Laser Orbital Debris Removal
Studies of Spacecraft Debris Removal Using Ground Based Lasers
Kajsa Eriksson Rosenkvist
Space Engineering, master's level (120 credits) 2019
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
Abstract
Overcrowding of the Low Earth Orbit (LEO) region is a growing problem. Decades of treating this part of space like a scrap yard has caused it to become a hazardous environment for operating satellites. At present, the largest pieces of debris are being continuously tracked and satellite operators avoid them by maneuvering their spacecrafts out of the way. This approach is not possible for pieces that are smaller than10 cm, since they are hard to detect and track as well as numerous. The exact number is not known but it is believed to be around190 000.
A number of different mitigation methods have been suggested. In this project the Laser Orbital Debris Removal (LODR) has been investigated and a basic sim- ulation model has been developed. Though many aspects have been studied, only a few have been implemented in this first version of the simulation program. The thesis has uncovered some limiting factors of the models and data that have been used to describe the physical phenomena that relate to this problem. These factors, and other suggestions, are mentioned in the chapter 5.
Though the model is far from perfected, it shows the technical feasibility of the suggested method, as well as some of the problems that need to be solved before it can be implemented. The fact that it would be possible to build a ground based LODR system, in no way assures that it is likely to occur. The political aspects of such a facility are too problematic at this day in age. How should it be operated?
Could we trust that it would not be used as a weapon? The questions are many and the answers are uncertain. For now, it seems best to focus on improving the understanding of the phenomena, the precision of the model and hope that there will come a time when this research will lead to an implementable solution.
Contents
List of Acronyms . . . iii
List of Constants . . . iii
List of Symbols . . . iv
1 Introduction 1 2 Theory 3 2.1 Laser Basics . . . 3
2.1.1 Laser Resonators . . . 3
2.1.2 Laser Beams . . . 10
2.1.3 Antennas and Telescopes . . . 15
2.1.4 Propagation . . . 19
2.2 Orbital Mechanics . . . 22
2.2.1 Object . . . 22
2.2.2 Orbit . . . 25
2.3 Interaction . . . 28
3 Model 31 3.1 Object Classes . . . 32
3.1.1 Laser . . . 32
3.1.2 Antenna . . . 33
3.1.3 Atmosphere . . . 33
3.1.4 Debris . . . 34
3.1.5 Orbit . . . 36
3.2 GUI . . . 37
3.2.1 Main Window . . . 37
3.2.2 Laser Widgets . . . 40
3.2.3 Debris Widgets . . . 43
3.2.4 Orbit Widgets . . . 44
4 Results and Discussion 49
5 Future Work 53
6 Conclusion 55
A Structural Python Files I A.1 Laser.py . . . I A.2 Antenna.py . . . II A.3 Atmosphere.py . . . II A.4 Debris.py . . . III A.5 Orbit.py . . . V
B GUI Python Files VII
B.1 Main.py . . . VII B.2 Laser_Widget.py . . . X B.3 Debris_Widget.py . . . XIV B.4 Orbit_Widget.py . . . XV
C Custom File Formats XVIII
C.1 Example .lodr File . . . XVIII C.2 Example .lcfg File . . . XIX C.3 Example .dcfg File . . . XIX C.4 Example .ocfg File . . . XX
Bibliography XXI
Index XXIII
List of Acronyms
CCD Charge Coupled Device.
cw continuous wave.
ELT Extremely Large Telescope.
GEO Geosynchronous Orbit.
gm gain medium.
GS Ground Station.
GUI Graphical User Interface.
HEO High Earth Orbit.
laser Light Amplified by Stimulated Emission of Radiation.
LEO Low Earth Orbit.
LODR Laser Orbital Debris Removal.
MEO Mid Earth Orbit.
NAN Not A Number.
TEM Transverse Electromagnetic.
UML Unified Modeling Language.
List of Constants
G Gravitational constant 6.6740831 · 10−11 [m3/kg s2]
RE Earth radius 6378 [km]
c Speed of light 299792458 [m/s]
ε Vacuum permittivity 8.85419 · 10−12 [A s/V m]
h Planck’s constant 6.6261 · 10−34 [J s]
µE Earth gravity constant 3.986 · 1014 [m3/s2]
List of Symbols
General
E energy [J]
W output energy [J]
α azimuth [rad]
β elevation [rad]
m mass [kg]
r radius [m]
t time [s]
z distance [m]
Antenna
D diameter [m]
Def f effective diameter [m]
f focal point
Mp primary mirror
Ms secondary mirror
Atmosphere
λref reference wavelength [m]
Tef f transmission efficiency
θatm atmospheric divergence angle [rad]
θbloom bloom divergence angle [rad]
θjitter jitter divergence angle [rad]
θturb turbulence divergence angle [rad]
θturbref reference turbulence divergence angle [rad]
Debris
~ag gravitational acceleration [m/s2]
Cm momentum coupling coefficient [N/W]
d diameter [m]
ηc combined efficiency factor
F~g gravitational force [N]
J~ impulse [N s]
md debris mass [kg]
ν true anomaly [rad]
o angular displacement from the equatorial plane
[rad]
~
p momentum [kg m/s]
ρ mass density [kg/m3]
~v velocity [m/s]
v speed [m/s]
Laser
Cd diameter diffraction coefficient Cdr radius diffraction coefficient
Cg gain coefficient
Cl loss coefficient
Cp pumping constant
Cs spontaneous coefficient
d0 aperture diameter [m]
ds spot size [m]
E complex amplitude
Ei energy level i [J]
frep repetition frequency [Hz]
G gain
gi gparameter i
k wavenumber [1/m]
L loss
lc length of the cavity [m]
lgm length of the gain medium [m]
λ wavelength [m]
Mi mirror number i
M2 beam quality factor
ni number of gain particles at energy level i p coordinate vector of a point
Φ fluence [J/m2]
P power [W]
q number of photons in the laser cavity
R reflectivity [%]
r0 aperture radius [m]
rc radius of curvature [m]
σ cross section area [m]
τ pulse width [s]
θ diffractrion limilted divergence angle [rad]
θef f effective divergence angle [rad]
θM2 M2divergence angle [rad]
Wp pulse energy [J]
w beam radius [m]
w0 beam waist radius [m]
z0 Rayleigh range [m]
Orbit
a semi-major axis [m]
b semi-minor axis [m]
δ angle between~r and ~z [rad]
EK kinetic energy [J]
EM mechanical energy [J]
EP potential energy [J]
ǫ eccentricity
γ angle between~v and ~r [rad]
L angular momentum [J s]
lf focal length [m]
ω argument of perigee [m]
ϕ latitudinal displacement between GS and
~r
[rad]
ra radius of apogee [m]
rp radius of perigee [m]
T period [s]
υ mean angular velocity [rad/s]
ζ angle between~v and ~z [rad]
Chapter 1
Introduction
Many of the things we take for granted, and use regularly, in today’s society require satellites to function. Data is transmitted across the globe at an incredible rate and the number of satellites is growing steadily. One might think that space is infinite and could therefore accommodate an infinite number of objects, which is true to an extent. The problem lies in the fact that, for us to make use of the object in the intended manner, it must be relatively close to the Earth, a region which is becoming increasingly crowded.
