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1Y.K. Bae Corporation, Corona, CA 92883, USA. Correspondence and requests for materials should be addressed to
Y.K.B. (email: ykbae@ykbcorp.com) 1
Demonstration of Amplified Radiation Pressure Propulsion with an Active Optical Cavity
Young K. Bae1
1Y.K. Bae Corporation, Corona, CA 92883, USA
The radiation pressure amplified in optical cavities by circulating laser beams has been applied in
optomechanics, gravitational wave detection, and km-sized sparse-aperture space telescopes.
Amplified Radiation Pressure Propulsion (ARPP) utilizing passive optical cavities has been
researched for several decades to overcome the limit posed by particle-based conventional
propulsion, but without success. Here, we report the first demonstration of ARPP using an active
cavity with a thin-disk gain medium by propelling a 0.75-kg object over 2 m in a Class-1000
cleanroom. The maximum achieved CW radiation pressure, radiation pressure force, and
circulating laser power were 90 Pa, 3.4 mN, and 510 kW respectively at 1.1-kW pump power.
Adaptation of a radiation pressure sensor proved the thermal convection effect on the demonstration
was negligible. The results establish the thrust/force generation efficiency of ARPP can be increased
to the same level of electric propulsion and pave the pathway for enabling propellant-free ARPP
applications beyond the reach of conventional propulsion.
The modern theory of radiation pressure was first established by Einstein who proved thermal
equilibrium between the radiation and kinetic energy of radiation-emitting gases by considering
the radiation pressure fluctuations acting on a movable perfectly reflective metal plate inside a
blackbody cavity1. At extremely high temperatures, radiation pressure competes with material
pressure in transport phenomena of astrophysics2 and nuclear fusion technologies3,4. In
astrophysics, the Eddington limit5 theorizes that the maximum luminosity of a stellar body, such
as an accreting black hole6, is controlled by the equilibrium between the radiation pressure acting
outward and the gravitational force acting inward on falling-in materials. The practical use of
radiation pressure for terrestrial scientific and technological applications, however, has only
emerged several decades after Einstein’s work1, especially after the invention of lasers. The
radiation pressure generated by laser beams has been extensively utilized in trapping and
manipulating atoms, molecules, and small particles7-8 resulting in many innovations including
demonstration of Bose-Einstein condensation in atomic gases9-10.
Manipulation of objects larger than small particles requires amplification of radiation pressure by
circulating laser beams in optical cavities. In particular, cavity optomechanics11-12 explores an
interaction between mechanical motion and radiation pressure amplified in micro passive cavities
with lengths on the order of the laser wavelength as illustrated in Fig. 1a. Cavity optomechanics
has played a critical role in a wide range of applications from optical cooling to quantum
computing11-12. Recently, cavity optomechanics with an active cavity was demonstrated to be
more advantageous than the passive cavity optomechanics13.
Precision manipulation of spacecraft with radiation pressure in km-sized optomechanical systems
with active cavities was proposed to construct large sparse-aperture space telescopes14-15. In the
proposed structure as illustrated in Fig. 1b, pairs of spacecraft were connected with tethers and
their equilibrium positions were maintained with nm accuracy by amplified radiation pressure in
an active cavity and opposing tether tension. The amplification of radiation pressure in km-long
high-finesse passive cavities has played a crucial role in the recent success of gravitational wave16
detection17. The shot noise in radiation pressure in the cavity could be sufficiently reduced to
2
achieve the position accuracy of the detector mirror on the order of 10-18 m by amplifying the
radiation pressure17. The experiment demonstrated 100-kW circulating laser power in 4-km high-
finesse passive optical cavities with a circulation-gain amplification factor of 5,000 and a radiation
pressure force of ~0.7 mN.
The adaptation of radiation pressure
for space propulsion has been
researched for almost a century, since
the solar radiation pressure sail was
proposed for interplanetary space
propulsion by Tsander in 192418. His
work established the foundation of
modern solar sail technologies19-20.
Forward proposed the laser sail for
interstellar roundtrip manned
missions21-22, in which a 1,000-km-
sized reflective sail is propelled by the
radiation pressure of a reflecting ultra-
high-power laser beam delivered from
the earth. A smaller scale laser sail was
also proposed to deliver 10-kg
payloads to Mars in 10 days23.
