U.S. patent application number 17/127616 was filed with the patent office on 2022-05-19 for optical local oscillator for all-optical time scales, and associated timekeeping methods.
The applicant listed for this patent is Government of the United States of America as represented by the Secretary of Commerce, The Regents of the University of Colorado, a body corporate, Government of the United States of America as represented by the Secretary of Commerce. Invention is credited to Tobias Bothwell, Terry Brown, Dhruv Kedar, Colin J. Kennedy, William R. Milner, Eric G. Oelker, John M. Robinson, Jun Ye.
Application Number | 20220155730 17/127616 |
Document ID | / |
Family ID | |
Filed Date | 2022-05-19 |
United States Patent
Application |
20220155730 |
Kind Code |
A1 |
Ye; Jun ; et al. |
May 19, 2022 |
OPTICAL LOCAL OSCILLATOR FOR ALL-OPTICAL TIME SCALES, AND
ASSOCIATED TIMEKEEPING METHODS
Abstract
The frequency stability of an optical local oscillator is
improved by locking a laser to a silicon Fabry-Perot cavity
operating at a temperature near 124 K, where the coefficient of
thermal expansion of silicon is near zero. The cavity is mounted
inside a cryostat housed in a temperature-stabilized vacuum system
that is surrounded by an isolating enclosure and supported by an
active vibration platform. Laser light is steered with a
superpolished mirror toward a superpolished focusing optic that
couples the laser light into the cavity. Light reflected from the
cavity is used to stabilize the laser via the Pound-Drever-Hall
technique, while light transmitted through the cavity is used to
stabilize the laser power. A resonant transimpedance amplifier
allows the laser power to be reduced, which reduces heating of the
cavity caused by residual absorption of the light.
Inventors: |
Ye; Jun; (Louisville,
CO) ; Oelker; Eric G.; (Boulder, CO) ; Milner;
William R.; (Boulder, CO) ; Robinson; John M.;
(Henderson, NV) ; Kennedy; Colin J.; (Boulder,
CO) ; Bothwell; Tobias; (Boulder, CO) ; Kedar;
Dhruv; (Boulder, CO) ; Brown; Terry;
(Lafayette, CO) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
The Regents of the University of Colorado, a body corporate
Government of the United States of America as represented by the
Secretary of Commerce |
Denver
Gaithersburg |
CO
MD |
US
US |
|
|
Appl. No.: |
17/127616 |
Filed: |
December 18, 2020 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
62949850 |
Dec 18, 2019 |
|
|
|
International
Class: |
G04F 5/14 20060101
G04F005/14; H03L 7/26 20060101 H03L007/26 |
Goverment Interests
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
[0002] This invention was made with government support under grant
number FA9550-19-1-0275 awarded by the U.S. Air Force; grant number
PHY1734006 awarded by the National Science Foundation; grant number
D18AC00037 awarded by DOD/DARPA; and grant number W911NF-16-1-0576
awarded by the U.S. Army Research Office. The government has
certain rights in the invention.
Claims
1. A timekeeping method, comprising: supporting a Fabry-Perot
cavity inside a cryostat housed in a temperature-stabilized vacuum
system that is surrounded by an isolating enclosure and supported
by an active vibration platform; stabilizing a temperature of the
Fabry-Perot cavity to a zero-expansion temperature at which a
coefficient of thermal expansion of a material forming a spacer of
the Fabry-Perot cavity is near zero; steering, with at least one
superpolished mirror, laser light toward at least one superpolished
focusing optic that couples the laser light into the Fabry-Perot
cavity; locking a frequency of the laser light to the Fabry-Perot
cavity based on laser light reflected from the Fabry-Perot cavity;
and stabilizing a power of the laser light based on laser light
transmitted through the Fabry-Perot cavity.
2. The timekeeping method of claim 1, further comprising actively
suppressing residual amplitude modulation produced by a sideband
generator modulating the laser light.
3. The timekeeping method of claim 1, further comprising:
supporting both the at least one superpolished mirror and the at
least one superpolished focusing optic with the active vibration
platform; and reducing, with the active vibration platform,
external vibrations of the cryostat, the at least one superpoli
shed mirror, and the at least one superpolished focusing optic.
4. (canceled)
5. The timekeeping method of claim 1, further comprising
cryogenically cooling the Fabry-Perot cavity with the cryostat.
6. The timekeeping method of claim 5, further comprising: thermally
isolating the Fabry-Perot cavity with at least one radiation shield
of the cryostat; and transmitting the laser light through an upper
window and a lower window of each of the at least one radiation
shield, each of the upper window and the lower window being
anti-reflection coated, wedged, and tilted at an angle relative to
an optical axis of the Fabry-Perot cavity.
7. The timekeeping method of claim 6, further comprising actively
stabilizing a temperature of the at least one radiation shield.
8-10. (canceled)
11. The timekeeping method of claim 5, wherein said cryogenically
cooling includes cryogenically cooling the Fabry-Perot cavity to a
temperature at or below 4 K.
12. The timekeeping method of claim 5, wherein said stabilizing the
temperature of the Fabry-Perot cavity includes stabilizing the
temperature of the Fabry- Perot cavity to a setpoint temperature
that is within 10 mK of the zero-expansion temperature.
13. The timekeeping method of claim 1, further comprising:
detecting the laser light reflected from the Fabry-Perot cavity
with a reflection photodetector whose optical window is removed;
and detecting the laser light transmitted through the Fabry-Perot
cavity with a transmission photodetector whose optical window is
removed.
14. The timekeeping method of claim 13, further comprising
converting, with a resonant transimpedance amplifier that includes
a phase shifter, a current outputted by the reflection
photodetector into a voltage.
15. The timekeeping method of claim 1, further comprising
reflecting one microwatt, or less, of the laser light from the
Fabry-Perot cavity.
16-17. (canceled)
18. The timekeeping method of claim 1, further comprising coupling,
with the at least one superpolished focusing optic, the laser light
into the Fabry-Perot cavity.
19. The timekeeping method of claim 1, further comprising actively
stabilizing a temperature of the temperature-stabilized vacuum
system.
20. The timekeeping method of claim 1, further comprising
periodically steering the frequency of the laser light based on an
atomic frequency standard.
21. An optical local oscillator, comprising: a Fabry-Perot cavity
comprising a spacer formed from a spacer material; a cryostat that
supports the Fabry-Perot cavity and stabilizing a temperature of
the Fabry-Perot cavity to a zero-expansion temperature at which a
coefficient of thermal expansion of the spacer material is near
zero; a temperature-stabilizable vacuum system housing the
cryostat; an active vibration platform that supports the
temperature-stabilizable vacuum system; an isolating enclosure that
surrounds the temperature-stabilizable vacuum system; at least one
superpolished focusing optic that couples laser light into the
Fabry-Perot cavity; at least one superpolished mirror that steers
the laser light toward the at least one superpolished focusing
optic; a transmission photodetector that detects laser light
transmitted through the Fabry-Perot cavity; and a reflection
photodetector that detects laser light reflected from the
Fabry-Perot cavity.
22. The optical local oscillator of claim 21, the at least one
superpolished mirror having a residual surface roughness of less
than one angstrom root-mean-square; and each of one or more
surfaces of the at least one superpolished focusing optic having a
residual surface roughness less than one angstrom
root-mean-square.
23. The optical local oscillator of claim 21, the cryostat further
comprising at least one radiation shield that thermally isolates
the Fabry-Perot cavity.
24. The optical local oscillator of claim 23, the at least one
radiation shield comprising copper.
25. The optical local oscillator of claim 21, wherein: the spacer
material comprises single-crystal silicon; and a wavelength of the
laser light is between 1500 and 1600 nm.
26. The optical local oscillator of claim 21, further comprising a
resonant transimpedance amplifier that converts a current outputted
by the reflection photodetector into a voltage, the resonant
transimpedance amplifier including a phase shifter and having a Q
of 10 of more.
Description
RELATED APPLICATIONS
[0001] This application claims priority to U.S. Provisional Patent
Application No. 62/949,850, filed Dec. 18, 2019 and titled "Optical
Local Oscillator for All-Optical Time Scales", the entirety of
which is incorporated herein by reference.
SUMMARY
[0003] A time scale is a type of timekeeping system that generates
a highly stable timing signal for metrological applications. Time
scales are typically operated by national metrology institutes
(e.g., the National Institute of Standards and Technology) as part
of a constructed time standard. For example, international atomic
time (TAI) is a constructed time standard formed from a weighted
average of timing signals generated by over 400 time scales located
in more than 50 laboratories around the world. TAI is the basis for
coordinated universal time (UTC), which is the primary time
standard used to regulate clocks and times throughout the
world.
