NIAC Phase I Final Report “Efficient Direct Conversion of Sunlight to Coherent Light at High

Average Power in Space” Richard L. Fork, Rustin L. Laycock, Dane J. Phillips, Wesley W. Walker, Spencer T. Cole

University of Alabama in Huntsville, Huntsville AL 35899 April 29, 2005

Description of the effort: We address technology for an optically based power infrastructure in the region extending from Earth to, and including, the lunar surface. The constraints imposed by physical laws are distinguished from constraints imposed by need for maturation of technology. We conclude such an infrastructure is feasible, in a prototypical sense, within 10-40 years. We develop a design for a scalable 100 kW solar pumped solid state laser power oscillator, located, e.g., near a lunar pole, supportive of such an infrastructure. Applications to exploration, alternative energy for Earth and space, and to defense of Earth, as from asteroid impact, are examined.

1.0 Introduction

The possibility of abundant safe alternative energy for Earth [1], the need to defend Earth, as from collisions with near Earth objects [2], and the desirability of exploring space are becoming more fully appreciated. In general these goals call for a “sustainable, affordable, and flexible” energy rich environment in Earth-Moon space [3] extending from Earth to our moon, including the lunar surface. We consider here fundamental physical laws and engineering issues as they relate to developing a prototypical optically based power infrastructure supportive of this region of space. The physical properties permit scaling in the long term to regions beyond Earth-Moon space.

1.1 Constraints set by physical laws vs. constraints set by need for technology maturation

We seek to distinguish constraints set by fundamental physical laws from constraints set by the need for maturation of technology. A useful technology must, of course, be allowed by fundamental physical laws, but may also be challenging as regards rapid maturation. We consequently consider advanced concepts and technologies that appear valuable, but which may take 10-40 years to develop [4]. We are particularly interested in advances that may be enabled by evolving technical capabilities, such as by nanostructured interfaces.

1.2 Fundamental physical laws allow a power infrastructure based on coherent light in Earth-moon space

In essence we seek a power infrastructure for Earth-Moon space that has properties similar to our existing electrically based terrestrial power infrastructure, but which serves Earth-Moon space and uses coherent light as a means of distributing power [5]. The goal is power in the needed form, as e.g., electrical, in the needed amount virtually anywhere and virtually anytime in Earth-Moon space. We conclude that an optically based power infrastructure in this region of space is allowed by fundamental physical laws and that a prototypical version might be realized within a 10-40 year time period.

1.3 Sunlight alone is not adequate, but can be transformed at high average power to coherent light

This proposed optical power infrastructure differs from deriving power from sunlight by use of photovoltaics. This latter approach is invaluable, but does not offer the amount of power or the accessibility of power virtually anywhere and virtually any time that is necessary to provide a “sustainable, affordable, and flexible” energy rich environment in Earth-moon space.

Transmission of power as by redirecting sunlight through the use of linear optical systems is not an adequate means of redistributing power in space. The LaGrange, or optical, invariant governing the imaging of light instructs us that fundamental physical laws limit the ability to concentrate sunlight to the high intensities needed for many applications. The optical invariant also limits the ability to redirect sunlight in significant amounts with adequate precision over the long distances characteristic of Earth-moon space [6].

The broad spectrum (~0.4 - 0.7 microns) of sunlight, while not preventing efficient transformation of solar power to alternate forms of energy from a fundamental point of view, complicates the practical task of converting solar power efficiently into alternative forms of power [3]. Coherent light can be produced as nearly monochromatic light. Monochromatic coherent light appears to be especially attractive as a means of delivering power over the long

distances characteristic of Earth-Moon space and transforming that optical power to electrical power at high efficiency in a relatively simple system.

1.4 Lowest order Gaussian mode is the most favored form of optical power in space

The lowest order Gaussian mode of free space is the simplest solution of the paraxial Helmholtz equation, which is most relevant to the propagation of optical beams over the long distances in space. This lowest order mode can be propagated with the maximum precision and the minimum aperture size physically allowed for optical systems [7]. We consequently seek to transform sunlight into this particular eigenmode of free space at high average power with reasonable efficiency.

2.0 Capabilities and strategies for transforming sunlight to coherent light at high average power

The principal enabling task for an optical power infrastructure is transformation of sunlight to coherent light at high average power. In general, we seek the simplest and most direct means of accomplishing this goal allowed by physical laws. This places a significant burden on maturation of technology, but allows us to focus on the strategies that could, in the long term, be the best strategies.

2.1 Optical switching point of view as providing an estimate of performance allowed by physical laws

As a means of identifying fundamental limits on efficiency we take an optical switching theory point of view [8]. We view the transformation of sunlight to coherent light as switching an optical quantum from a state in the optical field of our Sun in Earth-Moon space to a state in the optical field of a lowest order Gaussian mode of a particular near confocal resonator. The efficient coupling of these photons in the laser oscillator resonator into a given lowest order Gaussian mode of free space is a relatively simple exercise in linear optics.

We regard transmission through the vacuum of space as lossless since no switching process is involved. We view the transformation of the received quantum into an alternate form of energy, such as an electron-hole pair in a photovoltaic material, also as a switching process. Switching theory identifies the minimum amount of waste heat that must be generated in a switching event as k·T·ln2. Here, k is Boltzmann’s constant and T is the temperature of the switching material.

The amount of energy switched in transforming an optical quantum of frequency ν from one state to another state is hν, where h is Planck’s constant. We take the maximum possible efficiency as 1-kTln2/hν [8]. For T near 300K and a quantum of optical energy, e.g., wavelength of 0.5 microns, the maximum switching efficiency is >97%. The process of transforming sunlight to coherent power, transmitting the coherent power, and transforming the received power to electrical power, e.g., could ideally access efficiencies of 1-kT2ln2/hν, 95% or better using switching media at 300K. Lower temperature and higher quantum energy allow higher efficiencies. The point for this discussion is that there is substantial margin for any given optical system to depart substantially from the maximum efficiency allowed by fundamental laws and still be of practical interest.

In general, we seek to maximize the efficiency with which received coherent light is transformed to electrical power, but to not be particularly concerned about the efficiency with which the total bandwidth of sunlight is transformed to coherent light. The receiver efficiency is critical since heat in the receiving system will often be a major concern. The efficiency with which the total solar spectrum is transformed to coherent light is less critical such the portion of sunlight that is not useful can be removed in advance by spectral filtering and the total amount of solar power available in near Earth space is billions of terawatts. We do care about efficient use of the solar pump power that is delivered to the laser gain medium since generation of any unnecessary waste heat within the laser oscillator is almost always undesirable.