Recently, a silicon-chip-sized laser sail
was considered for interstellar flyby
missions24.
However, for routine space
endeavours, such as propellant-free
spacecraft manipulation in earth orbits
and commute between planets, a
propulsion should have the thrust-
generation efficiency orders-of-
magnitude larger than the laser sail25.
The thrust-generation efficiencies of
propulsions can be compared using
specific thrust: the ratio of the
thrust/force generated by reflecting
radiation or exhaust particles to input
energy25. The ratio Rst of the specific
thrust of radiation pressure propulsion
to that of conventional particle-based propulsion is approximated by
, (1)
where Nref is the number of reflections of radiation on the spacecraft mirror/sail, v is the exhaust
Figure 1. Schematic diagrams of amplified radiation
pressure devices. (a) Micro laser-wavelength-sized passive
cavity optomechanical device12. (b) Macro km-sized active
optomechanical device proposed for sparse-aperture space
telescopes14-15. (c) Earlier concept of Amplified Radiation
Pressure Propulsion (ARPP) system with a macro passive optical
cavity in which one mirror is mounted on the powering platform,
and the other mirror on the propelled spacecraft28.
3
particle velocity, and c is light velocity in a vacuum. The details of derivation of this equation are
presented in the method section below. For single-reflection radiation propulsion, such as the solar
sail, Nref = 1; and assuming v ~ 3x104 m for electric thrusters26, Rst ~ 10-4. Thus, the single-
reflection radiation pressure propulsion is ~104 times less efficient than electric propulsion and is
unsuitable for the routine applications. By adapting state-of-the-art high-power photonics
technologies, Nref can be increased by many orders of magnitude through reflecting the laser beam
numerous times between two HR mirrors: one located on the powering platform and the other on
the propelled spacecraft. Using a Herriot cell with two HR mirrors was first proposed for
increasing Nref 27. Later, it was found that the passive resonant optical cavity is more advantageous
than the Herriot cell, where after many reflections, laser beams overlap and become resonant28 and
injection and alignment of the laser beam is much easier in the former.
Amplified Radiation Pressure Propulsion (ARPP) is defined as the radiation pressure propulsion
with a circulating laser beam in an optical cavity as illustrated in Fig. 1c. The passive resonant
optical cavity, however, was found to be highly unstable against the movement of the mirror on
the propelled spacecraft, and thus unsuitable for ARPP14,15,29. To understand this aspect, we
consider an ARPP system with a passive optical cavity and a laser beam transmitted through the
first HR mirror as shown in Fig. 1c. The second mirror on the spacecraft is propelled by the
radiation pressure of the circulating laser beam in the cavity. The details of the calculation are
presented in the method section. Assuming the transmittance T and the reflectance R of both
mirrors, the kinetic energy imparted on the second mirror KE2 over a distance L-L0 is given by
, (2)
where Pinc is the incidence laser power, L0 is the initial length of the cavity, k’ is the laser
wavenumber of the circulating laser beam, and the coefficient of finesse FC is given by
, (3)
where F is the finesse of the cavity and F>>1. Assuming T ~ (1 – R) << 1 for simplicity, as shown
in the method section Eq. 2 can be transformed to
. (4)
Eq. 4 shows that the KE2 of the high-finesse passive cavity is smaller than the kinetic energy
imparted by the incident laser beam directly reflected off the second mirror without the first mirror
over L-L0. Since KE2 is independent of F and the circulation gain in the cavity is less than 1, the
passive optical cavity is unsuitable for ARPP.
Fortunately, our previous small-scale experiment29 discovered that the active cavity is stable
against the cavity mirror movement and can enable ARPP. The experiment used a Nd:YAG rod
gain medium and resulted in the maximum circulation power of 5.2 kW at 75-W pump power and
the corresponding radiation pressure force of 35 µN was directly measured with a digital weight
4
scale. Owing to the technical limits of the experiment, the actual propulsion of a macro object
could not be demonstrated. Therefore, the primary objective of the present work is demonstrating
the actual propulsion of a macro object by overcoming the technical limits and to establish the
technical and theoretical foundation for the active-cavity ARPP.