For decades, man has launched crafts into orbit, without considering how to remove them, and over time the number of unused objects have increased. In addition to this, upper stage rockets have, on multiple occasions, been deliberately detonated after disconnection, leaving a much lager amount of debris. A number of satellites have suffered a similar fate, though in most cases these explosions have not been deliberate. These objects, hereafter referred to as debris, are now causing problems, both by taking up space that is needed for new equipment, and by endangering the crafts that are currently in use. [1]
All crafts that orbit at heights close to 1000 km are now at risk. A collision with debris larger than10 cm would most likely cause irreparable harm to a satel- lite, however, the risk of such a collision is low. This is because these pieces of de- bris are continuously tracked, allowing the satellite operators to maneuver around them. The large number of debris in the1 – 10 cm size range, and the difficulty in detecting them, makes it unmanageable to keep track of them all. Despite this, a collision with one of these can cause significant damage to a satellite, or even render it inoperable. Any collision will also cause an increase in debris, further escalating the problem.
For years, this problem has been tackled by multiple projects, suggesting a wide range of solutions. Some of them include fitting satellites with protective shielding or launching a nuclear-powered debris sweeper to shoot down debris from orbit. Even though the latter of these methods would allow for interaction at a short distance, the cost of launching it into orbit and supplying it with enough energy would be enormous. There is also a risk that it might collide with the debris,
thereby creating more debris or even impairing the system.
Despite being much further from the debris, a ground based laser system has many advantages to one in orbit. Such a system can be maintained continuously and the options of power source are many. The use of adaptive optics and an arti- ficial laser guide-star can counteract the negative effects of the atmosphere signifi- cantly. A system like this has been suggested by, among others, Phipps et al.. The problem has been discussed and calculations have been performed, showing the feasibility of the idea, provided that the right technological hardware and software is available. In this thesis, the idea has taken another step towards reality.
This report highlights some of the physical concepts used for such a system to be built. First, the basic behind laser beams are treated, starting with how such a beam is made, transmitted and how it propagates and interact with matter. A brief description of orbital mechanics, including the Newton and Kepler laws which define such a trajectory, is then given, and the equations and assumptions used are presented. Following this, is a chapter which describes the model produced during the project, ranging from the overall structure used to the parameters and functions that are contained in the different sub-classes. The simulation, limitations and results are then discussed and conclusions regarding the validity of the model and outcome of the project are drawn. Questions such as; How accurate is the model?, Are the assumptions reasonable? and, Which are the main limitations?, are raised and debated before the suggestion of future work needed to increase the accuracy and reliability of the application. The thesis ends with a section drawing conclusions on the work done and the feasibility of using this as a starting ground for a more advanced simulation software in order to make an informed decision on how the orbital debris problem can be handled.
Chapter 2
Theory
To get a basic understanding of the physics that applies to this problem a short explanation of the main concepts is given in this chapter. As there are many aspects to consider, none of the concepts will be described in detail.
2.1 Laser Basics
The term laser has been used as an acronym since the 1960’s and was entered into the Oxford English Dictionary in 1976 [2]. The acronym stands for Light Amplified by Stimulated Emission of Radiation, and describes the crucial element of the system. The light is amplified while bouncing back and fourth inside a resonator.
2.1.1 Laser Resonators
In simple terms, a laser resonator can be described as an enclosed cavity with a mirror at each end. The distance between the mirrors is referred to as the length of the cavity (lc). A material, referred to as the gain medium (gm) is placed inside the cavity. Figure 2.1 shows a simple model of a laser resonator.
The two mirrors (M1andM2) have specific characteristics, which are not nec- essarily the same. In addition to shape and size each mirror surface has a reflectivity (Ri). For an ideal laserM1is perfectly reflective, i.e. R1 is100 %, and all energy is kept in the system. Though the ideal system has no lossesM2 is not perfectly reflective, i.e. R2 < 100 %. The energy that is not reflected by M2 is not a loss but represents the output energy from the laser. Another important parameter of the mirrors is the radius of curvature (rci). Even though the mirrors in Figure 2.1 are both plane, i.e. rc1 = rc2 = ∞, this is not always the case. In reality vari- ous setups are used in different systems, utilizing several combinations of convex, concave and plane mirrors. The choice of mirrors is key to getting the desired characteristics for the particular use intended for the laser.
Figure 2.1: Simplified model of a laser resonator. A gain medium of lengthlgmis placed between two mirrors (M1andM2) forming a cavity of lengthlc. It is charged by the pump energy to produce a laser beam through an aperture with diameter d0. The beam then propagates with a diffractrion limilted divergence angle (θ).
