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The rate of a network code is the ratio of the block size of the network's messages to that of its edge codewords. We compare the linear capacities and achievable rate regions of networks using finite field alphabets to the more general cases of arbitrary ring and module alphabets. For non-commutative rings, two-sided linearity is allowed. Specifically, we prove the following for directed acyclic networks: (i) The linear rate region and the linear capacity of any network over a finite field depend only on the characteristic of the field. Furthermore, any two fields with different characteristics yield different linear capacities for at least one network. (ii) Whenever the characteristic of a given finite field divides the size of a given finite ring, each network's linear rate region over the ring is contained in its linear rate region over the field. Thus, any network's linear capacity over a field is at least its linear capacity over any other ring of the same size. An analogous result also holds for linear network codes over module alphabets. (iii) Whenever the characteristic of a given finite field does not divide the size of a given finite ring, there is some network whose linear capacity over the ring is strictly greater than its linear capacity over the field. Thus, for any finite field, there always exist rings over which some networks have higher linear capacities than over the field., Comment: Submitted to IEEE Transactions on Information Theory June 4th, 2017 Revised January 29th, 2018

We prove the following results regarding the linear solvability of networks over various alphabets. For any network, the following are equivalent: (i) vector linear solvability over some finite field, (ii) scalar linear solvability over some ring, (iii) linear solvability over some module. Analogously, the following are equivalent: (a) scalar linear solvability over some finite field, (b) scalar linear solvability over some commutative ring, (c) linear solvability over some module whose ring is commutative. Whenever any network is linearly solvable over a module, a smallest such module arises in a vector linear solution for that network over a field. If a network is linearly solvable over some non-commutative ring but not over any commutative ring, then such a non-commutative ring must have size at least $16$, and for some networks, this bound is achieved. An infinite family of networks is demonstrated, each of which is scalar linearly solvable over some non-commutative ring but not over any commutative ring. Whenever $p$ is prime and $2 \le k \le 6$, if a network is scalar linearly solvable over some ring of size $p^k$, then it is also $k$-dimensional vector linearly solvable over the field $GF(p)$, but the converse does not necessarily hold. This result is extended to all $k\ge 2$ when the ring is commutative.

Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution over some finite commutative ring, then (i) the network is also scalar linearly solvable over a maximal commutative ring of the same size, and (ii) the (unique) smallest size commutative ring over which the network has a scalar linear solution is a field. We prove that a commutative ring is maximal with respect to the quasi-order if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the maximal ring's size. Finally, we prove that there is a unique maximal commutative ring of size $m$ if and only if each prime factor of $m$ has multiplicity in $\{1,2,3,4,6\}$. In fact, whenever $p$ is prime and $k \in \{1,2,3,4,6\}$, the unique such maximal ring of size $p^k$ is the field $GF(p^k)$. However, for every field $GF(p^k)$ with $k\not\in \{1,2,3,4,6\}$, there is always some network that is not scalar linearly solvable over the field but is scalar linearly solvable over a commutative ring of the same size. These results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but alternative rings may be needed when the alphabet size is fixed.

For each integer $m \geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when $m$ is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets., Comment: Submitted to IEEE Transactions on Information Theory January 14th 2016

The determination of radius of curvature (ROC) of optics typically uses either a phase measuring interferometer on an adjustable stage to determine the position of the ROC and the optics surface under test. Alternatively, a spherometer or a profilometer are used for this measurement. The difficulty of this approach is that for large optics, translation of the interferometer or optic under test is problematic because of the distance of translation required and the mass of the optic. Profilometry and spherometry are alternative techniques that can work, but require a profilometer or a measurement of subapertures of the optic. The proposed approach allows a measurement of the optic figure simultaneous with the full aperture radius of curvature.

Experiments measuring the effect of electron exposure on 1064nm transmission for optical sapphire were conducted. Detailed before and after inspections did not identify any resulting Litchenburg patterns. Pre- and post-exposure 1064nm transmission measurements are compared.

The authors report operational upgrades and streamlined data analysis of a commissioned electronic speckle interferometer (ESPI) in a permanent in-house facility at NASA's Goddard Space Flight Center. Our ESPI was commercially purchased for use by the James Webb Space Telescope (JWST) development team. We have quantified and reduced systematic error sources, improved the software operability with a user-friendly graphic interface, developed an instrument simulator, streamlined data analysis for long-duration testing, and implemented a turn-key approach to speckle interferometry. We also summarize results from a test of the JWST support structure (previously published), and present new results from several pieces of test hardware at various environmental conditions.