Experiments and Results
Radiation Pressure Measurement in Active Optical Cavities. The earlier experiment
successfully demonstrated radiation pressure amplification in an active cavity with a 5-cm-long
Nd:YAG rod in a laboratory
environment29. The extension of the
cavity length beyond 30 cm was
infeasible because of the difficulty in
alignment through the thin and long
gain medium as well as the large loss
from scattering and absorption in the
gain medium and by dusts in lab
atmosphere. In the present experiment,
a thin-disk laser system30 was employed
to significantly reduce the scattering
and absorption and to increase the
thermal management capacity resulting
in much larger circulating laser power.
A diamond wafer was used as a gain
medium heat sink to increase heat
conduction by several times. In
addition, the experiment was performed
in a Class 1000 room to decrease the
absorption and scattering by the
atmospheric dust.
Before actual propulsion
demonstration, we systematically
investigated radiation pressure
measurement in active cavities in
experimental setups illustrated in Fig. 2.
The details of the setups are presented
in the method section. We first started
with the setup similar to our eariler
experiment29. Briefly, as shown in Fig.
2a, a circulating laser beam was formed
between a R>0.999 mirror deposited on
the backside of the thin disk and a
R>0.999 mirror located on a digital
weight scale. We readily demonstrated
a
b
Figure 2. Schematic diagrams of the experimental setups for
directly measuring radiation pressure force. (a) Vertical
cross-sectional view of the radiation pressure measurement
setup with a digital scale as in our earlier setup29. Up to 10% of
thermal convection effect on the radiation pressure force
measurement at 150 kW intracavity laser power was observed.
(b) Horizontal cross-sectional view of the radiation pressure
measurement setup equipped with a radiation pressure sensor31.
Since the surfaces of mirrors faced horizontally, the effect of the
thermal convection on the radiation pressure force measurement
was reduced to less than 3% at the circulating laser powers up
to 510 kW.
5
a radiation force of ~1 mN, which equals to the circulating laser power of 150 kW, but encountered
the scale instability from laser heating in several seconds of operation. The instability could be
controlled with an aluminum heat sink sandwitched between the HR mirror and the scale pan.
However, even with the heat sink there still was a non-zero scale reading with the laser power off
after several miniutes of laser operation. The non-zero offset was deterimend to result from thermal
convection of air on the heated mirror and heat sink. In this setup, the maximum error from the
thermal convection in the radiation pressure measurement at 150 kW circulating laser power was
determined to be 10±3%.
To investigate the thermal convection effect further, we employed a radiation pressure sensor31
manufactured by Scientech Inc., which is essentially a horizontally operating digital scale as
illustrated in Fig. 2b. The experimental setup permitted the mirror to face horizontally and thus
reduced the error from the thermal convection on the radiation pressure measurement to less than
3 % at intracavity laser powers up to 510 kW. The details of experimental setup are presented in
the method section. Briefly, the radiation pressure force on the end HR mirror was directly
determined from the force on a flat 45o R > 0.999 mirror attached to an aluminum heat sink that
was mounted on the radiation pressure sensor head. In addition, the setup permitted the radiation
pressure force Frad on the end HR mirror with the circulating laser power Pcir to be estimated from
the transmitted laser output power Pt through the end HR mirror with transmittance T using the
equation:
. (5)
Fig. 3 shows the radiation pressure
force on the end HR mirror directly
measured with the radiation pressure
sensor as a function of Frad. Both
force measurements agree well at
radiation pressure forces up to 3.4
mN, which is equivalent to the CW
intracavity laser power of ~510 kW
over a laser beam diameter of 0.7 cm
and the radiation pressure of ~90 Pa.
In the active cavity, the laser beam is
generated within the cavity and the
amplification factor Aa is defined as
,
(6)
where Ppum and Pthr are the pump and
threshold powers, respectively. As
presented in the method section, over the operation laser power range, the threshold pump power
Figure 3. A comparison between two types of radiation
pressure force measurements at circulating laser powers up to
510 kW. The y-axis represents the radiation pressure force on the
end HR mirror directly measured with the radiation pressure
sensor31, while the x-axis represents the radiation pressure force
calculated with the circulating laser power estimated from the
transmitted laser power through the end HR mirror.