Setting up and aligning the mirrors is of utmost importance. If done correctly the result will be a highly focused and nearly unidirectional ray of high intensity light along the central axis of the gm. This medium is a material with the capabil- ity of producing a stimulated emission of light within a narrow range of a specific wavelength (λ), i.e. a near monochromatic light. A wide range of materials can be used for this, including solid-state, gas and semiconductor materials. The chosen material does not only determine theλ but also the power (P ) that can be reason- ably achieved. As the gm does not fill the cavity completely the length of the gain medium (lgm) is slightly shorter thanlc.
There are two main types of laser systems, continuous wave (cw) and pulsed systems. In the case of a cw laser the output energy (W ) is spread equally over time and the value is easily determined fromP . It is sometimes desirable to have a higher energy delivered in a shorter burst and this is exactly when a pulsed laser is used. The energy is no longer spread evenly but concentrated in shorter pulses with a specific pulse width (τ ). The number of pulses per second is specified by the repetition frequency (frep) and is limited to be less than1/τ. It is no longer of interest to know the mean energy. Instead we specify the pulse energy (Wp) which is found through (2.1). Though this equation holds true there are currently systems in which the user chooses a Wp andfrep, within specified ranges and P is then adjusted to achieve these values [3].
Wp = P
frep (2.1)
Independent of the laser type, energy needs to be added to the system in order to maintain operation. This is done through a pumping process. In this process the atoms or molecules, hereafter referred to as a gain particles, of the gm jump up
and down between different energy levels i (Ei) according to five specific energy conversions, see Figure 2.2.
(a) Absorption (b) Spontaneous Emission
(c) Stimulated Emission
(d) Photon Absorption (e) Deexcitation
Figure 2.2: The five specific energy conversions that take place in the gain medium (gm) of a laser resonator. As the gain particles transition from one energy level to another, energy is either absorbed or released in one form or another.
(a) A gain particle jumps to a higher energy level (E2) by absorbing energy∆E and moving an electron to a higher principal energy level.
E2 = E1 + ∆E
(b) A gain particle spontaneously falls to a lower energy level (E1) by releasing energy and emitting a photon with wavelength (λ).
hc
λ = E2 − E1
(c) An incident photon causes a gain particle to fall, similarly to item (b). The emitted photon has the exact sameλ as the stimulating photon.
hc
λ2 = E2 − E1 = hc λ1
(d) A gain particle jumps to a higher energy level (E2) by absorbing a photon with the exactλ required.
E2 = E1 + hc λ
(e) A gain particle falls to a lower energy level (E1) without emitting a photon.
The energy is converted to rotation- or vibration energy in the molecule, or rearranged to other electrons in the atom. This is called deexcitation.
There are different methods of pumping, including electric discharge-, optical- and diod pumping. Lasers are often classified according to the method used. An- other categorization of lasers is the number of levels they operate in. Until now the lasers have been assumed to be Two-Level lasers, in which the pumping and emission takes place in the same level gap. This is not always the case and there can be as many as four levels involved in the process. The lowest possible energy state level is called the ground state level. In this level the gain particles are at rest and have no excess energy. The pump energy lifts the gain particles to the pump levelwhich is the highest energy state level used. The photon emission takes place between the upper laser level and the lower laser level. A system that has all these individual levels is called a Four-Level laser. There are also two kinds of Three-Level lasers, in which either the pump level and upper laser level or the ground state level and the lower laser level coincide. Figure 2.3 shows the principal diagrams for these four laser categories.[4, 5]
The number of gain particles at the different energy levels i (ni) is changed by the pumping process. The active gain particles are those involved in the process, i.e.
the gain particles in the participating energy levels, and the total number of these is kept constant. In a resting gm close to all the gain particles are at their lowest possible energy level, i.e. n1≫ n2. In order to achieve laser operationn2must be high enough so that it is likely that a passing photon will cause stimulated emission.
In an optical Two-Level laser the photons used to pump gain particles fromE1 to E2 are identical to those causing stimulated emission. Due to this the best case scenario of such a laser setup is thatn2 ≈ n1. Ideally the distribution should be such that n2 ≫ n1. Having a Three-Level laser system with a separate ground state level is preferable. This third layer makes it possible to have an almost empty lower laser level which prevents photons with the desired λ from being absorbed as described in item (d). A mechanism which forwards any electrons placed inE1
by item (b) or item (c) to a lower ground state level must then exist, this is called a fast decay stage.
(a) Two-Level laser (b) Three-Level laser
(c) Three-Level laser (d) Four-Level laser
Figure 2.3: Schematics describing four different pumping processes, including two to four energy levels. The pumping raises gain particles from the ground state level to the pump level and the laser transition takes place between the upper and lower laser level.
Alternatively the third level can be a separate pumping layer above the upper laser level. Having a similar fast decay stage here allows the gain particles inE3to be transferred down to the upper laser level and thus the pumping energy can not cause them to drop back down to the ground state level. If the forwarding is fast enough the lower laser level is kept practically empty or the pumping effectively places gain particles in the upper laser level, depending on setup. Either of the Three-Level laser setups make it possible to achieve a positive population inver- sion which means thatn2 > n1. In a Four-Level laser the two fast decay stages are combined which further enhances the population inversion. Assuming that the stimulated emission is mush slower than the fast decay this type of setup will have a positive population inversion as long as the pumping rate is faster than the stim- ulated emission. A Four-Level laser requires a much lower power input to achieve laser operation than a Three-Level laser with equal conversion rates.
Another important quantity is the number of photons in the laser cavity (q).
