Abstract

We present the analysis of an unorthodox technique for locking a laser to a resonant optical cavity. Error signals are derived from the interference between the fundamental cavity mode and higher-order spatial modes of order two excited by mode mismatch. This scheme is simple, inexpensive, and, in contrast to similar techniques, first-order insensitive to beam jitter. After mitigating sources of technical noise, performance is fundamentally limited by quantum shot noise.

© 2014 Optical Society of America

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References

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2011 (1)

2000 (1)

1999 (1)

1996 (1)

R. A. Boyd, J. L. Bliss, and K. G. Libbrecht, Am. J. Phys. 64, 1109 (1996).
[CrossRef]

1994 (1)

1984 (1)

1982 (1)

1980 (1)

T. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

Adhikari, R.

Anderson, D. Z.

Arndt, M.

Asenbaum, P.

Bliss, J. L.

R. A. Boyd, J. L. Bliss, and K. G. Libbrecht, Am. J. Phys. 64, 1109 (1996).
[CrossRef]

Boyd, R. A.

R. A. Boyd, J. L. Bliss, and K. G. Libbrecht, Am. J. Phys. 64, 1109 (1996).
[CrossRef]

Camp, J.

Couillaud, B.

T. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

Gilbert, S. L.

Gray, M. B.

Hansch, T.

T. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

Libbrecht, K. G.

R. A. Boyd, J. L. Bliss, and K. G. Libbrecht, Am. J. Phys. 64, 1109 (1996).
[CrossRef]

Mavalvala, N.

McClelland, D. E.

Meers, B. J.

Morrison, E.

Mueller, G.

Reitze, D.

Robertson, D. I.

Shaddock, D. A.

Shu, Q.-Z.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sigg, D.

Tanner, D. B.

Ward, H.

Wieman, C. E.

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Figures (4)

Fig. 1.
Fig. 1.

Optical intensities of the Ψ0 and Ψ2 spatial modes (left and right axes, respectively). Each mode has an integrated power of 1 W.

Fig. 2.
Fig. 2.

Optical setup required to implement our technique. Only one of the two paths in reflection of the cavity, either 1 or 2, is required. Path 1’s bullseye photodiode consists of an inner circle of radius Rsplit surrounded by an annulus. The expanded view of the cavity depicts the fundamental cavity eigenmode using a dashed blue line and the mismatched input beam in solid red. Vertical lines indicate the waist positions and diameters.

Fig. 3.
Fig. 3.

Uppermost axes: theoretical error signal obtained using two conventional photodiodes and the cavity parameters Plaser=1W, λlaser=1064nm, Lcavity=1.3m, ra2=0.95, rb2=0.99, ROCa=ROCb=4m, and ϵ=0.01+i0.005. The width of the linear part of the error signal is set by the full-width half-maximum-power cavity linewidth. The feature near to ϕrt=π/2 is an error signal for the Ψ2 resonance. In this case the fundamental mode acts as the phase reference. Lower axes: error signals during tuning of iris position for various deviations, Δγ(z), from the optimal Gouy phase. At each location the radius of the iris is adjusted such that the error signal is zero far from resonance.

Fig. 4.
Fig. 4.

Output of the three photodiodes shown in Fig. 2 (upper axes) and the resulting error signal (lower axes) as the laser frequency is swept across Ψ0 resonance. Trans DC has been normalized and Photodiode B’s response has been scaled to match that of Photodiode A. At the time of this measurement, 70% of the input light was coupled into the Ψ0 mode. Such large mode mismatch is not a requirement of this technique.

Equations (8)

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Ψ0(r,z)=2/πω2(z)exp[r2/ω2(z)],
Ψ2(r,z)=2/πω2(z)[12r2/ω2(z)]×exp[r2/ω2(z)+i2γ(z)]=Ψ^2exp[i2γ(z)],
Ψ0in=Ψ0cavity+ϵΨ2cavity,
whereϵ=δω0ω0cavity+iδz2zRcavity,
r(ϕrt)=rata2rbexp(iϕrt)1rarbexp(iϕrt),
Preflcavity=A|r0Ψ0|2(i)+|ϵr2Ψ2|2(ii)+2R(ϵ*r0r2*Ψ0Ψ2*)(iii)dA,
2I|ϵr0r2|sin[θr2θr0+2γ(z)+θϵ+π/2],
I=0RsplitΨ0Ψ^22πrdrRsplitΨ0Ψ^22πrdr

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