6
Pthr << Ppum, and then Aa is given by
, (7)
where η is the pump efficiency32 and the Δ is the total roundtrip loss factor33 given by
, (8)
where δ0 is the roundtrip loss factor through the gain medium including the reflection loss factor
on the attached HR mirror and δ1,2,3… = (1 - R1,2,3…), where Rj is the reflectance of the jth mirror
surface33. Here, the scattering loss of air molecules and dusts is assumed to be negligible.
We systematically measured Aa as a function of the end HR mirror reflectance. Fig. 4a shows the
radiation pressure force data measured with the end HR mirror of R > 0.999 and 5-m radius of
curvature as a function of Ppum. The measured radiation pressure force on the end HR mirror was
linear with Ppum up to the maximum achievable radiation pressure force of 3.4 mN, which is
equivalent to Pcir~510 kW. The circulating laser beam was of multimode and its image on the end
HR mirror had a diameter of 0.7 cm and the maximum achievable radiation pressure and power
density were ~90 Pa and ~1.3 MW/cm2, respectively. The linear fitting of the data resulted in Aa
~ 450. Fig. 4b shows the radiation pressure force measured with R = 0.99985 with the radius of
curvature of 0.5 m as a function of Ppum. Although the measured radiation pressure force was
roughly linear with Ppum, it was unstable at higher powers and the corresponding data were
significantly scattered around the fitting line as in Fig. 4b. Here, the maximum achievable
radiation pressure force was 2.1 mN, which is equivalent to Pcir~315 kW and power density of
0.82 MW/cm2 at Ppum = 0.43 kW. The data fitting resulted in the maximum achievable Aa ~730.
Above the power density of 0.82 MW/cm2 the R=0.99985 mirror was damaged, and the circulating
a b
Figure 4. Radiation pressure force measurement data with different end HR mirrors. The graphs show
the directly measured radiation pressure force (mN) with the radiation pressure sensor31 as a function of the
pump laser power in kW. (a) The data with R>0.999 resulted in an amplification factor of 450. (b) The data
with R=0.99985 resulted in an amplification factor of 730.
7
laser beam abruptly died out. The δ0 of the Yb:YAG disk with a mirror coating was reported32 to
be ~1.7x10-4. Assuming that the reflection loss factor on the 450 HR mirror, which underwent
double reflections in the cavity, is δ1 = δ2 = 4.5x10-4, and the end mirror has δ3 = 1.5x10-4, we
obtain Aa ~ 740 that agrees with the measured value of 730. With the R>0.999 end mirror,
assuming δ3 = 1.0 x10-3 and other losses same as above, we obtain Aa=435, which agrees with the
measured value of 450. If the mirror losses can be further decreased to less than 10-5 using
available R>0.99999 mirrors, Δ ~ 2x10-4 and Aa~5,000 are achievable.
Demonstration of Amplified Radiaiton Pressure Propulsion (ARPP). This section reports the
demonstration of propelling a kg-
mass macro object with the
active-cavity ARPP. The
schematic diagram of the
experimental setup and an
infrared picture of demonstration
are presented in Fig. 5. The key
experimental components were
described in the previous section
and more details are provided in
the method section. The active
cavity was formed between the
R>0.999 HR mirror deposited on
the thin disk gain medium and an
end HR mirror with R>0.999 and
5-m radius of curvature mounted
on a mobile platform on a linear
air track. The circulating laser
power and radiation pressure
force were estimated from the
transmitted laser power through
the end HR mirror, as in the
previous section, and the
demonstration was recorded by an
infrared video camera.
The circulating laser power was
observed to be stable against not
only the motion along the laser
beam, but also the small side-to-
side swing. The mechanical force
Fmot was estimated from the
acceleration of the mobile
platform with the total mass
M=0.75 kg by analyzing the
recorded videos. An example of infrared photos of the demonstration is shown in Fig. 5b.
a
b
Figure 5. The ARPP demonstration that propelled a 0.75-kg
platform with a R>0.999 mirror by radiation pressure. (a)
Schematic diagram of the demonstration setup. The mobile platform
was floated on a linear air track and propelled by the radiation pressure
of the circulating laser power on the end HR mirror mounted on the
platform. (b) Infrared photo of the demonstration in Class 1000
cleanroom environment. The bright left portion was the thin disk laser
head. The CW power of the circulating laser beam (visible) here was
400 kW. The transmitted laser power through the HR mirror to the
right was ~250 W and invisible. The HR mirror mount area was
enclosed with a cube satellite enclosure, which is not shown in the
upper diagram. A spark from an exploding particle with a pop in the
beam path can be seen and resulted from laser-induced ignition and
combustion34.