[0004] A time scale typically includes an ensemble of atomic clocks
whose outputs are averaged together into a single timing signal
that is more stable than those generated by the individual clocks
themselves. Each atomic clock outputs a clock signal derived from
an atomic frequency reference that periodically steers the
frequency of a local oscillator (LO) to an atomic transition
frequency. Thus, the atomic frequency reference is a stable
oscillator that serves as the timing element for the atomic clock.
For averaging times less than that of the periodic steering (e.g.
10.sup.5 seconds), the stability of the atomic frequency reference
is determined by the LO. At longer averaging times, the stability
of the atomic frequency reference is typically limited by LO phase
noise, duty cycle of the atomic frequency reference,
signal-to-noise ratio of the atomic frequency reference, and/or
drift.
[0005] Many prior-art atomic frequency references, such as cesium
and rubidium fountains, use atomic transitions whose frequencies
lie in the microwave region of the electromagnetic spectrum.
However, optical frequency references, which use atomic transitions
in the infrared, optical, or ultraviolet regions of the
electromagnetic spectrum, typically achieve orders-of-magnitude
greater frequency stability and accuracy than their microwave-based
counterparts. Since all the atomic clocks currently used for TAI
utilize microwave frequency references, a transition to optical
clocks (which utilize optical frequency references) can greatly
enhance both the accuracy and stability of TAI, and thus UTC. Such
a transition can also advantageously reduce the size and complexity
of time scales by reducing the number of atomic clocks that are
needed in an ensemble to achieve a desired level of timing
error.
[0006] To fully incorporate the superior accuracy and stability of
an optical frequency reference into a time scale, the optical
frequency reference would ideally operate continuously (i.e., with
no periods of downtime). However, many prior-art optical frequency
references can only operate at a maximum duty cycle of
approximately 25% duty cycle (i.e., 6 hours/day). During each
period of downtime, which can last one day or more, the timing
error is determined primarily by the free-running frequency
stability of an optical local oscillator (OLO) whose frequency is
steered to an optical transition by the optical frequency
reference. Thus, one way to reduce the timing error of an optical
clock is to improve the frequency stability of the OLO so that less
timing error accumulates during the periods of downtimes.
[0007] The present embodiments feature an optical local oscillator
(OLO) with improved frequency stability, particularly at averaging
times around typical downtime durations (e.g., 10.sup.3 to 10.sup.6
seconds). As described in more detail below, the fractional
frequency instability of the OLO has been demonstrated to be below
10.sup.15 for averaging times up to 6.times.10.sup.5 seconds
(without steering to an atomic frequency reference), more than an
order of magnitude better than prior-art OLOs. The OLO may be
combined with an optical frequency reference (e.g., a .sup.87Sr
optical lattice) and optical frequency comb to produce an optical
clock that outperforms state-of-the-art microwave LOs steered by
either microwave or optical frequency standards. As such, the
present embodiments advantageously enable both the improvement of
prior-art microwave-based time scales, and the development of new
all-optical time scales.
[0008] Some embodiments of the OLO are based on a silicon
Fabry-Perot cavity operating at a temperature near 124 K. OLOs
based on cryogenic silicon reference cavities (e.g., Fabry-Perot
cavities) have demonstrated frequency stabilities better than all
other prior-art free-running OLOs at averaging times below 10.sup.4
seconds. To further improve the frequency stability, the present
embodiments use superpolished optics to reduce scatter, thermal
control of the environment (i.e., a vacuum chamber housing the
silicon Fabry-Perot cavity) to limit the effect of parasitic
etalons, and active power stabilization of light transmitted
through the Fabry-Perot cavity to reduce frequency excursions
arising from laser intensity fluctuations. The present embodiments
also include a resonant transimpedance amplifier that may be used
to reduce the amount of light used to lock a laser to the
Fabry-Perot cavity.
[0009] As an experimental demonstration of the present embodiments,
an OLO steered daily by an .sup.87Sr optical lattice clock over a
thirty-four-day data campaign accumulated an estimated time error
of only 48.+-.94 ps. During the campaign, the frequency stability
of the OLO surpassed that of hydrogen masers in the UTC(NIST) time
scale at all averaging times out to multiple days, demonstrating
the requisite stability for improved time-scale performance. The
continuous availability of the OLO coupled with the on-demand
performance of the optical lattice clock makes optical clocks
viable for future inclusion in existing time scales, thereby
allowing these time scales to advantageously harness the improved
accuracy and stability of optical frequency standards.
[0010] In embodiments, a timekeeping method includes stabilizing a
temperature of a Fabry-Perot cavity to a zero-crossing temperature
at which a coefficient of thermal expansion of a material forming a
spacer of the Fabry-Perot cavity is near zero. The Fabry-Perot
cavity is mounted inside a cryostat housed in a
temperature-stabilized vacuum system that is surrounded by an
isolating enclosure and supported by an active vibration platform.
The timekeeping method also includes steering, with at least one
superpolished mirror, laser light toward at least one superpolished
focusing optic that couples the laser light into the Fabry-Perot
cavity. The timekeeping method also includes locking a frequency of
the laser light to the Fabry-Perot cavity based on laser light
reflected from the Fabry-Perot cavity. The timekeeping method also
includes stabilizing a power of the steered laser light based on
laser light transmitted through the Fabry-Perot cavity.
BRIEF DESCRIPTION OF THE FIGURES
[0011] FIG. 1 is a functional diagram of an optical local
oscillator (OLO) that may be combined with an optical frequency
reference to form an optical clock, in an embodiment.
[0012] FIG. 2 is a cross-sectional side view of a Fabry-Perot
cavity mounted in a cryostat, in an embodiment.
[0013] FIG. 3 shows an optical clock that combines the OLO 100 of
FIG. 1 with an optical frequency reference to provide stable and
accurate timing for averaging times out to several months, or more,
in embodiments.
[0014] FIG. 4 shows an optical clock that is similar to the optical
clock of FIG. 3 except that the optical frequency reference
operates without frequency shifters and a Fabry-Perot cavity, in an
embodiment.
[0015] FIG. 5 shows a frequency record of the OLO of FIG. 1 over a
thirty-four-day data campaign.
[0016] FIG. 6 shows the integrated time error of optical clock of
FIG. 3, as determined from the residuals of the frequency record of
FIG. 5.
[0017] FIG. 7 is a plot of the fractional frequency instability of
the OLO of FIG. 1 as a function of averaging time.
[0018] FIG. 8 is a plot of the expected fractional frequency
instability of the optical clock of FIG. 3 as a function of
averaging time.
[0019] FIG. 9 is a schematic of a resonant transimpedance
amplifier, in an embodiment.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0020] FIG. 1 is a functional diagram of an optical local
oscillator (OLO) 100 that may be combined with an optical frequency
reference to form an optical clock (e.g., see the optical clock 300
of FIG. 3). The OLO 100 includes a single-frequency OLO laser 110
that is locked to a resonance of a Fabry-Perot cavity 106 to both
stabilize a frequency of the OLO laser 110 and narrow a linewidth
of the OLO laser 110. That is, the frequency of the OLO 110, when
free-running, is less stable than a length of the Fabry-Perot
cavity 106 (see cavity length 210 of FIG. 2). To increase the
stability of the length of the Fabry-Perot cavity 106, the
Fabry-Perot cavity 106 is mounted inside a cryostat 104 that cools
the Fabry-Perot cavity 106 to a cryogenic temperature. More details
about the cryostat 104 are presented below.
[0021] In FIG. 1, the OLO laser 110 is locked to the Fabry-Perot
cavity 106 via the Pound-Drever-Hall (PDH) technique. Specifically,
a laser output 111 of the OLO laser 110 passes through an optical
isolator 112, an amplitude modulator 114 controlled by an amplitude
feedback signal 123 to change an amplitude of the laser output 111,
and a frequency shifter 116 controlled by a frequency feedback
signal 121 to change a frequency of the laser output 111. The
frequency shifter 116 and the amplitude modulator 114 change the
laser output 111 into an OLO output 102 that is more phase-stable,
frequency-stable, and amplitude-stable than the laser output 111.
While FIG. 2 shows the OLO laser 110 being locked to the
Fabry-Perot cavity 106 using the PDH technique, the OLO laser 110
may be locked to the Fabry-Perot cavity 106 using another technique
(e.g., Hansch-Couillaud locking, frequency-modulation spectroscopy)
without departing from the scope hereof.
