1. Cavity Ring-Down Spectroscopy
PHYS Modern Atomic Physics
Presentation by:
Ruqayyah Askar
April 25, 2016

2. Outline Ring-down with a continuous wave laser
Ring-down with a pulse train
Applications and conclusion
Experimental setup in our Lab.
Future work

3. Continuous Wave in a Tunable Cavity: Model
Consider:
A monochromatic cw laser
Assume:
Distance between mirrors change Linearly
𝐿 𝑡 = 𝐿 𝑜 +𝑣𝑡
From: Pulse train in a cavity by Xiwen Zhang (2012)

4. Electric field inside cavity:
𝐸 𝑖𝑛 𝑧,𝑡 = 𝑛=0 ∞ 𝑇 𝑟 2𝑛 𝐸 𝑜 𝑒 𝑖(𝑘𝑧 −𝜔𝑡) 𝑒 𝑖 𝜙 𝑛
Where
T: transmission coefficient,
r: reflection coefficient,
𝐸 𝑜 : slow varying amplitude,
and the round-trip phase is:
Where
v: velocity of mirror’s movement
n: number of round-trips
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

5. Electric field inside cavity:
Assume the cavity is in resonance with the incident filed at t=0:
𝑘 𝐿 𝑜 =𝑁 𝜋 , 𝑁: 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
Assume round-trip time << than cavity decay time, then:
𝑡 𝑟𝑜𝑢𝑛𝑑−𝑡𝑟𝑖𝑝 = 2𝐿 𝑣 𝑡= 2 𝐿 𝑜 𝑐 𝑙 , 𝑙: 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
The round-trip phase becomes:
𝜙 𝑛 =𝑘 2𝑛 𝐿 𝑜 +𝑣𝑡 − 2𝐿 𝑜 𝑐 𝑣 𝑛 2
=𝑘𝑣 2𝐿 𝑜 𝑐 𝑛(2𝑙 −𝑛)
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

6. Intensity of the filed inside cavity:
𝐼 𝑖𝑛 𝑡 ∝ 𝑛=0 ∞ 𝑟 2𝑛 𝑒 𝑖 𝜙 𝑛 2 , 𝜙 𝑛 = 𝑘𝑣 2𝐿 𝑜 𝑐 𝑛(2𝑙 −𝑛)
If cavity scan speed is fast enough such that: 𝜙 𝑛 >> 1 at 𝑛=𝑙
𝑘𝑣 2𝐿 𝑜 𝑐 ≫ 𝑣≫ 𝑐 2𝑘𝐿 𝑜
Then the intensity is:
𝐼 𝑖𝑛 𝑡 ∝ 𝑟 4𝑙 𝑛=0 ∞ 𝑒𝑥𝑝 𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑛(2𝑙 −𝑛) ∝ 𝑅 2𝑙
=exp⁡( ln (𝑅 2𝑙 )) =exp⁡{ 2𝑙( ln 1− 1−𝑅 }
≈exp[−2 1−𝑅 𝑙] ≈ exp[−2 1−𝑅 ct/2L]
= exp(−t/ 𝑡 𝑐𝑎𝑣 )
(1-R) << 1
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

7. Intensity of the filed inside cavity:
If cavity scan speed is slow : 𝑣 ~ 𝑐 2𝑘𝐿 𝑜
𝐼 𝑖𝑛 𝑡 ∝ 𝑛=0 ∞ 𝑟 2𝑛 𝑒𝑥𝑝 𝑖𝑘𝑣 2𝐿 𝑜 𝑐 [ 𝑙 2 − (𝑛−𝑙) 2 ] 2
= 𝑛 ′ =−𝑙 ∞ 𝑟 2( 𝑛 ′ +𝑙) 𝑒𝑥𝑝 𝑖𝑘𝑣 2𝐿 𝑜 𝑐 [ 𝑙 2 − 𝑛 ′ 2 ] 2
= 𝑟 2𝑙 𝑒𝑥𝑝 𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑙 𝑛 ′ =−1 ∞ 𝑟 2 𝑛 ′ 𝑒𝑥𝑝 −𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑛 ′
= 𝑅 2𝑙 𝑛 ′ =−1 ∞ 𝑟 2 𝑛 ′ 𝑒𝑥𝑝 −𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑛 ′
= 𝑅 2𝑙 𝑛 ′′ =1 𝑙 𝑟 −2 𝑛 ′′ 𝑒𝑥𝑝 −𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑛 ′′ 𝑛 ′ =0 ∞ 𝑟 2 𝑛 ′ 𝑒𝑥𝑝 −𝑖𝑘𝑣 2𝐿 𝑜 𝑐 𝑛 ′
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