8
Fig. 6 displays Fmot as a function of the
radiation pressure force Frad calculated
from the transmitted laser power through
the end HR mirror. The fitting result
was Fmot = 1.1Frad – 0.5 mN. The
mechanical force data aagree with the
radiation pressure force data within 10%
except a constant deviation (-0.5 mN).
We suppose that the deviation mainly
resulted from the small friction in gliding
on the air track and the uneven flatness
of the track.
Discussion
Once implemented in space, the active-
cavity ARPP is projected to open doors
to innovative space applications beyond
the reach of the conventional thruster25.
The ARPP operates propellant-free14-15,
thus the mission of the ARPP-propelled
spacecraft is not bound by the propellant carrying capacity. Furthermore, the ARPP has a potential
to propel spacecraft to speeds orders of magnitude larger than that obtainable with conventional
thrusters25,29.
However, there are many issues that can limit the performance of the ARPP. These issues include
the stability of and the diffraction in the resonator35 over the operation distances greater than
100km. Here technical challenges in achieving the required surface flatness and radius of
curvature the mirrors must be overcome. In addition, the maximum obtainable spacecraft velocity
with the ARPP is limited by the Doppler shift of the reflecting laser beam in the cavity, which
shifts the resonance wavelength out of the gain bandwidth. The gain bandwidth of the Yb:YAG
laser system36-37 at 1,030 nm is ~ 10 nm, which is ~ 1 % of the wavelength. If the ARPP is mainly
limited by the gain bandwidth, the theoretical limit on the ARPP spacecraft velocity is ~3 x 106
m/sec, which enables transporting payloads from Earth to Mars in days rather than a year25.29. The
bandwidth of the gain medium and thus the spacecraft velocity limit can be further increased by
adopting different materials, such as Yb3+-Lu2O3 ceramic38.
Finally, we consider the mechanism of the active cavity in enabling the ARPP to propel spacecraft.
We assume that the active cavity of length L consists of two HR mirrors with identical reflectance
R and a gain medium in between. The details of the calculation are presented in the method section.
Briefly, by changing the laser phase rather than amplitude, the circulating laser beam sustains near
constant power, when the cavity length changes dynamically. Then the kinetic energy KE2
imparted on the mirror 2 by the radiation pressure over a distance L-L0 is given by
Figure 6. Comparison between two types of force
measurements. The y-axis represents the mechanical force
estimated from the motion of the mobile platform equipped
with an end HR mirror by analyzing the infrared videos on the
propulsion demonstration. The x-axis represents the radiation
pressure force on the platform estimated from the transmitted
laser power through the end HR mirror.
9
, (9)
where Aa is given by Eq. 7. Currently, the largest reported Ap of passive cavities is 245,000 in the
PVLAS experiment equipped with a 3.3-m evacuated cavity39. The maximum achievable Aa of
active cavities, however, is projected to be smaller owing to the loss through the gain medium.
The largest reported32 Aa is 2,500 with a projected achievable Aa of 5,000, which is 7 times larger
than that achieved in the present work. With Aa=5,000, the specific thrust– the thrust per unit input
energy– of the ARPP with the active cavity is projected to be ~35 mN/kW that approaches ~50
mN/kW of the Hall thruster26. It is foreseeable that an ARPP system with Aa=5,000 operating at
1-MW pump power can be implemented in the near future. Such ARPP system will be able to lift
a 3.5-kg object on Earth and a 20-kg object on the Moon.