This and the niquantities are highly interconnected and vital to the gain (G ) and
loss (L ) of the laser system. The gain is defined as the increase ofq due to stim- ulated emission, item (c). Assuming that the lower laser level decay rate is suffi- ciently high,n1 is close enough to 0 that it can be neglected. This is a reasonable assumption for many laser systems. The rate of gain (Gq) can then be estimated with a simple equation, (2.2), where the gain coefficient (Cg) is a constant depen- dent on, amongst other things, the length, cross section area (σ) and material of the gain medium. As can be seen in the equation, the rate of gain is not simply proportional ton2but toq as well, as it is due to the stimulated emission.
Gq= Cgn2q (2.2)
Similarly the decrease ofq is represented by the loss. The many processes that cause losses in the system, e.g. scattering, mirror transition and absorption, are combined into the loss coefficient (Cl). As for the rate of gain, a simple estima- tion of the rate of loss (Lq) can be computed through (2.3), however this is not dependent onn2. By combining these equations an expression for the total rate of change ofq is found, (2.4).
Lq= Clq (2.3)
dq
dt = Cgn2q − Clq (2.4)
It is not possible to fully solve this equation without an expression for the rate of change of n2, (2.5). As a stimulated emission of a photon consumes a gain particle atE2, this increase ofq corresponds to an equal decrease of n2. The spon- taneous emission of photons causes an additional decrease which is independent of q but proportional to n2 with the spontaneous coefficient (Cs). The increase due to pumping is independent of all quantities and represented by the pumping constant (Cp).
dn2
dt = −Cgn2q − Csn2+ Cp (2.5)
As stated above these equations are simple estimations and it is important to note that they do not take escaping photons or energy conversions to other levels into account, nor do they reveal how the coefficients and constant are determined.
There are no established methods for systematically solving nonlinear differential equations of this kind and no general solutions for these laser equations have been identified, but fortunately the physical aspects of the variables provide a number of well defined limiting cases. The most important of these cases is the steady state for which the change rates are both zero. By setting both (2.4) and (2.5) equal to 0 and remembering that neither of the quantities (n2,q) can be negative, expressions for threshold requirements ofn2 andCp are obtained, (2.6). Though these equations are not easily solved they do indicate a couple of important conditions for the sustainable operation of a laser, i.e. that Gq ≥ LqandCp > Cpt = n2tCs.
n2t= Cl Cg
Cpt= CsCl Cg
(2.6) The sustainability of the laser operation is not to be confused with the stability of the laser resonator. If a photon that is bouncing back and forth between the mirrors remains inside the resonator it is said to be stable. However, if the photon escapes from the cavity after a number of turns the resonator is unstable. The reason for this escape is often due to a misalignment of the mirrors, see Figure 2.4.
(a) Aligned mirrors (b) Misaligned mirrors
Figure 2.4: These two diagrams show the importance of proper alignment of the laser resonator mirrors. Even a slight misalignment can cause significant losses to the system.
A simple way to determine if this criteria is met is to use the g parameters (gi), (2.7). Each mirror in the resonator has its owngi parameter and their product is used in (2.8). If this condition is satisfied it is certain that the resonator is stable, if correctly aligned. As mentioned earlier, a system with two plane mirrors in parallel becomes unstable with the slightest misalignment and is therefor not used. It is not hard to grasp that the escaping photons in an unstable laser significantly increase the total losses of the system and that an unstable laser requires a higher gain to compensate for this. It may seem logical that all real laser systems are designed to be stable but this is not the case. Unstable resonators have qualities that make them preferable in some high-power lasers.
gi= 1 − lc
rci (2.7)
0 ≤ g1g2 ≤ 1 (2.8)
The stability condition assumes a perfectly aligned ideal system, however a practical system has impurities and output through one or both of the mirrors. One must keep in mind that none of the equations or conditions in this chapter alone can determine if the laser setup will be sustainable and that they are highly simplified approximations for a complex physical process.
2.1.2 Laser Beams
The amplified light leaves the laser resonator in the form of a laser beam. A beam consists of many rays. Even if the rays inside the resonator were perfectly unidi- rectional the output beam would not be. If the beam travels along the z-axis and the radial displacement from the axis at a given point isr(z), the slope of the beam isr′(z), see Figure 2.5. One way of understanding how the light travels is to view it as a wave. The wavefront shows the bending of the light and rays are drawn to show the direction of the light from this front. The electric field (E(p, t)) of a light wave is described by the (scalar) general wave equation, (2.9), where p is the coordinate vector of a point (x, y, z) and t is the time. Assuming that the wave is a monochromatic field, the solution to (2.9) has the form (2.10) leading to the Helmholtzequation (2.11).
Figure 2.5: As a laser beam leaves a resonator it propagates through the medium in a manner similar to a wave. The vectors on each wavefront show the direction of the light rays. At a specific distance (z) from the source the lateral displacement is r(z) which is dependent on the diffractrion limilted divergence angle (θ).
∇2E(p, t) − 1 c2
∂2
∂t2E(p, t) = 0 (2.9)
E(p, t) = E(p)e−ikct (2.10)
∇2E(p) + k2E(p) = 0 (2.11)
In these equationsE is a complex amplitude and the wavenumber k = 2π/λ. The plane wave and spherical wave solutions to (2.11) have the form (2.12) and (2.13) respectively. In the plane wave case it is assumed that the direction of the wavenumber vector is along thez-axis and for the spherical wave p = |p| is the distance from origin to the point p.
E(p) = E0eikz (2.12)
E(p) = E0
p eikp (2.13)
Neither of these solutions is a good approximation of a laser beam. The plane wave is unidirectional and has an infinite cross section and the spherical wave spreads in all directions and has a finite cross section. A better approximation of a laser is a paraxial beam. Such a beam has a small and constant slope (r′(z)), resulting in a nearly unidirectional beam with a limited cross section. The solution for this is found by combining (2.12) and (2.13) to (2.14) whereE0(p) indicates that the amplitude is not constant, i.e. that the cross section area is finite.