Experiments measuring the effect of electron exposure on 1064nm transmission for optical sapphire were conducted. Detailed before and after inspections did not identify any resulting Litchenburg patterns. Pre- and post-exposure 1064nm transmission measurements are compared.

The James Webb Space Telescope (JWST) is an infrared, space-based telescope scheduled for launch in 2013. JWST includes four scientific instruments, including the Near Infrared Spectrometer (NIRSpec), which operates in the wavelength range from 0.6 to 5 microns. NASA/Goddard Space Flight Center (GSFC) is responsible for two NIRSpec subsystems: the detector subsystem, with its focal plane array (FPA), and the micro-shutter subsystem, with its micro-shutter array (MSA). The FPA is comprised of two side-by-side Rockwell Scientific HgCdTe 2kx2k detectors, with detectors and readout electronics optimized for low noise. The MSA is a GSFC developed micro-electro-mechanical system (MEMS) that serves as a programmable slit mask, allowing NIRSpec to obtain simultaneous spectra of >100 objects in a single field of view. We present the optical characterization test plan of the FPA. The test plan is driven by many requirements: cryogenic operating temperature, a flight-like beam shape, and multi-wavelength photon flux from 1 to 60,000 per second, thus low stray light is critical. We use commercial optical modeling software to predict stray light effects at the FPA. We also present the optical contrast test plan of the MSA. Since each individual shutter element operates in on/off fashion, the most important optical metric is contrast. The MSA structure is designed to minimize stray and scattered light, and the test setup reduces stray light such that the optical contrast requirement is met.

The Infrared Multi-Object Spectrometer (IRMOS) is a principle investigator class instrument for the Kitt Peak National Observatory 4 and 2.1 meter telescopes. IRMOS is a near-IR (0.8 - 2.5 micron) spectrometer with low-to mid-resolving power (R = 300 - 3000). IRMOS produces simultaneous spectra of approximately 100 objects in its 2.8 x 2.0 arc-min field of view (4 m telescope) using a commercial Micro Electro-Mechanical Systems (MEMS) micro-mirror array (MMA) from Texas Instruments. The IRMOS optical design consists of two imaging subsystems. The focal reducer images the focal plane of the telescope onto the MMA field stop, and the spectrograph images the MMA onto the detector. We describe ambient breadboard subsystem alignment and imaging performance of each stage independently, and ambient imaging performance of the fully assembled instrument. Interferometric measurements of subsystem wavefront error serve as a qualitative alignment guide, and are accomplished using a commercial, modified Twyman-Green laser unequal path interferometer. Image testing provides verification of the optomechanical alignment method and a measurement of near-angle scattered light due to mirror small-scale surface error. Image testing is performed at multiple field points. A mercury-argon pencil lamp provides a spectral line at 546.1 nanometers, a blackbody source provides a line at 1550 nanometers, and a CCD camera and IR camera are used as detectors. We use commercial optical modeling software to predict the point-spread function and its effect on instrument slit transmission and resolution. Our breadboard and instrument level test results validate this prediction. We conclude with an instrument performance prediction for cryogenic operation and first light in late 2003.

The Infrared Multi-Object Spectrometer (IRMOS) is a facility instrument for the Kitt Peak National Observatory 4 and 2.1 meter telescopes. IRMOS is a near-IR (0.8 - 2.5 micron) spectrometer with low- to mid-resolving power (R = 300 - 3000). IRMOS produces simultaneous spectra of approximately 100 objects in its 2.8 x 2.0 arc-min field of view using a commercial Micro Electro-Mechanical Systems (MEMS) Digital Micro-mirror Device (DMD) from Texas Instruments. The IRMOS optical design consists of two imaging subsystems. The focal reducer images the focal plane of the telescope onto the DMD field stop, and the spectrograph images the DMD onto the detector. We describe ambient breadboard subsystem alignment and imaging performance of each stage independently, and the ambient and cryogenic imaging performance of the fully assembled instrument. Interferometric measurements of subsystem wavefront error serve to venfy alignment, and are accomplished using a commercial, modified Twyman-Green laser unequal path interferometer. Image testing provides further verification of the optomechanical alignment method and a measurement of near-angle scattered light due to mirror small-scale surface error. Image testing is performed at multiple field points. A mercury-argon pencil lamp provides spectral lines at 546.1 nm and 1550 nm, and a CCD camera and IR camera are used as detectors. We use commercial optical modeling software to predict the point-spread function and its effect on instrument slit transmission and resolution. Our breadboard test results validate this prediction. We conclude with an instrument performance prediction for first light.