[0022] To implement the PDH technique, sampled light 119 from the
OLO output 102 passes through a sideband generator 118 that, when
driven by a generator 124, creates frequency sidebands on the
sampled light 119. The sideband generator 118 is, for example, an
electro-optic modulator that phase modulates the sampled light 119.
As known in the art, optical materials (e.g., nonlinear crystals)
used for the sideband generator 118 may form an etalon that
produces residual amplitude modulation (RAM). Techniques known in
the art may be used to actively suppress (e.g., minimize) the RAM
produced by the sideband generator 118.
[0023] Modulated light 125 outputted by the sideband generator 118
is linearly polarized (e.g., horizontally) such that it reflects
off a polarized beamsplitter 126 into the Fabry-Perot cavity 106 as
cavity input light 150. A quarter-wave plate 128 converts the
polarization of the modulated light 125 from linear to circular. At
least one focusing optic 130 couples the cavity input light 150
into the Fabry-Perot cavity 106 such that the cavity input light
150 is mode-matched to the Fabry-Perot cavity 106. The cavity input
light 150 is preferably mode-matched to the TEM.sub.00 mode.
However, the cavity input light 150 may be coupled to another mode
(e.g., the TEM.sub.01 mode) without departing from the scope
hereof.
[0024] When the frequency of the sampled light 119 (i.e., a
frequency of the carrier of the modulated light 125) is resonant
with a resonance of the Fabry-Perot cavity 106, light builds up and
circulates inside the Fabry-Perot cavity 106. Due to the finite
reflectivities of the mirrors forming the Fabry-Perot cavity 106
(see mirror coatings 206 in FIG. 2), which introduce loss, some of
the light circulating inside the Fabry-Perot cavity 106 leaks out
of the Fabry-Perot cavity 106. In FIG. 2, transmitted light 154
leaks out of one end of the Fabry-Perot cavity 106, where it is
detected by a transmission photodetector 138. At the other end of
the Fabry-Perot cavity 106, cavity output light 152 propagates in a
direction opposite to that of the cavity input light 150. The
cavity-output light 152 has two components: (1) non-resonant cavity
input light 150 that reflects off an input of the Fabry-Perot
cavity 106, and (2) resonant cavity input light 150 that circulates
inside the Fabry-Perot cavity 106 and leaks out the input of the
Fabry-Perot cavity 106. The focusing optic 130 collimates the
cavity output light 152, and the quarter-wave plate 128 converts
the polarization of the cavity output light 152 from circular to
linear (e.g., vertically) such that it passes through the polarized
beamsplitter 126. The cavity output light 152 is then detected by a
reflection photodetector 134.
[0025] A modulated reflection signal 127 outputted by the
reflection photodetector 134 is demodulated by a frequency servo
120 (e.g., by a double-balanced mixer inside the frequency servo
120) at the same frequency used to drive the sideband generator
118. Thus, the frequency servo 120 is also driven by the generator
124, and a phase shifter (not shown) may be controlled to select
between in-phase and quadrature components of the modulated
reflection signal 127. The in-phase component, which crosses zero
where the frequency of the sampled light 119 equals a resonant
frequency of the resonance of the Fabry-Perot cavity 106, is
filtered and processed (e.g., via proportional, integral, and
derivative terms) to generate the frequency feedback signal 121
that controls the frequency shifter 116.
[0026] The frequency shifter 116 may be an acousto-optic modulator
(AOM) driven by a radio-frequency (RF) signal whose frequency is
selected according to the frequency feedback signal 121. The RF
signal may be generated by a voltage-controlled oscillator driven
by the frequency feedback signal 121, wherein the frequency
feedback signal 121 is an analog signal. Alternatively, the RF
signal may be digitally generated (e.g., via direct digital
synthesis) and frequency-controlled, wherein the frequency feedback
signal 121 is a digital signal. However, the frequency shifter 116
may be any other type of frequency shifter without departing from
the scope hereof, such as an electro-optical modulator driven to
implement serrodyne frequency shifting, or frequency shifting via a
time-varying Pancharatnam phase. Frequency shifter 116 may also be
implemented as a laser that is phase locked to the laser output 111
at an offset frequency determined by the frequency feedback signal
121.
[0027] The frequency shifter 116 responds to the frequency feedback
signal 121 quickly enough to ensure that the laser output 111 can
be locked to the Fabry-Perot cavity 106 with a bandwidth high
enough to narrow the linewidth of the laser output 111. Typically,
a locking bandwidth larger than the free-running linewidth of the
OLO laser 110 is selected. For example, when the OLO laser 110 is a
distributed feedback (DFB) fiber laser with a free-running
linewidth of 100 Hz, and the frequency shifter 116 is an AOM, a
locking bandwidth greater than 100 kHz can be achieved. When the
OLO laser 110 contains a mechanism for controlling the frequency of
the laser output 111, all or part of the frequency feedback signal
121 may be applied directly to the OLO laser 110. For example, the
OLO laser 110 may contain an internal lasing cavity whose length is
controllable via a piezoelectric transducer to change the frequency
of the laser output 111. In this case, the external frequency
shifter 116 may not be needed. However, the frequency-control
mechanism internal to the OLO laser 110 may not be fast enough to
achieve a sufficiently high locking bandwidth. In this case, the
frequency-control mechanism may be used for low-frequency steering
of the laser output 111, and the frequency shifter 116 may be used
for high-frequency steering of the laser output 111.
[0028] An amplitude servo 122 stabilizes a power of the transmitted
light 154, which advantageously reduces frequency drift of the OLO
output 102 that arises from an intensity-dependent frequency shift
of the cavity resonances. The amplitude servo 122 compares a
transmission signal 129 outputted by the transmission photodetector
138 to a stable DC reference signal (e.g., as generated by a
voltage reference) to generate an error signal. The error signal is
then filtered and processed (e.g., via proportional, integral, and
derivative terms) to generate the amplitude feedback signal 123
that controls the amplitude modulator 114. The amplitude modulator
114 may be, for example, an electrooptic amplitude modulator (i.e.,
an electrooptic polarization rotator followed by a polarizer).
Alternatively, the amplitude modulator 114 may be an AOM driven by
an RF signal whose amplitude is selected according to the amplitude
feedback signal 123. The amplitude of the RF signal may be
controlled with a voltage-controlled variable attenuator, wherein
the amplitude feedback signal 123 is an analog signal.
Alternatively, the amplitude of the RF signal may be controlled
digitally (e.g., via a digital step attenuator), wherein the
amplitude feedback signal 123 is a digital signal. In some
embodiments, both the amplitude modulator 114 and the frequency
shifter 116 are implemented as one AOM driven by one RF signal
whose amplitude is selected according to the amplitude feedback
signal 123, and whose frequency is selected according to the
frequency feedback signal 121.
[0029] An optical isolator 136 may be placed in front of the
transmission photodetector 138 to prevent scattered light (i.e.,
diffuse and specularly reflected light) from the transmission
photodetector 138 from propagating back into the Fabry-Perot cavity
106. Similarly, an optical isolator 132 may be placed in front of
the reflection photodetector 134 to prevent scattered light from
the reflection photodetector 134 from propagating back into the
Fabry-Perot cavity 106. Furthermore, the transmission photodetector
138 may be oriented at a slight angle with respect to an optical
axis of the transmitted light 154 (i.e., the transmitted light 154
does not impinge of the transmission photodetector 138 at normal
incidence) so that the specular reflection is steered away from the
Fabry-Perot cavity 106. The reflection photodetector 134 may be
similarly oriented.
[0030] The Fabry-Perot cavity 106 and cryostat 104 of FIG. 1 may be
provided as a stand-alone system that is configured to cooperate
with the remaining components of the OLO 100 (which may be provided
by a third party).
[0031] FIG. 2 is a cross-sectional side view of the Fabry-Perot
cavity 106 mounted in the cryostat 104. The Fabry-Perot cavity 106
is formed from a spacer 202 to which a first mirror 204(1) and a
second mirror 204(2) are affixed (e.g., via optical contact
bonding) in a counter-facing configuration. One or both of the
mirrors 204 may be convex with a radius of curvature selected such
that the Fabry-Perot cavity 106 forms a stable optical resonator.
Each of the mirrors 204 has deposited thereon a high-reflectivity
coating 206 that is selected to give the Fabry-Perot cavity 106 a
high finesse. For example, each high-reflectivity coating may be a
dielectric stack of alternating layers of SiO.sub.2 and
Ta.sub.2O.sub.5. The finesse may be, for example, 500,000 or more.
