formed by passing a KrF excimer laser beam through a phase mask and exposing the fiber to the resulting interference pattern. An InGaAs CCD was placed in the back focal plane of a 10cm focal length lens along the imaging axis. The propagation angle for this particular fiber was 31.7

*°*, resulting in the curvature seen in Fig. 3. The shape of the radiation pattern was shown to be in good agreement with the given parametric equations.

The Euclidean distance from the grating to any point along the curve shown in Fig. 3 on the object plane, after some simplification, is given by

*ϕ*, the phase is also:

## 3. Angular resolution

In the previous discussion, the cone angle was taken to be a single value. *θ _{c}*, however, has some width due to diffraction from a grating of finite extent. It is this angle spread that determines the resolution of the system in the dimension of interest. The angle spread function for a particular grating is calculated and confirmed through direct measurement.

Assuming the coupling is uniform over the length of the grating, the guided mode power decays exponentially with decay length *L _{c}*. Beyond the grating length

*L*, no energy is radiated. The complex amplitude of the out-coupled field,

_{g}*A*, can then be expressed by

*π*/2 −

*θ*, the sin

_{c}*θ*term is added to account for the projected lengths. The far-field pattern is calculated by taking a Fourier transform of Eq. (9) and substituting

_{c}*θ*/

_{x′}*λ*for the Fourier conjugate variable. By employing the small angle approximation, the angle spread function can be written as

*λ*is the wavelength. If

*L*is much greater than

_{g}*L*, Eq. (10) takes on the attributes of a Lorentzian function. As the coupling length (

_{c}*L*) approaches infinity, the angle spread function converges to the sinc function. To maximize coupling efficiency, the coupling and grating lengths should be approximately equal [14]. In all cases, the width of the angle spread function is inversely proportional to either the grating length or the coupling length, depending on which is shorter.

_{c}## 4. Experimental results

To validate Eq. (10), the angle spread function was measured using the system shown in Fig. 4. The beam from a 1.3 *μ*m laser diode (*LD*) was expanded by a microscope objective (*L*_{0}) and collimated by a 10cm focal length spherical lens (*L*_{1}). This beam was focused down to a line by cylindrical lens (*L*_{2}). An afocal system (*L*_{3} and *L*_{4}), with unity magnification, projected the apertured line onto the fiber. To mitigate the effects from the PSF curvature (resulting from
${\theta}_{c}\ne \frac{\pi}{2}$ in our experiment), the beam was apertured in the y direction in the afocal system’s Fourier plane. Finally, a detector measured the power exiting the fiber’s end-facet. The angle scanning was performed by translating *L*_{1} rather than rotating the fiber.

The normalized measured power with respect to angle *θ _{x}*

_{′}for an x aperture set to 1mm, 2mm, and 4mm is shown in Fig. 5a. Because of the 31° projection angle, the external apertures correspond to 2mm, 4mm, and 8mm at the fiber surface. The theoretical curves, given by the normalized square-magnitude of Eq. (10) with a coupling length

*L*= 10mm, are shown in Fig. 5b. The two sets of curves are in good agreement for all aperture widths. The measurement was repeated with no external aperture and is consistent with a grating width of 11mm. Because the exact grating length was not known, a corresponding theoretical curve was not calculated. While the exact cause of the asymmetry in the 8mm and no aperture curves is unknown, it is likely a consequence of the scanning technique rather than a feature of the grating.

_{c}## 5. Imaging example

Adding a chirp to the grating period or a physical curve in the fiber can focus the light outcoupling from the grating to a plane a finite distance away. This would correspond to traditional finite conjugate imaging. The simple TFBG of our system, however, maps a fiber mode to the point spread function in the far-field. To test the imaging performance of our infinite conjugate system, an optical projection system was designed to simulate a set of object points at an infinite distance from the fiber.