2.2 Solar pumped solid state laser power oscillator

We take as a model technology a solar pumped solid state laser power oscillator operating selectively in the lowest order Gaussian mode of a near confocal resonator. Such a solar pumped laser oscillator is the simplest most direct means of switching a quantum of sunlight to quantum of the lowest order Gaussian mode of free space. The solid state medium tends to be preferred for space where use of consumables is undesirable and a capability for long term use with minimal need for replacement is essential. Here power oscillator means simply a laser oscillator that operates at high average power.

In general, it is challenging to achieve high average power in a highly coherent spatial mode of a laser oscillator. Such goals are, however, consistent with fundamental physical laws. Given our strategy of seeking the simplest most direct strategies we consider the design and technology maturation requirements that may be needed for a solar pumped laser power oscillator that can generate high average power in a lowest order Gaussian mode.

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2.3 Four areas where technology maturation appears most needed

We identify four areas where technology maturation appears both necessary and achievable: (1) concentrating solar pump power at a power density per unit area integrated over the length of the gain medium approximating the saturation intensity of the laser transition in a spatial distribution approximating a lowest order Gaussian mode having a mean cross sectional area such that the product of the lowest order mode area and the saturation intensity approximate the required total power; (2) removing waste heat from this laser oscillator while avoiding the levels of thermal lensing, thermal stress induced birefringence, and thermal shock that prevent high beam quality, e.g. we seek an approximately diffraction limited output beam, negligible thermal lensing, negligible loss due to stress induced birefringence, and no thermally induced fracture of the laser medium; (3) achieving selective laser oscillation in an adequately large area lowest order Gaussian mode of a near confocal resonator of practical length; (4) transmitting the generated power over distances of several hundred megameters or more at high efficiency, e.g. >80%, to a useful alternative form of power, such as electrical power.

2.4 Key parameter is power density per unit area integrated over the length of the gain region

The key parameter is the solar pump power density per unit area integrated over the length of the gain medium. The point is that we are dealing with a highly non-equilibrium and nonlinear system. The laser field will collect the pump power in the excited gain medium as the laser field propagates through that gain medium. The important need is to collect sufficient power per unit area to enable the intraresonator optical field intensity to substantially exceed the saturation intensity of the laser transition.

The goal of the solar pump design is not achieving high power density per unit area at a given location, but rather achieving a power density per unit area, integrated over the length of the gain medium, that roughly approximates the saturation intensity for the laser transition. The saturation intensity is /sI hν στ= . Here σ is the cross section for the laser transition, ν is the laser frequency, and τ is the lifetime of the upper laser level.

Too low an intraresonator intensity will yield inefficient stimulated emission. Too high an intraresonator intensity will reduce the gain for the laser oscillator to unacceptably low levels. The intraresonator intensity will vary some from the integrated pump power density per unit area, depending on the resonator finesse, but typical practical concerns will favor an integrated pump power density that approximates the saturation intensity of the laser transition.

The saturation intensity for one of the favored transitions for solar pumping, Nd:YAG at 1.06 microns, is ~0.8-2.7 kW/cm2, (depending on doping and temperature) [9]. Another useful laser transition is that in Ti:sapphire near 0.8 microns which has Is~ 155 kW/cm2. We will take ~ 60 kW/cm2 as a measure of the solar pump power density per unit area integrated over the length of the gain material available to be transformed to coherent optical emission for this current discussion. A pump intensity that accesses this intensity will typically yield a single pass gain in the amplifying material that is adequate for laser oscillator action in a relatively high finesse resonator.

sI

3.0 Lowest order Gaussian mode of a near confocal resonator of practical dimensions

We seek the lowest order Gaussian mode of a near confocal resonator. The near confocal resonator is one of the most stable resonators against small misalignments of the mirrors forming the resonator. The lowest order Gaussian mode is the simplest solution of the paraxial Helmholtz equation which best describes the long slender optical beams which appear to be the preferred means of transmitting optical power in space.

Once generated in the laser resonator this lowest order Gaussian mode can be coupled relatively easily into the surrounding free space, expanded to the appropriate aperture and directed over long distances to a specific location using a linear optical system. The pointing precision and aperture size will be highly demanding for the distances in Earth-moon space. We note that current research is addressing large optical apertures for space applications, e.g. 150 m diameter optics [10], and precision pointing. We do not attempt to address these latter topics further in this current report.

3.1 Power extracted in the lowest order Gaussian mode for a given mode area

The power that can be extracted from a laser oscillator where wm is the mean radius of the lowest order Gaussian mode in the laser resonator and Io is the mean power per unit area that is extracted from the gain medium and coupled out of the resonator in the lowest order Gaussian mode is Po ~ 2

o mI wπ . For our nominal goal of 100 kW total power produced from the laser oscillator and ~ 30 kW/cm2, the mean radius of the lowest order mode must be wm ~ 1.03 cm. This is a relatively large radius for a confocal mode.

oI

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3.2 Problem of obtaining a near confocal resonator of practical length

The mirror spacing for the most commonly used confocal resonator, a symmetric confocal resonator is where λ is the laser wavelength and wo is the radius at the beam waist of the lowest order Gaussian

mode. We take this as an approximate estimate of the mean radius of the mode. For wo = 1.03 cm and λ=0.8 microns, as for Ti:sapphire, the length of a symmetric confocal resonator would be 405 m. This will typically be too long to be of practical interest. Below we discuss strategies for achieving the large radius and confocal character while still retaining a practical resonator length.

20 o2z w /= π λ

4.0 Solar concentrator design

We have an estimate of the pump power density per unit area integrated over the gain region needed for efficient transformation of solar power to coherent light. For an assumed quantum efficiency of 50% we need ~ 60 Watts/cm2 of integrated solar pump power per unit area. This is a power density per unit area well in excess of the intensity at the surface of the Sun. We also have an estimate of the cross sectional area of the pumped region required to achieve a given total power in a lowest order Gaussian mode. We can consequently explore design strategies for a solar concentrator that introduces solar power at the required density into a spatial region having the geometry of the lowest order Gaussian mode and the desired radius.