The Infrared Multi-Object Spectrometer (IRMOS) is a principle investigator-class instrument for the Kitt Peak National Observatory 2.1 m and Mayall 3.8 m telescopes. IRMOS is a near-IR (0.8 - 2.5 micron) spectrometer with low-to mid-resolving power (R = lambda/delta lambda = 300 - 3000). On the 3.8 m telescope, IRMOS produces simultaneous spectra of approximately 100 objects in its approximately 3 x 2 arcmin field of view using a commercial micro electro-mechanical systems (MEMS) digital micro-mirror device (DMD) from Texas Instruments. The multi-mirror array DMD operates as a real-time programmable slit mask. The all-reflective optical design consists of two imaging subsystems. The focal reducer images the focal plane of the telescope onto the DMD field stop, and the spectrograph images the DMD onto a large-format detector. The instrument operates at approximately 80 K, cooled by a single electro-mechanical cryocooler. The bench and all components are made from aluminum 6061-T651. There are three cryogenic mechanisms. We describe laboratory integration and test of IRMOS before shipment to Kitt Peak. We give an overview of the optical alignment technique and integration of optical, mechanical, electrical and cryogenic subsystems. We compare optical test results to model predictions of point spread function size and morphology, contrast, and stray light. We discuss some lessons learned and conclude with a prediction for performance on the telescope.

We describe the population, optomechanical alignment, and alignment verification of near-infrared gratings on the grating wheel mechanism (GWM) for the Infrared Multi-Object Spectrometer (IRMOS). IRMOS is a cryogenic (80 K), principle investigator-class instrument for the 2.1 m and Mayall 3.8 m telescopes at Kitt Peak National Observatory, and a MEMS spectrometer concept demonstrator for the James Webb Space Telescope. The GWM consists of 13 planar diffraction gratings and one flat imaging mirror (58 x 57 mm), each mounted at a unique compound angle on a 32 cm diameter gear. The mechanism is predominantly made of Al 6061. The grating substrates are stress relieved for enhanced cryogenic performance. The optical surfaces are replicated from off-the-shelf masters. The imaging mirror is diamond turned. The GWM spans a projected diameter of approx. 48 cm when fully assembled, utilizes several flexure designs to accommodate potential thermal gradients, and is controlled using custom software with an off-the-shelf controller. Under ambient conditions, each grating is aligned in six degrees of freedom relative to a coordinate system that is referenced to an optical alignment cube mounted at the center of the gear. The local tip/tilt (Rx/Ry) orientation of a given grating is measured using the zero-order return from an autocollimating theodolite. The other degrees of freedom are measured using a two-axis cathetometer and rotary table. Each grating's mount includes a one-piece shim located between the optic and the gear. The shim is machined to fine align each grating. We verify ambient alignment by comparing grating difractive properties to model predictions.

The Infrared Multi-Object Spectrograph (IRMOS) is a facility instrument for the Kitt Peak National Observatory Mayall Telescope (3.8 meter). IRMOS is a near-IR (0.8 - 2.5 micron) spectrograph with low to mid resolution (R=lambda/delta, lambda = 300 - 3800). The IRMOS spectrograph produces simultaneous spectra of - 100 objects in its 2.8 x 2.0 arc-min field of view using a commercial MEMS multi-mirror array device (MMA). The IRMOS optical design consists of two imaging systems, or "stages." The focal reducer, stage one, images the focal plane of the telescope onto the MMA. The spectrograph, stage two, images the MMA onto the detector. We describe the breadboard alignment method and imaging and scattered light performance for both the focal reducer and spectrograph. This testing provides verification of the optomechanical alignment method, and a measurement of the contribution of scattered light in the system due to mirror small scale surface error. After the stage I and 2 optics are integrated with the instrument, our test results will make it possible to distinguish between scattered light from the mirrors and the MMA. Image testing will be done at four wavelengths in the visible and near-IR. A mercury-argon pencil lamp will provide spectral lines at 546.1 and 1012 nm, and a blackbody radiation source lines at 1600 and 2200 nm. A CCD camera will be used as a detector for the visible wavelengths, and an IR photodiode will be used for the IR wavelengths. We compare our data with a theoretical analysis using a commercial software package. Mirror surface error is modeled by treating each surface as a superposition of various gratings (e.g., diamond turning tool marks, features due to the impurities of Al 6061, and periodic mid-frequency errors due to drift during machining).

