Passive Snapshot Remote Sensing of Orbital Velocity

Home , Atomic line filter, Fraunhofer lines

ABSTRACT

PANTALONE, BRETT ANTHONY. Passive Snapshot Remote Sensing of Orbital Velocity. (Under the direction of Michael Kudenov).

Techniques for tracking objects in low Earth orbit include line of sight angle measure-

ments and active range measurements using RADAR or laser reﬂection. Active ranging

techniques are less effective for high-altitude orbits because of the relatively large signal

loss from distant targets. Passive methods using only optical telescopes require a minimum

of three separate observations to solve the orbit equation. This thesis demonstrates how

incorporation of passive Doppler shift measurement of solar Fraunhofer lines can improve

initial orbit determination, while reducing the number of required observations to two.

First, a discussion of the current state of space situational awareness establishes the mo-

tivation for this work. Then, a brief review of conventional Doppler velocimetry highlights

the strengths and weaknesses of current methods.

In Chapter2, passive Doppler velocimetry is discussed, along with the rationale and

limitations of Fraunhofer lines as wavelength references. This is followed by a radiometric

analysis to estimate the optical power received from sunlight reﬂected off small orbiting

objects. Next, the topic of spatial heterodyning is introduced as a potential technique for im-

proving the signal-to-noise ratio of a passive velocimeter. Results of computer simulations

are presented to support this idea.

The derivation of a new mathematical algorithm for initial orbit determination incor-

porating Doppler shift measurement is presented in Chapter3. The new technique is a

modiﬁcation of Gooding’s method, which is also described to provide proper context. Next,

the results of orbital simulations using both methods are presented and compared. The

results indicate that in some orbital conﬁgurations, the new method can successfully solve an initial orbit using only two observations, proving the feasibility of the method.

The design and development of optical hardware for measuring the Doppler shift of

Fraunhofer lines is described in Chapter4. Trade-space analysis of several alternate designs

are considered, before concluding with a walk-through of the ﬁnal design of a dual-beam,

Doppler ratio, polarimetric, direct correlation spectrometer.

Chapter5 recounts the assembly, calibration, and preliminary testing of the prototype

instrument. Bench tests demonstrate a working spectral resolution of 0.04 nm, while radial velocity measurements of the planet Venus differ from theoretical calculations by only

0.59%.

by Brett Anthony Pantalone

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulﬁllment of the requirements for the Degree of Doctor of Philosophy

Electrical Engineering

Raleigh, North Carolina

2018

APPROVED BY:

Robert Kolbas Leda Lunardi

Kenan Gundogdu Michael Kudenov Chair of Advisory Committee DEDICATION

To my friend and mentor, Herbert Cooper, whose advice and encouragement have been

invaluable.

ii BIOGRAPHY

Brett Pantalone was born in Ravenna, Ohio to parents Paul and Virginia Pantalone. Brett graduated from the University of Akron with a B.S. in electrical engineering in 1992. For the next 20 years, he worked in the consumer electronics industry, writing embedded software for mobile phones and other wireless devices.

Brett moved to southern Sweden in 1999, working for the telecommunications giant

Ericsson. In 2001 he returned to Raleigh, North Carolina to work in the technology ofﬁce of the newly formed Sony Ericsson joint venture.

In 2006 Brett left Sony Ericsson to become an independent software consultant for technology companies in and around Research Triangle Park. After marrying his wife, Vicki, in 2008, he joined the full-time staff of Device Solutions, a small startup in Morrisville,

North Carolina.

Cursed with an insatiable curiosity and a desire to solve new types of problems, Brett returned to school in 2013 as a graduate student at North Carolina State University. Originally pursuing an M.S. in aerospace engineering, he eventually ended up as a doctoral student in electrical engineering, working in the Optical Sensing Laboratory under the supervision of

Dr. Michael Kudenov. Brett’s research interests include remote sensing and astronomical instruments.

Now, Brett lives with his wife and two spoiled cats in Pittsboro, North Carolina. He is just waiting for the next adventure.

iii ACKNOWLEDGEMENTS

Perhaps the most important thing I have learned as a graduate student is that no difﬁcult journey should be undertaken alone. There are many people who deserve my thanks for helping me on this trek. First, I would like to thank Dr. Kudenov for giving me the opportunity and support to do this work. I am also grateful to my current and former peers in the Optical Sensing Laboratory, especially Bryan Maione, Ruonan Yang, David Luo,

Ethan Woodard, Yifan Wang, and Brandon Ballance. In addition to their problem-solving assistance, they made a “non-traditional” student feel welcome in their group.

I owe a special debt of gratitude to my wife, Dr. Vicki Behrens, for her patience and encouragement. I also thank my parents, Paul and Virginia, for their unﬂinching belief in my ability. Finally, I thank Misha and Willa for reminding me when to take a break.

iv TABLE OF CONTENTS

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

Chapter 1 INTRODUCTION ...... 1 1.1 Review of Fraunhofer Lines...... 3 1.2 Review of Doppler Velocimetry...... 5

Chapter 2 PASSIVE DOPPLER VELOCIMETRY ...... 9 2.1 Radiometric Analysis ...... 11 2.1.1 Radiometric Transfer Model...... 12 2.1.2 Instrument Noise Model ...... 13 2.1.3 Signal to Noise Ratio...... 15 2.2 Spatial Heterodyning...... 16 2.3 Heterodyne Simulation...... 19 2.4 Velocity-Fitting Simulation...... 21 2.4.1 Single-Band Results...... 22 2.4.2 Two-Band Results...... 28 2.4.3 Three-Band Results...... 29 2.4.4 Manual Results...... 31

Chapter 3 ORBITAL SIMULATION ...... 34 3.1 Background...... 35 3.2 Mathematical development ...... 38 3.3 Simulation...... 40 3.3.1 Orbit generation...... 41 3.3.2 Orbit prediction ...... 41 3.4 Testing and results...... 48 3.4.1 Geostationary orbits...... 50 3.4.2 Mid-altitude orbits ...... 52 3.4.3 Eccentric and inclined orbits...... 55 3.4.4 Coplanar observer...... 58 3.5 Discussion...... 58

Chapter 4 INSTRUMENT DESIGN ...... 66 4.1 Full Spectral Imaging...... 67 4.1.1 Heterodyned Birefringent Interferometer...... 68 4.1.2 Heterodyned Sagnac Interferometer ...... 71 4.2 Dual-Band Doppler Ratio...... 74

v 4.2.1 Atomic Line Filters ...... 75 4.2.2 Birefringent Filters ...... 85 4.2.3 Fiber Bragg Grating...... 89 4.2.4 Optical Correlator...... 94 4.3 Trade-Space Summary...... 103 4.4 Direct Correlation Spectrometer...... 105 4.4.1 Spatial Light Modulators...... 106 4.4.2 Design of the Optical Correlator...... 108

Chapter 5 ASSEMBLY AND CALIBRATION ...... 113 5.1 Frequency Stability...... 117 5.2 Alignment and Calibration...... 119 5.3 Test Results ...... 124

Chapter 6 CONCLUSION ...... 130

REFERENCES ...... 133

vi LIST OF TABLES

Table 1.1 Major Fraunhofer line designations, sources, and wavelengths...... 5

Table 2.1 Values used for calculation of system SNR...... 16

Table 4.1 Filter parameters for the Voigt optical filter...... 84 Table 4.2 Simulation parameters for the fiber Bragg grating filter...... 92 Table 4.3 Optimized grating parameters for the fiber Bragg grating filter...... 93 Table 4.4 First order parameters at the Bragg condition for the VPH grating from Wasatch Photonics...... 100 Table 4.5 Design trade-space...... 105

Table 5.1 Key specifications of the direct correlation spectrometer...... 114 Table 5.2 Imaging sensor Apogee A694 specifications...... 127 Table 5.3 Santec spatial light modulator SLM-100 specifications...... 127 Table 5.4 Direct correlation spectrometer bill of materials...... 128

vii LIST OF FIGURES

Figure 1.1 The visible solar spectrum, from 380 nm to 710 nm, with locations of some major Fraunhofer lines shown...... 4 Figure 1.2 Laser Doppler interferometer...... 6 Figure 1.3 Power vs. range curve for a hypothetical RADAR system with 30 m antenna. The altitude of geosynchronous orbit is indicated by the vertical dashed line...... 8

Figure 2.1 Components of Doppler shift along the optical path...... 10 Figure 2.2 Michelson interferometer. (a) Conventional homodyne design centered at zero frequency. (b) Heterodyne design using diffraction gratings; the center frequency is chosen by selection of the blaze angle. . 18 Figure 2.3 Differential intensity vs. radial velocity for heterodyned frequency bands centered at 424 nm, 530 nm, and 628 nm...... 21 Figure 2.4 Overview of all single-band simulation results for a system SNR of 20. 23 Figure 2.5 Overview of all single-band simulation results for a system SNR of 10. 24 Figure 2.6 Overview of all single-band simulation results for a system SNR of 5. 25 Figure 2.7 Best-performing spectral bands for an SNR of 20. The legend indicates center wavelength values in nanometers...... 25 Figure 2.8 Best-performing spectral bands for an SNR of 10. The legend indicates center wavelength values in nanometers...... 26 Figure 2.9 Best-performing spectral bands for an SNR of 5. The legend indicates center wavelength values in nanometers...... 26 Figure 2.10 Histogram for the spectral band centered around 391.0 nm for simulated radial velocities of (a) 2000 m/s and (b) 8000 m/s...... 27 Figure 2.11 Overview of two-band simulation results...... 28 Figure 2.12 Best-performing pairs of spectral bands. The legend indicates center wavelength values of each band in nanometers...... 29 Figure 2.13 Overview of three-band simulation results...... 30 Figure 2.14 Best-performing triplets of spectral bands. The legend indicates center wavelength values of each band in nanometers...... 31 Figure 2.15 Simulation of three spectral bands taken separately and together. The legend indicates center wavelengths in nanometers...... 33

Figure 3.1 Illustration of Lambert’s orbital boundary-value problem...... 36 Figure 3.2 Geometric relationships between Earth, Sun, and target body...... 38 Figure 3.3 Block diagram of the orbit generation and prediction algorithm. . . . 42 Figure 3.4 Illustration of the admissible region (shaded) for an object in orbit. . 43 Figure 3.5 Block diagram of the coarse fit procedure...... 45 Figure 3.6 Block diagram of the fine fit procedure...... 47

viii Figure 3.7 Zero-noise performance of test case GEO-1 for (a) AO and (b) AV methods...... 51 Figure 3.8 Results for the GEO-1 test case using (a) three and (b) two observations. 51 Figure 3.9 Zero noise performance of test case MEO-1 for (a) AO and (b) AV methods...... 53 Figure 3.10 Results for the MEO-1 test case using (a) three and (b) two observations. 53 Figure 3.11 Results for the MEO-2 test case using (a) three and (b) two observations. 54 Figure 3.12 Results for the MEO-3 test case using (a) three and (b) two observations. 54 Figure 3.13 Zero-noise performance of test case GTO for (a) AO and (b) AV methods. 56 Figure 3.14 Results for the GTO test case using (a) three and (b) two observations. 56 Figure 3.15 Zero-noise performance of test case MOL for (a) AO and (b) AV methods. 57 Figure 3.16 Results for the MOL test case using (a) three and (b) two observations. 57 Figure 3.17 Results for the coplanar observer, using (a) three and (b) two observations...... 59 Figure 3.18 Orbital prediction error (circles) vs. phase angle (solid line)...... 59 Figure 3.19 Summary statistics for combined test cases GEO-1, GEO-2, and GEO- 3. (a) Error statistics for all phase angles. (b) Errors conditioned on

phase angle φ < 90◦...... 61 Figure 3.20 Summary statistics for test case MEO-1. (a) Error statistics for all

phase angles. (b) Errors conditioned on phase angle φ < 90◦...... 62 Figure 3.21 Summary statistics for test case MEO-2. (a) Error statistics for all

phase angles. (b) Errors conditioned on phase angle φ < 90◦...... 62 Figure 3.22 Summary statistics for test case MEO-3. (a) Error statistics for all

phase angles. (b) Errors conditioned on phase angle φ < 90◦...... 63 Figure 3.23 Summary statistics for test case GTO. (a) Error statistics for all phase

angles. (b) Errors conditioned on phase angle φ < 90◦...... 63 Figure 3.24 Summary statistics for test case MOL. (a) Error statistics for all phase

angles. (b) Errors conditioned on phase angle φ < 90◦...... 65 Figure 4.1 Snapshot hyperspectral imaging Fourier transform (SHIFT) spectrometer...... 68 Figure 4.2 Fourier imaging spectrometer based on a heterodyned birefringent interferometer...... 69 Figure 4.3 Zemax ray-trace of the heterodyned birefringent interferometer. . . . 71 Figure 4.4 Illustration of a simple Sagnac interferometer...... 73 Figure 4.5 Heterodyned Sagnac interferometer using blazed diffraction gratings. 73 Figure 4.6 Zemax ray-trace of the heterodyned Sagnac interferometer...... 74 Figure 4.7 Faraday anomalous dispersion optical filter...... 77 Figure 4.8 Electromagnetic simulation of a Faraday filter using longitudinally magnetized block magnets...... 79 Figure 4.9 Magnetic flux density along the vapor cell axis...... 79

ix Figure 4.10 Axisymmetric simulation of a Faraday filter using encircling ring magnets...... 80 Figure 4.11 Magnetic flux density along the vapor cell axis...... 80 Figure 4.12 Electromagnetic simulation of a Faraday filter using a solenoid. . . . 82 Figure 4.13 Magnetic flux density along the vapor cell axis...... 82 Figure 4.14 Electromagnetic simulation of a Voigt filter using NdFeB block magnets. 83 Figure 4.15 Magnetic flux density along the vapor cell axis for seven different permanent magnet separation distances...... 83 Figure 4.16 Simulated Voigt filter transmission for two Fraunhofer wavelengths of potassium, (a) D1, and (b) D2...... 85 Figure 4.17 Simple four-stage Lyot filter. The fast axis of each birefringent ele-

ment is oriented at 45◦ with respect to the linear polarizers (LP). . . . 87 Figure 4.18 Lyot filter transmission as a function of wavelength for (a) one stage, (b) two stages, (c) three stages, and (d) four stages. The elements are modeled as calcite; thickness of the first stage is 1 mm...... 87 Figure 4.19 Multi-band filter composed of three two-stage Lyot filter sections. . . 88 Figure 4.20 Fiber Bragg grating...... 91 Figure 4.21 Reflected power vs. radial velocity of the target object...... 92 Figure 4.22 Basic design of a dual-beam Doppler optical correlator...... 95 Figure 4.23 (a) Standard blazed grating. (b) Echelle grating...... 96 Figure 4.24 (a) Incoming light diffracted by an echelle grating and cross-disperser. (b) The resulting two-dimensional spectrogram...... 97 Figure 4.25 (a) Key physical parameters of a volume phase grating. (b) Modula- tion of the refractive index...... 98 Figure 4.26 First order diffraction efficiency versus wavelength. These results include an additional 3% loss in both S and P waves due to absorption.102 Figure 4.27 First order diffraction efficiency versus incident angle. These results include an additional 3% loss in both S and P waves due to absorption.102 Figure 4.28 First order diffraction efficiency versus index modulation (vertical axis) and DCG thickness (horizontal axis). Color maps to the average efficiency of unpolarized light. These results include an additional 3% loss due to absorption. The marker indicates the design parameters for the VPH grating described in the previous section...... 103 Figure 4.29 Transmissive liquid crystal (LC) cell. (a) In the absence of an electric field, the LC molecules orient themselves with the orthogonal alignment layers, forming a twisted helix structure. (B) When a voltage is applied to the electrodes, the molecules align with the electric field. 107 Figure 4.30 Final design of the optical correlator...... 109

x Figure 4.31 (a) Conceptual model of the photomask pattern used on the SLM. (b) A detail of the pattern, showing the relationship between wavelength of the input and phase modulation introduced by the SLM...... 112 Figure 4.32 Ratio of in-band optical power to total power as a function of simulated wavelength shifts...... 112

Figure 5.1 Laboratory prototype of the direct correlation spectrometer...... 114 Figure 5.2 Instrument platform with spectrometer and telescope on the EQ mount...... 115 Figure 5.3 Calibration section...... 116 Figure 5.4 Calibration image of iron lamp emission lines at the SLM focal plane. 116 Figure 5.5 Reference peak x -pixel position as a function of time...... 118 Figure 5.6 Instrument (blue) and ambient (red) temperatures during stability test.119 Figure 5.7 The SLM-100 spatial light modulator...... 121 Figure 5.8 Direct correlation spectra of the iron lamp (solid blue) and Venus (dashed red) obtained by the instrument...... 125 Figure 5.9 Subset of correlation data for (a) Venus, and (b) iron lamp, ﬁtted to Gaussian exponential curves (solid blue lines)...... 126 Figure 5.10 Theoretical radial velocity of Venus for all of 2018, calculated from ephemeris data. The red circle indicates the theoretical value at the time of the instrument test. The red cross indicates the measurement obtained from the instrument...... 129

xi CHAPTER

INTRODUCTION

The emergence of commercial space launch services and smaller, cheaper satellites has

accelerated the growing density of objects in orbit around the earth. These objects include

not just satellites, but spent rockets, cast-off tools, and debris [1, 2]. As early as 1978, researchers recognized the potential for catastrophic collisions between objects in orbit

above Earth. Kessler and Cour-Palais showed that when trafﬁc in a geocentric orbit reaches

a critical mass density, the debris generated by occasional collisions can cascade, resulting

in yet more collisions [3, 4]. Today, the amount of unwanted detritus in the skies above Earth is approaching this critical limit. The potential for cascading collisions, popularized

in the 2013 movie Gravity, is referred to as the Kessler Syndrome [5].