Methods
Specific thrust ratio. One of the important parameters for evaluating performance and efficiency
of thrusters is the specific thrust Fs: the ratio of the exhaust thrust or force to the energy required
for propelling the exhaust. The approximate comparison of Fs can be obtained by assuming that
the exhaust consists of identical particles including photons, and that the input energy to the
thruster is 100% converted to the particle kinetic energy. If the particle momentum, the kinetic
energy, and the rest mass are p, E, and m0; respectively, Fs is given by
, (10)
where dN/dt is the number of particles exhausted per second, and κ =1 for particle-based
propulsion and κ =2 for radiation propulsion because photons impart twice of the momentum upon
reflection. The Eq. 10 can be further transformed into25
. (11)
where β=v/c and v is the velocity of the exhaust particles. Eq. 11 is the unified equation for specific
thrust, which can be applied whether the exhaust consists of mass particles or reflected photons.
For non-relativistic particle propulsion, such as electric propulsion, with v<<c and β<<1, Eq.11 is
approximated as
. (12)
On the other hand, for radiation pressure propulsion with one reflection and v=c, β=1, the specific
thrust is given by
. (13)
10
If the photons are circulated Nref times between the photon source and the propelled spacecraft as
in ARPP, the ratio Rst of the specific thrust of the radiation pressure propulsion to the particle
propulsion is given by
. (14)
Experimental details for the radiation pressure measurement. The experimental setup of the
present measurement is illustrated in Fig. 2. Specifically, the present demonstration was based on
a thin disk laser system custom built by Dausinger-Guessen30. The active optical cavity was
structured with the dielectric HR mirror with R>0.999, which was deposited on the backside of a
thin disk medium and end HR mirrors with various reflectance and radius of curvature. The thin
disk was a 7% doped Yb:YAG crystal with 12 mm diameter and 0.22 mm thickness. The radius
of curvature of the HR mirror deposited on the backside of the thin disk was greater than 10 m.
The Yb:YAG crystal was pumped by 938 nm fiber-coupled pump laser diodes in a multipass
configuration30 as shown in Fig. 2. The HR mirror side of the thin disk gain medium was glued to
a diamond wafer heat sink that was cooled by water jets. The diamond heat sink is several times
more efficient in cooling the thin disk than a cupper heat sink and thus increases the maximum
achievable pump power density and circulating laser power.
We first investigated the effect of thermal convection of rising air heated by the HR mirror surface,
which was exposed to the circulating laser beam, on the radiation pressure force measurement. In
the experimental setup shown in Fig. 2a, the circulating laser beam path was directed vertically
from a HR mirror that was faced vertically and was placed on a vertically operated digital scale
similar to our earlier experimental arrangement29. In this configuration, the top surface of the HR
mirror on the scale can efficiently produce vertically rising heated air of which recoil can press the
mirror down. Radiation pressure force larger than ~1 mN, which is equivalent to the circulating
laser power of 150 kW, was readily demonstrated. The difference in the weights of the HR mirror
between laser power on and off was monitored to measure the total force from radiation pressure
and thermal convection pressure. Subsequently, it was observed that the mirror and the scale
heated up immediately, owing to circulating laser power absorption resulting in instability in scale
reading. By inserting a 450-g aluminum block as a heat sink between the HR mirror and the scale
pan, the instability was controlled. However after several minites of operation, the mirror and
heat sink were heated and there was a non-zero scale reading even after the laser beam was turned
off. The offset was interpreted to be resultant from the thermal convection effect of rising air on
the heated mirror and heat sink. The reading error at circulating laser powers of ~150 kW in this
setup was determined to be 10±3%.
We further investigated other experimental configurations that can minimize the thermal
convection effect. It was found that the thermal convection effect could be significantly reduced
by adopting a radiation pressure sensor31 manufactured by Scientech Inc., which consisted of a
horizontally operating digital scale as illustrated in Fig. 2b. The adaptation permitted the HR
mirror on the scale to face horizontally and thus significantly reduced the effect of the thermal
convection. The circulating laser power was determined by measuring the radiation pressure force
on a flat 45o HR mirror with R>0.999 mounted on the radiation pressure sensor. A small portion
11
of the circulating laser beam transmitted through the 45o HR mirror and absorbed by the mounting
head of the radiation pressure sensor heated the sensor assembly as illustrated in Fig. 2b. Once
the radiation pressure sensor was heated, it became unstable and malfunctioning within several
seconds of operation. The heating effects on the radiation power sensor was significantly reduced
by inserting a 450-g aluminum block as a heat sink between the HR mirror and the radiation
pressure sensor head. The long-lasting non-zero reading of the radiation power sensor after the
laser was powered off was not observable within the experimental error. The result shows that the
thermal convection effect on the horizontally faced mirror as in Fig. 2b is much less than that on
the vertically faced mirror as in Fig. 2a. The overall thermal convection effect on the radiation
pressure force measurement in this setup was determined to be less than 3% at circulating laser
powers up to 510 kW.