We have used ray tracing software, Advanced Systems Analysis Program (ASAP), to design a useful family of solar concentrators. This leads to solar concentrator designs, Figs. 1-4, that differ from previously reported solar concentrators. The principal distinction occurs in that we seek to maximize the integrated solar pump power density per unit area in a region having the geometry of the lowest order Gaussian mode having the needed cross sectional area. We only need to excite some portion of the gain path measured along the optical axis since the laser mode will sweep out the available gain in the optical path within the laser oscillator resonator.

4.1 Multiple non-imaging concentrators, tapered ducts and a spatially configured gain material

We illustrate one design we have identified in Fig. 1 composed of a titanium doped sapphire core and a surrounding structure composed of undoped sapphire. The core region (gold rod shaped region extending horizontally along the optical axis in the center of the figure) is simulated as doped with the titanium ions distributed with a concentration that decreases in the outward radial direction so as to approximate the Gaussian distribution of the lowest order Gaussian mode of the laser oscillator. This core region is coaxial with two tapered ducts that meet at the center of the structure. Non-imaging concentrators couple incident sunlight, focused by imaging optics into the large aperture end of the non-imaging concentrators, into the tapered duct.

Fig. 1 Wire frame image of a computer designed solar concentrator. The different colors indicate different parts of the concentrator and are not spectrally significant. The gold rod shaped region extending along the axis of the concentrator is doped with the laser gain material. Tapered ducts guide the solar pump power in the general direction of the optical axis. The incident solar rays are concentrated by the 16 cone shape non-imaging concentrators, pass multiple times through the titanium doped region and are absorbed with ~ 83% probability.

For the particular Ti:sapphire/sapphire solar concentrator shown in Fig. 1 the length of the gain region is 2m, the

radius of the gain region is 1.2 cm, the mean coefficient of absorption of the Ti:sapphire along the optical axis is α = 2 cm-1, The assumed percentage of useful sunlight (assumed to be pre-selected as by dichroic filters prior to imaging

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the incident sunlight into the concentrators) is 20%. The radius of the effective area normal to the incident sunlight of the collecting mirrors that direct the sunlight into the non-imaging concentrators is 4.8 m. Pump intensity=60kW/cm2, Pump density=300W/cm3, Laser output and power=135kW. Fig. 2 illustrates a several representative ray paths of incident solar rays that enter the non-imaging concentrator cones, pass into a section of the tapered duct, and then carry out a complex path that tends to favor multiple traversals of the doped core region in the center of the tapered ducts. These rays are absorbed with ~80% probability in the doped core region. A total of sixteen non-imaging concentrators (eight at each end of the structure) deliver concentrated solar pump power (filtered to exclude portions of the solar spectrum that are not useful for pumping) to these tapered ducts

4.2 Axial view of solar concentrator

The power of this strategy to couple substantial power into the spatially distributed gain region at the required density per unit mean cross sectional area is illustrated in axial views of the concentrator structure shown in Fig. 3. The view is as that of an observer looking along the optical axis from the mid-point of the structure toward one end of the concentrator. Each concentrator introduces solar pump power at the opening where solar power is coupled into the tapered duct at a power density slightly less, e.g., 90% of maximum, than that of an optimized non-imaging point solar concentrator.

The maximum intensity at the exit to an individual concentrator is not sought because we find it advantageous as regards coupling into the doped gain region to have slightly reduced angular divergence of the pump light as it enters the tapered duct. This does not compromise the efficiency of coupling solar power into the duct but tends to give the rays trajectories that cause a large number of traversals of the doped gain region and hence more efficient absorption. We find that this excitation strategy also avoids a tendency of the incident rays in the ducts to spiral around the doped gain region. The latter process contributes to the less efficient excitation obtained on using one single large concentrator to deliver the solar pump power to the doped gain region.

Fig. 2. Two axial views at different magnifications of the concentrator. The different colors distinguish features which are treated as individual objects by the software and are not spectrally significant. The orange disk in the centers of the figures identifies the doped core region.

The trajectories of the rays coupled into the tapered ducts cause the power coupled into the ducts from the non-imaging concentrators to be coupled with high probability, ~80%, into the doped core region. Adjusting the various parameters of the structure yields a relatively uniform excitation along the optical axis. Since the exit aperture to the individual conical non-imaging concentrators is comparable to that of an optimized non-imaging concentrator and the doped core region area is comparable to that of the exit aperture to each concentrator the net concentration of power in the doped core region integrated over the length of the core region is approximately 16 times that of the power density at the exit to each concentrator.

We have estimated some values for this 16 cone solar concentrator structure. We assume that 20% of the solar spectrum is useful for optical pumping and that 50% of this pump power is transformed into useful laser emission. Some of the key parameters are: overall output power of 123 kW; pump intensity at the exit to the individual concentrators is 0.885 kW/cm2; pump intensity in the core is 7.7 kWcm-2; the pump power density per unit volume in the core is 0.077 kW/cm3; and the heat left in the core is 0.0385 kW/cm3. The cross sectional area of the core is 15.97 cm2. The core radius is 2.25 cm. The radius of the individual collecting mirror area, as viewed from the sun, is 6.5 m. If we assumed a more efficient use of the solar spectrum, such as roughly 50% representative of recently reported work the radius of the individual collecting mirror would be reduced to 4.1 m.

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3.3 Collecting mirrors use conventional imaging optics The relatively large mirrors that collect the incident sunlight and redirect and focus that sunlight into the non-imaging concentrators use conventional imaging optics. We show a set of such mirrors for the simpler case of 8 non-imaging concentrators in Fig. 4. the output power from this system would be less than for the sixteen concentrator case. The view in Fig.4 is that of an observer looking from the direction of the solar illumination. The particular structure illustrated consists of flat turning mirrors which appear circular from the perspective of the observer, but are actually elliptical in outline and lenses designed to optimally focus each beam of incident sunlight into a particular one of the eight non-imaging concentrators. Our simulation program indicates that the pointing precision of the directing optics needs to be better than ~ 1± for efficient coupling into the gain medium.

36 m

Fig. 3. Collecting optics (note scale change from preceding figures) for a solar concentrator using 8 individual non-imaging concentrators. The white circular regions represent flat mirrors as perceived by an observer looking in the same direction as the incident solar illumination. The gray elliptical regions represent focusing lenses. The solar concentrator assembly appears in the center of the figure.