1 To reduce the risk of collisions, all known orbital objects are cataloged by a campaign of space situational awareness (SSA) [6]. When new objects are discovered, it is essential that their orbits be determined quickly. Accurate orbital prediction requires continual tracking of target objects and regular updating of their orbital parameters. However, when a new object is ﬁrst detected, a rough estimate of its orbit is required as a starting point for more accurate tracking protocols. This initial orbit determination (IOD) for potentially hazardous objects is the ﬁrst step toward generating useful orbit predictions and assessing potential risks to satellites and astronauts.

For objects close to Earth, IOD is facilitated by active ranging techniques such as RADAR and LIDAR. Low Earth orbit (LEO) objects with diameters of 10 cm or larger [1] are good targets for active ranging. High above the earth, in geosynchronous orbit (GSO), active ranging is hampered by signal attenuation; objects at a distance r from the transmitter return signals weakened by a factor of r 4. Since relatively little signal power is returned to the receiver, GSO objects are commonly tracked by passive optical telescopes rather than active ranging [1,7 ]. Although optical telescopes cannot measure the distance to an object in the same way as active instruments, they do excel at measuring the angular position of an object in units of right ascension and declination relative to the stellar background. But without range information, additional angle measurements are required to calculate an orbit, and multiple observations over several hours are often needed [8]. Because of changing weather conditions, limited availability of large telescopes, and the urgency of preventing collisions, it is desirable to calculate an IOD in the shortest possible time, with the fewest number of separate sightings at the telescope.

The earliest calculation of an IOD was Johannes Kepler’s solution for the orbit of Mars in 1605, using observations of the planet’s position in the sky. Since then, various solutions

2 have been proposed for solving an orbit from three angle measurements along the line of

sight. The two classic methods of Laplace and Gauss [9], while simple to implement, are limited in their accuracy and application. However, modiﬁcations of these two techniques

for improved accuracy and stability are still in use today [10]. Other methods leverage the availability of modern computing resources to implement iterative solutions, such as the

Double-r [11] and Gooding’s method [12]. More recently, new techniques, based on the use of Lagrange coefﬁcients, have been developed that circumvent problems contributing

to inaccuracies and instability in earlier methods [13]. All these angles-only methods are constrained by the requirement to have three independent observations for solving the

complete set of six orbital parameters.

To reduce the number of required observations and address some disadvantages of

active ranging, this dissertation describes a new method and instrument for calculating

an IOD that utilizes the Doppler shift of Fraunhofer lines in reﬂected sunlight. Doppler

analysis of Fraunhofer lines is a familiar technique for estimating the velocity of radiant

objects; the search for extra-solar planets is a well-known example of its application [14]. Within the solar system, Doppler line shifts have been shown to be a useful measurement for

improving orbital descriptions of interplanetary dust [15, 16] and asteroids [17]. To advance this technique, the present work demonstrates that passive Doppler measurements, using a

novel, high-resolution optical correlator [18], can provide additional information for solving the IOD problem with as few as two telescopic observations.

1.1 Review of Fraunhofer Lines

All light, emitted by the sun, passes through the solar atmosphere. The hydrogen, helium,

and other elements in the sun’s atmosphere absorb light at characteristic wavelengths,

3 creating dark Fraunhofer lines in the solar spectrum, as illustrated in Fig. 1.1. These lines

occur at speciﬁc, well-known frequencies and are typically very narrow in width, on the

order of 0.01 nm [19]. Their occurrence indicates the presence of speciﬁc chemical species, and they can act as reliable wavelength markers. The visible light that reaches Earth from

the sun contains thousands of Fraunhofer lines [20]. The common names of some especially prominent lines, along with their chemical sources, are listed in Table 1.1.

Fraunhofer lines are useful not only for their ability to identify unknown elements —

the basis for spectroscopic analysis — but also for providing information about the motion

of a radiant object. When two objects are in relative motion, waves traveling between them

are stretched or compressed due to the Doppler effect. The resulting shift in the wavelength

is given by v λ λ 1 , (1.1) 0 = + c where λ0 is the measured wavelength, λ is the original wavelength, v is the radial velocity between the source and observer, and c is the speed of light. (In general, this expression works for any non-relativistic object that emits or reﬂects wave energy, provided the con-

stant c is replaced by the propagation speed of the waves.) When the source of light is the

D KH G F b E 3-1 C B A h g f e d h c h 4-1 a

390 450 550 650 750 400 500 600 700 wavelength in nm

Figure 1.1 The visible solar spectrum, from 380 nm to 710 nm, with locations of some major Fraunhofer lines indicated [21].

4 Table 1.1 Major Fraunhofer line designations, sources, and wavelengths [22].

Designation Source λ (nm) Designation Source λ (nm)

CHα 656.28 b4 Mg I 516.73 D1 Na I 589.59 F Hβ 486.13 D2 Na I 589.00 d Fe I 438.36 D3 He I 587.56 G CH 431.42 E Fe I 526.96 g Ca I 422.67

b1 Mg I 518.36 h Hδ 410.17 b2 Mg I 517.27 i Fe I 404.58 b3 Fe II 516.91 H Ca II 396.85 Fe I 516.89 K Ca II 393.37

sun, the Doppler shift can be determined by comparing an image of shifted Fraunhofer

lines to an unshifted reference image. This is a common technique for determining the velocity of stellar objects; unseen planets orbiting other stars can be detected by the slight

changes in velocity they induce in the host star [14].

1.2 Review of Doppler Velocimetry

Conventional Doppler velocimetry makes use of a beamed RADAR (radio) or LIDAR (laser)

signal reﬂected from a moving target. For example, Fig. 1.2 illustrates the principle of a laser

Doppler interferometer (LDI). Laser light with frequency f0 enters beamsplitter BS1 and is

split into a test beam and a reference beam. The test beam passes through an acousto-optic

modulator (AOM), which uses sound vibrations to increase the frequency by an amount fb .

The AOM thus sets a measurable zero-velocity reference, so any ambiguity introduced by

the direction of the target velocity is eliminated [23]. The test beam is directed at the moving target, which imparts an additional Doppler

frequency shift ( fd ) to the beam. The reﬂected test beam then returns to the receiver, ±

5 Figure 1.2 Laser Doppler interferometer.

where it is recombined with the reference beam in beamsplitter BS3. The summation of

the test and reference beams is given by the identity

cos[2πf0] + cos[2π(f0 + fb + fd )] = 2cos[π(2f0 + fb + fd )]cos[π(fb + fd )]. (1.2)

The beat frequency fb + fd is easily isolated by the low-pass ﬁlter (LPF) and measured by the detector. The target’s radial velocity is then calculated from the beat frequency. While

LDI has been shown capable of relative accuracies down to 0.5% over short integration

times, the band-limited signal received at the detector results in a poor signal-to-noise

ratio (SNR) [24]. This problem becomes worse with distance from the target. Another common technique for velocity measurement, pulsed Doppler velocimetry,

uses short RADAR pulses with a variable duty cycle. The returning echoes are analyzed in

the frequency domain to extract information about the size, shape, range, velocity, and

rotation of a target. With a large enough antenna, even distant objects can be precisely

characterized. For example, the 70 m Goldstone antenna DSS-14 was able to resolve the

radial velocity of asteroid 4179 Toutatis with a precision of 0.15 mm/s [25, 26].

6 However, this precision comes at a cost. The classic RADAR equation [27] deﬁnes the SNR as P G 2λ2σ (SNR) t , (1.3) = 3 4 (4π) kT0BFR where

Pt is the output power of the transmitter,

G is the antenna gain,

λ is the wavelength of the signal,

σ is the effective RADAR cross section (RCS),

k is Boltzmann’s constant,

T0 is the input noise temperature of 290 K,

B is the signal bandwidth,

F is the receiver’s noise ﬁgure, and

R is the line-of-site range to the target.

Using Eq. (1.3) and typical values for these parameters, we can plot the average output

power necessary to obtain a desired SNR at any given distance. Fig. 1.3 illustrates the power vs. range curve for SNR values of 10, 20, and 30 dB. This curve was calculated using the

parameters for a hypothetical system with a 30 m antenna, 930 MHz frequency, 12 MHz

bandwidth, and typical noise values, pointing at a target with an RCS of 1 m2. Although an

overly simpliﬁed example, it demonstrates that resolving small satellites at GSO altitude

requires a large antenna and tens of megawatts of output power.

RADAR and LIDAR instruments are indispensable tools for accurate, long-term or-

bital tracking of LEO objects within about 20 103 km of Earth. Large RADAR instruments × can even make highly accurate orbital predictions for asteroid-sized objects at planetary

distances. However, neither technique is satisfactory for making rapid, snapshot measure-

7 Power vs. Range 120 SNR = 30 dB 100 SNR = 20 dB SNR = 10 dB

GSO altitude Power (dBW) Power 20

-20

-40 102 103 104 105 Range (km)

Figure 1.3 Power vs. range curve for a hypothetical RADAR system with 30 m antenna. The altitude of geosynchronous orbit is indicated by the vertical dashed line.

ments of small GSO objects in geosynchronous orbits. The current work develops a new tool for these situations.

8 CHAPTER

PASSIVE DOPPLER VELOCIMETRY

When sunlight is reﬂected by an orbiting body, the body is in motion relative to both the

sun and an observer on Earth. Consider a solar Fraunhofer line at some wavelength, λ,

depicted in Fig. 2.1 (a). When the light is reﬂected from an orbiting object, the observed wavelength is shifted twice. The ﬁrst shift, occurring along the optical path between Sun

and object, results in a wavelength of λ0 incident on the object, as illustrated in Fig. 2.1 (b). The second shift, introduced along the path of reﬂection between the object and a

ground-based observer, results in a measured wavelength of λ00, as illustrated in Fig. 2.1 (c). The two wavelength shifts are not necessarily in the same direction, and the ﬁnal, observed wavelength is dependent on both the geometry and relative velocities of Sun, object, and

9 observer. Applying Eq. (1.1) twice yields an observed wavelength of

ρ˙ + vs ρ˙ vs λ λ 1 , (2.1) 00 = + c + c 2

where vs and the range-rate, ρ˙, are the radial velocities of the object with respect to Sun

and observer, respectively.

(b)

λ' . ρ (a)

v ρ s λ

(c)

λ''

Figure 2.1 Components of Doppler shift along the optical path.

Although the above treatment considers only a single wavelength, a practical imple-

mentation would average the Doppler shift over many wavelengths by way of an optical

correlator [28, 18]. The accuracy of the ﬁnal measurement will improve with the number of Fraunhofer lines included in the average.

The accuracy of this technique also depends on the stability of the reference lines.

10 Short-term variations in Fraunhofer wavelengths are caused by thermal and magnetic

instability in the solar atmosphere, as well as the rotational velocity of the sun. However,

other research indicates that, when averaged over the entire disk of the sun, those effects

contribute an error of less than 2 m/s to the measured radial velocity of solar system objects

[29, 30]. This short-term error could be effectively eliminated by using a reference image of the solar spectrum obtained at the same time as observation of the target object.

An additional source of error is the Doppler contribution caused by rotation of the

target object. This situation is not considered by our model, since rotational motion of

an Earth-orbiting satellite is assumed to be small relative to its orbital velocity. Finally, it

should be noted that the required Doppler measurements must be performed at night to

avoid interference from ambient sunlight. The technique is also restricted to geometries where the target is not eclipsed by Earth’s shadow. Traditional RADAR measurements do

not suffer from these limitations.

2.1 Radiometric Analysis

To determine the feasibility of passive Doppler velocimetry for orbital measurements,

an overall signal-to-noise ratio (SNR) must be determined. First, a radiometric model is

established to estimate the amount of useful light energy available to a remote measurement

instrument. Second, a noise model for the instrument itself is developed. After deﬁning all

system parameters, these two models determine an optical system’s SNR.

11 2.1.1 Radiometric Transfer Model

2 Using the Planck law for blackbody radiation, the sun’s radiant exitance, in units of W/m , can be determined by integration over the spectral region of interest:

Z λ2 c1 dλ M = , (2.2) λ5 expc T 1 λ1 2/λ eff − where M is the solar exitance within the wavelength range λ1 to λ2; Teff is the effective tem-

perature of the sun; and c1 and c2 are the ﬁrst and second radiation constants, respectively.

Viewed from Earth orbit, the solid angle subtended by the sun, using the small-angle

approximation, is given by πR 2 Ω , (2.3) sun = D 2 where R is the radius of the sun and D is the Earth-Sun distance. The target body, e.g., a

small satellite in geosynchronous orbit, is modeled as a cube presenting a surface area of

1 m2. The incident irradiance on the satellite at this distance is

M Esat = Ω sun . (2.4) π

Since two common materials used in satellite construction, aluminum and mylar, both

have coefﬁcients of reﬂection above 0.90, it is assumed that the body as a whole has a

similar Lambertian (diffusive) reﬂectivity. Now, the reﬂected radiance from the satellite

can be modeled as a function of the incoming irradiance and the reﬂectivity, such that

Esat Lref = ρ cosθi , (2.5) π

where ρ is the reﬂectivity and θi is the incident angle of the sunlight. Observed from a

12 ground-based telescope, the satellite subtends a solid angle of

A Ω sat , (2.6) sat = d 2

where Asat is the area of the satellite and d is its distance from Earth. Now, the irradiance

falling on the telescope is given by

Etel = Lref Ω sat τatm , (2.7)

where τatm is the optical transmission of the atmosphere. At visible wavelengths, the

atmosphere is mostly transparent, and attenuation is dominated by Rayleigh scattering

[31]. Finally, the radiant ﬂux, in watts, striking the detector can be found as

2 Φw = Etel πRtel τopt , (2.8)

where Rtel is the telescope’s radius in meters, and τopt is the optical transmission of the

instrument. This radiant ﬂux represents the signal portion of the SNR.

2.1.2 Instrument Noise Model

The instrument noise is determined by consideration of the electrical currents generated

by the input signal and electronic noise in the detector. First, we convert the radiant ﬂux,

determined by the radiometric model, into the number of photons striking the instrument

during one second. The photon ﬂux is given by

Φ Φ w λ¯, (2.9) p = hc

13 where h is Planck’s constant, c is light speed, and λ¯ is the average wavelength of the photons.

For the signal current generated by this ﬂux,

isig = Φp η, (2.10) where η is the quantum efﬁciency of the detector. Arrival of the photons in discrete bundles

gives rise to a photon noise, or “shot” noise, that obeys Poisson statistics. The amount of

this noise is proportional to the square root of the integrated signal current,

Æ nshot = isig t , (2.11) where t is the integration time at the detector, in seconds. A second signiﬁcant source of

noise results from leakage (“dark”) current in the detector. Like shot noise, dark noise also

has a Poisson distribution, so its value can be found as

p ndark = idark t , (2.12)

where idark is the detector’s dark current leakage. The ﬁnal source of noise to be considered

by this model is read noise, nread , which is generated during read-out of the detector’s

pixel array. The total noise is calculated by adding these independent noise sources in

quadrature, such that

q 2 2 2 q 2 N = nshot + ndark + nread = (isig + idark )t + nread . (2.13)

Values for η, idark , and nread are provided by the detector’s manufacturer. For this analysis,

the Apogee Ascent A694 camera has been assumed.

14 2.1.3 Signal to Noise Ratio

Based on the radiometric transfer model and the noise model of the previous sections, the

signal-to-noise ratio can be found as

Φp ηt SNR . (2.14) = Æ 2 Φp ηt + idark t + nread

Substituting the parameter values listed in Table 2.1 yields the best-case SNR of 27 dB.

Looking at the problem another way, Eq. (2.14) can be rearranged to yield

2 2 2 2 2 2 Φp η t (Φp η + idark )(SNR) t nread (SNR) = 0, (2.15) − − which is a quadratic equation in t . We can now solve the integration time for any desired

SNR. For example, an average SNR of 20 dB requires an integration time of just over t = 11 seconds.

As with any practical radiometric analysis, this one represents an approximate estimate

of the attainable SNR under the assumed conditions. The choice and quality of the optics will also have an effect beyond the simple transmission coefﬁcient used in this model. How-

ever, even these approximate results are enough to show that the proposed measurements

are feasible.

15 Table 2.1 Values used for calculation of system SNR.

Parameter Description Value

λ1 Minimum wavelength 517 nm λ2 Maximum wavelength 537 nm Teff Solar effective temperature 5780 K R Solar radius 695,500 km D Mean Earth-Sun distance 150 106 km ρ Target reﬂectance× 0.70

θi Incident angle of Sunlight at target 45◦ 2 Asat Area of target 1 m d Target height above Earth (geosynchronous) 35,800‘km

τatm Optical transmission of the atmosphere 0.80 τopt Optical transmission of the instrument 0.20 Rtel Radius of telescope aperture 0.45 m t Integration time 300 s η Quantum efﬁciency of detector* 0.77 @580 nm

idark Camera dark current* 0.0041 e−/pixel/s nread Camera read noise* 5.9 e− (RMS) *Camera speciﬁcations are based on the Apogee Ascent A694 camera.

2.2 Spatial Heterodyning

To improve the SNR and sensitivity of a passive Doppler velocimeter, we have investigated

the use of spatial heterodyning. Consider a conventional Fourier transform spectrometer

(FTS) based on a Michelson two-beam interferometer, as illustrated in Fig. 2.2 (a). Mirror

M1 is ﬁxed, but M2 is movable along the z -axis. Coherent source light is collimated into

beamsplitter BS, which redirects the light toward two separate paths along the x and z -axes.