The Infrared Multi-Object Spectrometer (IRMOS) is a facility-class instrument for the Kitt Peak National Observatory 4 and 2.1 meter telescopes. IRMOS is a near-IR (0.8-2.5 micron) spectrometer and operates at approximately 80 K. The 6061-T651 aluminum bench and mirrors constitute an athermal design. The instrument produces simultaneous spectra at low- to mid-resolving power (R=lambda/delta lambda= 300-3000) of approximately 100 objects in its 2.8 x 2.0 arcmin field. We describe ambient and cryogenic optical testing of the IRMOS mirrors across a broad range in spatial frequency (figure error, mid-frequency error, and microroughness). The mirrors include three rotationally symmetric, off-axis conic sections, one off-axis biconic, and several flat fold mirrors. The symmetric mirrors include convex and concave prolate and oblate ellipsoids. They range in aperture from 94x86 mm to 286x269 mm and in f-number from 0.9 to 2.4. The biconic mirror is concave and has a 94x76 mm aperture, R(sub x)=377 mm, k(sub x)=0.0778, R(sub y)=407 mm, and k(sub y)=0.1265 and is decentered by -2 mm in X and 227 mm in Y. All of the mirrors have an aspect ratio of approximately 6:1. The surface error fabrication tolerances are less than 10 nm RMS microroughness, 'best effort' for mid-frequency error, and less than 63.3 nm RMS figure error. Ambient temperature (approximately 293 K) testing is performed for each of the three surface error regimes, and figure testing is also performed at approximately 80 K. Operation of the ADE Phaseshift MicroXAM white light interferometer (micro-roughness) and the Bauer Model 200 profilometer (mid-frequency error) is described. Both the sag and conic values of the aspheric mirrors make these tests challenging. Figure testing is performed using a Zygo GPI interferometer, custom computer generated holograms (CGH), and optomechanical alignment fiducials. Cryogenic CGH null testing is discussed in detail. We discuss complications such as the change in prescription with temperature and thermal gradients. Correction for the effect of the dewar window is also covered. We discuss the error budget for the optical test and alignment procedure. Data reduction is accomplished using commercial optical design and data analysis software packages. Results from CGH testing at cryogenic temperatures are encouraging thus far.

Challenges in fabrication and testing have historically limited the choice of surfaces available for the design of reflective optical instruments. Spherical and conic mirrors are common, but, for future science instruments, more degrees of freedom are necessary to meet challenging performance and packaging requirements. These instruments will be composed of unusual aspheres located far off-axis with large spherical departure, and some designs will require asymmetric surface profiles. In particular, single-surface astigmatism correction in spectrographs necessitates a toroidal surface, which lacks an axis of rotational symmetry. We describe the design, fabrication, optical testing, and performance of three rotationally symmetric, off-axis, aspheric mirrors and one toroidal, off-axis, biconic camera mirror on aluminum substrates for the Infrared Multi-Object Spectrograph (IRMOS) instrument. IRMOS is a facility instrument for the Kitt Peak National Observatory's Mayall Telescope (3.8 m) and an engineering prototype for a possible design of the Next Generation Space Telescope/Multi-Object Spectrograph. The symmetric mirrors range in aperture from 94x86 mm to 286x269 mm and in f-number from 0.9 to 2.4. They are various off-axis, convex and concave, prolate and oblate ellipsoids. The concave biconic mirror has a 94x76 mm aperture, Rx=377 mm, kx=0.0778, Ry=407 mm, and ky=0.1265 and is decentered. by -2 mm in x and 227 mm in y. The mirrors have an aspect ratio of approximately 4:1. The surface error fabrication tolerances are less than 63.3 nm RMS figure error and less than 10 nm RMS microroughness. The mirrors are attached to the instrument bench via a semi-kinematic, integral flexure mount. We describe mirror design, diamond machining, the results of figure testing using computer-generated holograms, and imaging and scattered light modeling and performance.