This collecting strategy offers the advantage that each of the redirecting flats and the associated lens can be specifically adjusted to optimize the direction and focusing of the incident sunlight into a given concentrator. A single reflecting element that both redirected and focused the sunlight into a given concentrator could also be used. The design of the optical element becomes more demanding; however, the simplification of using a single element could offset the more need for a more complex design, such as using an off-axis parabolic reflector.

The point for this discussion is that the solar concentrator needs to provide sufficient power density per unit area integrated over the length of the gain medium for the laser to access oscillation threshold and efficient operation; however, once that goal is achieved the non-linear non-equilibrium nature of the system determines the nature of the optical emission. The key parameters we will be concerned with are the net power per unit area produced in the laser field and the mean cross sectional area A00 of the lowest order Gaussian mode of the laser oscillator. The total power produced by the laser oscillator in this lowest order mode will be will be .

ALP

AL 00P AThis strategy does call for relatively large collectors; however, work on reducing the mass of collecting optics is

in process and suggests that relatively high collection efficiencies of e.g., 1 kW/kg are not forbidden by fundamental considerations. We suggest that the mass of this structure will tend to be dominated by the collectors implying the possibility of a relatively high power per unit mass ratio for eventual systems of this kind. In our model simulation we assume a high optical quality of the laser components and of the collectors. This high optical quality will not be trivial to access over the large areas involved in this system design, but also does not appear to be ruled out by fundamental physical laws.

5.0 Removing heat while limiting thermally induced lensing, birefringence, and shock

An unavoidable consequence of the successful concentration of solar pump power is a need to remove the resulting waste heat while limiting the thermally induced lensing, stress birefringence and shock to acceptable values. This

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need to remove waste heat has continued to set relatively low limits on the ability to scale solid state lasers to high average power while maintaining high beam quality. We discuss here strategies for handling these sources of distortion and stress designed to facilitate scaling of solid state lasers to high average power while retaining high beam quality.

5.1 “Gain/cool” unit cell

In an earlier paper we described a “gain/cool” unit cell strategy for removing waste heat that explores a strategy of utilizing two distinct elements in the gain region to reduce the undesirable consequences of thermally induced lensing, birefringence, and shock [11]. One element primarily performs the gain function and one element primarily performs the cooling function. By optimal design the negative consequences of thermally induced lensing, birefringence and shock can be significantly reduced. Considerable discussion of our strategy for minimizing thermal lensing, birefringence, and shock is provided in our prior reference [11].

5.2 Illustrative example

We illustrate one case here in Fig. 5 which shows the simulated temperature profiles for gain element composed of a series of Nd:YAG/diamond “gain/cool” unit cells. The temperature scale is indicated on the right hand side of the figure. The Nd:YAG is the thinner of the two materials and has the higher temperature, 235K at the extreme top and bottom of the figure. The temperature scale changes color every 0.5 K. The diamond is at 200K at the extreme top and bottom and increases in temperature by order of 0.5 degrees from the extreme top and bottom to the center of the figure.

235K

200K Fig. 4. Plot of temperature vs. location in an amplifying element using the “gain/cool unit cell”. The gain medium has an average temperature along the optical axis that increases with increasing radial position. The cooling medium, in blue, has an average temperature along the optical axis, that decreases with increasing radial position, r.

The principal points of interest are that the mean temperature in the gain material decreases with decreasing radial distance from the optical axis while the mean temperature of the diamond increases with decreasing radial distance from the optical axis. Taking into account the difference in the thermo-optic coefficient of the two materials we can adjust these thermal gradients of opposite sign so as to yield thermal lenses in the gain and the cooling components that cancel. In this case the “unit cell” has zero net thermally induced optical power. There is a noticeable thermal gradient in the Nd:YAG in the axial direction ( note the change in color in the axial direction); however, this thermal gradient has no particular consequence for the thermal lensing since it occurs along the direction of propagation of the laser field.

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We have assumed a Gaussian distribution of the pump power, and a corresponding delivery of waste heat under laser oscillator action, with a maximum on the optical axis. To achieve canceling thermal gradients under this pumping condition we postulate a thermal heat transfer coefficient at the interface between the two materials that increases with decreasing radial distance from the optical axis. We also postulate annular shaped interfaces at a series of discrete radial locations in the gain portion of the “unit cell” that reduce the rate of heat transfer in the radial direction in the gain material.

We recognize that the introduction of these position dependent interfaces within the gain material and at the interface between the “gain” and “cool” material is far from trivial to accomplish while leaving the necessarily excellent optical properties of the gain and cool materials largely unaltered. On the other hand our focus in this study is primarily on identifying fundamental limits and strategies for overcoming current limitations. We leave this demanding, but interesting, task to future efforts. In essence we hope that the profound differences between propagation of acoustic and optical excitations in solids can be used to advantage in achieving the desired unit cell properties.

Details of the above design are: diamond width is 1 mm and the radius is 6 mm; YAG width is 0.5 mm and the radius is 6 mm. The heat transfer across the interface between the YAG and the diamond is structured according to the pattern: 0<r<1.50 mm, h=0.48 ( )2W / Kcm ; 1.50<r<3.00 mm, h=0.29 ( )2W / Kcm ; 3.00 mm<r<4.50 mm,

h=0.103 ( )2W / Kcm ; and 4.50<r<6.00 mm, h=0.023 ( )2W / Kcm . Barriers to heat flow in the radial direction in

the YAG were also introduced according to the pattern: r = 0.5mm, h=0.1 ( )2W / Kcm ; r = 1.0 mm, h = 0.1

( )2W / Kcm ; r = 1.5mm, h = 0.1 ( )2W / Kcm ; r = 1.0 mm, h = 0.1 ( )2W / Kcm ; r = 1.0 mm, h = 0.1 ( )2W / Kcm ; r

= 1.5 mm, h = 0.1 ( )2W / Kcm ; etc. out to h portion of the optical pump power not removed by laser action was

given an approximately Gaussian distribution: 0< r <1.50 mm, γ = 634.878[W/cm3]; 1.50 mm < r < 3.00 mm, γ = 387.256[W/cm3]; 3.00 mm < r<4.50 mm, γ = 142.464[W/cm3]; and 4.50<r<6.00 mm, γ = 31.788[W/cm3]. The materials are otherwise regarded as having the conventional thermal conductivities, heat flow, and temperature distribution that follow from conventional thermal processes.