Alternatively, each of the mirrors 204 may be a crystalline mirror
that has reduced thermal noise by operating without a
high-reflectivity dielectric coating. The cavity length 210, equal
to a distance between the high-reflectivity coatings 206, is
typically between 1 cm and 50 cm. However, the cavity length 210
may be less than 1 cm, or more than 50 cm, without departing from
the scope hereof. In general, a longer cavity length 210 improves
the fractional frequency stability of the OLO 100, although the
corresponding larger size of the Fabry-Perot cavity 106 may
negatively increase sensitivity of the Fabry-Perot cavity 106 to
external vibrations.
[0032] The cryostat 104 uses liquid nitrogen to cool the
Fabry-Perot cavity 106 to a zero-expansion temperature where a
coefficient of thermal expansion (CTE) of the material forming the
spacer 202 crosses zero. Operating the Fabry-Perot cavity 106 near
the zero-expansion temperature advantageously reduces the impact of
temperature fluctuations on the cavity length 210, which increases
the frequency stability of the OLO 100 (i.e., reduces the
fractional frequency instability of the frequency of the OLO output
102, when locked to the Fabry-Perot cavity 106). For mirrors 204 to
be contact-bonded to the spacer 202, substrates of the mirrors 204
are formed from the same material as the spacer 202.
[0033] In one embodiment, each of the spacer 202, the first mirror
204(1), and the second mirror 204(2) is machined from one piece of
single-crystal silicon. As a material with a well-defined
crystalline structure, single-crystal silicon has several
advantages over other amorphous materials typically used to form
spacers and mirror substrates for Fabry-Perot cavities (e.g.,
ultra-low expansion glass, Zerodur, Pyrex, fused silica, etc.).
First, silicon has a zero-expansion temperature near 124 K (and
additionally at 17 K), a temperature easily accessible with the
cryostat 104. Second, crystalline silicon has a high Young's
modulus, and thus can be formed into a structure with a high
mechanical Q that suppresses fluctuations in the cavity length 210
due to Brownian motion. Third, silicon is transparent in the
infrared. When the OLO laser 110 is, for example operates between
1500 and 1600 nm, the substrates of the mirrors 204 will be
transparent, allowing the cavity input light 150 to couple into the
Fabry-Perot cavity 106, and allowing the cavity output light 152
and the transmitted light 154 to couple out of the Fabry-Perot
cavity 106, with minimal absorption by the substrates.
[0034] Each piece of single-crystal silicon may be machined with
the <111> axis parallel to a cavity axis of the Fabry-Perot
cavity 106 (i.e., parallel to the cavity length 210). The Young's
modulus of silicon is highest along this direction, and thus this
orientation yields the highest-Q mechanical structure, as compared
to other orientations (e.g., the <100> and <110>
orientations).
[0035] As shown in FIG. 3, the cryostat 104 stabilizes the
Fabry-Perot cavity 106 at a temperature near 124 K. At least two
thermal shields 212 increase the stability of the cavity length 210
by thermally isolating the Fabry-Perot cavity 106 from a
room-temperature vacuum chamber 218 that houses the cryostat 104
and the Fabry-Perot cavity 106. In FIG. 2, an inner thermal shield
212(1) and an outer thermal shield 212(2) are both maintained at
temperatures near 124 K using nitrogen gas that is heated from
liquid nitrogen (at 77 K) in a dewar, and transferred via hoses to
the vacuum chamber 218. Each of the inner and outer thermal shields
212 may be made of copper. The Fabry-Perot cavity 106 and thermal
shields 212 are mounted inside the vacuum chamber 218 with
thermally isolated, cryogenically compatible materials (e.g., G10)
to minimize heat flow between the thermal shields 212, between the
inner thermal shield 212(1) and the Fabry-Perot cavity 106, and
between the outer thermal shield 212(2) and the vacuum chamber 218.
Although FIG. 2 shows the cryostat 104 with only two thermal
shields 212, additional thermal shields 212 may be used without
departing from the scope hereof.
[0036] When the cryostat 104 is configured as described above, the
Fabry-Perot cavity 106 is thermally connected to the ambient
environment (i.e., room temperature) via an effective low-pass
filter whose time constant is 9.8 days, or 850,000 seconds. This
time constant is large enough to ensure that fluctuations of the
cavity length 210 due to the thermal fluctuations in the ambient
environment contribute less than 4.times.10.sup.-16 (for a one-day
averaging time) to the fractional frequency instability of the OLO
100, assuming that the temperature of the Fabry-Perot cavity 106 is
4.5 mK away from the true zero-crossing temperature.
[0037] The cryostat 104 may be alternatively configured to
stabilize the Fabry-Perot cavity 106 at a temperature near absolute
zero (e.g., 4 K), where the CTE of silicon asymptotically
approaches zero. In this case, the cryostat 104 is a closed-cycle
cryocooler that uses liquid helium. At least three thermal shields
212 are used to maintain successively lower temperatures from the
outside to the inside of vacuum chamber 218. For example, an outer
thermal shield 212 may be cooled to a temperature between 30 and 40
K, a middle thermal shield 212 may be cooled to 3.5 K, and an inner
thermal shield 212 may be held near 3.75 K. The middle thermal
shield 212 may be actively cooled via thermal contact with a
mechanically flexible cold finger (not shown in FIG. 2). In
addition, a large thermal mass may be placed between the cold
finger and the middle thermal shield 212 to passively suppress
temperature fluctuations of the cold finger. The large thermal mass
may be made, for example, from holmium copper (HoCu.sub.2), which
has a specific heat 2500 times larger than that of copper at 4
K.
[0038] To further increase the stability of the cavity length 210,
when operating near 4 K, the outer thermal shield 212 may be
actively temperature stabilized. Furthermore, the outer thermal
shield 212 may be made of copper, whose thermal conductivity at 40
K is 100 times larger than that of aluminum. A high thermal
conductivity helps ensure uniformity of the temperature of the
outer thermal shield 212, and thus reduces temperature gradients
across the Fabry-Perot cavity 106. Each of the inner and middle
thermal shields 212 may also be actively temperature stabilized by
controlling a resistive heater affixed thereto. Each of the inner
and middle thermal shields 212 may be made from copper, aluminum,
or another material with high thermal conductivity, high thermal
mass, and high emissivity.
[0039] When the outer thermal shield 212 is made of copper, and
each of the middle and inner thermal shields 212 is made from
aluminum, the thermal time constant of the Fabry-Perot cavity 106
is approximately 2000 seconds, which is smaller than the time
constant achieved when stabilizing the Fabry-Perot cavity 106 near
124 K. The discrepancy is due to the fact that most materials tend
to have smaller thermal masses at temperatures approaching absolute
zero.
[0040] Regardless of the temperature at which the Fabry-Perot
cavity 106 is stabilized, the vacuum chamber 218 may be evacuated
with a pump to a pressure below 10.sup.-9 mbar. To minimize
sensitivity to seismic motion and external vibrations caused by
vacuum equipment connected to the cryostat 104 (e.g., vacuum
pumps), it is preferable to mount the Fabry-Perot cavity 106
vertically (i.e., with the cavity length 210 parallel to gravity,
as shown in FIG. 2). When the Fabry-Perot cavity 106 is made of
<111> silicon, it is also preferable to support the
Fabry-Perot cavity 106 at three points near its midplane (i.e.,
halfway between the first and second mirrors 204) that align with
the 120 degree rotational symmetry of axes of the silicon. However,
the Fabry-Perot cavity 106 may be mounted in another configuration
without departing from the scope hereof.
[0041] To further reduce temperature fluctuations, the vacuum
chamber 218 may be actively temperature-stabilized. The vacuum
chamber 218 may also be surrounded by an isolating enclosure 270
that blocks air currents in the ambient environment from directly
impinging on the vacuum chamber 218, and thus improves the
temperature stabilization of the vacuum chamber 218 by isolating
the temperature of the vacuum chamber 218 from temperature
fluctuations of the surrounding air. The isolating enclosure 270
also improves the stability of the optical path length between the
OLO laser 110 and the vacuum chamber 218. The isolating enclosure
270 may have one or more holes 280 through which the cavity input
light 150 and the cavity output light 152 may pass. Although not
shown in FIG. 2, the isolating enclosure 270 may have additional
holes through which, for example, the transmitted light 154,
fiber-optic patch-cords, and/or electrical wires may pass.
[0042] To further minimize the effects of acoustic noise, the
vacuum chamber 218 may be mounted (e.g., via pedestals 262) on an
active vibration platform 260. The isolating enclosure 270 may be
lined with acoustic-damping foam to block acoustic noise
propagating through the surrounding air from directly impinging on
the vacuum chamber 218. The isolating enclosure 270 also helps
limit fluctuations of the optical path length between the OLO laser
110 and the vacuum chamber 218 arising from pressure fluctuations
in the surrounding air.