The transmittance of the end HR mirror was used for estimating the circulating laser power and
the radiation pressure force. To measure of the transmittance T of the end HR mirror, we first
generated a 100-W laser beam with power that was generated with an output coupler mirror with
R=0.99 from the thin disk laser head. Using this laser beam and a precision laser power meter, T
was determined by measuring the difference between the powers with and without the HR mirror
of interest. For example, T of the 0o R>0.999 HR mirror that was used for propulsion
demonstration below was determined to be 0.062±0.004%.
Experimental details for the demonstration of Amplified Radiation Pressure Propulsion
(AARP). The schematic diagram of the experimental setup and an infrared picture of the
propulsion demonstration is shown in Fig. 5. The circulating laser beam was generated in an active
cavity that consisted of the HR mirror deposited on the backside of the Yb:YAG thin disk gain
medium and a HR mirror with R>0.999 and 10 m radius of curvature, as described in previous
sections. Here, the end HR mirror was mounted on a platform with the total weight of 0.75 kg,
which was floated by air cushion on a linear air track to minimize friction in motion. The
circulating laser power was observed to be stable against the motion along the laser beam. During
the mirror movement, there was a side-to-side swing of the platform and the mirror, but the
circulating power was stable against it. In addition, deceleration of the moving platform to slow
down or stop was demonstration. Such a capability of ARPP is important for enabling propellant-
free manipulation and docking of spacecraft14,15,25.
The circulating laser power and radiation pressure force in the cavity druing the demonstration
was monitored by measuring the transmitted power through the end HR mirror with a laser power
meter. At circulating laser powers and power densities greater than 100 kW and 260 kW/cm2
respectively, numerous sparks with audible pops in the laser beam were observed. The sparks are
interpreted to result from laser induced ignition and combustion of micron-sized particles that were
emitted from the air track. A recent study34 showed that the laser induced ignition and combustion
requires CW laser power densities of several hundred kW/cm2 at 1064 nm for aluminum particles
with 1-10 µm diameter. The study results are in support of the present results. When the number
of particles increased, the amplification factor and circulating laser power decreased. Thus,
controlling the particle emission from the linear air track was crucial in achieving the high
amplification factors at recirculating powers greater than 100 kW. In Fig. 5b, the transmitted laser
power through the end HR mirror to the right was ~250 W and was too weak to be visible. The
12
HR mirror mount area on the mobile platform was enclosed with a cube satellite enclosure, not
shown in Fig. 5a. The enclosure is used to transform the platform to a mock cube satellite as well
as to cut off the scattered radiation from the HR mirror for improving the video quality.
The mechanism of the ARPP with a passive optical cavity. We consider the radiation-pressure
propelled motion of a HR mirror in a passive optical cavity of length L, which consists of two HR
mirrors. A laser beam is transmitted through the first mirror and builds a circulating laser beam in
the cavity and the cavity is in a vacuum. Here ri is the amplitude reflectivity, ti is the amplitude
transmission coefficient of the mirror i, where i =1 or 2. With the cavity operated in vacuum, the
roundtrip loss factor grt(k) of the laser signal amplitude is given by33
. (15)
The circulating laser signal amplitude, Ecir, is given by
, (16)
where Einc is the incidence laser signal amplitude. For a passive optical cavity without a gain
medium, the amplification factor Ap from circulating power gain is given by
. (17)
To illuminate the mechanism of the circulating power gain in the passive optical cavity, here we
assume that T1 = |t1|2 and R=|r1|2 = |r2|2 and
=
, (18)
where δr/2 is the amplitude phase change for one reflection. Then, Eq. 17 can be simplified to
, (19)
where
and FC is the coefficient of finesse that is given by
. (20)
For R~1, the finesse of an optical cavity F is approximated by33
. (21)
The kinetic energy KE2 imparted on the HR mirror 2 can be obtained by integrating the radiation
pressure force over L:
13
, (22)
where Pinc = |Einc|2 is the incident laser power, L0 is the initial length of the cavity before the laser
power is turned on, and c is the light velocity in a vacuum. Because the wavelength of the
incidence laser beam is fixed, the cavity transmits the incident laser power into the cavity only
when k’L is very near qπ, where q is some integer.