After reﬂection by M1 and M2, the two beams are re-integrated at the beamsplitter. At

this point the beams have traveled different distances; the optical path difference (OPD)

16 between them is

OPD = 2(D1 D2) . (2.16) | − | The two wavefronts, now offset by the OPD distance, interfere with each other, creating

bright and dark fringes on the focal plane array (FPA). An interferogram I (OPD) is created by recording the intensity at the FPA as mirror M2 is translated through its range of motion.

A Fourier transform of the interferogram,

I (σ) = F I (OPD) (2.17) { }

describes the intensity of the source as a function of wavenumber, where σ is the wavenum-

1 ber measured in inverse centimeters (cm− ). One major beneﬁt of an FTS is that the spectral resolution is limited only by the optical

path difference, such that 1 ∆σ = . (2.18) OPDmax

However, there is a practical limit to the spectral resolution. For any ﬁxed OPD, the central wavenumber of the Fourier transform always represents zero frequency; this is known as

a homodyne spectrometer. To get information about a non-zero frequency f , the spec-

trometer must process all frequencies in the range 0–f , seriously limiting throughput.

Furthermore, the relationship between OPD and interference order (i.e., fringe number) is

OPD = mλ, (2.19) where m is the order corresponding to the wavelength λ. For measurements at shorter wavelengths, the order of interference increases, and eventually the separation between

17 Figure 2.2 Michelson interferometer. (a) Conventional homodyne design centered at zero frequency. (b) Heterodyne design using diffraction gratings; the center frequency is chosen by selection of the blaze angle.

fringes becomes too small for most electronic detectors to resolve.

To remedy this situation, a spatial heterodyne spectrometer (SHS) replaces the two

plane mirrors of the Michelson with reﬂective diffraction gratings, as illustrated in Fig. 2.2

(b). The wavelength associated with the Littrow angle of the gratings is retro-reﬂected

toward the beamsplitter and becomes the zero wavenumber [32]. The Fourier transform can be centered around any desired wavelength according to the grating equation,

1 σ0 = , (2.20) 2Λsinθ

where σ0 is the zero wavenumber and Λ is the line density of the gratings. In this way, the

spatial frequency of the fringes is decreased, enabling measurement of larger wavenumbers, which translates to higher spectral resolution [33]. To take the heterodyning concept one step further, multiple-layer gratings can be used, where each layer retro-reﬂects a speciﬁc wavelength. This results in the overlay of two or

18 more narrow band spectra, one on top of the other. If spectral bands are chosen wisely,

such that the Fraunhofer lines in one band overlap, or nearly overlap, lines in another band,

the SNR will beneﬁt from the combination of intensities in each band. This overlapping

may also improve the sensitivity of Doppler velocity measurements, as discussed in the

next section.

2.3 Heterodyne Simulation

As previously mentioned, there are thousands of Fraunhofer absorption lines within the visible solar spectrum. The improvement in sensitivity afforded by overlapping two or

more spectra is not immediately obvious. To determine which spectral combinations

might be beneﬁcial, an empirical approach was tried. A MATLAB simulation was created

to overlap three spectral samples of solar intensity data, then record the signal power as

the wavelengths were Doppler shifted. The steps in the simulation were as follows:

1 1. Using a database of solar spectral intensities [34], three bandlimited (130 cm− ) spectra were extracted around randomly selected wavelengths in the range 400–700 nm. The

samples were interpolated onto a wavenumber axis such that the distance between

each sample was constant.

2. Each sampled spectrum was Doppler-shifted over a range of velocities from 0 to

10 km/s. The intensities of the three spectra were averaged after each shift. 3. The differential intensity vs. velocity, dI /dv , was calculated, and the wavenumber

corresponding to its maximum value was recorded.

The above steps were repeated for 5000 sets of randomly selected triplets. The results were sorted according to the maximum value of dI /dv for each set. Larger values imply

greater sensitivity to the target’s velocity.

19 A typical set of results is depicted in Fig. 2.3 for three spectral bands centered at 424 nm,

530 nm, and 628 nm. The x -axis represents the wavenumber of the heterodyned spectra; i.e.,

1 the three center wavelengths have been heterodyned to 0 mm− . The y -axis represents the radial velocity of a hypothetical target in geocentric orbit. Color represents the differential

intensity of the averaged light energy as the velocity is swept from 0 to 10 km/s in steps of

100 m/s. “Hot” locations on the map indicate a larger dI /dv . This translates to a higher sensitivity with respect to Doppler measurement. Maintaining a high sensitivity across

the entire velocity range requires observation at multiple wavenumbers. As heterodyned wavelengths shift relative to one another, some lines will overlap while some will separate.

1 The results of Fig. 2.3 indicate that even within a wavenumber range of 5 mm− there are enough sensitive regions to cover the entire velocity range. This conﬁrms that the use of

heterodyning to overlap several spectral regions can improve the sensitivity of Doppler

shift measurements.

20 Figure 2.3 Differential intensity vs. radial velocity for heterodyned frequency bands centered at 424 nm, 530 nm, and 628 nm.

2.4 Velocity-Fitting Simulation

Although the heterodyne simulations provided a clear indication of the potential beneﬁt of

heterodyning, additional simulations were needed to estimate the amount of sensitivity

gain. The effects of noisy measurement data and the overall signal-to-noise ratio also

needed to be considered. The main goal of additional simulation was to identify unique

features in the solar spectrum that might provide an improvement in sensitivity when

measuring Doppler shifts. The simulator code was designed to account for the following

21 situations:

Arbitrary selection of center frequency and bandwidth • Spatial heterodyning of two or more spectral bands • Variable aliasing of spectral information • Different SNR metrics for the optical system • Each simulation run was parameterized by speciﬁcation of the sample spectra, hetero-

dyne center wavenumber, spectral resolution, SNR, integration time, and radial velocity

of the target. One or more spectral bands, selected from reference [34], were Doppler- shifted, then heterodyned down to the speciﬁed wavenumber. The resulting superposed

signals were then ﬁtted to an unshifted reference of the expected signal, using MATLAB’s

fminsearch minimization function.

Heterodyning was controlled by a “shift” parameter that speciﬁed the wavenumber,

in inverse centimeters, that was mapped to the center of the Fourier transform. Initial

1 simulation runs considered only one spectral band shifted down to 0 cm− , which caused the band to overlap with an aliased (spatially reversed) version of itself. Other simulations

overlapped multiple spectral bands at various center wavenumbers.

For each scenario, input signals were corrupted with just enough noise to meet the

SNR speciﬁed in the simulation. Monte Carlo trials were run with random noise values to

determine the effect of noise on the accuracy of velocity estimation.

2.4.1 Single-Band Results

A high-level overview of all single-band simulation results is illustrated in Fig. 2.4, Fig. 2.5,

and Fig. 2.6 for systems with a simulated SNR of 20, 10, and 5, respectively. Each line in

these ﬁgures represents a single spectral band. The x -axis represents the radial velocity

22 of the target. The y -axis represents the standard deviation of the ﬁtted velocity, averaged over 1000 trials. There are two notable features apparent in these plots. First, there is a large range of standard deviation over the 40 spectral bands. For the system in Fig. 2.4, the standard deviation ranges from 120 m/s to over 2500 m/s. This indicates that some bands provide better sensitivity than others with respect to measuring the Doppler shift.

Second, standard deviation values seem to scale linearly with SNR, doubling with every

3 dB decrease in SNR.

Figure 2.4 Overview of all single-band simulation results for a system SNR of 20.

To better see details in the data, the next three ﬁgures show the seven best-performing spectral bands. System SNR values of 20, 10, and 5 are represented in Fig. 2.7, Fig. 2.8, and

Fig. 2.9, respectively. At this scale, several additional features become apparent. First, the overall shape of the response curve does not change with changes in system SNR. Second, there is no discernible trend in standard deviation vs. velocity. Both of these features seem to support the hypothesis that differences in sensitivity are due to various features in the

23 Figure 2.5 Overview of all single-band simulation results for a system SNR of 10.

spectral bands, and not a result of the random addition of noise.

It can be seen from these ﬁgures that, for Doppler velocities greater than 3000 m/s, selection of a spectral band around 391.0 nm might be an appropriate choice. For velocities less than 3000 m/s, the band with center at 397.6 nm might be a better choice. Both bands could be heterodyned to overlap in the frequency domain, creating a composite signal with good sensitivity in both velocity ranges.

Finally, Fig. 2.10 depicts histograms of the Monte Carlo results for the spectral band centered around 391.0 nm. Simulated radial velocities of 2000 m/s and 8000 m/s are plotted for a system SNR of 20. As expected, the shape of these plots resembles a normal distribution.

24 Figure 2.6 Overview of all single-band simulation results for a system SNR of 5.

Figure 2.7 Best-performing spectral bands for an SNR of 20. The legend indicates center wavelength values in nanometers.

25 Figure 2.8 Best-performing spectral bands for an SNR of 10. The legend indicates center wavelength values in nanometers.

Figure 2.9 Best-performing spectral bands for an SNR of 5. The legend indicates center wavelength values in nanometers.

26 Figure 2.10 Histogram for the spectral band centered around 391.0 nm for simulated radial velocities of (a) 2000 m/s and (b) 8000 m/s.

27 2.4.2 Two-Band Results

The results for 36 pairs of overlapping spectral bands are illustrated in Fig. 2.11. Since the

data trends are similar for all SNR values, only the results for SNR = 20 are discussed for the remaining simulations. Several of the results exhibit a concave shape with a minimum

standard deviation near 4000 m/s. This could be the result of multiple Fraunhofer lines crossing each other at the center wavenumber. The seven best results are reproduced

in Fig. 2.12. The lowest standard deviation is found from the overlap of the two center wavelengths at 514.97 nm and 387.83 nm. This pairing has an average standard deviation

of 365 m/s, which is slightly worse than the best single-band result.

Figure 2.11 Overview of two-band simulation results

28 Figure 2.12 Best-performing pairs of spectral bands. The legend indicates center wavelength values of each band in nanometers.

2.4.3 Three-Band Results

Three spectral bands were the largest number to be simulated. The results for 100 sets of

three overlapping spectra are depicted in Fig. 2.13. In general, the three-band trials do not

seem to show any improvement over two-bands. Some three-band trials demonstrate the

same convex shape seen for two-bands, with the minimum occurring near a slightly higher velocity of 4500 m/s. Approximately 20 of the trials also show a new feature: the standard deviation appears

linear with respect to velocity. In six of the trials, this relationship is consistent across the

entire range, with a slope of about 0.5. This might be explained by the fact that each band

experiences a different dλ/dv according to the relationship of Eq. (1.1). At higher velocities,

Fraunhofer lines will slide past one another, rather than adding in amplitude.

29 Figure 2.13 Overview of three-band simulation results.

Zooming in for a closer look, the best seven trials are shown in Fig. 2.14. At this scale the two-band trials in Fig. 2.12 seem to have a slight advantage. However, the best seven trials from both experiments lie in the same 250–500 m/s range of standard deviation. At this point a general conclusion might be drawn from all the simulations: Although stacking two spectral bands does seem to improve the accuracy of the measurement, addition of a third band does not afford any further advantage.

30 Figure 2.14 Best-performing triplets of spectral bands. The legend indicates center wavelength values of each band in nanometers.

2.4.4 Manual Results

To more directly compare the effects of spatial heterodyning, one group of spectral bands was chosen for simulation using two, three, and no overlapping bands. The following

simulations were performed:

A single band centered around 361.06 nm. • A single band centered around 375.26 nm. • A single band centered around 391.23 nm. • A heterodyned (“stacked”) combination of 361.06 and 391.23 nm bands. • A heterodyned (“stacked”) combination of all three bands. • These bands were chosen because they were among the best results of all the previous

single band simulations. The results are summarized in Fig. 2.15, which includes the ﬁve

31 simulations listed above. The red, blue, and yellow lines represent the single-band simula-

tions at center wavelengths of 361.06 nm, 375.26 nm, and 391.23 nm. The fourth simulation

(purple line) is a two-band overlap of center wavelengths 361.06 nm and 391.23 nm. Finally,

the ﬁfth simulation (green line) is an overlap of all three wavelengths.

The simulations using multiple overlapping bands show a deﬁnite improvement in

sensitivity compared to the single-band simulations. The best result is demonstrated by the

two-band simulation, with an average standard deviation of 276 m/s. This is signiﬁcantly

better than the best single-band result of 328 m/s. These results validate the idea that Doppler measurement sensitivity can be improved by overlapping multiple spectral bands.

Interestingly, the two-band overlap shows a better sensitivity than the three-band overlap.

This suggests that overlapping bands need to be carefully selected for maximum beneﬁt.

32 Figure 2.15 Simulation of three spectral bands taken separately and together. The legend indicates center wavelengths in nanometers.

33 CHAPTER

ORBITAL SIMULATION

Radial velocity alone provides insufﬁcient information for an IOD. A total of six independent

measurements are required to characterize an orbit with six degrees of freedom. Combining velocity with angle measurements of right ascension and declination is one way to estimate

an orbit using only two telescopic observations. However, this requires development of a

new method for solving orbital parameters from two line-of-sight (LOS) observations.

34 3.1 Background

The present method is a modiﬁcation of Gooding’s algorithm [12], ﬁrst published in 1996. The core of Gooding’s algorithm is a solution to Lambert’s problem, also known as the

two-body orbital boundary-value problem [35]. The geometry of Lambert’s problem is illustrated in Fig. 3.1, where the central body, typically Earth or Sun, is located at the origin.

An orbiting body is known to pass through points P1 and P2, and the time of ﬂight between

the two positions is measured as ∆t . The path traced by the orbiting body subtends an

angle θ at the origin. The problem is to discover the equation of the conic section c that

describes the Keplerian orbit [36]. The solution of Lambert’s problem, along with Kepler’s equations, is one of the basic

questions of orbital mechanics. The ﬁrst accurate solution, using three known positions

instead of two, was published in 1857 by Gauss [37]. The problem languished until the beginning of the space age in the mid-twentieth century, when several fresh solutions were

developed [35]. In 1990, Gooding published his own solution to Lambert’s problem [38], which he later used as the basis for his angles-only IOD method.

Gooding solved Lambert’s problem without knowing the distances to P1 and P2 — that

is, an “angles-only” solution. Initial guesses were assigned to the two range values, which were then improved, step by step, through the iterative method described below.

Gooding’s method starts with three observations of the target at times t1, t2, and t3,

respectively. Each observation comprises a pair of angles representing the target’s right

ascension (α) and declination (δ) coordinates on the celestial sphere. Each pair of α,δ 〈 〉 coordinates are converted to a unit vector L = x , y, z in Cartesian space, representing the 〈 〉 direction to the target. The three unit vectors corresponding to the observed direction at

35 c P1

Δ t =t −t 2 1 P2

r1 r θ 2

Figure 3.1 Illustration of Lambert’s orbital boundary-value problem.

times t1, t2, and t3 are deﬁned by

  cos(αi )cos(δi )   L  ; i 1,2,3. (3.1) i =  sin(αi )cos(δi )  =   sin(δi )

Four additional quantities are deﬁned as depicted in Fig. 3.2. The ﬁrst quantity is the

site vector, R, representing the location of the observer in x , y, z Earth-centered inertial 〈 〉 (ECI) coordinates. The second quantity is the range value, ρ, representing the unknown

distance between the station and the target. The third quantity, v, is the ECI velocity of the

target. Finally, the fourth quantity is the target position vector, r, deﬁned as

r = R + ρL. (3.2)

Using these deﬁnitions, a complete orbital solution can be described by a state vector r,v 〈 〉

36 comprising the target’s position and velocity at any point in the orbit. To estimate the state vector at time t2, Gooding used the following procedure:

1. Measure R i , L i , and ti for i = 1,2,3.

2. Assume values for the unknown ranges ρ1 and ρ3.

3. Calculate the estimated position vectors r1 and r3 using Eq. (3.2).

4. Solve Lambert’s problem to obtain velocity estimates v1 and v3.

5. Solve Kepler’s two-body problem for a prediction of r2,v2 . 〈 〉 6. Compute the error in the predicted LOS vector C at time t2.

7. Use Newton’s method to generate new estimates of ρ1 and ρ3.

8. Iterate (go to 3) until the error is below a desired tolerance.

The predicted LOS vector in step 6 is computed as C = r2 R2. Two orthogonal error − functions determine the difference between the predicted vector C and the angles measured

in step 1. Error functions f and g are deﬁned as [12]

P C f (x , y ) = · ; P = (L2 C) L2, and (3.3) P × × k k N C g (x , y ) = · ; N = L2 P, (3.4) N × k k where the independent variables x and y substitute for ranges ρ1 and ρ3, respectively. The

error functions are driven to zero by the two-variable Newton’s method,

      1  − x x fx fy f (x , y )   =      , (3.5) y y − g x g y g (x , y ) n+1 n where the subscript (n + 1) denotes the improved range estimates to be used in the next

iteration. The partial derivatives fx , fy ,g x , and g y are determined by repeating steps 3 - 5

37 Figure 3.2 Geometric relationships between Earth, Sun, and target body.

using incremental variations of the x and y range values.

Once the f and g errors have been minimized by application of Newton’s method, the

resulting state vector r2,v2 describes the instantaneous position and velocity of the target. 〈 〉 The state vector at any other time can be found by application of Kepler’s equations.

3.2 Mathematical development

Gooding solved Lambert’s problem using three independent observations of the target, as

did the solution proposed by Gauss. However, with the inclusion of Doppler shift as an

additional measurement, Lambert’s problem can be reduced — as in Lambert’s original

formulation — to a set of two observations. The following derivation demonstrates a new

solution to Lambert’s problem using only two independent observations of right ascension,

declination, and radial velocity, i.e., Doppler shift.