[0043] To allow light to enter and exit the cryostat 104, the
vacuum chamber 218 may include an upper window 228 through which
the transmitted light 154 can pass, and a lower window 238 through
which the cavity input light 150 and the cavity output light 152
can pass. Similarly, each thermal shield 212 may include a
corresponding upper window 222 and lower window 232. As shown in
FIG. 2, the upper windows 222, 228 and lower windows 232, 238 may
be coaxial with the Fabry-Perot cavity 106. Each of the upper
windows 222, 228 and lower windows 232, 238 may be fabricated from
a material that is transparent to the cavity input light 150 yet
blocks blackbody radiation to limit radiative heating of the
Fabry-Perot cavity 106. A size of each of the windows 222, 228,
232, and 238 may be selected to limit a solid angle through which
the blackbody radiation can pass.
[0044] Each of the windows 222, 228, 232, and 238 may be
anti-reflection coated and wedged to prevent the window from acting
as an etalon. In addition, each of the upper windows 222, 228 and
lower windows 232, 238 may be tilted at a small angle (e.g., a few
degrees or less), relative to the optical axis of the Fabry-Perot
cavity 106, to prevent a reflection from the window from coupling
back into the Fabry-Perot cavity 106.
[0045] At least one turning mirror 250 may be located near the at
least one focusing optic 130 on the active vibration platform 260.
The turning mirror 250 steers the cavity input light 150 toward the
Fabry-Perot cavity 106, and steers the cavity output light 152 away
from the Fabry-Perot cavity 106. Both the turning mirror 250 and
focusing optic 130 may be mounted in conventional optical mounts
that are bolted to, and thus supported by, the active vibration
platform 260. While FIG. 2 shows only one turning mirror 250, more
than one turning mirror 250 may be used. For example, two turning
mirrors 250 may be used to control all four degrees of freedom of
the cavity input light 150 (i.e., tip, tilt, sideways translation,
and vertical translation).
[0046] Since the at least one turning mirror 250 and at least one
focusing optic 130 are located close to the Fabry-Perot cavity 106,
some light scattered off these components may couple into the
Fabry-Perot cavity 106, where it can distort the error signal used
to lock the OLO laser 110 to the cavity resonance. Such distortion
introduces another temperature-dependent source of frequency drift.
To minimize scattered light, the front optical surface of the
turning mirror 250 and both optical surfaces of the focusing optic
130 may be superpolished, i.e., each of these optical surfaces is
polished such that its residual roughness is less than 1 .ANG. rms
across the optical surface. This level of surface roughness
corresponds to less than 10 ppm of optical scatter per surface.
Although the windows 222, 228, 232, and 238 were not superpolished,
it is expected that superpolished windows 222, 228, 232, and 238
would further enhance frequency stability. Accordingly, any of the
windows 222, 228, 232, and 238 may be superpolished.
[0047] The spacer 202, and the substrates of the mirrors 204, may
be made from a material other than silicon. As described above, the
CTE for many materials approaches zero as temperature approaches
absolute zero, and thus can be cryogenically cooled such that the
cavity length 210 is insensitive to thermal fluctuations. Materials
that are stiff (i.e., have a high Young's modulus) and transparent
(at least partially in the visible and infrared parts of the
electromagnetic spectrum) include sapphire, quartz, silicon
carbide, silicon nitride, diamond, germanium, and gallium arsenide.
Any of these materials, or others, may be used to form the spacer
202, and the substrates of the mirrors 204, without departing from
the scope hereof.
[0048] The cryostat 104 of FIG. 2 may be provided as a stand-alone
system configured for the Fabry-Perot cavity 106, as provided by a
third party, to be mounted therein. Alternatively, the cryostat 104
and the Fabry-Perot cavity 106 may be provided as a stand-alone
system configured to cooperate with the remaining components of
FIG. 2, which may be provided by a third party.
[0049] Optical Clock Embodiments
[0050] FIG. 3 shows an optical clock 300 that combines the OLO 100
of FIG. 1 with an optical frequency reference 304 to provide stable
and accurate timing for averaging times out to several months, or
more. The optical clock 300 may be used as one of an ensemble of
atomic clocks forming a time scale. In fact, the frequency
stability of the optical clock 300 is high enough that it may be
used by itself as a time scale (i.e., without additional atomic
clocks as part of an ensemble). The OLO 100 maintains the time of
the optical clock 300 over short and intermediate averaging times
(e.g., 10.sup.5 seconds and less), while the optical frequency
reference 304 determines the stability over long averaging times
(e.g., greater than 10.sup.5 seconds). Due to the enhanced
stability of the OLO 100 at short and intermediate averaging times,
the optical clock 300 can advantageously operate at less than a
100% duty cycle while providing a lower fractional frequency
instability than prior-art microwave LOs (e.g., masers). For
example, the optical frequency reference 304 may operate for as
little as 1 out of every 24 hours (i.e., a 4.2% duty cycle). For
clarity, only the cryostat 104 (with the Fabry-Perot cavity 106),
OLO laser 110, and frequency shifter (FS) 116 of the OLO 100 are
shown in FIG. 3.
[0051] The optical frequency reference 304 may be a one-dimensional
lattice of laser-cooled neutral atoms 314, as shown in FIG. 1.
Here, a portion of the output of a clock laser 310 is used to
pre-stabilize the clock laser 310 to a Fabry-Perot cavity 316. A
remaining portion of the output of the clock laser 310 is
frequency-shifted by a frequency shifter 312 to produce a reference
output 318. A portion of the reference output 318 is
frequency-shifted by a frequency shifter 313 to generate probe
light 322 that probes an optical clock transition of the atoms 314.
A clock correction signal derived by probing the atoms 314 is
applied to the frequency shifter 313 to steer a frequency of the
probe light 322 towards a clock frequency of the optical clock
transition. Each of the frequency shifters 312 and 313 may be, for
example, an acousto-optic modulator.
[0052] In one example of the optical frequency reference 304,
neutral .sup.87Sr atoms 314 are laser-cooled to microkelvin
temperatures and trapped in a one-dimensional optical lattice. In
this case, the clock transition is the
.sup.1S.sub.0.fwdarw..sup.3P.sub.0 transition at 698 nm, and the
clock laser 310 may be, for example, an external cavity diode laser
at this wavelength. However, the optical frequency reference 304
may be based on a different atomic or molecular transition in the
infrared, visible, or ultraviolet regions of the electromagnetic
spectrum. Thus, while FIG. 3 shows the optical frequency reference
304 as a one-dimensional lattice of laser-cooled neutral atoms 314,
the optical frequency reference 304 may be alternatively based on a
two-dimensional or three-dimensional lattice of laser-cooled
neutral atoms 314. Other species that may be used for laser-cooled
neutral atoms 314 include Yb, Ca, and Hg. Alternatively, the
optical frequency reference 304 may be based on one or more atomic
ions that are laser-cooled and trapped. Examples of such atomic
ions include Hg+, Yb+, Al+, and Sr+.
[0053] An optical frequency comb 302 with a plurality of comb teeth
306 may be used to transfer the linewidth and frequency stability
of the OLO 100 to the frequency of the clock transition, as shown
in FIG. 3. For example, when the optical frequency reference 304 is
based on the .sup.1S.sub.0.fwdarw..sup.3P.sub.0 transition of
.sup.87Sr at 698 nm and the OLO 100 operates at 1542 nm, the
optical frequency comb 302 may be configured with a spectrum that
extends below 698 nm and above 1542 nm. A first comb tooth 306(1)
may be phase-locked to the OLO output 102 at 1542 nm at a first
offset-locking frequency .DELTA.f.sup.(1), thereby transferring the
phase noise and frequency stability of the OLO 100 to the first
comb tooth 306(1). Although not shown in FIG. 3, the optical
frequency comb 302 has a carrier-envelope-offset (ceo) frequency
f.sub.ceo that may be detected with an f-2f interferometer and
phase-locked to a microwave frequency reference. With f.sub.ceo
phase-locked in this manner, phase-locking the first comb tooth
306(1) to the OLO output 102 transfers the phase noise and
frequency stability of the OLO 100 to all other comb teeth 306 at
all other wavelengths in the spectrum of the optical frequency comb
302.
[0054] Using a heterodyne beat .DELTA.f.sup.(2) between the
reference output 318 and a second comb tooth 306(2) at 698 nm, the
reference output 318 may be phase-locked to the second comb tooth
306(2) by controlling the frequency of the reference output 318 via
the frequency shifter 312. This phase lock transfers the frequency
stability of the OLO output 102 to the reference output 318, for
all averaging times.
