Thermal Noise Limit in Frequency Stabilization of Lasers with Rigid Cavities

1
Thermal Noise Limit in Frequency Stabilization of
Lasers with Rigid Cavities
Kenji Numata, Amy Kemery, and Jordan Camp
Exploration of the Universe Division, Code 663,
NASA/Goddard Space Flight Center, Greenbelt,
Maryland 20771
We evaluated thermal noise (Brownian motion) in a
rigid reference cavity used for frequency
stabilization of lasers, based on the mechanical
loss of cavity materials and the numerical
analysis of the mirror-spacer mechanics with the
direct application of the fluctuation dissipation
theorem. This noise sets a fundamental limit for
the frequency stability achieved with a rigid
frequency-reference cavity, of order 1Hz/rtHz
(0.01Hz/rtHz) at 10mHz (100Hz) at room
temperature. This level coincides with the
world-highest level stabilization results. (Phys.
Rev. Lett. 93 (2004) 250602)

• Experiment to measure Q

• Frequency stabilization of lasers

• The cavity material is polished on every surface
  and suspended to avoid any surface and support
  loss. The internal mode is excited by an
  electrostatic actuator or a PZT actuator, and its
  decay is measured by a Michelson interferometer
  (miniature version of LISA, shown in the photo).
  The decay time gives us the Q factor of the
  material.

• Lasers have a wide range of applications
  including
• Optical frequency standards
• High resolution spectroscopy
• Fundamental physics tests
• Interferometric measurement ex.) LISA, LIGO...
• The use of a rigid cavity is a very common
  technique in many fields. This technique is
  required to suppress laser frequency noise, which
  hinders us from performing sensitive
  measurements. The use of a rigid cavity is a
  basic design for LISA as well.
• LISAs frequency stabilization scheme
• Pre-stabilization with rigid cavity gain106
  (30Hz/rtHz)
• Arm locking gain104
• Post processing (TDI) gain103

• Sample preparation the samples are coated with
  gold to get reflected light, which is required
  for interferometric measurement, from the
  surface.
• We measured several low CTE materials. They
  showed much lower Q factors than, for example,
  fused silica. The low Q enlarges thermal noise,
  if these materials are used as cavity materials.
  In the measured frequency range, from a few kHz
  to 100kHz, a frequency dependence of loss was not
  observed. Therefore, the structural damping model
  was adopted for the analysis.
• ULE Q6.1x104
• Zerodur Q3.1x103
• Clearceram (Z-HS) Q2.3x103

• A Thermal noise in a rigid cavity

• Thermal noises are one of the fundamental
  phenomena in physics. Based on statistical
  physics, Callen related the spectrum of random
  motion to loss in a system, obtaining the
  fluctuation dissipation theorem (FDT).
• Frequency stability achieved with these FP
  cavities is dependent on the stability of cavity
  length. However, thermal noise (Brownian motion)
  of the cavity has never been evaluated
  rigorously.
• We evaluated this noise by using the
  Fluctuation-Dissipation Theorem (FDT). To apply
  the FDT, we needed to calculate dissipated energy
  in the cavity under an applied force. (It is
  similar to measuring Joule heat from a resistance
  under a applied voltage to estimate its thermal
  noise.) We calculated it based on
• Experimentally measuring the Q factor of cavity
  materials
• Calculation of strain energy using FEM (dynamical
  Levins approach)

• Calculation using FEM

• To obtain the cavity's thermal noise spectrum, we
  calculate the dissipated energy by applying
  cyclic forces with Gaussian profiles at the
  required frequency onto mirrors connected by a
  spacer. The dissipated energy is proportional to
  both the loss and strain energy. To get the
  differential displacement between
  beam-illuminated areas, we apply two opposing
  Gaussian forces simultaneously.
• This procedure intrinsically includes the loss
  distribution and correlation between the two
  areas.
• We modeled a cavity used in the experiment that
  achieved the world-highest level frequency
  stability (shown in the photo). The experiments
  were done by the NIST and VIRGO groups. Their
  limiting noise was not identified at that time.
  The figure to the right is the FEM model used in
  our analysis.

• Rough order estimation

• We discuss the calculation of thermal fluctuation
  in an Fabry-Perot cavity where two mirrors are
  optically contacted to the ends of a rigid
  spacer. To get a rough order estimate, we
  calculate thermal fluctuation in the spacer and
  mirror with coating separately and then add the
  results, assuming that every noise component is
  uncorrelated.
• Thermal noise contributions from each components
  can be estimated by the figures and the table
  shown below. What we need to consider are a
  stressed volume and a stress under a force.
• As an example, we assume a typical cavity made of
  ULE (fsub1/60000). We also use following
  parameters T300K, R4cm, L24cm, w0240um,
  d2um, Ta2O5 coating (fcoat1/104) and frequency
  f1Hz. The total thermal noise becomes
  6x10-17m/rtHz. The spacer mirror substrate, and
  coating provide 1, 84, and 15 of it,
  respectively. If we assume 563nm light, it
  corresponds to frequency noise of 0.1Hz/rtHz and
  Allan variance of 5x10-16. This is the estimated
  thermal noise limit.

• Results discussion

• The mirror is the dominant thermal noise source.
  This is because, at a frequency region well below
  the mechanical resonance, only the losses around
  the beam spot contribute to thermal noise.
  Therefore, the cavity's overall shape and/or the
  loss of the spacer do not greatly affect thermal
  noise. Because we cannot greatly alter the
  coating thickness, the coating loss, or the beam
  radius, thermal noise from the coating becomes
  the practical limitation.
• Use of low mechanical loss mirrors, cooling, and
  a larger beam radius are expected to renew the
  world-highest level frequency stability. We
  believe that our result has significance for many
  precision measurement communities.

• Gaussian force on substrate with coating

• Our calculation results agree very well with the
  experimental stabilization results. We can see
  that, once frequency stability reaches 0.1Hz/rtHz
  level at 1Hz, the thermal noise of the cavity has
  a significant contribution.

• Force on one-dimensional bar (spacer)

• Estimation of each thermal noise contribution

We would like to thank Dr. P.Bender in JILA for
his very useful comments. For further
information, please contact Kenji Numata
(numata_at_milkyway.gsfc.nasa.gov).