5.3 Negative thermal lens in the gain region despite a Gaussian distribution of waste heat

The configuration described above produces a negative thermal lens in the gain region despite a lowest order Gaussian mode distribution of the waste heat which is maximum on the optical axis and decreases with increasing radial distance from the optical axis. In effect we use the thermal interfaces to influence the heat flow in such a manner as to achieve a temperature profile which is a maximum at maximum radial distance from the optical axis and increases slightly as the radial distance decreases. Because the YAG thermo-optic coefficient is large and the diamond thermo-optic coefficient small and thermal conductivity large, relatively small thermal gradients in the both the diamond and the YAG are adequate to both yield a cancellation of the induced thermal lens and remove the needed quantity of waste heat.

5.4 Reduction of stress induced birefringence as well as thermal lensing

The configuration described above also assists in reducing the stress induced birefringence. The stress induced birefringence is closely related to the thermal lensing; however, canceling the thermal lensing in two different materials does not necessarily cancel the stress induced birefringence. On the other hand the weak thermal lenses in the two materials and the opposite signs of the thermal gradients tend to yield a relatively small net stress induced birefringence. In particular, for two different materials a temperature can typically be found where both the thermal lensing and the stress induced birefringence are both minimized.

If the materials are identical, as in the case of Ti:sapphire and undoped sapphire, e.g.,, the conditions that yield cancellation of the thermal lensing in the two materials also tend to lead to cancellation of the stress induced birefringence in the two materials because of the change in sign of the thermal gradients. The model we address that includes introduction of specifically designed interfaces complicates the analysis to a degree that more detailed analysis of the stress birefringence issue appears inappropriate at this writing.

6.0 High power large area lowest order Gaussian mode near confocal laser oscillator

A laser oscillator operating at high average power in a relatively large area lowest order TEM00 Gaussian mode in a near confocal resonator appears to be the simplest means of efficiently transforming concentrated incident sunlight into optical power having the high degree of spatial coherence and power needed for an optical power infrastructure in space. Such laser power oscillators do not exist at this time. However, we find no fundamental laws forbidding

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realization of such laser power oscillators. We discuss here some of the practical barriers and explore several strategies for overcoming those barriers.

6.1 Need for a large area mode

In our model system the need for high power and high spatial coherence tend to call for a large TEM00 mode area. The argument is that the intraresonator intensity needs to approximate the saturation intensity sI . A lower intensity will not extract the power stored in the gain medium efficiently. A much higher intensity will saturate the gain medium so strongly the intraresonator losses will have to be unreasonably small to maintain oscillation threshold. The power that can be extracted from the laser power oscillator is consequently ~ 2

s oI wπ where is the radius of the beam waist in the resonator for the TEM00 mode.

ow

The saturation intensity is ~0.8-2.7 kW/cm2 for Nd:YAG at 1.06 microns, e.g. so the beam radius needed to extract order of 100 kW is 3.4 cm < < 6.3 cm. this is not an extremely large radius; however, the simplest resonator supporting the TEM00 mode, a symmetrical confocal resonator has a mirror spacing

ow22 o oz w /= π λ where

is the optical wavelength, 1.06 microns. This would require a mirror spacing in the range of 3.43 km > >11.76 km. These are impractical dimensions for foreseeable systems. λ 2 oz

6.2 Near confocal ring resonator

A near confocal resonator is strongly preferred since such resonators tend to be stable against small misalignments. The lowest order Gaussian mode is also strongly preferred as the simplest solution of the paraxial Helmholtz equation. A resonator we have explored in some detail appears to offer a solution. We refer to this as a double confocal configuration. We show such a resonator in Fig. 5 Here two curved mirrors of radius R are positioned so that a closed path having four path segments with two of the path segments approximately satisfying the confocal condition (mirror separation along the particular path segment is equal to the radius of the two mirrors) and each exhibiting a relatively small beam waist. The beam is approximately collimated over each of the two remaining path segments. The beam radius in each of the two collimated segments can be large, satisfying the condition on the beam radius identified above, while the mirror spacing can be relatively small, e.g. a few meters. The resonator retains much of the desirable stability characteristics of a confocal resonator.

Fig. 5 Double confocal resonator offers a large lowest order mode cross sectional area, small beam waist, and near confocal alignment properties in a resonator having practical dimensions.

A related ring near confocal resonator has been used for the free electron laser at Jefferson Laboratory [12]. That laser differs from the configuration shown here and in Fig. 6 and Fig. 7 in that the gain is introduced at the beam waist of the free electron laser and flats are introduced so the optical path is essentially half of that shown in Fig. 6. The purpose of this resonator in the case of the free electron laser is to produce a relatively large mode area on the

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mirrors so as to keep the heating of the mirrors at an acceptably low level given high average power, e.g. 1 MW, of average power. Our configuration yields similar large mode areas on the mirrors, but differs in that the gain area in our laser is also relatively large. This is a necessary consequence of the difference between the free electron laser and our solid state laser technology.

6.3 Parabolic mirrors and Gaussian soft apertures

Additional requirements on the laser oscillator design are recommended. We find that the relatively large beam waist and relatively large angles that appear to be needed for this resonator and the gain media we are considering recommend that the reflectors be parabolic as means of retaining good mode quality. We prefer to retain the option of using Brewster angle surfaces for the gain media. We show in Fig. 6 one way of introducing gain media having Brewster angle surfaces for the entrance and exit. One to four gain elements of the type shown could be introduced. Flats could, e.g. be used to complete the optical paths with fewer gain elements in the system.

Fig. 6. Double near confocal ring resonator including gain media and spatially selective loss for causing selectively operation only on the lowest order Gaussian mode of the resonator.

Another need is means of selectively favoring the lowest order Gaussian mode as illustrated in Figs. 6 and 7. We show in Fig. 7, as well as sample gain elements, means for introducing spatially selective loss in the regions near the beam minima. Given the symmetry of this configuration we have avoided the precise center of the resonator since the beams cross in this example and each aperture would tend to interfere with both beam paths. The spatial selective loss should ideally have a Gaussian distribution about the individual beam axis that discriminates between the lowest order TEM00 mode and the next higher order mode, e.g., the TEM01 mode while introducing only a relatively small loss for the preferred TEM00 mode.