A Dammann grating, designed to produce six diffraction orders with approximately equal efficiency, was illuminated with a collimiated 1.3*μ*m laser diode. The diffraction orders of the Dammann grating simulate a finite number of distinctly separated points located at infinity without needing a Fourier transforming lens. An independent measurement of the object intensity pattern as a function of angle, captured by a CCD in the back focal plane of a lens inserted behind the projection system, is shown in Fig. 6a. To increase coupling efficiency, a cylindrical lens focused the expanded beam down to the fiber. It is important to note that this lens had no focusing power in the imaging direction, but did allow for phase matching in the transverse direction. Scanning was performed by translating the laser collimating lens, effectivley tilting the angle of incidence of the object with respect to the imaging thread. In an actual system, the imaging thread itself would be tilted through a small set of angles to produce a one-dimensional image. Angle rotation could be accomplished by mounting one end of the fiber to a piezoelectric actuator while leaving the other end fixed. For this particular grating and wavelength, a 100*μm* translation range would correspond to approximately one hundred resolvable points.

The image captured by the imaging thread is shown in Fig. 6b. The one-dimensional scan data was expanded to two-dimensions in software for comparison to Fig. 6a. The relative intensities and locations of the peaks are in reasonable agreement. The geometrical distortion apparent in part (b) of the figure is due to our angle scanning technique and can easily be removed by post-processing.

## 6. Discussion

Whereas simple, high-resolution imaging has been demonstrated using an optical system with a one-dimensional form factor, this extreme geometry comes at a cost. The TFBG, like other grating couplers, introduces linear chromatic dispersion. Light from a single polychromatic object point is spread across many image points. To maintain high resolution, the object must have narrow spectral width or one must be able to measure the power in narrow spectral bands. Alternatively, an active system that uses a tunable monochromatic source could be scanned by changing the wavelength of the illumination. It should be possible with such a system to use the TFBG as both the illuminator and the imager which would maximize optical throughput. However, high coupling efficiency is not possible for all wavelengths. The grating can only be Bragg matched over a spectral range around a design wavelength.

In addition to the narrow-spectral-band requirements, the imaging thread has reduced throughput compared to lens-based imaging systems. The etendue of an optical system is proportional to the area of its entrance pupil, which approaches zero as the system becomes increasingly one-dimensional. Within the restricton of the radiance theorem, one can increase the effective aperture in the non-imaging direction by using cylindrical microoptics. Thus, one can trade transverse width for optical throughput to suit the specific application. In addition, one can provide modest resolution in the orthogonal direction by using these same cylindrical microlenses.

The one-dimensional form factor, and the resulting asymmetric point spread function, are best suited for one-dimensional objects. In theory, it would be possible to form a high resolution image of an arbitrary two-dimensional object. By rotating the fiber while scanning, one would capture slices which can be synthesized into an image using computer tomography. A rotating fiber, however, is no longer one-dimensional.

Finally, whereas a conventional lens forms all points at once, the imaging thread requires scanning. Scanning reduces the dwell time available per pixel and can be problematic in a low-light environment. This can be partially overcome by multiplexing multiple fiber modes at several different angles to reduce the total scanning time and consequently increasing the dwell time per pixel.

## 7. Conclusions

It has been shown that a tilted fiber Bragg grating incorporated into a single-mode fiber can be used as a one-dimensional imaging system with a far-field resolution ultimately limited by the length of the grating along the the fiber. The shape and width of the imaging point spread function was calculated for arbitrary grating geometries. The resolution of the optical system was experimentally measured and shown to correspond to theoretical predictions. Some of the limitations of this system including chromatic dispersion and reduced optical throughput have been identified. A test image formed by the imaging thread was compared to one made by traditional lens imaging, thereby demonstrating a telephoto imaging system can be made with a one-dimensional form factor.

## Funding

National Science Foundation (NSF) (ECCS-0925731).

## Acknowledgments

The authors wish to thank Jacques Albert and Albane Laronche for fabricating the tilted fiber Bragg gratings, and CorActive for the fiber material.

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