[0055] As another example of how the optical frequency comb 302 may
be used when the optical frequency reference 304 is based on the
.sup.1S.sub.0.fwdarw..sup.3P.sub.0 clock transition of .sup.87Sr at
698 nm, the optical frequency comb 302 may be configured with an
octave-spanning spectrum that does not extend to 698 nm. In this
case, a portion of the optical frequency comb 302 near 1396 nm may
be frequency-doubled to generate the second comb tooth 306(2) at
698 nm.
[0056] The comb teeth 306 are equally spaced by a repetition rate
f.sub.rep that can be detected with a photodetector 308 to produce
an electronic clock output 320 whose frequency stability, for all
averaging times, is determined by the OLO 100. Although not shown
in FIG. 3, the clock correction signal used to control the
frequency shifter 313 may also be used to correct the clock output
320 so that at long averaging times, the clock output 320 displays
the long-term stability of the optical frequency reference 304
rather than the OLO 100.
[0057] To improve the frequency stability of the optical clock 300
when the optical frequency reference 304 is unavailable (i.e.,
during downtime), the frequency of the OLO 100 may be steered using
a predictive model. The predictive model may be based on a Kalman
filter that estimates, at a given time, a next frequency of the OLO
100 based on prior measurements of the frequency. As described in
more detail below, drift in the OLO frequency between daily
measurements can be modeled using a quadratic function. The model
prediction is a state vector that is updated epoch-by-epoch when
the optical frequency reference 304 is running.
[0058] In one embodiment, the OLO output 102 is coupled into a
fiber-optic cable 324 that forms part of a phase-stabilized
fiber-optic link that transfers the OLO output 102 over long
distances (e.g., several kilometers). An output 326 of the
fiber-optic link may then be used with a remote optical frequency
comb that is similar to the optical frequency comb 302 of FIG. 3.
Thus, the OLO 100 need not be physically proximate to the optical
frequency comb 302 or the optical frequency reference 304 to form
part of an optical clock or time scale. This embodiment is
straightforward to implement when the OLO laser 110 operates at a
wavelength compatible for optical communication, such as the C-band
(1530-1565 nm) or L-band (1565-1625 nm).
[0059] In another embodiment, the clock correction signal is used
to adjust the first offset-locking frequency .DELTA.f.sup.(1). In
this case, the optical frequency reference 304 may operate without
the frequency shifter 313, wherein the reference output 318 and
probe light 322 have the same frequency. One benefit of this
embodiment is that the clock output 320 displays the long-term
stability of the optical frequency reference 304 without
correction.
[0060] FIG. 4 shows an optical clock 400 that is similar to the
optical clock 300 of FIG. 3 except that the optical frequency
reference 304 operates without the frequency shifters 312, 313 and
Fabry-Perot cavity 316. In this case, the clock laser 310 is
directly phase-locked to the second comb tooth 306(2) at a second
offset-locking frequency .DELTA.f.sup.(2) with a bandwidth large
enough that the clock laser 310 inherits the narrow linewidth of
the OLO 100. The second offset-locking frequency .DELTA.f.sup.(2)
may be selected so that the probe light 322 is resonant with the
clock transition. The clock correction signal may then be used to
adjust the first offset-locking frequency .DELTA.f.sup.(1) so that
the clock laser 310 is steered toward the clock frequency. In this
case, the clock output 320 displays the long-term stability of the
optical frequency reference 304 without correction.
[0061] In other embodiments, the stability of the OLO 100 is
transferred to the optical frequency reference 302 without the
optical frequency comb 302, wherein the optical frequency comb 302
is used only to generate the clock signal 320. For example, when
the clock laser 310 and the OLO laser 110 have frequencies that are
close (e.g., <100 GHz), the clock laser 310 can be directly
phase-locked to the OLO output 102 without the optical frequency
comb 302. In this case, the OLO 100 operates similarly to the
Fabry-Perot cavity 316 of FIG. 3. The second comb tooth 306(2) may
then be phase-locked to either the reference output 318 or the OLO
output 102 to generate the clock output 320. Alternatively, when
the OLO output 102 has a frequency sufficiently close to that of a
harmonic (e.g., a second harmonic) of the clock laser 310, the
clock laser 310 may be controlled so that its harmonic is
phase-locked to the OLO output 102. Alternatively, when a harmonic
(e.g., a second harmonic) of the OLO output 102 has a frequency
sufficiently close to that of the clock laser 310, the clock laser
310 may be phase-locked to the harmonic of the OLO laser 110. For
example, when the OLO 110 operates at 1396 nm and the clock laser
310 operates at 698 nm, the OLO output 102 may be frequency-doubled
to 698 nm, wherein the clock laser 310 is phase-locked to the
frequency-doubled OLO output 102.
[0062] Optical Clock Performance
[0063] FIGS. 5-8 demonstrate performance of the OLO 100 over a
thirty-four-day data campaign spanning from a modified Julian date
(MJD) of 58430 to 58464. For this data, the optical frequency
reference 304 was based on laser-cooled neutral .sup.87Sr atoms
trapped in a 1D optical lattice. A clock laser 310 at 698 nm probed
the .sup.1S.sub.0.fwdarw..sup.3P.sub.0 clock transition at this
wavelength. The Fabry-Perot cavity 106 of the OLO 100 was made of
single-crystal silicon, and was cryogenically cooled to 124 K in
the cryostat 104. The Fabry-Perot cavity 106 and cryostat 104 were
configured as described above for FIGS. 1 and 2. The optical
frequency comb 302 was based on a Er:fiber laser with a repetition
rate f.sub.rep of approximately 250 MHz, and f.sub.ceo phase-locked
to 35 MHz. RAM generated by the sideband generator 118 was actively
suppressed. The clock output 320 was compared, via a microwave
fiber link, to AT1, a free-running microwave time scale at the
Boulder campus of the National Institute for Standards and
Technology (NIST). AT1 features a 95% uptime, allowing nearly
continuous monitoring of the OLO 100 over the data campaign.
[0064] FIG. 5 shows a frequency record 502 of the OLO 100 over the
thirty-four-day data campaign. Specifically, data points 504 of the
frequency record 502 (shown as circles) represent measurements of
the frequency of the OLO 100 relative to the optical frequency
reference 304. The optical frequency reference 304 ran daily except
for MJDs 58444 and 58447. Three days before the first measurement,
the optical power incident on the Fabry-Perot cavity 106 was
changed to reset an intensity noise servo. The frequency evolution
of the OLO 100 after adjusting the incident optical power is well
modeled by a constant linear drift plus an exponential relaxation
term, i.e., a +bt+ce.sup.-t/d. A best-fit line 506 shows the data
points 504 fit to this drift model, where a=24.16 Hz, b=-9.632
Hz/day, and d=7.813 days. A residuals plot 510 show the residuals
512 of the best-fit line 506. A line 514 in the residuals plots 510
shows similar residuals based on frequency measurements made
relative to AT1 and corrected with the best-fit line 506.
[0065] During the interval between clock operation on MJD 58441 and
58442, two frequency jumps on the OLO were identified with a
combined amplitude of 5.02.times.10.sup.-15. A correction of the
same magnitude is applied to all data after this step when
performing the analysis presented herein. No significant change in
the long-term drift trend of the OLO 100 was observed following
these excursions.
[0066] The performance of a time scale is typically compared to a
reference time scale with significantly lower timing uncertainty.
Since no such time scale exists, the .sup.87Sr optical frequency
reference 304 is treated as an ideal frequency reference, and the
performance of the OLO 100 is inferred by examining the fractional
frequency offset between the steered OLO frequency record and the
clock transition frequency, hereafter referred to as the prediction
error. The time error of the optical clock 300 is defined as the
integral over time of the prediction error.
[0067] If the frequency record 502 was continuous, the time error
could be determined to within the measurement precision of the
optical frequency reference 304. However, a finite gap of time
separates the measurements in the frequency record 502, ranging
from a five-second interrogation cycle of the optical frequency
reference 304 to twenty-four hours between daily measurements. Most
of the time error accumulates during the longer gaps, when the
Kalman filter must accurately predict changes in the OLO frequency
without new measurement data from the optical frequency reference
304. The time error contribution from a gap is simply the gap
duration multiplied by the mean prediction error during this
interval. However, the latter quantity cannot be determined exactly
from the available data. Instead, the mean prediction error is
estimated by averaging the values before and after the gap and
multiplying the average value by the gap duration to compute an
estimated time error. A 1.sigma. confidence interval is calculated
for the estimated time error through repeated simulations of the
OLO frequency during each gap to determine the uncertainty in the
estimation of the prediction error.