E(p) = E0(p)eikz (2.14)
Using this solution in the Helmholtz equation and assuming that the variation of amplitude within a wavelength inz-direction is sufficiently small and slow for the∂2/∂z2 term of the Laplacian to be neglected gives the paraxial wave equation (2.15) in which∇T is the transverse Laplacian.
∇2TE0+ i2k∂E0
∂z = 0 (2.15)
For a paraxial beam z ≫ r and r′(z) can be said to be approximately the same as the beam’sθ, (2.16). The ideal laser has a θ which is defined solely by λ, the aperture radius (r0) and a radius diffraction coefficient (Cdr), (2.17). This radius diffraction coefficient is determined by the type of beam or source used.
Two approximations that are commonly used will now be described.
r′(z) = tan θ ≈ sin θ ≈ θ (2.16)
θ = Cdrλ
r0 (2.17)
Gaussian Beam
The Gaussian beam is a paraxial beam in which the rays bend close to the point where the radial displacement is the smallest, creating a beam waist at z = 0. In Figure 2.6w0is the beam waist radius. The length of the waist region is known as the Rayleigh range (z0) and computed through (2.18). The beam radius (w) at the distancez is determined by w0 andz0 according to (2.19), from which it follows thatw(z0) = w0√
2. The radius of curvature of the wavefront at z depends on w0 andλ and can be found from z0, (2.20).
z0 = πw02
λ (2.18)
Figure 2.6: This diagram shows a Gaussian beam in the vicinity of the beam waist. In this region the rays of the beam bend, resulting in a minimum radiusw0atz = 0. The bending only occurs close to the waist and outside of the Rayleigh range (z0) the beam is considered to spread linearly.
w(z) = w0 s
1 + z2
z02 (2.19)
rc(z) = z +z02
z (2.20)
As the Gaussian beam is a paraxial beam it must satisfy the paraxial equation (2.15) and the general solution for this is (2.21), where Hn(u) is the nth-order Hermite polynomialof parameteru according to (2.22).
Emn(p) =Aw0 w(z)Hm
√ 2 x
w(z)
Hn
√ 2 y
w(z)
ei[kz−(m+n+1) arctan(z/z0)]
× eik(x2+y2)/2rc(z)e−(x2+y2)/w2(z)
(2.21)
Hn(u) = (−1)neu2 dn
dxne−x2 (2.22)
It is important to note thatw(z) and rc(z) are independent of m and n. These indices represent the transverse mode of the beam, also known as the Transverse Electromagnetic of order (m, n) (TEMmn). The mode determines the intensity pat- tern, (2.23), of the beam and Figure 2.7 shows a few low order patterns.
I(p) = cε
2 |E(p)|2 (2.23)
In the special case of the zero-order mode (TEM00) (2.21) is simplified to (2.24). In this mode the intensity pattern is one spot with radius w(z) which is preferable for the application treated in this thesis. For distances z ≫ z0 the divergence angle can be approximated by (2.25), i.e. Cdr andr0in (2.17) are1/π
andw0respectively.
E00(p) = Aw0
w(z)ei[kz−arctan(z/z0)]eik(x2+y2)/2rc(z)e−(x2+y2)/w2(z) (2.24)
(a) TEM00 (b) TEM10 (c) TEM20
(d) TEM01 (e) TEM11 (f) TEM21
(g) TEM02 (h) TEM12 (i) TEM22
Figure 2.7: These images depict nine lower order Transverse Electromagnetic (TEM) modes. The mode numbers indicate the number of intensity minimums that occur within the beam in the x and y direction respectively. As shown, these two numbers are not interconnected.
θ ≈ w(z) z ≈ w0
z0 = λ
πw0 (2.25)
Airy Disk
Another common approximation of a laser beam is that of the Airy disk. Tech- nically this is not an approximation of a beam itself but of its intensity pattern.
It originates from the diffraction of a plane wave. Huygen’s principal states that each point on a wavefront can be viewed as a point source of a spherical wave that contributes to the total wave. This principal makes it possible to estimate how the wave travels through different media and around obstacles.
Consider the simple case of two planes separated by a distancez. If the values ofE in the first plane are known those in the second plane can be determined. This is done by letting the contribution of each small area ∆A1 in the first plane be
denoted∆E according to (2.26). Assuming that z ≫ k(x21+ y21) holds true for all points in the first plane, the totalE in the second plane can be approximated by the Fraunhofer diffraction integral (2.28).
∆E(x2, y2, z) = −i λ
eikr
r E(x1, y1, 0)∆A1 (2.26) r =q(x2− x1)2+ (y2− y1)2+ z2 (2.27)
E(x2, y2, z) ≈ −ieikz
λz eik(x22+y22 )/2z
× Z Z
E(x1, y1, 0)eik(x2x1+y2y1)/z
dx1dy1
(2.28)
This assumption hold for the case of an Airy disc as it is a result of a plane wave diffracted by a circular aperture of radiusr0. Further assuming that the aperture is uniformly illuminated with a field of amplitudeE0 and that the screen around the aperture is perfectly absorbing means thatE(x1, y1, 0) = E0 on all points on the aperture and 0 everywhere else. As the aperture is circular it is advantageous to use circular coordinates instead of Cartesian. Applying this to (2.28), the integrals be- come (2.29). Recognizing that the integral overφ is a zeroth-order Bessel function of the first kind (J0), according to (2.30) withn = 0, the integral is easily solved, (2.31), and the field at plane two is given by (2.32).
Z r0
0
r1dr1 Z 2π
0
e−ikr2r1 cos(φ1−φ2)/zdφ2 (2.29)
Jn(u) = 1 2π
Z 2π
0
ei(u cos(φ)−nφ)dφ (2.30)
2π Z r0
0
J0
kr2r1 z
r1dr1= πr022J1(kr0r2/z)
kr0r2/z (2.31) E(r2, z2) =−ieikz2eikr22/2z2πr02
λz2
2J1(kr0r2/z2)
kr0r2/z2
(2.32) The Airy disc intensity pattern at distancez from the aperture is thus defined by (2.33). The radius of the central spot is found from the first r for which I(r, z) = 0. This occurs at the first zero of the J1 functions which gives r ∼= 1.22λz/2r0. Assuming that the radius changes sufficiently slow asz increases the θ of the Airy disc can be approximated by (2.34) which means that Cdr in (2.17) is0.61 in the case of an Airy disc.