This “soft” aperture is technically demanding in that the beam intensity will be high near this focus region. This is the inverse problem of the free electron laser system at Jefferson Laboratory. There the laser has an approximately Gaussian gain in this focused region. The net result can be similar in our case; however, the mode selection in our case is assisted both by a large area Gaussian gain region in the region where the beam has a relatively large radius and this Gaussian loss region near the focus. Both of these mechanisms tend to assist in the stabilizing and selecting out the preferred TEM00 mode [4].

6.4 Role of nonlinear mode competition as a means of selecting lowest order Gaussian mode

Preferential selection of the lowest order Gaussian mode out of the many modes that could be supported by the resonators shown in Fig.s 5 and 6 is essential to a useful optical power infrastructure. We can describe the competition of the different modes, denoted here by the subscripts n and m, using a formalism developed by Willis Lamb and his students [13]. If we let the intensity of a given Gaussian mode of the resonator mode be described as

jI where j varies to denote the full range of possible resonator modes, we can describe the time dependent evolution of a given mode intensity using a set of expressions of the general form

/ 2 ( )n n n n n n nmm

mI I I I Iα β θ= − − ∑ (2)

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Here jα denotes the net gain experienced by mode j within the resonator, n nIβ denotes the net decrease in the gain available to mode n due to self saturation by mode n, and nm mIθ denotes the net decrease in gain available to mode n due to cross saturation by mode m. The point is that by choosing the spatial distribution of the gain and loss we can favor operating conditions where only the lowest order Gaussian mode oscillates and is consequently the only mode populated. In the particular case discussed here one would, e.g., shape the gain (upper path) so as to provide maximum gain for the Gaussian TEM00 profile (approximately a Gaussian distribution of the gain with a maximum on the optical axis) and the loss (lower path) so as to cause greater loss for higher order transverse modes (rapidly increasing loss for the region having a radial distance larger than 2wo where wo is the radius of the focused lowest order Gaussian mode at the beam waist in the lower path in Fig. 6).

7.0 Imaging and power beaming

Once the lowest order Gaussian mode has been generated in the laser oscillator resonator it is a relatively straightforward matter to use conventional optics to expand the beam to the dimensions are needed for optimal transmission over the long distances of Earth-moon space. In effect the optical power has been produced in simplest eigenmode of free space and the power can be transformed using linear optics to the particular transverse dimensions that are most useful for precise and efficient transmission.

Relay mirror at gravitational minimum

Earthmoon

Fig. 7 Scale diagram of Earth, our moon, geostationary orbit around the Earth and coherent power beaming from an optical power source on the moon via a relay mirror at one of the gravitational minima (a LaGrangian point, such as L4 or L5 of the Earth moon system) to an otherwise obscured satellite in geostationary orbit .

7.1 Coherent power beaming

Coherent light in the form of the lowest order Gaussian mode of free space appears to be the preferred means of beaming power over the long distances characterizing Earth-moon space, e.g., the mean distance between the earth and the moon is ~ 384 Mm. A confocal imaging system needs an aperture diameter d of ~ 6.5 times the optical beam waist, for efficient transmission of a high quality electromagnetic beam over a distance [4]. The beam

waist is related to the distance L by ow 2 oL z=

ow L / 2= λ π . For the average Earth-moon distance and an optical wavelength of 1.06 microns = 8.04 m the mirror diameter needs to be 6.5 or ~52.3 meters. ow ow

This large beam diameter is an unavoidable consequence of the laws of diffraction and the optical wavelength. Mirrors of these dimensions and substantially larger, e.g. 150 m diameter, are, however, at this time being explored for space applications [5]. Microwave emission for transmission is essentially ruled out for these distances since microwave wavelength and hence the dimensions of the transmitting antenna will need to be from 1000 to 10,000 times the dimension of these optical apertures.

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7.2 Intrinsically safe power beaming

One point for this study is that because of the unavoidable need for relatively large beam dimensions even substantial amounts of power, e.g. 3.5 MW, will necessarily be transmitted at intensities comparable to that of natural sunlight, ≤ 1.4 kW/m2. Given the diffraction limited character of spatially coherent light the intensity of the transmitted optical power can be increased, at the discretion of the receiver, at the location where the power is received. This offers the option to the receiver of accessing the high intensities needed, for example, for nonlinear optical phenomena or novel physical processes. Also as we discussed in an earlier paper, by suitable design of the transmitting optics optical power can be transmitted to Earth while working well within widely accepted safety standards [1].

8.0 Optical power infrastructure

The achievement of substantial amounts of power in coherent optical beams of the dimensions discussed above would provide a means of distributing power over distances characteristic of Earth-moon space. In general, an optical power infrastructure should be capable of delivering needed power in the needed form virtually anywhere and anytime in the region supported by the infrastructure. We discuss here one possible strategy.

8.1 Sources on the lunar surface and relay mirrors at the LaGrangian points

A variety of strategies for constructing an optical power infrastructure can be imagined. One approach would be to locate the coherent power sources on the lunar surface and direct the coherent power via relay mirrors located at the L4 and L5 LaGrangian points. We show in Fig. 8 a schematic diagram of the power beaming paths and in Fig. 9 a perspective view as might be recorded by a sensitive camera located slightly beyond a satellite in geostationary orbit. (The optical beam would only be observable with instruments much more sensitive than the human eye given the extremely small amount of scattering material in space.)

Fig. 8. A perspective view of a coherent optical beam transmitted from the lunar surface to a satellite in geostationary orbit as recorded by a sensitive camera located slightly beyond a satellite in geostationary orbit.

Locating the coherent sources on the lunar surface has advantages of a stable platform, means of dissipating waste heat, and the other resources that would tend to be found at a lunar base. As regards accessing the LaGrangian points from the lunar surface most of the side of the moon facing Earth can access the LaGrangian point L4 and L5. Roughly one third of the locations on the far side of the moon can also access the LaGrangian points L4 and L5 at any time,.

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8.2 Absence of atmosphere

The absence of atmosphere on the moon facilitates the transmission and reception of optical power. There is, in effect, an absence of the many difficulties encountered in optical transmission through the Earth’s atmosphere. The moon also provides a stable platform with a degree of gravitational pull that is roughly one fourth that on Earth. The lower gravitational pull makes the moon a favorable platform for launching spacecraft and satellites.