Given an estimated state vector r,v , we calculate the following two velocities, which 〈 〉

38 are illustrated in Fig. 3.2: r S vs = (v vsun ) − , and (3.6) − · r S k − k ρ˙ = (v vobs ) L, (3.7) − · where vs is the target’s radial velocity with respect to the sun, and the range-rate ρ˙ is the

target’sradial velocity with respect to an Earth-bound observer. The vectors vsun and vobs are

the known ECI velocities of the sun and the observer, respectively, and S is the sun’s position vector. The predicted Doppler shift, ∆λ = λ00 λ, can now be calculated by substitution of − vs and ρ˙ into Eq. (2.1), which yields

ρ˙ + vs ρ˙ vs ∆λ . (3.8) = c + c 2

Because the Doppler shift measurements are independent of the angle measurements, this

method can ﬁnd an orbital solution using only two sets of observations. In other respects,

the procedure is like the one used by Gooding:

1. Measure R i , L i , ti , and ∆λi for i = 1,2.

2. Assume values for the unknown ranges ρ1 and ρ2.

3. Calculate the estimated position vectors r1 and r2 using Eq. (3.2).

4. Solve Lambert’s problem to obtain velocity estimates v1 and v2.

5. Generate independent predictions of ∆λ1 and ∆λ2 using Eq. (3.8).

6. Use Newton’s method to generate new estimates of ρ1 and ρ2.

7. Iterate (go to 3) until the error is below a desired tolerance.

The new algorithm employs a multivariate Newton’s method to minimize the prediction

error but uses a distinct set of error functions from Gooding’s method. The new error

functions, f and g , are based on the difference between the predicted and measured

39 wavelength shifts at times t1 and t2,

˜ f (x , y ) = ∆λ1 ∆λ1, and (3.9) −

˜ g (x , y ) = ∆λ2 ∆λ2, (3.10) − where ∆λ and ∆λ˜ represent the measured and predicted values, respectively.

Optimization of the estimated range values proceeds as described in Eq. (3.5). However,

the method is easily extended from two observations to three or more, with an attendant

improvement in accuracy. Using three variables, Newton’s method becomes

1      −   x x fx fy fz f (x , y, z )                , (3.11)  y  =  y   g x g y g z   g (x , y, z )      −     z z hx hy hz h(x , y, z ) n+1 n where the additional variable z represents the estimated range value ρ3 at time t3, and

h(x , y, z ) is its associated error function.

3.3 Simulation

To investigate the performance of the Doppler shift IOD method, software was developed

to perform two main functions: (1) orbit generation, and (2) orbit prediction. The goal was

to use the new algorithm described in section 3.2 to estimate orbital parameters based on

simulated orbit data.

40 3.3.1 Orbit generation

A simpliﬁed block diagram of the simulation procedure is depicted in Fig. 3.3. Keplerian

parameters for the simulated orbit, reference coordinates of the observing station, and a

Julian date were provided as inputs to the simulation. Based on these inputs, state vectors were computed for two points in the orbit. Simulated observation data comprising right

ascension, declination, and Doppler shift were then extracted from the state vectors. To

simulate real measurements, these values were corrupted by the addition of Gaussian noise.

Angle measurements were randomly perturbed by 1 arcsec of standard deviation around

the true value. Doppler shift measurements were corrupted by values of 2, 5, and 10 m/s standard deviation around the target’s expected radial velocity as determined by Eq. (3.7).

3.3.2 Orbit prediction

Gooding’s method and the present algorithm both use Newton’s method to minimize the

error in successive estimates of orbital range values. Because Newton’s method is sensitive

to the initial estimate, better results can be obtained if the starting values are chosen wisely.

To this end, the simulation was implemented as two separate procedures:

1. “Coarse ﬁt,” which attempts to eliminate invalid starting values, and

2. “Fine ﬁt,” which applies the new algorithm to optimize the solution.

Each step is discussed separately in the following sections.

3.3.2.1 Coarse ﬁt

The coarse ﬁt procedure narrows the solution’s search-space by mapping an “admissible

region” over a set of range and range-rate values, as illustrated in Fig. 3.4. The admissible

41 Figure 3.3 Block diagram of the orbit generation and prediction algorithm.

region is deﬁned as that portion of the range/range-rate plane within which the two-body energy of the Earth-satellite system is less than zero [39, 40]. Positive values of energy imply an unstable orbit that is not bound to the central body, so only regions of negative energy contain valid orbits. For example, the curve labeled εE in Fig. 3.4 represents zero geocentric energy; values of range and range-rate outside of this curve are not conﬁned to

Earth orbit. The lines labeled ρmin and ρmax are limits on the range values to further reduce the search-space. The intersection of the constraint regions, shown by the shaded area, indicates the region of potential orbital solutions. Implementation of admissible region estimation reduces the CPU time of the simulation by a factor of ten and improves accuracy by 3%.

For best results, admissible region estimation requires accurate measurement of the rate of change of right ascension and declination. However, to obtain a rough outline of

42 Figure 3.4 Illustration of the admissible region (shaded) for an object in orbit [41]. the admissible region, the angular rate of change can be approximated by

α2 α1 α˙ − and (3.12) ≈ t2 t1 −

˙ δ2 δ1 δ − (3.13) ≈ t2 t1 − for right ascension and declination, respectively. For three or more observations, the approximation could be improved by using Lagrange interpolation or similar methods. The accuracy also improves as t2 t1, the time between observations, decreases. − A block diagram of the coarse ﬁt procedure is depicted in Fig. 3.5. The ﬁrst step is estimation of an admissible region, as just described. In the next step, a sampling grid is overlaid on the admissible region. The grid intervals have been chosen as 1000 km for the range and 1000 m/s for the range-rate. These values result in a relatively small set of 400-900 sample pairs, depending on the orbit. In the last step of the procedure, each

43 sample pair is combined with the measured direction angles to generate an estimated state vector for time t1. A second state vector is predicted for time t2 by solving Kepler’s problem.

The predicted position is then compared to the measured angles, and the error is recorded.

After iteration over the sampling grid, the range value associated with the smallest error is

chosen as the starting point for the ﬁne ﬁt procedure.

44 Figure 3.5 Block diagram of the coarse ﬁt procedure.

45 3.3.2.2 Fine ﬁt

The fine fit procedure is illustrated in Fig. 3.6. The output of the coarse fit is used as the

initial guess for range estimates x and y at times t1 and t2, respectively. The process then

proceeds according to the steps outlined in section 3.2. The ﬁnal output is an orbital state vector that represents the best ﬁt to the given (noisy) measurement values.

46 Figure 3.6 Block diagram of the ﬁne ﬁt procedure.

47 3.4 Testing and results

To analyze the performance of the new IOD algorithm, simulation results were compared

to those of Gooding’s method. A variety of orbital conﬁgurations were used to test different

aspects of each algorithm:

1. Three geostationary equatorial orbits were selected to test performance with respect

to the angle of observation. These orbits were identiﬁed as GEO-1, GEO-2, and GEO-3

and were stationed at longitudes of 0◦, 45◦E, and 45◦W, respectively.

2. Three mid-altitude, circular orbits with inclinations of 56◦ were selected to test performance with respect to orbital height. These orbits were identiﬁed as MEO-1, MEO-2,

and MEO-3 and were given semi-major axes of 12 000 km, 22 000 km, and 33 000 km,

respectively.

3. One geostationary transfer orbit (GTO) was chosen to test performance with respect

to eccentricity. The transfer orbit was given a semi-major axis of 24 500 km, an

eccentricity of 0.72, a 0◦ argument of perigee, and no inclination. 4. One Molniya orbit (MOL) was chosen to test performance in case of high inclination.

This orbit was given a semi-major axis of 26 660 km, an eccentricity of 0.72, an

ascending node at 192◦, a 270◦ argument of perigee, and an inclination of 63.5◦. 5. One coplanar equatorial orbit was chosen such that the point of observation lies

on the orbital plane. This geometry cannot be solved by the unmodiﬁed Gooding’s

method. The orbital parameters of this test case are identical to the GEO-1 orbit, but

the observing site has been moved to the equator.

Scenarios 1–4 are like those used elsewhere [42] for comparison of different IOD methods. Gooding’s angles-only method (AO) requires a minimum of three observations to solve

48 an orbit. The new “angles and velocity” method (AV) requires only two observations, but it was also tested with three observations for a more direct comparison to Gooding. Each

simulated observation consists of one right ascension angle, one declination angle, and

one Doppler shift value. The time between observations was varied among 4, 8, 16 and 32

minutes to test the effect of angular separation on algorithm performance. However, shorter

separations yielded consistently worse predictions for both methods, because noise has

a proportionately greater effect on smaller separation angles. In the interest of efﬁciency,

only the results for 16-minute separation times are reported, although comparable results were obtained for the other separation times.

Since the angle of reﬂection affects the Doppler measurement accuracy, simulations were repeated for multiple target positions, covering a minimum of one complete orbit.

Results were recorded at 20-minute intervals over a simulated 24-hour period. Although

some of these simulations included infeasible daytime observations, the resulting test suite

comprehensively characterizes the performance of the algorithm over a range of scenarios.

To verify the implementation of each algorithm, the simulations were ﬁrst tested using

noiseless data. These results also served to generate a baseline reference that reﬂects the

best case accuracy of each method. Next, to simulate real-world conditions, the data was

corrupted by noise values randomly chosen from a normal distribution. Angle measure-

ments were perturbed by 1 arcsec of standard deviation, and Doppler shift measurements were perturbed by amounts equivalent to 2, 5, and 10 m/s standard deviation of the radial velocity. For each orbital scenario, 1000 Monte Carlo simulations were executed using this

noisy data.

Errors were averaged over the entire orbit. Both the true and estimated orbits were

sampled at 100 points, starting at perigee, and evenly distributed in time over one orbital

49 period. The error in the predicted orbit was then calculated as

1000 100 1 X 1 X rm rˆm,n err(%) 100, (3.14) = 1000 100 −r n 1 m 1 m · = = k k

th where rm is the true position vector at the m sample point, and rˆm,n is the estimated

position vector at the m th sample point of the n th Monte Carlo trial.

3.4.1 Geostationary orbits

The zero-noise performance of the GEO-1 test case is depicted in Figs. 3.7 (a) and (b) for the

AO and AV methods, respectively. Note that the AV data includes results obtained using both

two and three observations. In this conﬁguration, the target is stationary above the equator,

16.6◦ east of the ground station. For the hypothetical noiseless situation, AO outperformed AV by a factor of 1000. However, in absolute terms, the AV method’s worst-case result was

accurate to within 21 cm. Test cases GEO-2 and GEO-3 produced comparable results and

are not shown here to avoid redundancy.

Figs. 3.8 (a) and (b) illustrate the GEO-1 results after noise was added to the simulated

measurements. The AO method exhibited an average error of 6.9% with very little variation

over one period. The average error for the AV method, using three observations with a noise

level of 2 m/s, was 4.4%. With only two observations, the AV error increased to 8.9%. These plots demonstrate that the accuracy of the AV method is strongly dependent on the orbital

position of the target. The large error peak occurring 280 minutes after perigee coincides with the maximum solar phase angle between Earth and Sun. Test cases GEO-2 and GEO-3

produced identical results, with error peaks straddling the maximum phase angle in each

case. These results suggest an overall dependence on the solar phase angle, rather than the

angle of observation.

50 Figure 3.7 Zero-noise performance of test case GEO-1 for (a) AO and (b) AV methods.

Figure 3.8 Results for the GEO-1 test case using (a) three and (b) two observations.

51 3.4.2 Mid-altitude orbits

Figs. 3.9 (a) and (b) depict the zero-noise performance of test case MEO-1. Although the

AO method again outperformed AV by a factor of 1000, the maximum AV error was less

than 4 cm in absolute terms. The noiseless performance for test cases MEO-2 and MEO-3 were similar.

Figs. 3.10 (a) and (b) illustrate the MEO-1 results after noise was added to the data.

The average error of the AO method was 0.12% compared to 0.46% for AV, assuming a

measurement error of 2 m/s. In the worst case, the average error for the AV method, using

two observations with a noise level of 10 m/s, increased to 4.4%. Results for the mid-altitude orbits MEO-2 and MEO-3 are illustrated in Figs. 3.11 and 3.12, respectively. In all cases,

the accuracy of the predicted orbit decreased with orbital height. This is because the

measurement error has a relatively larger impact when the angular separation between

subsequent observations is small, which is true for the larger orbits. As with the GEO test

orbits, the AV method produced error peaks during periods when the solar phase angle was

largest.

52 Figure 3.9 Zero noise performance of test case MEO-1 for (a) AO and (b) AV methods.

Figure 3.10 Results for the MEO-1 test case using (a) three and (b) two observations.

53 Figure 3.11 Results for the MEO-2 test case using (a) three and (b) two observations.

Figure 3.12 Results for the MEO-3 test case using (a) three and (b) two observations.

54 3.4.3 Eccentric and inclined orbits

Noiseless performance for the GTO test case with highly eccentric orbit is depicted in

Figs. 3.13 (a) and (b). The AO method outperformed AV by three orders of magnitude,

although the absolute error for the worst-case AV was less than 2 cm.

Figs. 3.14 (a) and (b) illustrate the GTO results after the addition of noise. As in the

previous test cases, the errors in the AV method peaked near the maximum phase angle.

However, the errors produced by the AO method were dominated by the angular separation

of the three observations.

Noiseless results for the MOL test case with highly inclined orbit are depicted in Figs. 3.15

(a) and (b). Once again, the AO method was more accurate for the noiseless case, but the

absolute worst-case error for the noiseless AV method was less than 16 cm.

Finally, Figs. 3.16 (a) and (b) illustrate the results for the MOL test case with noise added.

All of the AV results follow the familiar pattern of maximum error near the maximum solar

phase angle.

55 Figure 3.13 Zero-noise performance of test case GTO for (a) AO and (b) AV methods.

Figure 3.14 Results for the GTO test case using (a) three and (b) two observations.

56 Figure 3.15 Zero-noise performance of test case MOL for (a) AO and (b) AV methods.

Figure 3.16 Results for the MOL test case using (a) three and (b) two observations.

57 3.4.4 Coplanar observer

Angles-only methods are known to perform poorly, or not at all, when the observing station

is located on, or very near, the plane of the orbit. With a coplanar observer, all angle measure-

ments are constrained to a single plane and are therefore not truly independent, resulting

in insufﬁcient information for solving the orbit. However, measurement of wavelength

shift is not affected by coplanar geometry. Figs. 3.17 (a) and (b) depict AV method results

from the GEO-1 test case, with the observing station located on the equator. These results were identical to the non-coplanar case illustrated in Fig. 3.8. No results were available

for the AO method, since the algorithm would not converge to a solution under coplanar

conditions.

3.5 Discussion

The results illustrated in the previous sections exhibit large variations in accuracy across

the simulated 24-hour orbital period. However, a pattern emerged when the results were

compared to the solar phase angle. Fig. 3.18 illustrates the pattern for the MEO-1 test case.

The prediction errors were higher when the phase angle was large and lower when the

phase angle was small. This pattern was typical for all test cases and is explained by the

fact that the measured wavelength shift is a function of the reﬂection angle. The situation

can be considered analogous to the geometric dilution of precision experienced by global

positioning satellites [43]. The pattern depicted in Fig. 3.18 suggests a way to determine when the AV method

might have an advantage over conventional AO methods. When results corresponding

to large phase angles were ignored, the average error of the remaining results decreased

58 Figure 3.17 Results for the coplanar observer, using (a) three and (b) two observations.

25 3

2.5 20

2 15 1.5 10

Percent error Percent 1 Phase angle (rad) angle Phase 5 0.5

0 0 0 500 1000 1500 Time since perigee (minutes)

Figure 3.18 Orbital prediction error (circles) vs. phase angle (solid line).

59 signiﬁcantly. Although the exact phase angle (φ) cannot be known at the time of observation,

the elongation angle (θ ) between Sun and target can be measured directly. For targets in

geosynchronous orbit, the approximation

φ 180◦ θ (3.15) ≈ −

is always within 2◦ of the true phase angle. Based on this approximation of the phase angle, we can estimate the relative accuracy of the Doppler shift measurement.

Figs. 3.19- 3.24 illustrate the effectiveness of using the phase angle to qualify individual

AV results. Each plot summarizes the error distribution for a particular test case; box plot

symbols indicate the minimum, maximum, median, and interquartile range of errors over

one orbital period.

Figs. 3.19 (a) and (b) combine the results for test cases GEO-1, GEO-2, and GEO-3.

Fig. 3.19 (a) depicts the full set of error statistics including all results, regardless of phase

angle, while Fig. 3.19 (b) depicts only the results obtained from observations where the

phase angle was less than 90◦. From left to right in each chart, the ﬁrst box plot represents

Gooding’s AO method. The next three boxes represent the three-point (N = 3) Doppler AV

method with radial velocity measurement errors of 2, 5, and 10 m/s, respectively. The ﬁnal

three boxes represent the two-point (N = 2) AV method. Compared to Fig. 3.19 (a), the subset of results in Fig. 3.19 (b) shows signiﬁcant improve-

ment. After excluding observations corresponding to large solar phase angles, outlying

Doppler results were removed from the statistics, and the error variance was reduced.

Conversely, the Gooding results remained unchanged.

Figs. 3.20- 3.22 summarize the results for test cases MEO-1, MEO-2, and MEO-3. As with the GEO test cases, application of the phase angle condition has ﬁltered out large,

60 Figure 3.19 Summary statistics for combined test cases GEO-1, GEO-2, and GEO-3. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦ outlying Doppler errors and reduced the variance of the results. However, the effect on the

Gooding results were mixed. For the MEO-1 case in Fig. 3.20, the results were unchanged.

Fig. 3.21 depicts elimination of a large error outlier for the MEO-2 case, but little or no change in the variance. Finally, the error variance of the Gooding method increased after the phase angle condition was applied to the MEO-3 results in Fig. 3.22.