[0068] FIG. 6 shows the integrated time error 602 of the optical
clock 300, as determined from the residuals 512 of FIG. 5. After
thirty-four days of integration, the optical clock 300 accumulates
an integrated time error 602 of 48.+-.94 ps. FIG. 6 also shows a la
confidence interval 608 for the integrated time error 602. For
comparison, FIG. 6 also shows an integrated time error 604
estimated for a time scale consisting of a hydrogen maser steered
to a .sup.133Cs fountain for 24 hours/day, and an integrated time
error 606 estimated for a time scale consisting of a hydrogen maser
steered to an .sup.87Sr optical lattice clock for 6 hours/day. The
integrated time errors 604 and 606 were estimated by computing time
errors from repeated simulations, and are represented in FIG. 6 as
the RMS spread over a thirty-four-day window. Each of the
integrated time errors 604 and 606 is larger than the integrated
time error 602 of the optical clock 300.
[0069] FIG. 7 is a plot of the fractional frequency instability of
the OLO 100 as a function of averaging time. Data points 710
(circles) are gap-tolerant Allan variances calculated from the
residuals 512 of FIG. 5, and correspond to the frequency of the OLO
100 measured relative to the optical frequency reference 304. Data
points 710 were fit to a model 712 that includes a known thermal
noise floor and a random-walk (RW) frequency noise term, resulting
in a fractional frequency instability at long averaging times
consistent with .sigma..sub.RW=1.3.times.10.sup.-18 {square root
over (.tau.)}, where .tau. is the averaging time in seconds. Data
points 708 (triangles) are gap-tolerant Allan variances calculated
from the residuals 512 of FIG. 5, and correspond to the frequency
of the OLO 100 measured relative to AT1. For averaging times of
10.sup.5 seconds and higher, the data points 708 and 710 agree
within their statistical uncertainties. For shorter averaging
times, the data points 708 are consistent with a noise model (not
shown) that accounts for instability from the microwave link, the
OLO 100, and AT1.
[0070] FIG. 8 is a plot of the expected fractional frequency
instability of the optical clock 300 as a function of averaging
time. Data points 810 are modified Allen deviations calculated from
a time-scale frequency record that was generated by simulating the
OLO 100 being steered by the optical frequency reference 304 for an
uptime of 12 hours/day. To generate the time-scale frequency
record, a local oscillator frequency record was first generated
using the model 712 of FIG. 7 to represent noise, and including the
drift model represented by the best fit 506 of FIG. 5. The local
oscillator frequency record was then steered according to a
simulated optical frequency reference 304, using the same Kalman
filtering techniques described above, to generate the time-scale
frequency record. Data points 808 are similar to the data points
810, except that the OLO 100 is simulated as being steered by the
optical frequency reference 304 for an uptime of only 1
hour/day.
[0071] For comparison, FIG. 8 also shows simulated fractional
frequency instabilities of atomic clocks that use a hydrogen maser
as a local oscillator instead of the OLO 100. Specifically, data
points 806 correspond to a maser steered to the optical frequency
reference 304 for an uptime of 12 hours/day, data points 804
correspond to a maser steered to the optical frequency reference
304 for an uptime of 1 hour/day, and data points 802 correspond to
a maser continuously steered to a .sup.133Cs fountain clock for 24
hours/day. The stability of the optical clock 300 improves with
increased uptime (i.e., higher duty cycle) and reduced noise of the
OLO 100, and is reasonably consistent with the expected instability
limit from aliased LO noise past 10.sup.6 seconds. When the optical
frequency reference 304 is run at a fixed duty cycle, the steered
OLO 100 significantly outperforms the steered maser at all
averaging times. Even for an uptime of only one hour/day, the
optical clock 300 is more stable than a maser steered at a 50% duty
cycle.
[0072] The improved stability of the optical clock 300 allows for
competitive time-scale performance with significantly relaxed
uptime requirements. For example, based on the analysis presented
above, a fractional frequency instability of approximately
1.8.times.10.sup.17 is expected for a thirty-four-day campaign with
an average uptime of 6 hours/day. For an uptime of 12 hours/day,
the frequency instability of the optical clock 300 is expected to
remain at or below the 2.times.10.sup.-16 level at all averaging
times, and is projected to reach an instability below 10.sup.-17
after 85 days of operation. Additional effort on automation should
increase the clock duty cycle to above 50%. Furthermore, averaging
the outputs of N independent OLOs 100 can reduce the LO instability
by a factor of 1/ {square root over (N)}.
Kalman Filtering
[0073] To minimize the time error during periods when an optical
frequency reference is unavailable, a time-scale local oscillator
must have a predictable frequency. The frequency of the local
oscillator can then be steered using a predictive model to keep it
as close to the value of the optical frequency reference as
possible. Kalman filtering techniques are often used to construct
such a model based on periodic measurements of the local oscillator
frequency against an atomic frequency reference. For a hydrogen
maser, a linear model of the form f (t+.DELTA.t)={circumflex over
(k)}.sub.0+{circumflex over (k)}.sub.1.DELTA.t is typically used to
track its frequency drift over time, where {circumflex over
(k)}.sub.0 is an estimate of the maser frequency at time t and
{circumflex over (k)}.sub.1=df/dt is an estimate of the
linear-frequency drift rate.
[0074] For the OLO 100, a linear model is sufficient to form a time
scale with competitive performance. However, the presence of
random-walk frequency noise and the exponential term in its
frequency drift tend to add a slight curvature to the OLO frequency
evolution. Better predictive performance can be obtained by
modeling the OLO frequency with a quadratic function of the form
f(t+.DELTA.t)={circumflex over (k)}.sub.0+{circumflex over
(k)}.sub.1.DELTA.t+{circumflex over (k)}.sub.2.DELTA.t.sup.2/2,
where the additional term {circumflex over (k)}.sub.2 estimates the
time derivative of the linear drift rate. As shown below, this
function is typically written in matrix form, and the coefficients
{circumflex over (k)}.sub.0, {circumflex over (k)}.sub.1, and
{circumflex over (k)}.sub.2 form what is known as the state
estimate vector.
[0075] For the analysis presented in FIG. 6, the separation time
between updates of the Kalman filter state vector was .DELTA.t=1 s.
For the n.sup.th epoch, the local oscillator frequency at time
t.sub.n is predicted using the state estimate vector at time
t.sub.n-1 and its expected evolution over the interval .DELTA.t.
Mathematically,
( k ^ 0 .function. [ n | n - 1 ] k ^ 1 .function. [ n | n - 1 ] k ^
2 .function. [ n | n - 1 ] ) = ( 1 .DELTA. .times. t .DELTA.
.times. t 2 / 2 0 1 .DELTA. .times. t 0 0 1 ) .times. ( k ^ 0
.function. [ n - 1 ] k ^ 1 .function. [ n - 1 ] k ^ 2 .function. [
n - 1 ] ) . ( 1 ) ##EQU00001##
The vector {right arrow over (k)}[n.ident.n-1] is known as the
prior estimate of the state at epoch n. The local oscillator
frequency is steered by applying a frequency correction of
{circumflex over (k)}.sub.0[n|n-1] during each epoch. When no new
measurement with the optical frequency reference 304 occurs during
an epoch, the new state estimate vector is simply set equal to the
prior estimate, or
( k ^ 0 .function. [ n | n - 1 ] k ^ 1 .function. [ n | n - 1 ] k ^
2 .function. [ n | n - 1 ] ) = ( k ^ 0 .function. [ n - 1 ] k ^ 1
.function. [ n - 1 ] k ^ 2 .function. [ n - 1 ] ) . ( 2 )
##EQU00002##
[0076] Equations 1 and 2 summarize the behavior of the steering
algorithm when the optical frequency reference 304 is offline or
non-operational. When the optical frequency reference 304 is
operational, and a new frequency measurement is available during
epoch n, the fidelity of the model prediction may be assessed by
comparing the prior prediction of the OLO frequency {circumflex
over (k)}.sub.0[n|n-1] with the measurement:
.DELTA.f[n]=y[n]-{circumflex over (k)}.sub.0[n|n-1] (3)
where y[n] is the measurement of the frequency detuning of the
free-running OLO 100 relative to the atomic transition and
.DELTA.f[n] is the prediction error (i.e., the residual detuning
after steering the OLO 100). If the optical frequency reference 304
is assumed to be ideal, .DELTA.f[n] represents the residual
frequency fluctuation of the OLO 100 after steering. Following the
measurement, the state vector may be updated according to:
( k ^ 0 .function. [ n | n - 1 ] k ^ 1 .function. [ n | n - 1 ] k ^
2 .function. [ n | n - 1 ] ) = ( k ^ 0 .function. [ n - 1 ] k ^ 1
.function. [ n - 1 ] k ^ 2 .function. [ n - 1 ] ) + .DELTA. .times.
f .function. [ n ] .times. K n . ( 4 ) ##EQU00003##
The matrix K.sub.n, known as the optimal Kalman filter gain for
epoch n, determines the relative weight of the measurement and the
prior estimate when computing the new state estimate vector, and is
optimized using knowledge from past measurements and the known
noise properties of the OLO 100.