8. Experiment: measured cavity decay for various scan speeds and analyzed results
Probe laser: 791 nm
Mirror spacing: L0 ~ 1 mm, could be varied by a piezoelectric transducer.
𝑘𝑣 2𝐿 𝑜 𝑐 𝑙 𝑚 2 ≈2𝜋𝑚 𝑎𝑛𝑑 𝑡 𝑚 = 2𝐿 𝑜 𝑐 𝑙 𝑚
Τ12 = 𝑡 2 − 𝑡 1 ≈ 2 − 𝐿 𝑜 𝑐 λ 𝑣 1/2
Time interval between first and second minima:
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

9. Experiment: measured cavity decay for various scan speeds and analyzed results
Probe laser: 791 nm
Mirror spacing: L0 ~ 1 mm, could be varied by a piezoelectric transducer.
Cavity finesse: F = 1.03 × 106
Cavity decay time: 𝑡 𝑐𝑎𝑣 = 1.14 μs
Τ12 ~ 170 ns
Mirror velocity: 𝑣 = 32 μm/s
Τ12 ~ 380 ns
Mirror velocity: 𝑣 = 6.4 μm/s
Kyungwon An, C. Yang, R. Dasari, and M. Feld, Optics Lett. 20, (1995)

10. Pulse train in a tunable cavity: Model
From: Pulse train in a cavity by Xiwen Zhang (2012)

11. Pulse train in a tunable cavity: Model
Interference condition:
Resonance condition:
Size of the resonant cavity:
Resonant wavelength:
From: Pulse train in a cavity by Xiwen Zhang (2012)

12. Intensity of field inside cavity
A similar approach yields:
𝐼 𝑖𝑛 𝑙=𝑣 𝑝 𝑜 + 𝑝 ′ ∝ 𝑙 ′ =1 𝑙 𝑅 − 𝑐 2𝐿 𝑜 𝑇 𝑙 ′ 𝑒𝑥𝑝 −𝑖𝑘𝑣 𝑐 2𝐿 𝑜 𝑇 2 𝑙 ′
Time interval between two adjacent minima:
𝑇 𝑚,𝑚−1 = 𝑡 𝑚 − 𝑡 𝑚−1 ≈ 𝑚 − 𝑚− 𝐿 𝑜 𝑐 λ 𝑣 ,
𝑡 = 𝑙𝑇
From: Pulse train in a cavity by Xiwen Zhang (2012)

13. A Numerical result 𝑇 𝑚,𝑚−1 = 𝑡 𝑚 − 𝑡 𝑚−1 ≈ 𝑚 − 𝑚−1 2𝐿 𝑜 𝑐 λ 𝑣 1 2 ,
𝑇 𝑚,𝑚−1 = 𝑡 𝑚 − 𝑡 𝑚−1 ≈ 𝑚 − 𝑚− 𝐿 𝑜 𝑐 λ 𝑣 ,
From: Pulse train in a cavity by Xiwen Zhang (2012)

14. Applications Conclusion Cavity ring-down with a cw probe laser:
Measuring the relative velocity of the mirrors
Determining the linewidth of the probe laser
Cavity ring-down with a pulse train:
Measurement of group velocity dispersion
Determining the cavity loss factor
Conclusion
Fast scan of the cavity results in the usual decay signal.
Slow scan of the cavity shows oscillations due to interference between the incident wave along the decay signal.
Similar oscillations should be seen when we use a pulse train.

15. Experimental setup in our Lab.

16. Experimental results showing oscillations due to acoustic noise:
The fitting is not accurate enough due to such oscillations
Fields build up inside cavity,
Then a trigger signal turns off the AOM
Fitting to exponential function to get the decay constant

17. Decay constants(µs) vs. detector’s position(mm)
The Baseline is not stable.
Different sets of data were collected and showed similar results.

18. Cavity Ring-Down with a Pulse Train
Current challenges to consider if we want to apply the slow scan technique:
A narrower linewidth than the one we are currently using.
Suppressing the acoustic noise from the environment.
Using lighter cavity mirrors and piezoelectric transducer to avoid vibrations.
J. Wojtas, J. Mikolajczyk, and Z. Bielecki, Sensors 13(6), (2013)