Next, Fig. 3.23 depicts statistics for the GTO test case. The AV and AO statistics both exhibited a reduction in median error and variance after the phase angle condition was applied. Because the GTO orbit is highly eccentric, the AO statistics beneﬁt from removal of observations near apogee.

Finally, Fig. 3.24 depicts statistics for the MOL test case. In this scenario, a large orbital inclination maintained the phase angle within a relatively small range near 90◦, so eliminat- ing results above that value had little effect on the statistics. Other than a slight increase in the median error, the conditional results in Fig. 3.24 (b) demonstrate no major differences

61 Figure 3.20 Summary statistics for test case MEO-1. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦

Figure 3.21 Summary statistics for test case MEO-2. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦

62 Figure 3.22 Summary statistics for test case MEO-3. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦

Figure 3.23 Summary statistics for test case GTO. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦

63 when compared to the full set of statistics in Fig. 3.24 (a).

64 Figure 3.24 Summary statistics for test case MOL. (a) Error statistics for all phase angles. (b) Errors conditioned on phase angle 90 . φ < ◦

65 CHAPTER

INSTRUMENT DESIGN

Several concepts were considered prior to selection of the ﬁnal Doppler camera instrument

design. The concepts fall into two broad categories of operation: (1) full spectral, and

(2) dual-band Doppler ratio. The full spectral concept represents a more conventional

imaging spectrometer, but depending on the choice of implementation, it suffers from

calibration difﬁculties and/or cost issues. The dual-band Doppler ratio concept beneﬁts from simpler construction and improved spectral resolution but gives up spatial imaging of

the target. Both concepts, along with alternate instrument designs for each, are discussed

in the following sections.

66 4.1 Full Spectral Imaging

The full spectral concept is based on the snapshot hyperspectral imaging Fourier transform

(SHIFT) spectrometer [44]. The main advantage of this design is that in addition to the spectral information, it also provides a spatial image of the target.

One basic design for a SHIFT spectrometer1 is illustrated in Fig. 4.1. Twolinear polarizers,

LP1 and LP2, are oriented with their optical axes at 45◦ with respect to the x -axis. Two

Nomarski prisms, NP1 and NP2, are each constructed of two wedge-shaped birefringent

crystals, with the fast axis of one crystal tilted by an angle β. The fast axis of each prism is

offset by a small angle δ with respect to the x -axis, but the prisms themselves are oriented

180◦ opposite to each other. As light enters the instrument, the lens array (LA) creates a two-dimensional set of

N M separate beams. Linear polarizer LP1 generates a uniform polarization state. Then, × the birefringent prism NP1 creates an optical path difference (OPD) between the horizontal

and vertical polarization components of each beam. The variation in thickness caused by

the wedge shape and the tilt of the fast axis results in a different OPD for each beam.

Next, the half-wave plate (HWP) between the two prisms rotates the polarization state

by 180◦. The combination of the half-wave plate and prism NP2 compensates for any introduced refraction, such that all interference fringes are imaged in a common xy plane,

coincident with the focal plane array (FPA). The ﬁnal polarizer, LP2, recombines the or-

thogonal beam components and creates the actual interference fringes, which overlay the

sub-images on the FPA.

The theoretical spectral resolution of a general Fourier transform spectrometer (FTS) is

1 Technically, this is an interferometer, which creates optical interference fringes. The device becomes a true spectrometer when the fringe spacing is Fourier-transformed by appropriate software into frequency domain information.

67 LA LP1 HWP LP2

β α α β

x FPA NP1 NP2 y z

Figure 4.1 Snapshot hyperspectral imaging Fourier transform (SHIFT) spectrometer.

limited only by the optical path difference of the system, such that

λ2 ∆λ , (4.1) ≈ OPD where λ is the wavelength of light. In practice, resolution is limited by size constraints

and detector sensitivity. The designs documented in the following sections incorporate

additional techniques to increase the practical spectral resolution.

4.1.1 Heterodyned Birefringent Interferometer

Fig. 4.2 illustrates the design of a spatially heterodyned interferometer (SHI) constructed

from birefringent crystal prisms [32]. From left to right, light from the telescope is ﬁeld- narrowed by the stop, then collimated into the instrument. The N M lens array (LA) creates × multiple sub-images of the target. The beams formed by the lens array are collimated again,

then ﬁltered to restrict their bandwidth. The linear polarizer LP, oriented at 45◦ to the system’s x -axis, generates a coherent polarization state for the light.

68 Figure 4.2 Fourier imaging spectrometer based on a heterodyned birefringent interferometer [32].

The three Wollaston prisms (WP) are monolithic versions of the simple SHIFT spec-

trometer depicted in Fig. 4.1. Light leaving WP1 has experienced a phase delay between its

horizontal and vertical polarization components, due to the optical path difference within

the birefringent prism. The orthogonally polarized beams now travel separately through

the instrument.

A slab of dielectric material with refractive index ng , along with enclosing mirrors

M1 and M2, serves as a waveguide. The waveguide introduces an optical shear (lateral

displacement) S between the two beams, effectively increasing the OPD. Wollaston prisms

WP2 and WP3 compensate for chromatic dispersion introduced by WP1 and the waveguide.

Spatial heterodyning is created by polarization gratings PG2 and PG3. These are thin-

film liquid crystal devices that diffract the polarized light into a single first-order beam [45]. Notwithstanding their birefringence and high-efficiency, they perform the same role as the

69 conventional blazed gratings in the Michelson interferometer of Fig. 2.2 (b).

The second waveguide directs the beams to PG1, which effectively un-diffracts the

beams and collimates them along the optical axis once again. The subsequent linear

polarizer reintegrates the orthogonal beams, and the resulting interference fringes are

imaged onto a detector array.

A ray-trace diagram for the birefringent interferometer is illustrated in Fig. 4.3. Based

on the size and geometry of this design, the optical shear S = 222 mm. The maximum OPD

is given by [45] x OPD S max , (4.2) max = f where f is the focal length of the re-imaging lens LR, and xmax is the maximum x -coordinate

of the detector array. Assuming f = 100 mm and xmax = 13 mm, the maximum OPD is 28.86 mm. For a wavelength of 518 nm, the spectral resolution is given by Eq. (4.1) as

∆λ = 0.009297 nm. Although this design meets the target requirement of 0.01 nm spectral resolution, the

cost of construction was found to be prohibitive. To attain the required optical shear, the

prisms must be highly birefringent. Assuming the use of calcite, each prism requires a 30◦ wedge angle and 50 mm of clear aperture. The total weight of the crystals alone is estimated

at well over 1 kg. Alternate conﬁgurations were unsuccessful in signiﬁcantly reducing the

size of the prisms while maintaining the necessary shear. The cost estimate for a single

prism of the required size was $3000, leading to a total cost of $36,000 for all 12 prisms.

Thus, this design was simply not feasible within the constraints of the budget.

70 Figure 4.3 Zemax ray-trace of the heterodyned birefringent interferometer.

4.1.2 Heterodyned Sagnac Interferometer

The second design for a full spectral imaging camera is based on the Sagnac interferometer

depicted in Fig. 4.4. Unlike the birefringent interferometer, the Sagnac uses mirrors to

create the optical shear. Incident light is redirected by beamsplitter BS toward two angled

mirrors. Mirror M1 is located a distance d from the beamsplitter in the ( z ) direction, and − M2 is located a distance (d + α) in the x direction. As long as the difference α is not zero,

the two beams will be sheared by an amount [46]

S = p2α, (4.3)

and the resulting OPD is given by

OPD = S cosθ , (4.4)

71 where θ is the angle subtended by M1 and M2 with respect to the z and x axes, respectively.

After traversing the interferometer in opposite directions, the two sheared beams are rein-

tegrated at the beamsplitter, and the resulting interference fringes are imaged onto the

focal plane array (FPA).

Fig. 4.5 illustrates a complete design incorporating the spatial heterodyning effect. Light

from a telescope is collimated into the instrument by lens L1. Next, a set of N M sub- × images are created by a lens array (LA) at the array’s common focal plane. Lens L2 collimates

these sub-images into a Sagnac interferometer.

The interferometer is identical to the simple Sagnac device of Fig. 4.4, except for the

addition of diffraction gratings G1 and G2. These gratings shift the zero-order interference

fringe in the same way as the polarization gratings did for the birefringent interferometer. To

capture multiple spectral pass-bands, making this a true multi-spectral imager, the gratings

are speciﬁed to have a large phase depth, such that multiple orders may be observed on

the FPA [46]. A ray-trace diagram for the Sagnac interferometer is illustrated in Fig. 4.6. The resulting

optical shear S = 200 mm, which gives a maximum OPD of 26 mm and a corresponding spectral resolution of 0.01032 nm. Although this is slightly worse than the birefringent

design, the result can easily be improved by increasing the separation of mirror M2.

This design is similar to an unrelated project [47] undertaken by our lab, and the following conclusions are informed by that earlier work. Although the requirement of 0.01 nm

spectral resolution can be met without difﬁculty, there are questions of stability and cali-

bration. Unlike the birefringent design, in which most of the components are cemented

together in a monolithic unit, the Sagnac interferometer contains more than a few me-

chanical parts that must be precisely aligned. Earlier attempts at calibrating a similar

device produced non-uniform fringe images, which may have been a result of temperature

72 Figure 4.4 Illustration of a simple Sagnac interferometer.

Figure 4.5 Heterodyned Sagnac interferometer using blazed diffraction gratings.

73 Figure 4.6 Zemax ray-trace of the heterodyned Sagnac interferometer.

dependence of individual parts. In light of those earlier results, it was decided that the risk was too great.

4.2 Dual-Band Doppler Ratio

In comparison with the full spectral imaging solution, dual-band Doppler is a different

concept for measuring Doppler shift. Rather than measure interference fringes across

a range of wavelengths, this method ﬁlters incoming light through one or more narrow

pass-bands corresponding to individual Fraunhofer lines. Similar techniques have been

used in the measurement of air speed [48], in which laser light is scattered and Doppler shifted by a sample volume of moving air. Returning laser light is narrow-band ﬁltered and

74 compared to an unshifted reference source. Air velocity is determined from the wavelength

shift.

When the input spectrum is unshifted, optical ﬁlters at Fraunhofer wavelengths transmit very little in-band energy, where “in-band” is deﬁned as light falling within the width of

a Fraunhofer line. However, if the incident spectrum has been Doppler shifted, some

adjacent energy will appear at Fraunhofer wavelengths, and this energy will pass through

the ﬁlters. The ratio of in-band to out-of-band power provides a measure of the Doppler

shift. The main advantage of this concept is that none of the received light is discarded —

both the in-band and out-of-band wavelengths are used to determine the Doppler shift,

thus providing an improvement in the SNR.

For the dual-band Doppler concept to work, the ﬁlters must have a very narrow pass-

band, on the order of the width of a Fraunhofer line, or approximately 0.01 nm [19]. The following sections describe various optical ﬁlters that were pursued as potential core design

elements for a dual-band Doppler instrument.

Although this technique can offer potentially higher spectral resolution, spatial imaging

becomes a slightly more complex problem because of the narrow-band ﬁltering involved.

Thus, a decision was made to forego the spatial imaging feature in the following initial

designs.

4.2.1 Atomic Line Filters

Two types of atomic line ﬁlters (ALF) were investigated for use in detecting the Doppler

shift of solar Fraunhofer absorption lines. The ﬁrst type is known as a Faraday anomalous

dispersion optical ﬁlter (FADOF). This device uses Faraday rotation to implement an ex-

tremely narrow-band optical ﬁlter. As shown in Fig. 4.7, an optically transparent gas cell is

75 placed between two crossed polarizers, and a magnetic ﬂux is applied along the optical axis.

After passing through the ﬁrst polarizer, the incoming light is linearly polarized. Optical

frequencies near the atomic resonances of the gas experience Faraday rotation as they pass

through the cell. Faraday rotation is caused by linear birefringence due to the presence of

the magnetic ﬁeld. Rotated light passes through the second, orthogonal polarizer, while

the other, non-rotated frequencies are completely blocked. The ﬁnal bandpass ﬁlter (BPF)

blocks resonance frequencies outside the region of interest.

FADOF ﬁlters rely on the phenomenon of Faraday rotation to change the polarization

of transmitted light. The rotation occurs only at speciﬁc resonance frequencies of the

gas inside the ﬁlter; therefore, the selectivity of these ﬁlters is very high. When light at a

resonance frequency is Doppler shifted before entering the ﬁlter, the measured intensity at

the detector changes. The amount of optical rotation is determined by

β = V Bd , (4.5) where B is the magnetic ﬂux density, d is the length of the gas cell, and V is the Verdet

constant for the gas under a given set of conditions. The Verdet constant is dependent on

the gas temperature and wavelength of light. If the Doppler width of the absorption line is

large compared to its Zeeman splitting, the transmission of the gas cell is [49]

1 αR d αL d (αR +αL )d /2 IT = I0 e− + e− + 2cos(2β π)e− , (4.6) 4 −

where IT is the transmitted intensity, I0 is the polarized incident intensity, and αR and

αL are the absorption coefﬁcients for right and left circularly polarized light, respectively.

For absorption coefﬁcients αR d 1 and αL d 1, the transmission approaches 100% for

76 Figure 4.7 Faraday anomalous dispersion optical ﬁlter.

optical rotation β = 90◦. A second type of atomic ﬁlter, the Voigt ﬁlter, was also investigated. This device is

constructed like a FADOF with its magnetic ﬁeld perpendicular to the optical axis and

rotated 45◦ with respect to the polarizers. With the magnetic ﬁeld oriented perpendicular to the axis, the effect on the incoming light is that of linear birefringence. Compared to

the FADOF, this conﬁguration is simpler to implement using permanent magnets. The

transmission of the gas cell in this case is given by [50]

¨ «2 1 X X I I sin2 θ cos2 θ exp α ν d /2 exp α ν d /2 , (4.7) T = 2 0 [ i ( ) ] [ i ( ) ] i =π − − i =σ − where θ is the angle formed by the magnetic ﬁeld vector and either polarizer axis. The

absorption coefﬁcients αi are functions of frequency ν and are speciﬁc to each individual

Zeeman line. The indices i = π and i = σ indicate summation over all (anomalous) Zeeman splittings of the atomic line. Note that Eq. (4.7) represents the complete ﬁlter transmission, with I0 representing the unpolarized incoming light.

Efficient atomic line filters require magnetic fields on the order of 0.2 Tesla. The software

77 package Finite Element Method Magnetics (FEMM) [51] was used to estimate the magnetic ﬁeld strength associated with four different hardware conﬁgurations. In each case, the gas

cell was modeled as a 2.75-inch-long transparent tube with 0.5-inch radius. The interior of

the cell, which contains ionized gas, was modeled as “air.” The permanent magnets used in

the simulations were based on off-the-shelf products from K&J Magnetics.

Faraday ﬁlters are difﬁcult to construct using block magnets. To maximize the magnetic

ﬂux, the magnets should be positioned with their poles pointing along the optical axis.

However, in this conﬁguration, the magnets would obstruct the ends of the optical tube.

Furthermore, off-the-shelf block magnets tend to be magnetized through their thickness,

rather than along their length, making it difﬁcult to obtain a constant ﬂux along the length

of the tube.

The magnetic ﬁeld lines of a simulated Faraday ﬁlter are depicted in Fig. 4.8. This design

uses 42 MGOe neodymium blocks that have been magnetized along their length, creating

ﬂux along the optical axis without obscuring the optical path. However, the ﬁeld strength

along the optical axis, plotted in Fig. 4.9, is barely 0.2 T at the center of the gas cell and falls

off rapidly on both sides.

The second design uses 12 ring magnets, magnetized through their thickness, to encircle

the gas cell located along their axis. In contrast to the previous simulation, this model

is axisymmetric; to visualize the geometry, Fig. 4.10 should be rotated around the axis.

Although a large ﬂux is generated inside the ring material, there is less ﬂux along the optical

axis, where air, glass and inert gas provide high magnetic resistance. One advantage of this

conﬁguration is a nearly constant ﬂux density along the length of the gas cell, as plotted

in Fig. 4.11. With more ring magnets stacked at both ends, this ﬂat ﬂux density could be

extended across the entire length of the cell.

The third design, depicted in Fig. 4.12, replaces the permanent magnets with a wire-

78 Figure 4.8 Electromagnetic simulation of a Faraday ﬁlter using longitudinally magnetized block magnets.

Figure 4.9 Magnetic ﬂux density along the vapor cell axis.

79 Figure 4.10 Axisymmetric simulation of a Faraday ﬁlter using encircling ring magnets.

Figure 4.11 Magnetic ﬂux density along the vapor cell axis.

80 wound solenoid. As with the previous simulation, this one is also axisymmetric, with the

optical axis and gas cell located along the axis of the coil’s air core. A major advantage

of this design is that the ﬂux density is greatest along the axis, not within the coil itself.

Another advantage of the coil is the ability to control the ﬂux density by adjusting the

current. A 1000-turn coil driven by 20 amperes of current generates a maximum ﬂux of

0.2 T within the gas cell. The variation of the ﬂux density is plotted in Fig. 4.13. The ﬂux is

fairly constant, but could be ﬂattened even more by lengthening the coil. Construction of

a pair of Helmoltz coils would guarantee uniformity of the ﬂux, but would require much

larger coils to generate the same maximum ﬂux density.

The fourth design is a Voigt filter. This design uses large, 0.75 1.5 3.0-inch neodymium × × block magnets with a B H product of 52 MGOe. In contrast with the Faraday filters, the × magnetic flux B is perpendicular to the optical axis, as indicated by the field lines depicted

in Fig. 4.14. In general, this conﬁguration results in less optical rotation within the gas cell,

but is mechanically easier to construct. Flux density B along the axis of the gas cell is | | plotted in Fig. 4.15. Although the ﬂux is constant near the center of the tube, it falls off

quickly at both ends.