I(r, z) = I0 πr02
λz
![2J1(2πr0r/λz)]2
(2πr0r/λz)2 (2.33) θ = arctan
r z
≈ r
z = 0.61λ
r0 (2.34)
Beam Quality
The beams described above are ideal cases which means that their divergence an- gles are caused solely by diffraction. In reality imperfections in the construction, such as irregularities in the mirrors or a slight misalignment, cause addition of some higher TEM modes or random phase variations across the output wavefront which leads to increased divergence. As the divergence angle increases so does the spot size, and since the energy is then spread across a larger area the intensity decreases.
These effects of the imperfections need to be taken into account. A commonly used measure for this is the standard beam quality factor (M2), which is specified by the manufacturer. It is measured by comparing the output of a constructed laser to the ideal theoretical version of the same setup and wavelength. The value ofM2 has the lower limit one, which would mean a perfectly constructed laser. Depending on gain medium and setup some lasers have values close to this whereas others have values around ten or higher. The M2 factor affects the divergence resulting in a beam withM2divergence angle (θM2).
θM2 = M2Cdr λ
r0 (2.35)
It is possible to improve the divergence angle by applying a spatial filer, see Figure 2.8. As shown the beam is focused on an aperture which is slightly larger that the focused beam while deviating rays are focused on the screen surrounding the aperture. After passing this pinhole the second lens is used to realign the rays to a paraxial beam. This is often done simultaneously with beam expansion.
Figure 2.8: This diagram shows the concept behind a spatial filter. Correctly aligned rays are focused on a small aperture in an absorbing screen, passed through the aperture and finally realigned by a second lens. Any rays with deviating direction or wavelength (λ) are obstructed by the screen and thus removed from the beam.
2.1.3 Antennas and Telescopes
In order to transmit the beam over long distances a well designed antenna is re- quired. An optical antenna and a telescope are basically the same, except for that a telescope is used to look at objects far away. Both are constructed using either re- fractive or reflective optical elements, see Figure 2.9. As the size of these structures is increasing, Extremely Large Telescopes (ELTs) have diameters of20–42 m, re- flective systems are commonly preferred. This is mainly due to the physical prop- erties of lenses in comparison to mirrors. In order for a large lens to have the same
focal point as a mirror, it has to be thick. Light is then absorbed by the material in the lens and its excess mass leads to an increase in gravity deflection. Contrary to reflection, the refraction angle is dependent on λ, which significantly restricts the bandwidth of a system. Another benefit of mirrors is that they only have one surface which needs to be polished whereas a lens has two.
(a) Refractive Telescope (b) Reflective Telescope of Cassegrain design.
Figure 2.9: The two basic techniques for telescopes used in the optical region.
One of the most commonly used designs of modern telescopes is the Cassegrain design, shown in Figure 2.9b. This design classically has a paraboloidal primary mirror (Mp) and a hyperboloidal secondary mirror (Ms) but today telescopes using a concaveMp a convexMsare often designated as Cassegrain telescopes. In this setup the focal point (f ) is often behind Mp. As for all asymmetric telescopes with two or more mirrors theMswill cause an obstruction for the center of the ray. This obstruction is especially small in the Cassegrain design, hence its frequent use.
The size of a single mirror is limited by the difficulty to make, transport and handle a large mirror. When the diameter (D) reaches approximately 8 m the mir- ror is usually split into smaller segments to simplify the process, however the fit- ting of these segments to an underlying structure is no easy task. Regardless of the number of segments, the reflective surface must be precise enough not to cause distortions to the wave. The required precision is inversely proportional to the λ used, meaning that longer wavelengths are preferable as the precision can then be lower. Due to gravitational and wind forces, among other things, the high precision of the surface can not be kept without continuous adjustments.
Active optics is an automated system used to control the form of a single pri- mary mirror, allowing it to be larger and thinner whilst still keeping its shape. A wavefront sensor is placed at the focal point of the telescope and analyzes the image quality at certain intervals. This information is interpreted by a computer, using ei- ther an open-loop system with look-up tables or a low-bandwidth closed loop with a guide star system, and used to determine the support forces needed for each ac-
tuator. This system was originally intended to be used to compensate aberrations from the telescope itself but it can also correct some slow atmospheric aberrations.
In the future active control could be used to compensate for any aberration of up to
≈ 0.05 Hz that typically has a high amplitude.
In a telescope where the primary mirror is comprised by multiple segments a similar system is called a segment control system. It controls the tip, tilt and piston of each individual segment to form one large, smooth surface area. Each segment needs three actuators to control the three degrees of freedom. Many edge sensors, each with an accuracy better than 20 nm, are used to detect offsets between the segments. The positioning error must be in the order of10 nm or less to achieve a sufficiently precis surface. This is made more difficult due to the fact that the segments are often not rotationally symmetric but may have off-axis paraboloids or hyperboloids. A wavefront sensor can be used to measure the global radius of curvature. If a segment has a radius of curvature that differs significantly from its nominal value it will not match the form of the global mirror resulting in scalloping.
Even when using active optics or segment control aberrations that are fast vary- ing are not corrected. One system that does this is adaptive optics, not to be con- fused with active optics. It was originally designed to compensate for aberration from the atmosphere but also corrects for fast telescope aberration, from about 0.05 Hz. Unlike the previous systems it does not control the main mirrors of the telescope, instead it has its own dedicated mirrors. Figure 2.10 shows a typical setup of this system with its four main parts.