9.0 Efficient transformation of beamed optical power to electrical power

Perhaps the most important part of an optical power infrastructure is the transformation of the received optical power into a form useful to the receiver of the power. We examine here, primarily the fundamental limits on the efficiency with which highly coherent optical power can be transformed to electrical power. The point of interest is that given the capability of the stimulated emission process to produce light which is both temporally and spatially coherent we can prepare the optical power as a monochromatic field that can be imaged with optimum spatial distribution on a photovoltaic device. This offers an opportunity to approach fundamentally limited performance in transforming the coherent optical power to electrical power. Our goal here is to separate the fundamental limitations from the practical limitations as a means of guiding device design.

9.1 Expression for efficiency of idealized photovoltaic cell

If we neglect practical limitations, such as grid shadowing, incomplete absorption, surface recombination, reflection, etc. the efficiency η of a photovoltaic cell can be written [8]

1 1 ln L sT T T

oc oc oc oc

J rV V VV V V V

η⎡ ⎤⎛ ⎞ ⎛ ⎞

= − + −⎢⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

⎥ (3)

In general, in our optimal case we will make the open circuit voltage Voc match the band gap voltage Vbg. We introduce the parameter

From a fundamental point of view optical power can be transformed to electrical power with high efficiency. The basic need is one of minimizing loss mechanisms that

In addition to the critically important spatial coherence gained by transforming sunlight to light having the spatial distribution of the lowest order Gaussian mode, temporal coherence can also be gained. One way this temporal coherence can be utilized is by transforming the energy available as sunlight to nearly monochromatic light. The use of monochromatic light having a particular frequency can assist in maximizing the efficient with which received optical power is transformed to electrical power.

In general the efficiency of transforming power from one form to another is influenced by what might be termed “matching” criteria. Of particular importance in transforming optical power to electrical power through the use of a photovoltaic, such as the semiconductor material GaAs, is the matching of the frequency of the incident quantum energy to the energy of the band gap of the semiconductor to within a precision of the order of qkT where q is number of the order of 3. For purposes of discussion we will use a highly simplified expression for the efficiency η of a photovoltaic receiver.

(1 )(1 ln(1 ))

/qkT hη ξ ξ ξξ ν

= − + −≅

(3)

In the interest of exploring fundamentally limited performance we have made the approximation that a number of practical mechanisms, such as internal resistance, grid shadowing, incomplete absorption, surface recombination, etc. that typically reduce photovoltaic efficiency can be neglected. The point of this exercise is that we treat the non-zero temperature of the photovoltaic as the source of inefficiency that is most fundamental in character and hence one of the more important parameters to which attention should be given.

9.2 Density of states issues

We recognize that one of the implicit assumptions in the above discussion is that the energy of the incident quanta is assumed to closely approximate the band edge energy of the photovoltaic. Also implicit in our addressing of relatively large average power is the assumption that one using this technology is interested in transmitting relatively

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large amounts of power while minimizing the dimensions of the receiver. This tends to imply a sharply defined band edge and a density of states at the band edge adequate to handle substantial power.

In general the precision with which the band edge can be defined will depend on the temperature of the photovoltaic and the intensity of the incident light. In particular the density of states for bulk semiconductor material is small precisely at the band edge and only increases as the square root of the energy above the band edge. These are important concerns that we, however, choose not to address in detail at this stage of discussion. We do recognize that reduced dimensionality of the semiconductor device is an attractive strategy for increasing the density of states and hence presumably handling a large energy flux.

9.3 Significance of quantum energy and temperature of the photovoltaic receiver

Given the above definitions and assumptions the key parameters determining the maximum efficiency with which coherent optical power can be received and transformed into electrical power are the quantum efficiency and the temperature of the photovoltaic material. We give a plot of the maximum achievable efficiency for a photovoltaic receiver operating under the assumed approximations in Fig. 9 . The principal points of interest are the potential for high efficiency and the importance of maximizing the quantum energy and minimizing the temperature of the photovoltaic.

Fig. 9. Efficiency of photovoltaic receiver vs. wavelength and temperature for monochromatic power beaming

10.0 Deflection of near Earth objects using coherent optical power One potential application for coherent optical power produced at high average power in space is the deflection of a near Earth object (NEO) that threatens Earth as through an Earth crossing orbit. A review of the literature, e.g., shows a popular notion that the coherent optical power needed to cause significant deflection of a typical NEO, e.g., 2004 MN4 at 4.6 x 1010 kg (46 million metric tons) would require a laser source far beyond foreseeable laser technology. We suggest here, to the contrary, that the laser design outlined here would be more than adequate to cause the needed deflection. This conclusion is arguably counterintuitive; however, we develop some simple analytical equations here and use well established data to estimate the needed average laser power.

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10.1 Equations describing deflection of a near Earth object by coherent light The equation describing the force applied to a near Earth object by illumination by optimized coherent light can be developed using a simple model. We use parameters from one of the original experiments characterizing the ejection of material from a material surface caused by illumination by ultrashort optical pulses at high intensity. We assume an ideal process in which all of the incident optical power is transformed into kinetic energy of the ejected material and ejection of the material that produces a force in the preferred direction. The actual force will be somewhat less; however, the information from the experimental work indicates that this ideal example can serve as a useful estimate of the upper limit on the deflection that can be achieved.

yFc

mneo, vneo z

Fig. 10 Trajectories of a 2004 MN4 relative to Earth for different levels of coherent optical power. This model is for a continuous series of optical impulses delivered at maximum efficiency and optimum direction.

The equations characterizing the conservation of momentum and energy can be written for the near Earth object

(NEO) as , which are the equations expressing the conservation of energy and momentum respectively. Here mneo and ∆vneo are the mass and change in velocity (in the y direction), respectively, of the NEO, and mej and vej are the mass and mean velocity in the –y direction of the ejected material from the NEO. Let P be the optical power delivered to the NEO and τ is the time over which power is transferred. Assuming constant generation of momentum in the direction transverse to the unperturbed orbit the shift in the location of the NEO relative to the unperturbed orbit will be given by

( )2 2neo ej ejneo

m v m v∆ = neo neo ej ejm v m v∆ =

22Py

neo ejm vτ

∆ = (4)

The power required to deflect a NEO by a distance of the order of the Earth radius over a time τ can be estimated from the relationship . The interesting observation is that even though the NEO is relatively massive, ( e.g., 4.6 x 1010 kg) and the radius of the Earth is relatively large, 6.37 Mm, because the duration of the optical impulse can be relatively long, e.g. 109 sec and because the duration of the optical impulse enters the expression quadratically the average coherent optical power required is within an accessible range, ~ 1 kW. Here we have used a value for the velocity of the ejected material of 10,000 m/s. We have, of course, assumed maximum efficiency of all the processes, however even allowing for substantial departure from ideal efficiency the average power required appears to be technically accessible in a reasonable time frame.