The software package ElecSus [52] was used to simulate the optical transmission of a Voigt ﬁlter with the parameters listed in Table 4.1. The angles B-θ and B-φ refer to the angle

of the magnetic ﬁeld with respect to the optical axis and crossed polarizer axes, respectively.

Fig. 4.16 illustrates the results for two prominent ground-state lines of potassium, the

D1 and D2 Fraunhofer lines. The top portion of Fig. 4.16 plots the total transmission S0

through the gas cell, i.e., excluding losses in the polarizers. The bottom portion plots the

transmitted intensity of the x and y polarization components. Based on these plots, the

ﬁlter bandwidths for the K D1 and K D2 lines have a FWHM of 0.01147 nm and 0.01255 nm,

respectively. Due to the its relative simplicity compared to the Faraday ﬁlter and its potential

81 Figure 4.12 Electromagnetic simulation of a Faraday ﬁlter using a solenoid.

Figure 4.13 Magnetic ﬂux density along the vapor cell axis.

82 Figure 4.14 Electromagnetic simulation of a Voigt ﬁlter using NdFeB block magnets.

Figure 4.15 Magnetic ﬂux density along the vapor cell axis for seven different permanent magnet separation distances.

83 to realize the 0.01 nm bandwidth goal, the Voigt ﬁlter design would seem to be preferred.

Both the solenoid and the permanent magnet designs have advantages and disadvan-

tages. The solenoid can create very strong ﬁelds, and the ﬁlter is easily tuned by adjusting

the current. However, this design requires an additional power supply, the coils are heavy

and bulky, and the heat generated by the coils must be efﬁciently carried away from other

temperature-sensitive components.

The permanent magnets are compact, require no power, and generate no heat. Simula-

tions indicate that by adjusting the distance between pairs of magnets, the ﬁeld strength

can be tuned just as precisely as the solenoid design. However, tuning the ﬁeld requires

incremental adjustment of the mechanics, and mounting the magnets in proper alignment with adequate precision could be difﬁcult.

Table 4.1 Filter parameters for the Voigt optical ﬁlter.

Parameter Value

Cell temperature 98◦C Cell length 68.6 mm B-ﬁeld 0.12 T

B-θ 90◦ B-φ 45◦ Input polarization 90◦

84 Figure 4.16 Simulated Voigt ﬁlter transmission for two Fraunhofer wavelengths of potassium, (a) D1, and (b) D2.

4.2.2 Birefringent Filters

Birefringent ﬁlters have the advantage of simplicity of design and sub-angstrom frequency

discrimination. The classic design was introduced by astronomer Bernard Lyot in 1933. A

simple, four-stage Lyot ﬁlter is illustrated in Fig. 4.17. Each birefringent element bi has a

thickness di equal to twice that of the previous element, such that di +1 = 2di . For a single stage of the ﬁlter, the optical retardance in radians is

2πβdi δi = , (4.8) λ

where λ is the wavelength of incident light, and β = (ne no ) is the birefringence, i.e., the − difference between the ordinary and extraordinary indices of refraction. Invoking Malus’

Law, the intensity transmitted through a single stage of the ﬁlter is

2 δi 2 πβdi Ti = cos = cos . (4.9) 2 λ

85 When all four stages are considered sequentially, the total transmission is

2 πβd1 πβ2d1 πβ4d1 πβ8d1 T = τ cos cos cos cos , (4.10) λ λ λ λ where τ models any reﬂection and absorption losses in the stack. The theoretical trans-

mission of this ﬁlter at the output of each stage is illustrated in Fig. 4.18. The birefringent

elements are modeled as calcite, with a 1 mm thickness of the ﬁrst stage. The response

demonstrates that while the width of the last (thickest) stage sets the ﬁlter bandwidth, the

ﬁrst (thinnest) stage determines the free spectral range.

Although the center frequency of the pass-band can be moved by changing the thickness

of the retarders, this is not a mechanically practical solution. A more convenient method is

to insert a quarter-wave plate (QWP) immediately after each retardation element [53]. The QWP converts the linearly polarized light to a circular polarization state. The combination

of the QWP and rotating polarizer act as an adjustable phase-shifter. For a single stage, the

transmission becomes 2 πβdi Ti = cos + ξi , (4.11) λ where ξi is the additional phase shift. The center wavelength λ can now be selected by

choosing the phase shift such that

βdi + ξi = an integer. (4.12) λ

The optical system illustrated in Fig. 4.19 comprises three independently tunable Lyot

ﬁlters. The two beamsplitters (BS) direct incident white light into the instrument’s three

arms. The ﬁlter in each arm is preceded by its own bandpass ﬁlter, which reduces the

extraneous light outside of the Lyot pass-band. Continuing along the x axis, light passes

86 Figure 4.17 Simple four-stage Lyot ﬁlter. The fast axis of each birefringent element is oriented at 45 with respect to the linear polarizers (LP). ◦

Figure 4.18 Lyot ﬁlter transmission as a function of wavelength for (a) one stage, (b) two stages, (c) three stages, and (d) four stages. The elements are modeled as calcite; thickness of the ﬁrst stage is 1 mm.

87 through a linear polarizer (LP1) oriented parallel to the y axis. Next, a birefringent retarder

(R1) is oriented with its fast axis at an angle of 45◦ from the y axis. This introduces a phase shift between the two polarization components. The thickness of R1 is chosen such that the

phase shift is minimum near the desired pass-band frequency. Next, a quarter wave plate

(QWP1) converts the linearly polarized light to a circular polarization state. Rotation of the

subsequent linear polarizer (LP2) controls an additional phase shift, providing ﬁne-tuning

of the pass-band frequency. This is followed by a second Lyot ﬁlter stage to narrow the

output bandwidth.

Figure 4.19 Multi-band ﬁlter composed of three two-stage Lyot ﬁlter sections.

The other two arms implement identically arranged two-stage Lyot ﬁlters; only the

88 bandpass ﬁlters and retarder thicknesses are optimized for different spectral pass-bands.

This allows detectors FPA1, FPA3, and FPA4 to monitor three separate pass-bands, each

corresponding to one or more distinct Fraunhofer features.

Finally, a fourth detector (FPA2) collects half the light rejected by bandpass ﬁlters BPF2

and BPF3. If the bandwidth of these ﬁlters is narrow, the light collected at FPA2 represents

the unﬁltered intensity incident on each arm of the instrument. Comparing the ﬁltered

intensities measured at FPA1, FPA3, and FPA4 to the unﬁltered intensity measured at FPA2,

the amount of Doppler shift can be determined.

There are many advantages to the Lyot filter; notably, the ability to fine-tune the center wavelength and to adjust the bandwidth by the addition of more stages. However, the filter

characteristics are sensitive to temperature, and as segments are added — each one twice

the width of the previous segment — the instrument quickly becomes large, heavy, and

expensive.

4.2.3 Fiber Bragg Grating

Fiber Bragg gratings (FBG) offer another way to implement ultra-narrow-band notch ﬁlters.

These ﬁlters are created by modulating the refractive index along a length of optical ﬁber.

The resulting structure behaves like a conventional Bragg grating, such that reﬂected light

interferes constructively when

mλ = 2d sinθ , (4.13) where m is the order of diffraction, d is the grating pitch, and θ is the angle of incidence.

Modulation of the refractive index can be performed by a number of methods. Typically,

the ﬁber is doped with germanium, then exposed to an ultraviolet laser source. The source

is varied as the ﬁber is pulled through the beam, creating stepped index variations along

89 the length of the ﬁber, as illustrated in Fig. 4.20. Using this technique, multiple ﬁlters can

be created within a single length of ﬁber. The narrow stop-band of each ﬁlter results from

the Bragg reﬂection at a particular wavelength. The center wavelength can be found by [54]

λB = 2neff Λ, (4.14)

where λB is the reﬂected wavelength, neff is the effective index of refraction of the ﬁber

mode, and Λ is the period of the grating. The full-width half-maximum (FWHM) bandwidth

of a single grating is given by [55]

p 2 2 ∆λ = s (δn/2n¯) + (1/N ) λB , (4.15) where s is the “strength” of the grating, with s = 1 representing 100% reﬂection at the Bragg wavelength; n¯ is the average refractive index; δn is the index variation, or modulation

depth as a function of exposure time to the ultraviolet light; and N is the number of index

steps written to the ﬁber.

Consistent with the in-band to out-of-band ratio concept, the light reﬂected by an

FBG can be directly compared to the light transmitted through the ﬁber. If an FBG were

constructed with the Bragg wavelength matched to a Fraunhofer wavelength, the ratio

of reﬂected to transmitted power would change in proportion to the Doppler shift. Fur-

thermore, if multiple gratings were incorporated within a single ﬁber, each matched to

a different Fraunhofer line, the SNR and sensitivity would improve with each additional

grating.

To this end, MATLAB simulations were created to investigate FBG performance in

multiple-grating ﬁlters [56]. The simulations were based upon solving coupled-mode wave

90 Λ Optical Fiber n0

Fiber Core n3

Core Refractive Index

n n3

Spectral Response λ B I I I

Input λ Transmitted λ Reflected λ

Figure 4.20 Fiber Bragg grating.

equations [57] using a transfer matrix technique [58]. The ﬁlter model included 20 gratings with Bragg wavelengths corresponding to signiﬁcant Fraunhofer features in the range of

500–600 nm. The simulated ﬁlter was excited with an input signal consisting of a band-

limited version of the solar spectrum [20]. Table 4.2 lists the basic assumptions used in the model.

Simulations were performed using Doppler shifted input signals. The amount of

Doppler shift was equivalent to radial velocities in the range of 1–10 km/s, which is the approximate range of orbital velocities expected to be observed from Earth. Results are

plotted in Fig. 4.21, which depicts the total output power reﬂected from the ﬁber with

respect to the Doppler shifted input signal. Using a data set like this, unknown radial velocities could be determined by a simple power measurement followed by interpolation

or a parametric ﬁt to the data.

91 Table 4.2 Simulation parameters for the ﬁber Bragg grating ﬁlter.

Parameter Value Grating length 1 cm Expected reﬂectivity 99% Fringe visibility 1 Number of transfer matrix sections 100 Core diameter of ﬁber 4 µm Numerical aperture 0.2 Gap length between two gratings 10 cm Attenuation loss 30 dB/km

Reflected power (photons/s) 2

1 0 2000 4000 6000 8000 10000 Radial velocity (m/s)

Figure 4.21 Reﬂected power vs. radial velocity of the target object.

Table 4.3 summarizes the optimized FBG parameters generated by the simulation. The grating length (1 cm), grating pitch, number of sections (N ), and index modulation (δn) represent all the information required for construction of a real ﬁlter.

Although the FBG filter design has potential, our lab currently lacks the expertise and equipment to fabricate it. Several manufacturers were contacted, but none were willing to provide a quotation for construction of an optical fiber containing 20 individual Bragg gratings. Although there is no theoretical barrier to construction of such a filter, state-

92 Table 4.3 Optimized grating parameters for the ﬁber Bragg grating ﬁlter [56].

λB Pitch N δn Max Strength Side- FWHM 5 1 (nm) (µm) (steps) ( 10− ) reﬂect. (cm− ) lobes (nm) × 501.202 0.1705 58660 4.7752 96.23 % 2.9932 20.30 % 0.0118 501.839 0.1707 58590 4.7812 92.41 % 2.9932 15.40 % 0.0118 504.172 0.1715 58310 4.8035 87.78 % 2.9932 12.11 % 0.0119 505.160 0.1718 58200 4.8129 98.98 % 2.9932 28.79 % 0.0120 508.331 0.1729 57840 4.8431 95.34 % 2.9932 18.68 % 0.0121 516.732 0.1758 56900 4.9231 98.94 % 2.9932 29.29 % 0.0125 517.269 0.1759 56840 4.9282 95.94 % 2.9932 19.78 % 0.0126 520.451 0.1770 56490 4.9586 97.62 % 2.9932 23.38 % 0.0127 520.843 0.1772 56450 4.9623 98.98 % 2.9932 29.35 % 0.0127 522.720 0.1778 56240 4.9802 98.80 % 2.9932 28.41 % 0.0128 526.956 0.1792 55800 5.0205 98.47 % 2.9932 26.66 % 0.0130 537.154 0.1827 54730 5.1177 87.56 % 2.9932 11.95 % 0.0136 540.583 0.1839 54400 5.1504 94.27 % 2.9932 17.24 % 0.0137 540.984 0.1840 54350 5.1542 98.90 % 2.9932 29.03 % 0.0137 558.886 0.1901 52600 5.3247 98.43 % 2.9932 26.50 % 0.0147 559.457 0.1903 52550 5.3302 98.92 % 2.9932 28.58 % 0.0147 561.576 0.1910 52350 5.3504 98.76 % 2.9932 28.13 % 0.0148 589.014 0.2003 49910 5.6118 87.53 % 2.9932 11.88 % 0.0163 589.612 0.2005 49860 5.6175 89.58 % 2.9932 13.19 % 0.0163 589.668 0.2006 49860 5.6180 96.88 % 2.9932 21.58 % 0.0163

93 of-the-art manufacturing techniques are not yet mature enough to provide multi-notch

commercial-grade ﬁlters.

4.2.4 Optical Correlator

The ﬁnal design considered for the instrument was a dual-beam optical correlator, illus-

trated in Fig. 4.22. Incident light is collimated by lens L1 into beamsplitter BS. Half of the

light is deﬂected by the beamsplitter through focal lens L4 and onto detector FPA2. The

intensity measured at FPA2 is half the total incident power.

The undeﬂected portion of the light passes through the beamsplitter and is incident on

diffraction grating G. The beam is dispersed by the grating onto lens L2, which collimates

the diffracted beam onto a reﬂective photomask.

The photomask, a spatial ﬁlter, is patterned with an image of the solar spectrum. Areas

on the mask corresponding to Fraunhofer wavelengths reﬂect light; other portions of

the mask absorb or transmit light. In an ideal system, an unshifted spectrum is perfectly

correlated to the mask, and no light is reﬂected. As the sampled spectrum is Doppler shifted,

out-of-band light “leaks” onto the reﬂective parts of the mask, generating a non-zero output

signal.

The output signal follows a retro-reﬂective path through L2 and the grating, which

un-diffracts the dispersed wavelengths. At the beamsplitter, the beam is partly reﬂected

through focal lens L3 and onto detector FPA1. The ratio of the intensity of the in-band

signal measured at FPA1 to the reference intensity at FP2 provides the information needed

to determine the Doppler shift.

Two different grating types were researched for the correlator’s dispersive element. The

following sections present an analysis of each.

94 Figure 4.22 Basic design of a dual-beam Doppler optical correlator.

4.2.4.1 Echelle Grating

With a simple wire grating, much of the incident light is diffracted into the zero-order, which

shows no dispersion, and therefore provides no information about the spectral content of

the light. This reduces the total light power available for analysis, decreasing the overall

signal-to-noise ratio (SNR). Blazed gratings use angled facets instead of a wire grid to create

diffraction, as shown in Fig. 4.23 (a). The facets provide a reﬂective or refractive surface that

shifts the diffracted beam into higher orders where the spectral dispersion dβ/dλ is greater,

resulting in higher spectral resolution [59]. All blazed gratings obey the same equation that applies to standard wire gratings,

mλ sin(α0) sin(β0) = , (4.16) − d where α0 is the angle of incidence, β0 is the angle of diffraction, λ is the wavelength, d is

the grating period, and m is the diffraction order.

95 Figure 4.23 (a) Standard blazed grating. (b) Echelle grating.

Unfortunately, blazed gratings suffer from two disadvantages. First, although some of the zero-order light is redirected to higher orders with greater dispersion, the efﬁciency, deﬁned as the ratio of input power to the power in a given order, decreases with higher orders. Second, the free spectral range (FSR) decreases with higher orders, until eventually the orders begin to overlap, resulting in loss of spectral information.

The ﬁrst issue can be solved using an echelle grating, as shown in Fig. 4.23 (b). An echelle is a type of blazed grating with a large blaze angle and long spatial period, which shifts even more of the zero-order energy into higher orders than a standard grating, resulting in improved efﬁciency [59]. This, however, does not eliminate overlapping orders. To solve the second issue of decreasing FSR, an additional grating or prism is usually employed, as shown in Fig. 4.24 (a). The second grating or prism, called a cross-disperser, is oriented at right angles to the echelle. This provides dispersion in the orthogonal direction, effectively separating the overlapping orders and creating a two-dimensional spectrogram, as shown in Fig. 4.24 (b).

96 Figure 4.24 (a) Incoming light diffracted by an echelle grating and cross-disperser. (b) The resulting two-dimensional spectrogram. (Adapted from [60].)

4.2.4.2 Volume Phase Grating

Volume phase gratings are constructed from a layer of dichromated gelatin (DCG) or other

photosensitive medium sandwiched between glass plates, as shown in Fig. 4.25 (a). Expo-

sure to a UV interference pattern creates a sinusoidal variation of refractive index within

the DCG. Because this process, illustrated in Fig. 4.25 (b), is similar to the construction

of holographic plates, these devices are often called volume-phase holographic (VPH)

gratings.

Diffraction in a VPH grating follows Eq. (4.16), where the period d is taken as the spacing

between modulation planes (fringes) at the surface of the DCG. However, reﬂection at the

fringe planes insures that diffraction occurs only for beams that satisfy the Bragg condition.

97 Figure 4.25 (a) Key physical parameters of a volume phase grating. (b) Modulation of the refractive index.

When the fringes are orthogonal to the surface, the Bragg condition becomes

mλB sin(θB ) = , (4.17) 2Λ

where θB is the incident angle corresponding to the Bragg angle, m is the Bragg order,

λB is the Bragg wavelength, and Λ is the perpendicular distance between fringe planes within the DCG. Diffraction will occur for a small bandwidth of wavelengths and angles

near the Bragg condition. The full-width, half-maximum (FWHM) spectral bandwidth can

be approximated as [61]

∆λ λΛ/d tan(α2B ), (4.18) ≈ where α2B is the incident angle between the incoming beam and the fringe plane. The

FWHM angular bandwidth is approximately [61]

∆θ Λ/d , (4.19) ≈

98 where Λ is the perpendicular distance between fringe planes within the DCG. Rearranging

Eq. (4.16) and differentiating the diffracted angle with respect to wavelength yields an

expression for angular dispersion as a function of wavelength:

1/2 2 2 dβ m m λ 2mλ − 0 1 sin2 α sin α . (4.20) = ( 0) 2 + ( 0) dλ −Λg − − Λg Λg

1 Using a VPH from Wasatch Photonics with a fringe spacing of 1800 mm− and a Bragg wavelength of 532 nm, we obtain the results shown in Table 4.4.

To determine the “off Bragg” behavior of the VPH grating requires use of coupled-wave

theory. Kogelnik’s 1969 paper [57] uses a coupled-wave analysis to derive five parameters for use in the characterization of VPH gratings. The first parameter Γ quantifies the dephasing

of the incident wave from the Bragg condition. It is calculated as

2π∆αcos(π/2 + α ϕ) π∆λ Γ , (4.21) = − 2 Λ − n2Λ where ∆α is the difference (in radians) between the Bragg angle and angle of incidence, ∆λ

is the difference (in nanometers) between the Bragg wavelength and incident wavelength,

ϕ is the slant angle of the fringe planes with respect to grating normal, and n2 is the bulk

refractive index of the DCG layer. Next, Kogelnik deﬁnes two “obliquity factors,”

cR = cosα, and (4.22)

λ cS = cosα cosϕ, (4.23) − n2Λ where cR and cS represent coefﬁcients of the coupled wave equations for the reference

(incident) beam and signal (diffracted) beam, respectively.

99 Table4.4 First order parameters at the Bragg condition for the VPH grating from Wasatch Photonics.

Parameter Symbol Value Units

Grating period d 1/1800 mm Fringe plane angle ϕ 0 deg Fringe plane spacing Λ 1/1800 mm Bragg wavelength λB 532 nm Bragg angle θB 28.61 deg Thickness t 3 µm Spectral bandwidth ∆λ 180 nm Angular bandwidth ∆θ 10.6 deg Dispersion dβ0/dλ 0.117 deg/nm

Kogelnik then derives a general expression for the diffracted wave in a lossless medium,

p jξ p 2 2 j cR /cS e− sin ν + ξ S = − p , (4.24) 1 + ξ2/ν2 where ν and ξ are the ﬁnal two Kogelnik parameters, deﬁned as

π∆nd ν = , and (4.25) λpcR cS

Γ d ξ = , (4.26) 2cS with ∆n being the modulation amplitude of the refractive index. Now, the ﬁrst-order

diffracted beam efﬁciency for both S-polarized and P-polarized waves can be expressed as

2 p sin ν2 + ξ2 ηS = , and (4.27) 1 + ξ2/ν2

2 q sin ν2 cos2 π + 2α 2ϕ + ξ2 ηP = − . (4.28) 1 + ξ2/ ν2 cos2 π + 2α 2ϕ −

100 The diffraction efﬁciency for both polarizations is plotted against wavelength in Fig. 4.26.

Values include an estimated 3% loss of efﬁciency due to absorption in the VPH. This data

shows high efﬁciency (above 90%) for wavelengths in the range of 500–550 nm.

Fig. 4.27 depicts the diffraction efﬁciency as a function of incidence angle. Both polar-

izations exhibit maximum diffraction efﬁciency at an incident angle near 28◦. This was

expected given the calculated Bragg angle of 28.61◦. Finally, Fig. 4.28 plots diffraction efﬁciency at the Bragg condition as a function of both

grating strength (i.e., amplitude of the refractive index modulation) and thickness of the

DCG material. The marked data point indicates best case VPH efﬁciency of 92% when the

DCG has a thickness of 3.024 µm and an index modulation of 9%.

101 Figure 4.26 First order diffraction efﬁciency versus wavelength. These results include an additional 3% loss in both S and P waves due to absorption.

Figure 4.27 First order diffraction efﬁciency versus incident angle. These results include an additional 3% loss in both S and P waves due to absorption.

102 Figure 4.28 First order diffraction efﬁciency versus index modulation (vertical axis) and DCG thickness (horizontal axis). Color maps to the average efﬁciency of unpolarized light. These results include an additional 3% loss due to absorption. The marker indicates the design parameters for the VPH grating described in the previous section.

4.3 Trade-Space Summary

Table 4.5 summarizes the advantages and disadvantages of each instrument design dis-

cussed in the previous section. Each design has been given a partially subjective pass/fail score based on the following criteria:

1. Size and weight

2. Cost of materials

3. Availability of parts

4. Difﬁculty of calibration

5. Stability over time and temperature

6. Complexity of construction

103 7. Spectral resolution

8. Spatial resolution

The most critical requirement is spectral resolution, since that is directly linked to the

accuracy of any Doppler measurement. Without the beneﬁt of spatial heterodyning, the two

full-spectral designs would not meet the required resolution of 0.01 nm. Based on results

from a previous heterodyne project in our lab [47], it was expected that the difﬁculties with calibration and overall complexity of these designs would preclude their successful

completion within the scope of this project. Additionally, the cost of the birefringent SHI was expected to exceed the project’s budget. Unfortunately, only the two full-spectral

designs were immediately capable of spatial imaging. Although the remaining designs

could potentially be enhanced to include imaging at a later date, the additional complexity

placed it outside the scope of this project.

The atomic line ﬁlters offer the potential for very high resolution. However, they suffer

from stability issues because the optical response is highly dependent on the temperature

and pressure both inside and outside the gas tube. Also, experiments with hollow-cathode

lamps indicated that the number of Fraunhofer lines that could be incorporated into a

single measurement was limited.

The birefringent Lyot ﬁlter provided both high spectral resolution and selection of pass-

bands with relative ease. However, spectral resolution is dependent on the number of ﬁlter

stages, and each pass-band requires its own ﬁlter. Ultimately, the size, weight, and cost of

such a design was prohibitive.

The ﬁber Bragg grating was a unique idea that offered good spectral resolution, multiple

pass-bands, negligible size and weight, and simplicity of construction and calibration. A

number of manufacturers produce commercial FBG sensors for temperature and strain

104 measurement for a wide variety of applications. However, no vendors were willing or able

to manufacture an FBG ﬁlter to our speciﬁcations. Apparently, at this early stage in develop-

ment, manufacturing processes are not ﬂexible enough to handle custom conﬁgurations.

Finally, the design based on the dispersive optical correlator provided good spectral res-

olution, the ability to concurrently measure dozens of Fraunhofer lines, and construction

from off-the-shelf parts. For the ﬁnal design, a VPH grating was chosen over an echelle grat-

ing as the dispersive element because of its comparative simplicity. Several design changes

and enhancements were applied during development, including the use of polarization

elements to improve the SNR. This ﬁnal design is described in detail in the next section.

Table 4.5 Design trade-space.

Design Size Cost Availability Calibration Stability Complexity Spectral Res. Spatial Res. Birefringent SHI Sagnac SHI Atomic line ﬁlter Birefringent ﬁlter Fiber Bragg grating (*) Direct correlation *Cost data unavailable.

4.4 Direct Correlation Spectrometer

A direct correlation spectrometer (DCS) was chosen as the ﬁnal design. A key aspect of the

DCS is direct optical correlation of the target spectrum with known, unshifted, Fraunhofer

105 absorption lines. The dual-band Doppler concept ratio of in-band (within the width of an

absorption line) to out-of-band signal power provides a measurement of the amount of

Doppler shift. After investigation of several different optical ﬁlters (described in section 4.2)

a ﬁlter based on a VPH grating was chosen for the ﬁnal design. Advantages of the design

include

Improved SNR with high spectral resolution on the order of 0.025 nm • Direction of in-band and out-of-band light into orthogonal polarization states • Simultaneous integration of 155 deep Fraunhofer lines spanning 520–540 nm • The main advantage of this method is the simultaneous integration of many lines across

a 20 nm region of the solar spectrum. This provides an order of magnitude improvement in

the SNR.

One major component of the correlator is a spatial light modulator (SLM) used to map

a shifted spectral image to a reference spectrum. Because of its importance to the design, a

brief review of SLM technology is provided in the next section.

4.4.1 Spatial Light Modulators

A spatial light modulator is a device that can modify the intensity, phase, and/or polarization of incident light over a two-dimensional optical wavefront. The active area of a typical

device is divided into horizontal and vertical pixels, each having an individual birefringence

controlled by a programmed operation T (x , y ). This results in modulation of the wavefront

properties across the surface of the SLM [62]. Depending on the construction and intended application, SLMs can be either trans-

missive or reﬂective. The physical method of modulating the pixel birefringence can be

mechanical, chemical, or optical [62]. This description considers only optical-mode de-

106 vices, in which pixels are electrically programmed, because they are the most common

type and are employed in the present work. In this type of device, each pixel is composed

of a liquid crystal (LC) cell, which is illustrated in Fig. 4.29.

Figure 4.29 Transmissive liquid crystal (LC) cell. (a) In the absence of an electric ﬁeld, the LC molecules orient themselves with the orthogonal alignment layers, forming a twisted helix structure. (B) When a voltage is applied to the electrodes, the molecules align with the electric ﬁeld.

The liquid crystal molecules are bipolar, so that their orientation can be controlled with

an applied electric ﬁeld. The molecules are also birefringent, with the slow optical axis

oriented along the polar axis of the molecule. With no electric ﬁeld applied, the molecules

at either end of the cell self-align with crossed layers of a highly polar polyimide ﬁlm. This

self-alignment creates a twisted stack of LC molecules, as shown in Fig. 4.29 (a). Light

entering the cell experiences a rotation of its polarization state as it travels along the stack.

When an external ﬁeld is applied between a pair of transparent electrodes, the molecules in

107 the cell line up end-to-end, as shown in Fig. 4.29 (b). In this conﬁguration, the optical axis

of the cell is parallel to the direction of wave propagation, and the wavefront polarization is

unaffected [63]. Light passing through each pixel of the SLM experiences a different index of refraction

depending on the voltage applied to each cell. Programming the cells with a speciﬁc voltage

pattern creates a phase modulation across the pixel array. When polarizing ﬁlters are used

in the path of the input and output wavefronts, the polarity or intensity of the output can be

controlled in the same way. This makes SLMs useful for a variety of optical tasks, including

beam shaping, pulse shaping, and aberration correction [64]. Although Fig. 4.29 illustrates a transmissive cell, the preceding description also ap-

plies to reﬂective cells, which are constructed with a dielectric mirror at one end. The

incoming wavefront is reﬂected by the mirror and passes through the LC cell a second time,

experiencing additional phase shift.

4.4.2 Design of the Optical Correlator

When programmed with a reference pattern of interest, a spatial light modulator (SLM)

can be used as a photomask for an optical correlator. If the SLM is placed at an image

plane, the portions of the image coincident with the pattern on the SLM will experience a

phase shift relative to the rest of the image. When the incoming light is linearly polarized, it will experience a rotation of polarization state as described above. Our correlator design,

illustrated in Fig. 4.30, makes use of this polarization difference to measure the wavelength

change caused by Doppler shift.

Starting in the upper left of Fig. 4.30, light from a telescope passes through the entrance

slit and is linearly polarized by LP1. Next, the polarized light is collimated into a VPH grating

108 Figure 4.30 Final design of the optical correlator.

109 and diffracted.

To make the best use of the limited space on the breadboard and ensure that the optical

design is telecentric, mirrors M1 and M2 fold the ﬁrst-order diffracted beam and direct

it toward a relay lens. It should be noted that M2 is mounted against a piezo-electric

transducer (PZT). The PZT has an angular sensitivity of 5 µrad/V over its control range of 0–150 V, enabling it to smoothly tilt the mirror through a maximum angle of 750 µrad.

Tilting M2 serves two purposes: it facilitates calibration, and it allows the instrument to be

used in either a “scanning” or a “one-shot” mode of operation. Calibration and operating

modes are discussed in more detail in chapter5.

Next, the relay lens directs the beam through an achromatic quarter-wave plate (AQWP)

before creating an image of the diffraction pattern on the surface of the SLM. The AQWP

imparts a circular polarization to the already linearly-polarized beam.

The SLM can be programmed with any desired spectral image as a correlation reference.

Fig. 4.31 (a) illustrates a sample pattern for wavelengths in the range 533.8–535.2 nm. The

black line illustrates an unshifted reference spectrum [20]. The red line represents a pattern on the SLM. Pixels adjacent to absorption lines are programmed to induce a polar rotation

of 90 degrees. Fig. 4.31 (b) provides a close-up of the mask pattern, which more clearly

illustrates the concept behind the correlator. If the spectral image slides to the right due to

Doppler shift, the in-band light energy (spatially overlapping the SLM pattern) is subject to

polar rotation. Conversely, if the image shifts to the left, the out-of-band energy (the rest of

the spectral continuum) is unaffected.

Next, the light is returned by the reﬂective SLM. The telecentric conﬁguration of the

optics permits an image of the diffracted beam to be re-formed on the VPH. Passing through

the VPH a second time, the original dispersion is compensated, and the diffracted beam is

re-integrated.

110 Following the VPH, a birefringent Wollaston prism (WP) splits the light into two beams of orthogonal polarization states. These two beams contain the in-band (IB) and out-of- band (OB) energy. Next, linear polarizer LP2 attenuates some of the out-of-band energy to insure that the in-band to out-of-band intensity ratio does not exceed the camera’s dynamic range.

Next, a polarization grating (PG) with a large spatial period provides a small amount of diffraction, creating a low-resolution spectrum of each beam. This helps to normalize low-frequency ﬂuctuations of the spectrum when operating in one-shot mode.

Finally, mirror M3 redirects the beams toward a reimaging lens, which forms two separate images on the focal plane array (FPA). The ratio of the intensity measured in each area is related to the Doppler shift observed at the telescope. This power ratio is plotted in

Fig. 4.32 as a function of simulated wavelength shifts in the range -0.005 – 0.005 nm. Fitting a polynomial to this curve allows the Doppler wavelength shift to be determined.

111 Figure 4.31 (a) Conceptual model of the photomask pattern used on the SLM. (b) A detail of the pattern, showing the relationship between wavelength of the input and phase modulation introduced by the SLM.

Figure 4.32 Ratio of in-band optical power to total power as a function of simulated wavelength shifts.

112 CHAPTER

ASSEMBLY AND CALIBRATION

The prototype of the direct correlation spectrometer (DCS) is pictured in Fig. 5.1. Some key

speciﬁcations of the instrument are listed in Table 5.1. Located at the end of the chapter,

additional details about the imaging sensor, SLM, and bill of materials can be found in

Table 5.2, Table 5.3, and Table 5.4, respectively.

To guard against thermal drift caused by ambient temperature ﬂuctuations, the optics

depicted in Fig. 5.1 are assembled on a water-heated breadboard and enclosed in an insu-

lating foam box. The interior of the box is maintained at 30◦C by a separate thermal control unit. The entire system – including water circulator, computer controller, and power supply

for the calibration lamp – ﬁts on a small, custom lab cart. The ﬁnal assembly, mounted on

113 Figure 5.1 Laboratory prototype of the direct correlation spectrometer.

Table 5.1 Key speciﬁcations of the direct correlation spectrometer.

Bandwidth 20 nm Resolution R 0.004 nm Imaging sensor Apogee∼ A694, 2750 x 2200 pixels Photomask (SLM) Santec reﬂective LCOS, 1440 x 1050 pixels Dispersive element Wasatch Photonics VPH, 1800 lines/mm Temperature stabilization Thorlabs water-cooled breadboard

114 an Orion Atlas Pro equatorial mount behind the telescope, is depicted in Fig. 5.2.

Figure 5.2 Instrument platform with spectrometer and telescope on the EQ mount.

A closer view of the system assembly reveals the calibration section, depicted in Fig. 5.3.

This small space between the telescope and spectrometer holds an iron lamp, a back-

lit iodine gas cell, and a removable fold mirror. The mirror, located just forward of the

spectrometer’s entrance slit, can be tilted to illuminate the slit from either light source. The

iron lamp produces many of the same atomic lines present in the solar spectrum, and there

are dozens of prominent lines in the range of 500–550 nm, as evidenced by calibration

image of Fig. 5.4. For this reason, the iron lamp was typically used for bench-top calibration, when a large number of markers inspired more conﬁdence in the alignment procedure.

Just prior to observation, the mirror was rotated to the temperature-stabilized (100 0.1◦C) ± iodine cell, which was used to determine the zero-correlation point.

115 Figure 5.3 Calibration section.

Figure 5.4 Calibration image of iron lamp emission lines at the SLM focal plane.

116 5.1 Frequency Stability

Measurements of a known source (halide lamp) were taken over a period of 16 hours to

provide an estimate of the instrument’s wavelength stability over time and temperature.

