1. Noise in cavity ring-down spectroscopy (CRDS) by transverse mode coupling
Haifeng Huang and Kevin K. Lehmann
Chemistry Department, University of Virginia
62nd International Symposium on Molecular Spectroscopy
Columbus OH, June 21, 2007

2. Introduction to CRDS Independent of laser power fluctuations
Long effective absorption length
Single mode excitation in CW-CRDS
Detector
Laser
τ =154.66±0.08µs
χ2 =0.99 µs mV

3. Our system Laser λ = 1652nm Cavity length 39.5cm
Lens
He-Ne laser
DFB diode laser
Laser control board
AOM
AOM driver
Detector
Faraday isolator
Computer
3PZTs
Flat mirror
Curved mirror
Mode matching optics
Cavity
Trigger signal
Laser λ = 1652nm
Cavity length 39.5cm
Mirror reflectivity %
Mirror curvature 1m
Free spectrum range 379.5MHz
Transverse mode spacing 82.1MHz
Mode size: 0.507mm at flat mirror and 0.652mm at curved mirror

4. Experimental results
Xenon: ~ 1.83torr pressure change = one FSR
Nitrogen gas or empty cavity
~ 50V change corresponds to one FSR
In both scans, laser current was modulated at 10Hz ~ 1.1FSR to generate resonance.
These peaks are not absorptions in the cavity.
The reduced χ2 >> 1, the decay signals of dropouts are no longer single exponential decays.

5. Two-mode beating model
With this equation, the reduced χ2 always ~ 1.

6. Analysis of noise peaks
~ 50V change corresponds to one FSR, which corresponds to the cavity length change of λ/2 = 0.83μm.
Δν tuning rate ~ 120Hz/V, or 7.215kHz/μm
Xenon: 81% scan from refraction index change
Cavity length changes ~ 0.1μm/torr
Δν tuning rate ~ 30.7kHz/torr, or 307kHz/μm
Resonance by tuning!

7. TEMnm modes excitation?
Mode size of TEMnm
Mode A was captured in both PZT and pressure scans.
Mode B was only seen in pressure scans.
More excitations in pressure scans than in PZT scans.
~ 1mm/pixel by calibration
Mode size of A and B: ~ 4mm and 3.6mm
Index n+m: ~ 60 for A and ~ 50 for B
Hermite-Gaussian function

8. Problem solved!
Cavity PZT and pressure scans with 4mm diameter aperture in the cavity, at distance ~13.5cm to the flat mirror. The size of TEM00 mode is 0.526mm at this position.
The slope of τ in the pressure scan is from the alignment of the cavity.
No dropouts of τ and excitations of higher order transverse modes were observed.

9. |νqnm – νq’00| Resonance frequency νqnm:
For our cavity, the frequency difference of two nearly resonant modes will shift at a rate of
Resonance frequency νqnm:
A change in n0 holding L constant will not lead to any mode crossings but changing L can.
PZT scans: kHz/μm n+m ~ 59
Pressure scans: 307kHz/μm n+m ~ 2500 ?
Cavity length L=39.604cm n+m ~ 60
Higher order mode images n+m ~ 60, 50
Limiting aperture (r = 7.6mm) n+m < 135
4mm diameter aperture n+m < 14

10. Coupling between TEM00 & TEMnm
Why the excitation of TEMnm modes generates beating in the decay signal?
The scattering loss of mirrors and transmission are not spatially uniform because of damaged spots on mirror surfaces.
Detector radius is 150μm. The TEM00 mode is focused to 28.4μm. Only n+m < 27 modes are detected. 54% of the energy of TEMnm with n+m ~ 60 is not collected by the detector.
Moreover, TEMnm modes with n+m ~ 60 can not be excited directly. Coupling between TEM00 and TEMnm modes is needed in order to excite TEMnm modes.
Tuning parameter
Mode frequency
Anti-crossing caused by coupling 00 nm
The coupling will generate two new eigenmodes. They are mixtures of TEM00 and TEMnm modes. The new fitting equation can be derived by assuming all the energy of TEMnm mode is not detected by a small size detector.
B1 and B2, τ1 and τ2 are amplitudes and lifetimes of these two new modes. Δν is the frequency difference between these two new modes.

11. Surface scattering coupling
Image of the damaged spots near the center of the surface of one old super-mirror (image size 146μm×100μm)
Why we almost always observe τ dropouts?
The TEM00 mode can decay through both the cavity intrinsic loss and this coherent coupling, and the energy in the TEMnm mode is not collected efficiently. The TEMnm mode decays faster than the TEM00 mode (τnm < τ00).

12. TEM10/01 μs mV
The degeneracy between TEM10 and TEM01 modes is lifted. A small difference of 0.26% between the radius of curvature of the mirror in x direction and that in y direction will generate this splitting.
B3 is not zero. The TEM10 and TEM01 modes are no longer orthogonal with each other, possibly because the mirror scattering loss, transmission or the quantum efficiency of the detector are not spatially uniform.

13. Conclusions
■ We observed dropouts of decay time constant in both cavity length and pressure scans. Images of higher order transverse modes excited at certain cavity conditions were captured by an IR camera. One 4mm diameter intracavity aperture will remove this noise.
■ The excitation of higher order transverse modes caused by surface scattering coupling explains experimental results successfully. Why τ decreases is explained.
■ However, why pressure scans have much larger tuning rate and denser dropout peaks is still unexplained.
■ The degeneracy between TEM01 and TEM10 modes is lifted and can be explained by weak astigmatism in the cavity. These two modes are no longer orthogonal with each other.
Submitted to Optics Express

14. Acknowledgements Paul Johnston and Robert Fehnel
Dr. Brooks Pate Group of UVA
Princeton Institute for the Science & Technology of Materials (PRISM), Princeton University

15. Surface scattering coupling
Klaassen T. et al. Optics Letters (2005)

16. Pressure scans with new mirrors
Nitrogen: ~ 3.26torr pressure change = one FSR
59% scan from refraction index change
Xenon: ~ 1.83torr pressure change = one FSR
81% scan from refraction index change
In both scans, PZT voltage was modulated at 12Hz ~ 1.1FSR.
New mirrors give smaller dropout depth and reduced χ2 number.

17. Peak intensity of TEMnm modes excited through scattering coupling when on resonant: