1. Introduction

Modal filters are an essential technology for nulling interferometry. Nulling interferometry is an approach to starlight suppression that will allow the detection and characterization of Earthlike exoplanets at midinfrared wavelengths ($6–20\mathrm{\mu m}$) [1, 2, 3, 4]. Modal filtering of the light received through an interferometer significantly reduces the effect of optical aberrations in telescope systems, allowing the starlight to be suppressed, or nulled, below other sources of systematic noise. The most basic filter is a simple pinhole, and pinholes have indeed been used to achieve the deepest laser nulls so far at midinfrared wavelengths. However, pinholes only operate well over a narrow bandwidth and so are ill-suited for broadband spatial filtering for science instruments. In addition, they do not reject low spatial-frequency wavefront aberrations. The development of improved broadband techniques for modal filtering at midinfrared wavelengths will be crucial to the success of future missions.

The goal of this technology development is to demonstrate modal filters that have a single-mode throughput of near 50% and a modal suppression of $25\mathrm{dB}$ (a factor of 300) for nonfundamental modes, as required for NASA’s Terrestrial Planet Finder (TPF) project [1, 2]. The desire is also to cover the wavelength range of 6 to $20\mathrm{\mu m}$ using no more than two spatial filters, each with their own separate wavelength coverage within that range. We have previously reported results of chalcogenide fibers produced by the US Naval Observatory [5]. Chalcogenide fibers should operate successfully over the wavelength range of $6-12\mathrm{\mu m}$, which satisfies part of our requirements.

Here we report the measurement of the modal filtering properties of a silver halide fiber fabricated at the Tel Aviv University at the $10.5\mathrm{\mu m}$ wavelength. The filtering properties of the reported fiber are excellent: we measured the 15000-fold suppression of nonfundamental modes at $10.5\mathrm{\mu m}$, far exceeding the required factor of 300. We know of no other device delivering comparable performance. The reported result implies that the silver halide fiber technology can be used for modal filtering in at least the $10.5–17.5\mathrm{\mu m}$ spectral range since the fiber will retain single mode behavior at wavelengths longer than $10.5\mathrm{\mu m}$ and the material has high transparency in this spectral range [6].

The use of single mode fibers for modal filtering in interferometers was proposed by Clark and Roychoudhuri [7]. We are interested in using them as modal filters in nulling stellar interferometers as they can help achieve deeper interferometric nulls by improving the wavefront quality. Ollivier and Mariotti [8] have proposed that spatial filters, such as pinholes, be used for this purpose. Mennesson et al. [9] later pointed out that, while spatial filters only remove high-frequency components of the wavefront disturbance, modal filters also remove the low-frequency components, and would therefore provide superior performance.

After passing through an ideal modal filter, such as an ideal single mode fiber, the light emerges with a field distribution identical to the fiber’s fundamental mode irrespective of the shape of the input field [9, 10, 11]. The rejected field components are radiated out. As Wallner et al. [11] pointed out, this rejected light may appear at the fiber’s output, and one must take measures to separate it from the useful signal.

While the ideal modal filter completely rejects all components of the input field that have no overlap with the filter’s fundamental mode, in real-life filters there will be some leakage. We have developed a procedure for measuring this leakage in infrared single mode fibers [5]. To determine the leakage we use an antisymmetric input field that should have no overlap with the symmetric fundamental mode of the fiber and, ideally, should be completely rejected. The filter’s quality is proportional to its capability to reject the antisymmetric input mode and to its capability to transmit the fundamental mode. Accordingly, we have introduced the filter quality metric $Q_f$, defined as the ratio of the power transmittance for the fundamental mode $T_F$ to the power transmittance (leakage) $T_A$ of the antisymmetric input field: $Q_f=T_F/T_A$ [5]. We have shown that, if the wavefront aberrations are the only factor causing the finite null depth in a two-beam interferometer, then, exclusive of coupling losses, the use of a modal filter leads to improvement of the rejection ratio by a factor of $Q_f$.

Previously we have reported the modal filtering properties of an infrared fiber produced by the Naval Research Laboratory (NRL) [5]. The measurement yielded $Q_f$ of approximately 1000, but it was not clear what limited the filter’s quality. The theory developed in Refs. [10, 11] suggests much higher $Q_f$, but the model does not include the interface between the cladding and air. In this model, the input field components that do not couple into the fundamental mode are radiated out. In real fibers, the air/cladding interface serves as the boundary of secondary waveguide, so the rejected light, if not attenuated, can be guided to the output end. This may have been the reason for the $Q_f$ limit. While an absorbing coating was applied to the outside of the cladding, we had no way to measure its effectiveness.

Since the Tel Aviv University (TAU) fibers reported here have much thicker cladding than the NRL fibers, in addition to measuring the leaked light intensity, we were able to image the leaked intensity distribution at the output end of the fiber with an infrared camera. The images reveal that in all four fiber segments reported here, rather than being the result of multimode behavior of the core, the leakage is dominated by the light propagating through the cladding. This result is rather encouraging— previously only much longer fibers made of silver halides demonstrated single mode behavior [12].

The light in the cladding results from the rejected (radiated) components of the input beam that, rather than escaping into the free space, propagate along the cladding. This was anticipated [11], and an absorbing outer cladding layer was incorporated into the fiber design. To further reduce the propagation of the radiated light to the detector it is possible to apply an aperture that will transmit most of the fundamental mode but obscure the rejected light propagating through the cladding. Detailed measurements show that, if a proper aperture is used, a filtering quality $Q_f$ of 15,000 at least can be achieved in fiber segments as short as $10.5–20\mathrm{cm}$. Without the aperture the filtering quality falls at least sixfold.

2. Single Mode Silver Halide Fibers

2A. Background

The fibers described in this paper were manufactured from silver halide crystals. These $\mathrm{Ag}{\mathrm{Cl}}_x{\mathrm{Br}}_{1-x}$, crystals (where x is the molar fraction of $\mathrm{Cl}$ in the compound) are highly transparent in the $3–30\mathrm{\mu m}$ spectral range. The refractive index $n(x)$ of $\mathrm{Ag}{\mathrm{Cl}}_x{\mathrm{Br}}_{1-x}$ varies between $n(0)=2.16$ (pure AgBr) and $n(1)=1.98$ (pure AgCl) at $\lambda =10.6\mathrm{\mu m}$. Fibers are fabricated by first making a “rod-in-tube” preform by inserting a rod of higher refractive index ($n_{\text{core}}$) into a tube of lower refractive index ($n_{\text{clad}}$) and then extruding it to form a core-clad fiber. In order to achieve single mode operation, the fiber parameter V should satisfy the critical value of $V<V_c=2.405$ [10]. The fiber parameter is a function of the core diameter and the refractive index difference $\mathrm{\Delta }n=n_{\text{core}}-n_{\text{clad}}$. To achieve single mode operation, small $\mathrm{\Delta }n$ as well as small core diameter are required. Both of these requirements present a challenge.

The first silver halide single mode fibers operating at $\lambda =10.6\mathrm{\mu m}$ were demonstrated by Shalem et al. in 2004 [12]. In order to fabricate fibers with small values of $\mathrm{\Delta }n$, single crystals from which the preform was made had to be extremely homogeneous in their chemical composition. Toward this goal, the crystal growing techniques were improved, achieving inhomogeneity better than 2% within a single crystal $150\mathrm{mm}$ long and $10\mathrm{mm}$ in diameter for all crystal compositions used. The fabricated fibers [12] had step index structure, with core and clad diameters of 60 and $900\mathrm{\mu m}$, respectively, and with $\mathrm{\Delta }n$ of 0.0024, resulting in V of 2 at $10.6\mathrm{\mu m}$ wavelength. Single mode operation at $10.6\mathrm{\mu m}$ wavelength was observed for fiber sections longer than $2\mathrm{m}$ and the measured losses were $4–10\mathrm{dB}/\mathrm{m}$.

Further improvements led to the demonstration of single mode operation in fibers approximately $50\mathrm{cm}$ long [13]. The improvements included further optimization of the extrusion parameters, which allowed TAU to fabricate fibers with smaller cores. Also, in order to strip off cladding modes, the external surface of the fibers was exposed to UV light, forming an absorbing layer approximately $100\mathrm{\mu m}$ thick. This was expected to cause a significant reduction in the minimum length required to observe single mode operation. TAU then fabricated step index fibers with core diameters of $50\mathrm{\mu m}$ and clad diameters of $900\mathrm{\mu m}$. The cladding and core compositions were $\mathrm{Ag}{\mathrm{Cl}}_{0.30}\mathrm{Br}{0}_{.70}$ and $\mathrm{Ag}{\mathrm{Cl}}_{0.32}{\mathrm{Br}}_{0.68}$, respectively, leading to a $\mathrm{\Delta }n$ of 0.0047 and a fiber parameter V of 2.1 [13]. These step index fibers exhibited single mode operation for fiber sections longer than $50\mathrm{cm}$, with losses of $10–20\mathrm{dB}/\mathrm{m}$. Moreover, the output field pattern of these fibers was greatly improved in comparison to the field observed in previous work [12], exhibiting extremely smooth and symmetrical Gaussian distribution.

2B. Fiber Under Test

The fiber used for this study has a two-cladding design. The core is $50\mathrm{\mu m}$ in diameter and is made of $\mathrm{Ag}{\mathrm{Cl}}_{0.30}{\mathrm{Br}}_{0.70}$. The first (inner) cladding is $250\mathrm{\mu m}$ in diameter and is made of $\mathrm{Ag}{\mathrm{Cl}}_{0.32}{\mathrm{Br}}_{0.68}$. The second (outer) cladding is made of $\mathrm{Ag}{\mathrm{Cl}}_{0.05}{\mathrm{Br}}_{0.95}$ and is $900\mathrm{\mu m}$ in diameter. The fiber was exposed to UV light to form an absorbing layer on the outside of the cladding to facilitate the stripping of cladding modes. The measured refractive indices of the layers are summarized in Table 1. Notice that the outer cladding has a greater refractive index than the core. This so-called “M-fiber” design is being investigated for increasing the extinction of the light propagating through the cladding. This investigation is not covered in this report.

3. Measurements

The technique for measuring the leakage was described in Ref. [5]. In the following we describe the measurement setup to provide background for further discussion and describe changes introduced into the setup. The setup is shown schematically in Fig. 1. In order to produce an antisymmetric field that would not couple to the fundamental mode, the collimated beam from a quantum cascade laser operating at $10.5\mathrm{\mu m}$ was reflected off a mirror with a step in the middle. The step height was calculated and the angle of incidence was adjusted to produce a π phase difference between the two halves of the beam. The beam was focused on the fiber under test. The calculated cross section of the field in the focal plane is shown in the inset of Fig. 1. Such a field should not couple into the fundamental mode of a cylindrical fiber when the field’s node is aligned with the fiber’s center.

The output end of the fiber was imaged by an infrared camera. To subtract the background, a difference of images taken with the laser on and off was calculated. In addition to imaging, we used the camera in the detector mode, whereby a single number was obtained by summing pixels within an area around the fiber’s core. We have introduced a digital equivalent of aperture by choosing areas around the fiber core for data summation. The signal for “digital aperture of radius $N_R$” was collected (summed) from areas around the core such that

For testing, we mounted fibers on an XYZ translation stage with an additional fast electrostrictive scanner in the X direction—the direction normal to the step in the incident wavefront. In order to align the fiber for the best extinction, a one-dimensional scan in the X direction was performed using the scanner. The result of such a scan performed on the $20\mathrm{cm}$ TAU fiber segment with $N_R$ set to 2 pixels is shown in Fig. 2. The result is consistent with the single mode behavior for the fiber: the minimum in the middle corresponds to the alignment of the fiber’s core to the node, while the two maxima correspond to the alignment of the core with the two peaks of the input field distribution. The signal at the minimum is the measure of the leaked power $P_L$. To obtain this data we used the slow XYZ stage to (1) find the approximate X position to enable the use of the electrostrictive scanner which has a limited range, and (2) find the Y position that maximizes the signal to ensure that the fiber is moved through the center of the input beam.

In order to find the leakage $T_A$, in addition to leaked power $P_L$, we needed to measure the total power incident on the fiber $P_I$: $T_A=P_L/P_I$. To that end, a $500\mathrm{\mu m}$ pinhole was mounted within approximately $75\mathrm{\mu m}$ from the input end of the fiber. At the end of the measurement, the fiber was removed and the power through the pinhole was measured using the same infrared camera. The image of the input beam was intentionally blurred on the imager so that no pixels were saturated, and then all the pixels in the blurred image were summed.

In our measurement setup we identified three main parameters affecting the rejection of the input field. The first one is the angle of incidence on the stepped mirror. This angle was adjusted in small increments until the best rejection (smallest $P_L$) was obtained. The $20\mathrm{cm}$-long fiber segment was used for this adjustment. The second parameter is the angle of incidence of the input beam on the fiber tip. This parameter is important because of the presence of multiple reflections in the system. Since the coherence length of the laser source ($\sim 1\mathrm{cm}$) is much smaller than the length of the beam train, the multiple reflections add incoherently. If a fiber is not normal to the input beam, the phase step in the reflected wavefront will not align exactly with the step in the mirror; the wavefront reflected from the laser back to the fiber will also be misaligned. The multiple-reflection problem was alleviated by introducing the optical isolator—the quarter wave plate and the linear polarizer arrangement [14] shown in Fig. 1. In addition, the fiber was mounted on a tilt/tip stage so that the angle of incidence could be adjusted for the best rejection. This alignment was unique for each fiber since the fiber tip angle varied somewhat from fiber to fiber. The third parameter is the finite linewidth of the laser source which cannot be adjusted. As discussed in Section 5, the laser linewidth is likely the limiting factor in our measurement.

4. Results

Images of the fiber’s end for all four fiber segments were obtained with the fiber positioned at the point of the best extinction. This location of the central minimum is shown in Fig. 2. The images of the segments’ ends are shown in Fig. 3. The intensity in all four images was normalized to unity. The $50\mathrm{\mu m}$ diameter core takes up one pixel on the image. Its position is marked by white cross hairs. It was obtained by marking the brightest pixel in the image taken with the fiber positioned away from the point of best extinction, so the light coupled into the fundamental mode dominated the output.

Inspection of Fig. 3 reveals that the light intensity in the cladding area is significant. In fact, the brightest pixel in the image of the $4.5\mathrm{cm}$ long segment is not even at the center of the image (i.e., it is not at the core). To achieve the best filtering, the light propagating through the cladding should be removed: an aperture placed near the fiber’s end or at an intermediate focus could do the job. Alternatively, one could use a camera for data collection and use the digital aperture as described in Section 3. The aperture should not block the fundamental mode so that the useful signal is not significantly reduced. In the following we will find the optimal size of the digital aperture and evaluate the best filtering quality that can be achieved in the tested fiber segments.

For an apertured output we need to modify the quality metric $Q_f$. Since the aperture is attenuating the fundamental mode, we will use a new definition for the fundamental mode transmission: $T_F^{\text{'}}=\gamma T_F$, where the obscuration factor γ accounts for the additional intensity reduction in the fundamental mode caused by the aperture. The original definition of $Q_f$ is still valid: $Q_f=T_F^{\text{'}}/T_A$, where $T_A$ and $T_F^{\text{'}}$ are the leakage and the transmittance for the fundamental mode, respectively, as defined in Section 1. The physical meaning of the $Q_f$ is unchanged: it is proportional to the filter’s capability to reject the asymmetric input mode and to the filter’s transmittance for the fundamental mode.

The transmittance for the fundamental mode $T_F$ can be found by combining the results of the cutback measurement of optical loss [15] and the calculated Fresnel reflections of the fiber’s ends by using the well-known expression for the incoherent transmission through a lossy slab of material:

The dependence of the obscuration factor γ on the radius of the digital aperture was found experimentally. The image of the output end of the fiber was recorded and the summed intensity was plotted versus the size of digital aperture. One potential problem we endeavored to minimize was the appearance at the output of light propagating via the cladding modes. The stepped mirror in the setup was exchanged for a regular, plane mirror so as to improve the coupling at the input end and minimize the generation of cladding modes. To maximize the extinction of any cladding modes that were still generated, we chose to perform the measurement on the $20\mathrm{cm}$ long segment. The resulting dependence of γ on the radius of the digital aperture $N_R$ is presented in Fig. 4a.

The leakage $T_A$ was found as described in Section 1. Measurements similar to the one depicted in Fig. 2b were performed for each size of digital aperture, and the leaked photon flux was obtained from a parabolic fit near minimum. The leakage $T_A$ was then obtained by dividing the minimum value of the fitted curve by the measured photon flux through the input pinhole. The results of multiple measurements for each value of $N_R$ were averaged to improve accuracy. The dependence of $T_A$ on the digital aperture radius for all studied fiber segments is shown in Fig. 4b. Notice that the $4.5\mathrm{cm}$ segment has much greater leakage than all other segments and is plotted on a separate scale.

The values of $T_F^{\text{'}}$ were obtained by multiplying the obscuration factor γ shown in Fig. 4a by values of $T_F$ computed for each fiber segment (Table 2). The quality metric $Q_f$ was then found from $Q_f=T_F^{\text{'}}/T_A$. The dependence of $T_A$ and $Q_f$ on the digital aperture radius for all studied fiber segments is shown in Figs. 4b, 4c, respectively.

The maxima in $Q_f$ occur at the digital aperture radius $N_R$ of 2 for $15\mathrm{cm}$ and $20\mathrm{cm}$ long segments and at the radius of 1 for the $10.5\mathrm{cm}$ long segment. Note that the maximum values of $Q_f$ for the three longer fiber segments are fairly close. While the best $Q_f$ was recorded at $N_R=1$ for the $10.5\mathrm{cm}$ long section ($Q_f=17,000\pm 900$), the results for the three longer fiber sections have overlapping error bars at $N_R=1$ and $N_R=2$. The best values of $Q_f$ for the 15 and $20\mathrm{cm}$ long sections were $\text{16,000}\pm 3000$ and $\text{15,000}\pm 2000$, respectively. They were recorded at $N_R=2$. For convenience, the results obtained on $10.5\mathrm{cm}$ and longer segments for $N_R$ in the range from 1 to 4 are summarized in Table 3.

5. Discussion

5A. Limit of the Measurement Setup Capability

There is strong evidence that the best-measured value of $Q_f$ is limited by the capabilities of the measurement setup. The closeness of the maxima for the $10.5\mathrm{cm}$, $15\mathrm{cm}$, and $20\mathrm{cm}$ long segments is consistent with this point of view—otherwise the $Q_f$ should increase exponentially with the segment’s length. In addition, the measured values of $T_A$ for the three longer sections of the fiber are consistent with the limit placed on the measurement by the linewidth of the laser source. The manufacturer’s data sheet shows the linewidth $\delta \lambda$ of approximately $10\mathrm{nm}$, while the operating wavelength λ is $10.5\mathrm{\mu m}$. It can be shown that the uncertainty $\delta \lambda /\lambda$ leads to residual intensity $\delta I$ that can couple into the fundamental mode of the fiber at the input such that $\delta I/I\approx (\pi \delta \lambda /\lambda {)}^2$, where I is the intensity of the input beam. In our setup this corresponds to the residual intensity of the order of ${10}^{-5}$. This is the limit of our ability to measure $T_A$ with the current setup. It is consistent with the values of $T_A$ measured at $N_R=2$ (i.e., for the aperture that transmits most of the fundamental mode and rejects most of the cladding light) for the $10.5\mathrm{cm}$, $15\mathrm{cm}$, and $20\mathrm{cm}$ long segments, which are $2.0\times {10}^{-5}$, $2.4\times {10}^{-5}$, and $2.8\times {10}^{-5}$, respectively.

5B. Influence of the Pinhole at the Fiber Entrance on the Measurement Results

It should be noted that the pinhole at the fiber’s entrance is used for calibration only, and of itself does not act as an additional spatial filter. It is used as an aid in properly pointing the camera when measuring the intensity of light incident on the fiber. Indeed, this $500\mathrm{\mu m}$ pinhole is much too large to alter the measurement. We computed the intensity distribution in the focal spot for our setup configuration ($2.2\mathrm{cm}$ diameter mirror, $8\mathrm{cm}$ focal length) assuming flattop illumination profile. The distance between the first zeros in the wide direction (see cross section in the inset in Fig. 1) is approximately $150\mathrm{\mu m}$. For the $500\mathrm{\mu m}$ diameter pinhole, only a small fraction of the input beam power ($\sim {10}^{-2}$) is being cut by the pinhole. This is consistent with the images recorded by the camera, which show the double-spot intensity pattern confined to the central area far away from the pinhole edges. Incidentally, the same double-peaked intensity profile scanned with the single mode fiber made by the NRL is shown in Fig. 4 of [5]; most of the intensity is confined between the first zeros in the wide direction, which are approximately $150\mathrm{\mu m}$ apart, consistent with our calculation.

5C. Influence of Aperturing the Fiber Output on the Modal Filter Performance

While we used a digital aperture in this study, in practical systems a metal pinhole placed close to the fiber’s end (well within Rayleigh range) will provide the rejection of the cladding light. Such a circular aperture is a much better match to the circular shape of the fundamental mode than the digital aperture, which provides only a very rough approximation. Therefore one might be able to achieve even better separation of the fundamental mode from the cladding light.

Two potential adverse effects have to be addressed. Firstly, the cladding light rejected by the pinhole will be reflected into the fiber and can, after multiple reflections, exit through the pinhole. This effect should not significantly alter the performance of the $10.5\mathrm{cm}$ and longer segments for the following reason. Shalem et al. [12] found that for fibers treated with high-intensity UV exposure, such as the one investigated in this report, the light in the cladding is attenuated at the rate of approximately $2\mathrm{dB}/\mathrm{cm}$. Therefore, in the $10.5\mathrm{cm}$ long segment every double reflection will be attenuated by approximately $40\mathrm{dB}$, or the factor of ${10}^{-4}$. According to Fig. 4b, the transmittance $T_A$ for the antisymmetric mode through the unapertured fiber is approximately $6\times {10}^{-4}$ for this segment, so the additional double pass will add less than $6\times {10}^{-8}$ of the input intensity to the output, i.e., it will have negligible effect on the parasitic leakage. For comparison, the value of $T_A$ for the digital aperture $N_R=1$, for which the value of $Q_f$ is optimized, is approximately $2\times {10}^{-5}$. Even weaker multiple reflections are expected for longer fiber segments.

Another concern is that diffraction at the pinhole may increase the divergence angle of the output beam, leading to the loss of light that otherwise would fall on the detector. Using the analytical expression for a truncated Gaussian beam developed by Drège et al. [16] we found that, for the light exiting the fiber, the truncation caused by round pinholes sized to be same area as the digital apertures with $N_R$ values of 1 and 2 causes only very modest variation of the divergence angle.

A potential concern is that aperturing the fiber’s output amounts to introducing an additional spatial filter, so the ultimate performance measured in this report is that of the aperture. However, the input field we use for testing is antisymmetric and has the node in the center. For such disturbance an output aperture cannot act as a filter, because the field components that need to be rejected are at the center of the aperture and are not modified by its presence. In our setup, the aperture is needed to prevent the field components rejected by the fiber from reaching the detector. Wallner et al. [11] pointed out that the rejected (radiated) light, if not absorbed, appears at the output end, so measures must be taken to prevent it from reaching the detector. They proposed to achieve this by properly sizing the detector to effect the angular separation of the guided and the radiated waves.