For these tests, the camera was placed at the focal plane of the SLM so that the dispersed

spectrum could be imaged directly. The thermal control unit (TCU) for the instrument was set to circulate water through the breadboard at a constant temperature of 30 0.1◦C. ± An image of the spectrum was captured once every minute. Each of the resulting 1000

spectral images was averaged along the y -pixel axis to generate a set of intensity vs. x -pixel

relationships. Based on comparison with a spectral image of the lamp captured with a

4 Thorlabs FTS, a calibration constant of 4.23 10− nm/pixel was calculated. × To estimate the frequency stability, an isolated intensity peak near x = 1353 pixels was chosen as a reference line. For each image, a Gaussian curve was ﬁtted to the reference line,

and the parameters of height (peak amplitude), width (standard deviation), and center pixel

(mean) were extracted. The center pixel values for every image are plotted with respect to

time in Fig. 5.5. The extremes of the graph show a maximum swing of less than 0.6 pixels, while the smoothed data (yellow line) is well within a range of 0.2 pixels, corresponding to

5 a wavelength stability of 4.23 10− nm. For the reference line at 542.4 nm, this implies a ± × maximum radial velocity error of

5 4.23 10− ± × c = 23.4 m/s. (5.1) 542.4 ±

Extensive testing has indicated that the setting of the camera’s thermoelectric cooling

system (TEC) can signiﬁcantly affect the wavelength stability. For best results, the TEC was

set to a temperature of 18◦C, or 2◦C below ambient. A chart of the ambient and instrument

117 temperatures measured during the test is depicted in Fig. 5.6.

Figure 5.5 Reference peak x -pixel position as a function of time.

118 Figure 5.6 Instrument (blue) and ambient (red) temperatures during stability test.

5.2 Alignment and Calibration

Alignment of the optical elements proceeded as follows. The reader may ﬁnd it helpful to

refer to Fig. 4.30 for context.

1. Position the slit at the focal point of the collimator. A camera, focused at inﬁnity, was

placed at the output of the VPH to capture images of the zero-order beam. The slit’s

position was adjusted until the sharpest image was obtained.

2. Adjust mirrors M1 and M2. This was accomplished using two diode lasers, a single-

mode at 532 nm and a multi-mode at 520 nm, which roughly deﬁne the extreme ends

of the system’s optical bandwidth. A beamsplitter was used to couple both lasers

into the entrance slit, and the beams were traced (using a white card) from the VPH,

through one half of the relay lens, to the SLM. The mirrors were adjusted so that the

ﬁrst-order diffracted beam was reﬂected through the relay lens and onto the SLM

119 with no vignetting.

3. Rotate the SLM to achieve retro-reﬂection. With the same setup as the previous step,

the beams were traced in the reverse direction, from the SLM, through the other half

of the relay lens, back to the VPH. The SLM was rotated within the x z plane until − the returning beams were un-diffracted after passing through the VPH.

4. Position the relay lens for approximate focus. The SLM was programmed with a

pattern to produce a half-wave of retardance across its surface. Using only the 532 nm

laser, the retardance resulted in two sheared beams, which interfered at the exit of the

relay lens. The distance between the relay lens and the SLM was adjusted, by moving

the lens in its slip mount, until the interference fringes were vertically oriented.

5. Position the SLM for precise focus. Both lasers were again coupled into the system.

The SLM was programmed to scan a single column of pixels across its surface. At

the end of the scan, the in-band/out-of-band power ratio measured by the camera was used to construct a spectrum of the laser diodes. Since the multi-mode laser

generates a comb of lines on the SLM, it was particularly useful for this step.

Strange behavior of the SLM resulted in some unanticipated challenges during alignment. It was expected that pixels would be rendered on the display (Fig. 5.7) in the same order as the programming data. Instead, every other column of pixels was transposed, pos- sibly because of a byte-order mismatch. Although unnoticeable by casual inspection, this pixel reversal was decidedly unhelpful to the alignment process. The error was eventually detected, and the programming data was modiﬁed to accommodate the SLM.

After alignment of the instrument, the following calibration procedure was followed:

1. Calculate the camera’scoefﬁcient of linearity. First, a 532 nm LED was focused through

a linear polarizer onto the entrance slit. The polarizer was rotated until its optical axis

120 Figure 5.7 The SLM-100 spatial light modulator.

was parallel with system polarizer LP1. Next, the image exposure time was adjusted

so that the camera output was nearly saturated. Then, the output intensity was

measured as the polarizer was rotated in 10◦ increments from 0 to 180◦. This data was plotted against the ideal transmission through the two polarizers, as per Malus’s

Law. The second-order polynomial ﬁtted to this plot was used to linearize all data

subsequently read from the camera.

2. Calibrate the wavelength axis. The same pixel scanning technique used during system

alignment was repeated to generate a spectrum of the iron lamp. This data was then

compared to a reference spectrum of an iron lamp [65]. The x -axis location of peaks in the measured spectrum was manually ﬁtted to peaks in the reference data. The

resulting second-order polynomial was used to linearize the wavelength axis for all

subsequent measurements.

3. Calibrate the piezo-electric transducer (PZT). First, a 532 nm LED was directed

through the iodine calibration cell and into the entrance slit. The angle of mirror M2

was adjusted using the PZT, causing the image of the spectrum to be translated along

121 the width of the SLM. This created an effective Doppler shift of the iodine spectrum.

Meanwhile, the SLM was programmed with a reference spectrum for iodine [66] that had been Doppler up-shifted (using linear interpolation) by 0.1 nm. Next, the

voltage on the PZT was swept through its speciﬁed range of 0 to 150 V. When the

in-band/out-of-band ratio of the correlator reached a minimum, the voltage on the

PZT was recorded as V1. The SLM was then re-programmed with the same reference

spectrum, but this time it was down-shifted by 0.1 nm. The sweep of the PZT was

repeated, and the voltage corresponding to the minimum correlation value – i.e.,

the best match between sample and reference – was recorded as V2. The ﬁrst-order

calibration coefﬁcient for the PZT was then calculated as

∆λ1 ∆λ2 Γ = − , (5.2) V1 V2 −

where ∆λ1 and ∆λ2 are the interpolated Doppler shifts of –0.1 nm and +0.1 nm,

respectively. Substituting the measured voltages of V1 and V2, the value of Γ was found

to be 0.003571 nm/V. This value was used to ﬁnd the Doppler shift of the sampled spectrum, dλ, through the relationship

dλ = Γ dV. (5.3)

4. Perform a low-resolution spectral calibration. The polarization grating (PG) illustrated

in Fig. 4.30 disperses the in-band and out-of-band beams into low resolution (ap-

proximately 2 nm) spectra on the FPA. This effectively normalizes any low-frequency

spectral components between the two beams. Although this calibration is not re-

quired when the instrument is used with the PZT in scanning mode, it is a necessary

122 step for single-shot mode. To calibrate the x -axis of the low resolution spectra, the

SLM was programmed to display 100-pixel-wide columns of half-wave retardance.

The low resolution features captured by the FPA were then calibrated with reference

to the calibration of the SLM.

5. Calibrate DCS measurements. Just prior to obtaining experimental measurements

of a target object, images of the iron calibration lamp were captured as the PZT was

scanned through voltages of 35–115 V. The same sequence was repeated for images

of the target object. Both data sets were separately ﬁtted to a Gaussian function,

V V 2 I 1 exp ( 0) , (5.4) = − 2 − − 2σ

where

I was the intensity measured by the camera,

V was the control voltage driving the PZT,

V0 was the ﬁtted value of voltage at the correlation minimum, and

σ was the ﬁtted standard deviation, roughly corresponding to the line width.

The voltage difference between ﬁtted values of V0 for the iron lamp and the target

was calculated. Next, this value was used as dV to solve for the Doppler shift, dλ,

using Eq. (5.3). Finally, the radial velocity corresponding to this shift was found by

rearranging Eq. (1.1) to yield dλ v = c . (5.5) λ

Calibration step5 is only necessary when the instrument is operated in scanning mode, much like a conventional DCS. When used in snapshot mode, step4 enables spectral resolution of the in-band and out-of-band beams. This helps to ensure that the intensity ratio of the two beams is constructed from identical regions of the spectrum.

123 Based on data collected using this calibration procedure, the full-width half-maximum

spectral resolution was determined to be 0.04 nm.

5.3 Test Results

The planet Venus was chosen as a target object for testing the instrument. Because the

prototype unit has limited mobility, it was desireable to select a target that could be seen

through the westward facing window adjacent to the lab. Also, due to the limited aperture

of the 8-inch Schmidt-Cassegrain telescope, a fairly bright object was needed to collimate

enough optical power into the instrument. At the time of testing on June 29, 2018 at 9:30

PM EST, Venus was a convenient target on both accounts.

Making astronomical observations indoors, through a window, was a challenge. At-

mospheric seeing was relatively poor. The seeing problem was compounded by heat

conduction through the window, from an outdoor temperature of 27◦C to the 21◦C interior of the building. Additionally, indoor polar alignment of the telescope was sketchy at best;

the target had to be manually tracked using the guidescope. Nevertheless, the instrument

performed well. Direct correlation spectra were obtained for both the iron lamp calibration

source and the sunlight reﬂected from the planet. For each step of the PZT, the camera was

set to integrate the spectral image for 20 seconds.

The raw measurement data is plotted in Fig. 5.8 (a). The correlation values for Venus

and the iron lamp exhibit an inverse relationship because the spectra are absorptive and

emissive, respectively. Fig. 5.8 (b) depicts the same data after inverting the iron spectrum,

ﬁltering through a Hamming window, then normalizing to a range of correlation values

between 0 and 1.

To ﬁnd the true value of the voltages corresponding to the correlation minima, the

124 Figure 5.8 Direct correlation spectra of the iron lamp (solid blue) and Venus (dashed red) obtained by the instrument.

data points of Fig. 5.8 (b) were cropped around their minima, then ﬁtted to the Gaussian

function of Eq. (5.4). The resulting data and ﬁtted curves are plotted in Fig. 5.9. The residual

3 4 mean-square error of the ﬁt was 1.39 10− and 1.36 10− for Venus and the iron lamp, × × respectively. The PZT voltages (V0) corresponding to the correlation minima were found to

be 74.237 V and 67.978 V for the Venus and iron data, respectively.

Using Eq. (5.3) to solve for the Doppler shift, followed by substitution into Eq. (5.5),

yielded a value of 12.714 km/s for the observed radial velocity. The theoretical value of − the radial velocity of Venus, with respect to Raleigh, North Carolina, during 2018 is plotted

in Fig. 5.10. These values were calculated from NASA ephemeris data [67]. The sinusoidal variation is casued by the rotation of the earth. At the time of the experiment, the theoretical value was 12.789 km/s. The difference between theory and measurement is 75 m/s, which − is an error of 0.59%.

Although the theoretical model used to calculate radial velocity accounts for Earth’s

rotation, it does not include the rotation of Venus or the movement of clouds. However,

125 Figure 5.9 Subset of correlation data for (a) Venus, and (b) iron lamp, ﬁtted to Gaussian exponential curves (solid blue lines).

Venus’ rotation rate is approximately 6.5 km/h, and average wind speeds are less than 1 m/s

[67]. These effects are negligible compared to atmospheric turbulence and systemic errors in the equipment.

126 Table 5.2 Imaging sensor Apogee A694 speciﬁcations.

Sensor Sony ICX694 Pixels 2750 x 2200 Sensor size 12.5 x 10 mm Pixel pitch 4.54 x 4.54 µm

Well depth 21,000 e−/pixel Read noise 5.9 e− Binning 1 x 1 to 8 x 200 Quantum efﬁciency 77% @580 nm

Temperature stability 0.1◦C ± Dark current 0.0041 e−/pixel/s Digital resolution 16 bits Exposure time 100 ms – 183 min Computer interface USB 2.0

Table 5.3 Santec spatial light modulator SLM-100 speciﬁcations.

Type Reflective liquid crystal on silicon Operation mode Amplitude and phase modulation Bandwidth 500–700 nm AR coating BBAR, R < 0.5% average, R < 1.0% peak Active area 10 x 10 mm Resolution 1440 x 1050 pixels Pixel fill factor 90% Panel reflectivity > 80% Gray levels 1024 Phase depth > 2π rad Phase fluctuation < 0.002π rad Operating temperature range 15–35◦C Temperature stability < 0.001π rad/◦C Repeatability r < 0.002π rad

127 Table 5.4 Direct correlation spectrometer bill of materials.

Item Description Qty. Price Total SLM-100 Spatial Light Modulator 1 12500 12500 WP-1800/532-25.4 Volume Phase Hologram 1 800 800 Apogee A694 CCD Camera 1 3281 3281 KPC064AR.14 Newport Lens (Main Barlow) 1 81 81 SLB-50.8B-200NM Optosigma Lens (Alternative Barlow) 1 134 134 SLB-50.8-1000PM Optosigma Lens (Collimator / Reimgr) 3 86 257 49665 Aspherized Achromat (Objective) 1 115 115 49287 Edmund Optics Achromat (Collimator) 1 140 140 48249 Edmund Optics PCX (Reimgr) 1 68 68 49289 Edmund Optics Achromat (Reimgr) 1 140 140 5GCT0 Foam Sheet, 220 Poly, 1 x 24 x 18” 6 19 116 AC254-050-A 50 mm Achromat Lens (FPA Obj.) 1 73 73 LPVISE100-A Polarizing Filter 2 89 178 MBC2412 Water-Cooled Breadboard 1 814 814 VA100 Adjustable Mechanical Slit 1 248 248 RS3 1" Pillar Post 6 24 141 RSH3 1" Post Holder with Flexure Lock 6 34 204 XT95P11 Drop-On Rail Carriage for 95 mm Rails 5 80 400 XT95SP-1000 95 mm One-Sided Construction Rail 1 201 201 RSHT3 1" Post Holder with Flexure Lock 1 34 34 LMR2 Lens Mount with Retaining Ring 5 25 123 PF175 Clamping Fork 6 17 105 SM2L10 SM2 Lens Tube 1 29 29 PF20-03-P01 Protected Silver Mirror 1 101 101 POLARIS-K2F3 Polaris Low Distortion Mirror Mount 1 290 290 PLS-P150 1” Post for Polaris Mirror Mounts 1 30 30 PFR10-P01 Protected Silver Mirror 1 80 80 FMP1 Fixed 1" Optical Mount 1 15 15 LMR1 Lens Mount with Retaining Ring 1 15 15 PH3E 1/2" Pedestal Post Holder 7 24 168 CF125C Clamping Fork 7 11 74 TR3 1/2" Optical Post 7 5 38 XE25L09 25 mm Construction Rail 4 15 59

128 Figure 5.10 Theoretical radial velocity of Venus for all of 2018, calculated from ephemeris data. The red circle indicates the theoretical value at the time of the instrument test. The red cross indicates the measurement obtained from the instrument.

Table 5.4 Direct correlation spectrometer bill of materials, continued.

Item Description Qty. Price Total MB612 Aluminum Breadboard 1 130 130 XE25-CUST-22 25 mm Construction Rails, 22" Long 4 85 340 RM1G Counterbored Construction Cubes 4 17 67 MB4 Solid Aluminum Breadboard 2 41 82 CAM2 Right-Angle Bracket 2 97 194 AP90 Right-Angle Mounting Plate 2 78 156 TB4 Black Hardboard, 24 x 24” 2 63 125 5GCT0 Foam Sheet, 220 Poly, 1 x 24 x 18” 6 19 116 P826A Hollow Cathode Lamp (Fe) 1 264 264 P209-USB HCL Power Supply 1 2487 2487

129 CHAPTER

CONCLUSION

This thesis has demonstrated an alternate method for initial orbit determination, based

on measurement of angles and radial velocity, that can produce an orbital solution with

only two passive measurements of the target. The design, construction, and preliminary

test results of a prototype instrument — a dual-band Doppler ratio, direct correlation

spectrometer (DCS) — for obtaining Doppler shift measurements in reﬂected sunlight have

also been presented.

Although the main beneﬁt of the new method is its ability to provide IOD results using

fewer target observations, a second advantage is its ability to generate solutions under

conditions of target-observer co-planarity, which is beyond the capability of conventional

130 angles-only methods without employing four or more points of observation.

Computer simulations of the new technique have demonstrated that in some situations,

incorporation of Doppler shift measurements can generate IOD solutions with smaller

position errors than angular measurements alone. Due to the effect of orbital geometry

on the observed wavelength shift, the accuracy of the technique suffers as the solar phase

angle increases. However, large phase angles are unlikely to be encountered under real

observing conditions, since this typically implies extreme geometries where the target is

either lost in the glare of the sun or not within the telescope’s line of sight. Also, a simple

estimation of the current phase angle can be used to determine whether the new technique

is likely to provide an advantage over conventional methods.

Advantages of the hardware design include high spectral resolution, simultaneous

integration of more than 150 Fraunhofer absorption lines, and the use of orthogonal po-

larization states for further improvement of SNR. Another unique aspect of the device

is its ability to operate in both a traditional DCS mode, and a “one-shot” mode offering

potentially higher throughput.

The measured full-width half-maximum spectral resolution of the hardware prototype was found to be 0.04 nm. Although this result was short of the 0.01 nm design goal, velocity

measurements of the planet Venus resulted in an error of only 75 m/s or 0.59%. The potential beneﬁts to space situational awareness are obvious, but instruments

based on this design could ﬁnd applications in other areas of remote sensing where sub-

angstrom resolution and high throughput is required. The ability of this device to obtain a

snapshot correlation value based on an arbitrary, programmable spectral reference offers

enormous potential for chemical sensing, agriculture, and other applications of chemical

spectrophotometry.

Additional work remains. In the near term, more detailed investigation of one-shot

131 mode is needed. This should include a comprehensive analysis of one-shot calibration, and performance metrics should be compared to DCS mode. More ﬁeld testing is also needed to exercise both operational modes of the instrument, not just DCS mode. Outdoor measurements of artiﬁcial satellites, rather than a bright celestial object, will provide a better assessment of the instrument’s capability in low SNR conditions. This will require upgrading the mobile platform for outdoor service.

Two additional features are likely candidates for future development. The ﬁrst is real- ization of measurement accuracies of 10 m/s or better, with short integration times. The second is incorporation of spatial imaging, which was an original design goal because of its potential utility for space situational awareness.

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