[0077] When computing the optimal Kalman filter gain K, two
covariance matrices Q and R, corresponding to process noise and
measurement noise, must be specified. Process noise represents
uncertainty in the future state of the OLO 100, which was found to
be dominated by the random-walk frequency noise which limits the
frequency stability of the OLO 100 at 1-day averaging intervals.
The process noise matrix Q is defined as:
Q = ( Q 11 0 0 0 Q 2 .times. 2 0 0 0 Q 3 .times. 3 ) , ( 5 )
##EQU00004##
where the parameter Q.sub.11, which corresponds to the random walk
noise of the silicon Fabry-Perot cavity 106, is set to
5.1.times.10.sup.-36 (s/s).sup.2, as based on the random-walk
coefficient from the local oscillator noise model in FIG. 7. The
parameters Q.sub.22 and Q.sub.33, corresponding to random run noise
and higher-order noise, are negligible. In practice, Q.sub.22 and
Q.sub.33 are set to 2.2.times.10.sup.-46 (s/s.sup.2).sup.2 and
3.5.times.10.sup.-57 (s/s.sup.3).sup.2, respectively. The
prediction error is not particularly sensitive to the values of
Q.sub.22 and Q.sub.33. It was found that only coarse tuning of
these parameters is required to guarantee the performance of the
Kalman filter.
[0078] Measurement noise represents uncertainty in the current
frequency measurement due to relative noise between the optical
frequency reference 304 and the OLO 100. The measurement noise
matrix R has a single non-zero entry with a value of
2.5.times.10.sup.-33 (s/s).sup.2, corresponding to the Allan
variance of the thermal noise floor of the OLO 100.
[0079] Resonant Photodetection
[0080] FIG. 9 is a schematic of a resonant transimpedance amplifier
(RIA) 900 that may be used to resonantly convert a current
outputted by the reflection photodetector 134 into a voltage. The
RIA 900 is designed to resonantly detect light (e.g., cavity output
light 152 or transmitted light 154) at the frequency of the
sidebands added to the light by the sideband generator 118 (i.e.,
near the frequency of the generator 124). The RIA 900 is similar to
many conventional transimpedance amplifiers in that it uses
feedback to enhance linearity. However, unlike other resonant
detection circuits, the RIA 900 uses a phase shifter 914 at the
sideband frequency to ensure loop stability. For the PDH technique,
the sideband frequency is typically between 1 and 100 MHz. However,
the RIA 900 may be used to resonantly detect sidebands at a
sideband frequency less than 1 MHz, or more than 100 MHz, without
departing from the scope hereof.
[0081] Resonant photodetection reduces the amount of active gain
needed to detect light. Thus, the RIA 900 has a lower input noise,
within the bandwidth of its resonance, as compared to a
non-resonant transimpedance amplifier. With this lower input noise,
the power of the detected light can be reduced while still ensuring
the light is detected at the photon shot-noise level. Specifically,
for shot-noised limited detection, the signal-to-noise ratio (SNR)
scales as the square-root of the light power. Therefore, decreasing
the light power by a scale factor k decreases the SNR by k.sup.1/2.
However, for amplifier-limited detection (i.e., when the noise
floor is set by the electronics), the SNR scales linearly with the
power, in which case decreasing the light power by k will also
decrease the SNR by k.
[0082] Reducing the amount of light used with the OLO 100
advantageously reduces residual heating, thereby improving thermal
stability. Residual heating may occur from absorption of light
propagating through the substrates of the mirrors 204(1) and
204(2), and is a likely contributor to the linear drift shown in
the frequency record 502 of FIG. 5. Accordingly, reducing the
amount of light used with the OLO 100 may reduce the slope of the
best fit 506, which can improve how this linear drift is
corrected.
[0083] The RIA 900 includes a bias source 902 that injects a DC
bias current 922 into the cathode of a photodiode 906. The
photodiode 906 may be any type of photodiode, such as an avalanche
photodiode, PN photodiode, or PIN photodiode. To detect light at
the photon shot-noise level, the photodiode 906 should have no
internal gain. The RIA 900 also includes a resonator 908 that
resonates at the sideband frequency. In the example of FIG. 9, the
resonator 908 is an inductor and capacitor connected in series. In
this case, the resonator 908 will typically have a Q between 10 and
1000, and a full-width at half-maximum (FWHM) of 10 kHz to 10 MHz.
The FWHM may define the bandwidth of the RIA 900.
[0084] The RIA 900 also includes a bias monitor 904 with an output
916 that provides a low-bandwidth (e.g., DC-2.5 kHz) signal for
monitoring the DC bias current 922. The input resistor of the bias
monitor 904 sinks the DC bias current 922 to a virtual ground. The
op-amp U1 ensures that the output 916 has a relatively low
impedance, and is therefore isolated from the resonator 908. Where
the bias monitor 904 is not needed, it may be replaced with a
resistor to ground that sinks the DC bias current 922.
[0085] The RIA 900 also includes an input stage 912 that amplifies
the resonantly-enhanced voltage 924 between the resonator 908 and
the anode of the photodiode 906. To increase gain and bandwidth, an
input transistor 910 of the input stage 912 may be a cascode pair,
as shown in FIG. 9. The cascode pair be constructed from a
dual-gate gallium arsenide (GaAs) metal-semiconductor field-effect
transistor (MESFET), such as the NE25139 manufactured by NEC
Corporation. However, another type of gain device may be used for
the input transistor 910 without departing from the scope
hereof.
[0086] The RIA 900 also includes the phase shifter 914, which
shifts the phase of an amplified signal 926 outputted by the input
stage 912. An output 928 of the phase shifter 914 is the same as
the amplified signal 926, but shifted by a phase that is adjustable
via a trimpot 930. The phase may be selected to ensure stability of
the RIA 900.
[0087] The output 928 of the phase shifter 914 connects to a
feedback path 918 that returns to the input stage 912. The
transimpedance gain of the RIA 900 is set by the components in the
feedback path 918 which, unlike conventional transimpedance
amplifiers, contains no resistive element. The output 928 also
connects to an output conditioner 920 that uses a transformer to
convert the single-ended output 928 into a differential signal 932.
The differential signal 932 may be used as the input to the
frequency servo 120 of FIG. 1.
[0088] The factor k by which the optical power can be reduced
approximately equals the Q of the resonator 908. For non-resonant
transimpedance amplifiers, an optical power of tens of microwatts,
or more, is usually needed to reach the photon shot-noise limit.
When the RIA 900 is used with a resonator 908 having a Q of 100,
optical powers below 100 nW can be detected at the photon
shot-noise limit. With an even higher Q, the optical power could be
reduced more while still ensuring photon shot-noise-limited
detection.
[0089] One drawback to a higher Q is that the narrower bandwidth of
the RIA 900 may reduce the servo bandwidth of a feedback loop that
uses the RIA 900. For example, when locking the OLO laser 110 to
the Fabry-Perot cavity 106 using the PDH technique, the servo
bandwidth may need to be as high as 1 MHz (depending on the
free-running linewidth of the OLO laser 110). However, when the
resonator 908 has a Q of 1000, the bandwidth of the RIA 900 is only
a few tens of kilohertz, which is likely too small to support a
servo bandwidth of 1 MHz. In this case, the linewidth of the OLO
laser 110 may be narrowed (e.g., to a few hundred hertz) by locking
to a second Fabry-Perot cavity. With this narrower linewidth, the
OLO laser 110 can then be locked to the Fabry-Perot cavity 106 with
a servo bandwidth much less than 1 MHz.
[0090] Changes may be made in the above methods and systems without
departing from the scope hereof. It should thus be noted that the
matter contained in the above description or shown in the
accompanying drawings should be interpreted as illustrative and not
in a limiting sense. The following claims are intended to cover all
generic and specific features described herein, as well as all
statements of the scope of the present method and system, which, as
a matter of language, might be said to fall therebetween.
* * * * *