We also explored the influence of inserting an aperture at the fiber’s output on the broadband (approximately 50% bandwidth, from $10.5\mathrm{\mu m}$ to $17.5\mathrm{\mu m}$) operation of the modal filter. If a single mode fiber is optimally matched to a lens at $10.5\mathrm{\mu m}$, at $17.5\mathrm{\mu m}$ the coupling will decrease to $\sim 90\%$ of the optimal value [10]. In addition, as the wavelength increases, so does the mode field diameter, so the aperture at the exit cuts off larger fraction of the output. We have calculated the mode field diameter increase over the specified wavelength range using a simplified model—a core in an infinite slab of cladding material. We found that it increases from $57\mathrm{\mu m}$ to $92\mathrm{\mu m}$. We then computed the obscuration factor at $10.5\mathrm{\mu m}$ and $17.5\mathrm{\mu m}$ for all values of $N_R$ in Table 3 using a simple Gaussian model for the fiber’s fundamental mode. This allowed us to compute the values of $Q_f$ at $17.5\mathrm{\mu m}$. We found that, if the $Q_f$ is of ultimate importance, the best choice is the $10.5\mathrm{cm}$ long segment, which provides the $Q_f$ of approximately 11,900 at $17.5\mathrm{\mu m}$ when the aperture of $183\mathrm{\mu m}$ diameter ($N_R=2$) is used. The power throughput at $17.5\mathrm{\mu m}$ in this case is reduced to approximately 74% of its original value at $10.5\mathrm{\mu m}$. One can choose a compromise based on the particular system requirement. For example, choosing the $324\mathrm{\mu m}$ aperture ($N_R=4$), one obtains 91% of the original power, but lower $Q_f$ (6000 for the $10.5\mathrm{cm}$ segment and 8500 for the $20\mathrm{cm}$ segment).

6. Conclusions

We characterized modal filtering in a midinfrared silver halide fiber produced by Tel Aviv University. The $10.5\mathrm{cm}$, $15\mathrm{cm}$, and $20\mathrm{cm}$ long sections of the fiber can reduce the undesirable components of the wavefront by a factor of 15,000 at least. This experimental value is limited by the noise properties of our measurement setup and far exceeds the requirements of a space-based nulling interferometer. Poor cladding mode suppression seems to be the main obstacle to obtaining short fibers with good modal filtering performance. To achieve high rejection, an aperture must be placed at the output of the fiber to block the cladding modes. In fibers even as short as the $4.5\mathrm{cm}$ long segment that we tested, a rejection of approximately 600 can be achieved.

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA).

Table 1. Refractive Indices and Core/Cladding Diameters of the Fiber

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Table 2. Values of $T_F$ for the Measured Fiber Segments

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Table 3. ${\mathsf{Q}}_{\mathsf{f}}$ Obtained on $\mathsf{10.5}\mathsf{cm}$ and Longer Fiber Segments^a

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Fig. 1 Measurement setup. The stepped mirror is used to produce the input field pattern with the node at the center. The optical isolator helps to minimize the multiple reflection problem. Inset: cross section of the field in the middle of the input beam. Note the sizes of the pinhole and the input beam pattern. The majority of the power ($\sim 99\%$) is transmitted by the pinhole.

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Fig. 2 (a): X scan of $20\mathrm{cm}$ long fiber segment. The dots represent experimental results. The solid curve is drawn as a guide to the eye. (b) X scan of the same segment near minimum. The parabolic fit is shown as the solid curve. The leaked power $P_L$ is determined from the minimum of the fit curve.

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Fig. 3 Images of the output end of the four fiber segments. Images are taken with the segments aligned for best extinction. The white cross hairs mark the position of the core. One pixel corresponds to approximately $45\mathrm{\mu m}$ by $45\mathrm{\mu m}$ area on the fiber’s tip.

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Fig. 4 (a) Obscuration factor γ versus digital aperture radius $N_R$. (As defined in Eq. (1), $N_R=0$ corresponds to data collection from a single pixel.) The dots show experimental results. The solid curve is a guide to the eye. (b) Leakage $T_A$ versus the digital aperture radius. (c) Modal filter quality factor $Q_f$ versus the digital aperture radius. One pixel corresponds to approximately $45\mathrm{\mu m}$ by $45\mathrm{\mu m}$ area on the fiber’s tip. Note that in (b) and (c) the data for the $4.5\mathrm{cm}$ long segment are plotted on a separate (right) scale.

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