With high finesse cavities, F>>1, the integration in Eq. 22 is highly complex, but can be
approximated by summing the areas under resonance peaks of the function in the integral with the
height of 1. The number of peaks NP is given by
, (23)
where ΔLf is the free-range bandwidth in cavity length, which is given by
. (24)
By approximating the resonance peak as a triangle, the area S under each peak is given by
, (25)
where ΔLc is the cavity resonance bandwidth in length, which is given by
. (26)
Then, the integral part of Eq. 19 can be approximated by using Eqs. 23-26 as
. (27)
For large F, the KE2 can be approximated by
. (28)
Assuming T ~ (1 – R) <<1, Eq. 28 can be further simplified to
. (29)
Eq. 29 shows that the kinetic energy of spacecraft delivered by the ARPP with a passive cavity is
smaller than that by the single-reflection laser sail with the incident laser beam. Therefore, it is
14
proved that the ARPP with a passive cavity is impractical.
The mechanism of the ARPP with an active optical cavity. We consider the radiation-pressure
propelled motion of a HR mirror in an active optical cavity of length L, which consists of two HR
mirrors and a gain medium in between. We further assume that the cavity operates in a vacuum.
A circulating laser beam is generated by the gain medium within the cavity. The circulating laser
signal amplitude, Ecir, of the active cavity is given by
, (30)
where Epum is the pump laser signal amplitude and ε is the pumping efficiency in amplitude of the
gain medium. The roundtrip gain factor in the laser signal amplitude grt is given by33
, (31)
where
, φ is the sum of all roundtrip phase changes from reflections and through the
gain medium, δm is the roundtrip power gain factor, and Δ is the total roundtrip loss factor including
that through the gain medium. Δ is given by
, (32)
where δ0 is the roundtrip loss factor through the gain medium including the reflection loss on the
HR mirror deposited on the backside of the thin disk, and δ1,2,3… = (1 - R1,2,3…), where Rj is the
reflectance of the jth mirror surface.
The amplification factor Aa from circulating power gain in the active cavity is given by
, (33)
where and η =ε2, the pump efficiency in power. The δm in turn is a function of Pcir and given by33
, (34)
where
is the unsaturated power gain factor and Psat is the saturation power of the gain medium.
Then, the kinetic energy KE2 imparted on the mirror 2 of the active cavity can be obtained by
integrating the radiation pressure force over L as
. (35)
15
Here, Aa is a function of not only L, but also Pcir or Ppum, and thus evaluating the integral in Eq. 35
is highly complicated. The detailed quantum electronics theory on the circulating power stability
of the active optical cavity against the mirror movement has yet to be established. The integration,
however, can be simplified by noting that at steady-state when the cavity length changes in the
active cavity, the resonance condition and the circulating power is adiabatically sustained by
changing the phase of the laser beam, yet keeping the laser amplitude constant33. In this case, the
conservation of energy states that the energy loss out of and the energy pumped into the cavity
should be the same at steady state:
. (36)
Then, the amplification factor Aa is given by
, (37)
and
. (38)
Therefore, the kinetic energy of the moving HR mirror is larger by a factor of 1/Δ than that of the
single-reflection laser sail, and thus the ARPP with the active cavity is viable, when Δ<<1.
Acknowledgements
The work was supported by NASA Innovative Advanced Concepts (NIAC) grant, NNX13AR27G
and IR&D program of Y.K. Bae Corp. The author acknowledges the enthusiastic contributions
and discussions by Dr. Hagop Injeyan, Mr. Lawrence Williams, Mr. Joseph Harkenrider, Dr. Joerg
Neuhaus, and Mr. Robert Lee.
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