Figure 2.10: A schematic representation of a simple adaptive optics system. The Tip/Tilt and Deformable mirror are designated compensating mirrors regulated by a control system in order to reconstruct a plane wavefront.
Firstly the wave is reflected by the compensating mirrors. This part often uses
two separate mirrors in combination. The first mirror is a slow tip/tilt mirror which has a large stroke and is used to compensate for tip/tilt phase errors. After this there is a fast deformable mirror which has a smaller stroke and is used to compensate for higher order errors, including phase variations introduced by the atmosphere and the telescope. A thin flexible mirror with actuators at specific points is commonly used here. The number of actuators and their positioning determines the upper spatial frequency that is correctable. The deformation patterns corresponding to a single actuator determine the ability to remove certain aberrations. For longer λ it is possible to achieve nearly diffraction limited imaging whereas shorter λ can have partial compensation. The length of the stroke decides the maximum amplitudes that can be fully corrected whereas temporal characteristics are set by the bandwidth, which can be100 Hz or more.
Once the wave has passed the compensating mirrors a beam splitter is used to relay the image to the science focus but also to the wavefront sensor. This compo- nent measures the shape wavefront. To do this it requires a bright guide star near the object. There are few natural light sourced on the sky that are bright enough to be used for this, thankfully artificial ones can be generated at approximately 90 km altitude by aiming lasers in the desired direction. The wavefront phase er- ror is converted to intensity variations and recorded by a focal plane camera, e.g.
a Charge Coupled Device (CCD). Parameters such as read-out noise, integration time and number of pixels per sub image affect the sensitivity of the sensor and its temporal behavior. To avoid aliasing it is important to adapt the temporal sampling interval to the temporal characteristics of the atmosphere and the bandwidth of the deformable mirror.
The information provided by the wavefront sensor is passed to the wavefront reconstructor and controller. This unit uses the information to reconstruct the wavefront error and from the shape of it determines the actuator command errors.
These errors are then given as input to a controller, often a PI-controller, which computes the new command signals to the actuators. The accuracy of this stage might cause significant delays in a large telescope and it is therefor necessary to make a trade-off between accuracy and speed.
If all the components of the adaptive optics operate adequately the quality of the wave passed from the beam splitter to the science focus should be sufficient.
There are adaptive optics systems of different configurations, using one or more deformable mirrors in combination with natural or artificial guide star. These dif- ferent configurations have different benefits and drawbacks and it is vital to choose a setup that is best suited for the particular telescope.
The telescope also needs to be dirigible, in order to track and follow detected debris. With modern technology it is easy to compute and define position and movement in any orthogonal coordinate system. Therefor most modern telescopes now use azimuth (α) and elevation (β) to indicate the angle along and above the horizon, respectively, as these are intuitive to understand. Most telescopes use actuators to turn the mirror in the desired direction. However, there are some tele-
scopes which use an un-uniform surface with a large diameter asMp. The direction and effective diameter (Def f), i.e. the illuminated diameter, is then controlled by changing the position ofMsso that different parts ofMpare used.
2.1.4 Propagation
Even with the best possible laser and antenna setup, the laser beam will still be affected by the media it travels through. The effects depend on the properties of the media, which in turn are connected to the wavelength, and the distance the wave has to travel. In space science the media is the earths atmosphere in which the signal is affected through composition, clouds, rain, and interaction with objects like insects or birds. In the case of a ground based laser system and orbital debris, the media can not be changed. Instead it is vital to choose a wavelength which is resilient in this media and minimize the distance as much as possible. Selecting a location for the ground station at a high altitude has two advantages. First, the distance to the debris is lowered, and second, the effects that the lower atmosphere has on the laser beam are partly avoided. This second advantage is due to the differences in the atmospheric layers. Phipps et al. have suggested the Uhuru site on Kilimanjaro which is at an altitude of 5888 m1, thus significantly diminishing the effects of the lowest layer. In addition to the high altitude, the Uhuru peak is situated near the equator, and will thus be passed by almost all orbiting debris [1].
Each layer is referred to as a sphere with an upper boundary called a pause. The lowest layer is called the troposphere and it reaches from the surface to about10–
16 km. It has a roughly uniform composition of 78 % N2,21 % O2,1 % Ar fused with trace amounts of H2O, O3, CO2and other elements. In total about80 % of the total mass of the atmosphere is in this layer. Another significant characteristic of the troposphere is that the temperature decreases with higher altitude, generally at a rate of about6.5◦C/km. The tropopause separates this region from the stratosphere which extends to about55 km. Unlike in the troposphere the temperature here is increased due to absorption of UV rays resulting in a nearly isothermal region.
Above the stratosphere is the mesosphere. In this region, which reaches al- titudes of approximately80–85 km, the temperature again decreases with higher altitudes, dropping as low as around180 K at the mesopause. As this layer is sit- uated above the highest reach of balloons and below the lowest reach of satellites, it is the least studies. It is however known to have extremely strong winds around 65 km altitude. The fourth layer is a high temperature region called the thermo- sphere. Here O3 and O2 molecules are rapidly broken down to atomic oxygen.
This is the most absorbing layer and the temperature is increased by the absorption of UV rays and quickly rises with higher altitude. The modeling of these layers is not simple but there are a number of standard models, each with different benefits and drawbacks.[7, 8]
1The altitude of the Uhuru peak is of some dispute as the measurement results differ. The altitude 5888 mis a fairly recent measurement and thus the one chosen for this thesis, [6]
There are four main mechanisms that affect a signal propagating through the atmosphere. Gases with lossy properties absorb some of the signal energy and converts it to heat, as described in item (d) and item (e) on page 6. The magnitude of the absorption is a direct function of particle density in the gas and increase with higher frequencies and longer distance. Most attenuation of the signal strength is caused by absorption and scattering. Scattering is the reflection of rays in the signal away from the intended direction. This can occur either off of particles or the boundary of a volume of particles, e.g. a cloud, but also off of larger objects such as dust, insects or birds. Depending on the surface scattering is divided into particle and volumetric scattering. Normal scattering increases with higher frequencies and larger particles as the particle needs to be bigger than the wavelength for the ray to scatter, however that is not the case for Rayleigh scattering which is common for signals in the optical range.
Attenuation is not the only issue with propagation through the atmosphere, the general direction of the beam is also affected. Fluctuations in pressure and tem- perature in the different layers of the atmosphere cause variation in the refractive index. As mentioned in subsection 2.1.3 the refraction angle is different for differ- ent wavelengths. In fact the index is inversely dependent on the phase velocity, i.e.
the speed the wave travels with, and thus varies less for higher frequencies. Though it does not cause a vast change in the direction of an optical beam, it can be sig- nificant enough if the target is small. Another mechanism that affects the direction of the beam is turbulence. This is the irregular and seemingly random fluctuation of the speed and direction of air currents, both in time and space. It can induce fluctuations in refractive index, as in the chase of the blurry, dancing area over a heat source, e.g. a hot road or a toaster. It is also what can be felt as bumps on a flight, though then it is through wind eddies. Similarly to how water flows around rocks in a stream, the air swirls around in eddies. It is caused by the atmosphere being unevenly heated by solar radiation. The air is then mixed through wind and convection to form what can be regarded as a mixture of eddies of all scales and sizes, approximately in the range 1–100 m. Contrary to what one might expect, turbulence is the most severe on a clear, hot and humid day. It is not a very well understood phenomenon and requires vast simplification to be formulated.
The manner in which the atmospheric turbulence affects the beam is deter- mined by the size of the eddy. Eddies that are smaller than the beam diameter lead to phase fluctuations in the wavefront. These fluctuations cause the intensity dis- tribution to broaden, i.e. beam spreading. The amount of spreading is defined by the turbulence divergence angle (θturb), estimated according to (2.36). Using1 µm as reference wavelength (λref), the turbulence reference angleθturbref is9.0 µrad for propagation from sea level, and 2.0 µrad from an altitude of 6 km, to space [1].
This is part of the advantage of placing the laser at a high altitude.
θturb = θrefturb λref λ
!0.2
(2.36)
If the eddies are larger than the beam diameter they will cause the well known, random wandering of the beam spot often referred to as “jitter”. This is when the spot seems to be dancing around its intended position. The wandering increases the difficulty of aiming the beam. For slow systems, where this jitter is rapid in comparison to the laser pulse duration, this contributes to the beam spread accord- ing to the jitter divergence angle (θjitter). Turbulence can also increase the thermal blooming. This is a relatively slow phenomenon caused by the atmosphere be- ing heated by the beam. As the gases along the route absorb part of the energy thermal distortion arises leading to an added spread called the bloom divergence angle (θbloom).
These atmospheric effects combine into the atmospheric divergence ang- le (θatm), (2.37), and the resulting effective divergence angle (θef f) of a beam which has propagated through the atmosphere is given by (2.38). In this project, only the turbulence spread has been accounted for as it is assumed that the interac- tion between laser and debris is fast enough andτ is short enough, for θjitter and θbloomto be neglected.
θatm2 = θturb2+ θjitter2+ θbloom2 (2.37)
θef f2 = θM22+ θatm2 (2.38) The diameter of the diffraction limited illuminated spot or spot size (ds) at dis- tancez from an antenna with the effective diameter Def f is defined by (2.40). Here the diameter diffraction coefficient (Cd) is4Cdr, i.e.4/πand2.44 for the Gaussian beam and the Airy disk respectively. Taking the beam quality and atmospheric spreading into account, i.e. usingθef f, gives (2.40).
dslim= Cdλz
Def f = 2Cdrλ
ref f z = 2θz (2.39)
ds= 2θef fz (2.40)
The fluence (Φ) of a laser beam is a measure of the energy delivered per unit area, (2.41). As discussed above the beam is attenuated in by a number of mecha- nisms in the atmosphere. To simplify the model, transmission efficiency (Tef f) is used to account for all of them. Φ is also affected by θef f, as a larger divergence angle will result inWpspreading over a larger area. These equations are key to this project.[1, 5, 7–9]
Φ = WpTef f
πθef f2z2 (2.41)
2.2 Orbital Mechanics
Orbital mechanics is the physical principles that explain how an object orbits a mass and it has its roots in kinematics. In this section some laws of physics will be used to describe this first from the object point of view and then from the orbit point of view, giving some defining parameters for the shape of the orbit.
2.2.1 Object
An object in orbit can be thought of as a projectile on which external forces act.
The trajectory of the projectile follows Newton’s three laws of motion.
1st An object continues in a state of rest or in a state of motion at a constant velocity, unless compelled to change that state by a net force.
2nd When a net external forcePF acts on an object of mass m, the acceleration~
~a that results in directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force.
~a = PF~
m or XF = m~a~
3rd When a body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body.
In addition to these, Newton also defined a law for the interaction of two bodies that are not in physical contact with each other. This is known as Newton’s law of gravity which states that:
Every particle in the universe exerts an attractive force on every other parti- cle. For two particles that have masses m1 andm2 and are separated by a distancer, the force that each exerts on the other is directed along the line joining the particles and has a magnitude given by
F = Gm1m2
r2 (2.42)
Imagine an isolated two body system with a projectile traveling over the surface of the Earth. According to the laws presented above the projectile is subjected to a gravitational force directed towards the Earth causing it to accelerate towards the surface, see Figure 2.11. Assuming that the projectile is traveling in perfect vacuum, the only force acting on it is the gravitational force ( ~Fg). This force is directed towards the center of the Earth and its magnitude is found by using the Earth mass (mE), debris mass (md) and the radius (r) between the two bodies’
center of mass in (2.42) whereG is the gravitational constant resulting in (2.43).