ER2

neo ej EP m v R / 2= τ

10.2 Importance of precise knowledge of the trajectory of near Earth objects

An important conclusion that can be drawn from equation (4) and Section 9.1 is that the duration τ of the “optical push” is a major factor enabling a moderate power laser oscillator source to deflect a near Earth object by a sufficient margin to prevent a collision. In particular, to access reasonable laser powers the duration of the deflection time is preferably of the order of 20 years or more. This has the consequence that finding threatening near Earth objects sufficiently long before the potential collision would occur is essential. Also knowing the trajectory with extremely high precision is also critical since it is obviously extremely important that the “optical push” be exerted in the correct direction and have adequate magnitude and be correctly applied over this long time period with an end result error small compared to the Earth radius.

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10.3 Capability of moving a large NEO substantial distance using moderate coherent optical power

In general, the ability to cause a massive object to move a substantial distance under optical illumination may seem counterintuitive. There are a number of factors that enter into the dynamics of this system that bear on this result. One is that the optical power transmitted to the object is transformed quite efficiently into kinetic energy of material ejected from the NEO. This occurs in our model because we assume that the optical power reaches the object concentrated in both space and time so that threshold for efficient material heating and ejection is accessed.

Because the optical power is largely transformed into kinetic energy of material and because that material is ejected, in our model, in the direction opposite to the direction which we seek to accelerate the NEO, and because the material is ejected with a relatively large velocity, 104m/sec we achieve a relatively efficient acceleration of the NEO. The long period of illumination is also extremely important to achieving this large effect from relatively modest optical power.

Also by using ultrashort pulses we can deliver the full optical power to the object in a time short compared to the ejection of the material from the object. This is helpful since once the material is ejected the ejected material can itself interfere with subsequent optical power reaching the particular illuminated region of the object. In practice the optical power can be pulsed and the illuminated region changed with each event so that this plume of ejected material does not interfere with efficient use of the optical power.

10.4 Location of final focusing element

The coherent laser source can be located in a spacecraft that is co-orbiting with the near Earth object. This strategy has the advantage that the optics can be relatively small in dimension. Let the laser pulse energy be pU A representative requirement for efficient ablation of the illuminated NEO material is an energy density of 1 J/cm2. The effective diameter of a focused Gaussian beam is od 2f / D≈ λ where D is the diameter and f the focal length, respectively, of focusing element.

11. Conclusions

We conclude that the near vacuum of space, the low temperature of the surrounding universe, 2.7K, the pervasive nature of sunlight in near Earth space, and presence of LaGrangian points in the Earth-moon and Earth-Sun systems, provide basic features that will be useful for an optically based power infrastructure. The lunar surface, in particular, appears to offer a favorable location for locating solar pumped solid state lasers of the type discussed here. There would, of course, be a need for solar tracking and multiple lasers as a means of providing continuous power. We have examined a number of strategies for such systems and our initial conclusion is that such a lunar based optical power infrastructure is a favorable approach.

We also conclude that the task of constructing a solid state laser oscillator directly pumped by sunlight that will operate at multi-kilowatt power and higher while producing high quality optical fields is demanding, but possible. In particular, we are able to identify technical strategies that appear to offer means of collecting power available as sunlight in near Earth space and converting that power to coherent light at multi-kilowatt levels and higher while achieving the degree of spatial coherence needed for beaming that power over the multi-megameter and longer distances required of an optically based power infrastructure in near Earth space.

Peer reviewed publication: We are writing one, or possibly, two papers for Optics Express (which allows the use of color and computer based movies). These are expected to be completed this month. References:

1. Matthew H. Smith, Richard L. Fork and Spencer T. Cole, “Safe delivery of optical power from space,” Opt. Express 8, 537-546, http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-537 (2001) .

2. Nigel Hawkes, “Earth’s gravity may lure deadly asteroid”, http://www.timesonline.co.uk/article/0,,2-1573790,00.html (2005).

3. “The Vision for Space Exploration”. http://www.nasa.gov/pdf/55583main_vision_space_exploration2.pdf (2004).

4. NASA Institute for Advanced Studies, http://www.niac.usra.edu/ (2005). 5. Richard L. Fork, “High-energy lasers may put power in space”, Laser Focus World 37, 113-117 (2001). 6. Roland Winston, Non-imaging Optics, … 7. A.E. Siegman, Lasers, University Science Books, Mill Valley CA (1986). 8. R.L. Fork, “Physics of optical switching”, Physical Review A 26, 2049-2064 (1982).

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9. W. Koechner, Solid-State Laser Engineering, 5th Ed, (Springer-Verlag, Berlin, 1999). 10. Project Lasso. http://www.darpa.mil/baa/baa04-36.htm 11. Richard L. Fork, Wesley W. Walker, Rustin L. Laycock, Jason J.A. Green, Spencer T. Cole, “Integrated

Sapphire Diamond Laser”, Optics Express 11, 2532-2548 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2532.

12. Stephen Benson, George Neil, Michelle Shinn, “Lasing With a Near-Confocal Cavity in a High Power FEL”, Proceedings of SPIE Vol. 4632, 115-121 (2002).

13. Murray Sargent III, Marlan O. Scully, and Willis E. Lamb, Jr., Laser Physics, (Addison Wesley, Massachusetts, 1974.)

14. Taku Saika, Shigeaki Uchida, Kazuo Imasaki, Shinji Motokoshi and Masahiro Nakatsuka, "Solar-Pumped Nd doped Multimode-Fiber Laser with a D-Shaped Large Cladding", pp 378-389, Second International Symposium on Beamed Energy Propulsion, Edited by K. Komurasaki, American Institute of Physics (2004).

15. Verdeyen, Laser Electronics, Prentice-Hall, Upper Saddle River, NJ (2000). 16. "Multipass Optical Amplifier Using a Double Confocal Resonator Geometry", R. L. Fork, F. A. Beisser, and

D. K. Fork, Revue de Physique Appliquee 22, 1665-1671 (1987). 17. M. C. Downer, R. L. Fork, and C.V. Shank, “Femtosecond imaging of melting and evaporation at a

photoexcited silicon surface” JOSA B2, 595-599 (1985).

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