NICE-OHMS of Oxygen at Cryogenic Temperatures
8.1 Optical frequency modulation
In contrast to cavity enhancement, the use of frequency modulation (FM) tech-niques in spectroscopy increases the detection sensitivity by reducing the
detec-121
122 NICE-OHMS OF OXYGEN AT CRYOGENIC TEMPERATURES tion noise. Small resonance information is recovered from a noisy background by pushing the associated signal to higher frequency regions where both the source and the detector have relatively low noise amplitudes. The ultimate detection sensitivity is then in principle only limited by fundamental quantum noise, such as shot noise.
The high bandwidth associated with common radio frequency (rf) modu-lation enables a fast signal recovery in the subsequent demodumodu-lation process, often referred to aslock-in detection. An absorption type resonance is thereby typically converted into a dispersion-like lineshape, whose almost linear be-havior around the resonance peak is also ideally suited for feedback purposes.
Since their first demonstration in 1980 [113, 114], optical FM techniques have therefore been very popular not only in laser spectroscopy, but also in laser stabilization [111]. Both aspects have been extensively used in this thesis.
8.1.1 FM principles
A typical setup for the implementation of optical frequency modulation is sketched in Figure 8.1. The output of a quasi-monochromatic light source at os-cillation frequencyωLis phase-modulated at frequencyωm, so that the resulting optical field may be written as
E(t) =E0exp[−iωLt]·exp[−iβsin(ωmt)]
=E0· ∞ k=−∞
Jk(β) exp[−i(ωL+kωm)t] ,
using the convention J−n(β) = (−1)nJn(β) withn∈Nfor the Bessel functions Jn. The amplitude of the modulation β is called the modulation index. Since the light oscillation frequency at any time is ω(t) = ˙ϕ(t) per definition, the above is exactly equivalent to a frequency modulation of ωL. While both thus
ACDC
R L
I
Phase modulator Optical system RF detector
Light source
H( )!
Mixer
Demodulated DC signal
!m
RF amplification RF oscillator
phase lock
Low pass
Figure 8.1: Standard setup for optical frequency modulation.
OPTICAL FREQUENCY MODULATION 123 have the same effect, one would in general use the latter term to refer to a mod-ulation method directly acting on the laser emission frequency. External phase modulation is usually accomplished with electro-optic modulators (EOMs).
The frequency-modulated light field propagates through an arbitrary opti-cal system, characterized by its transfer function H(ω), which has to account for any attenuation and phase shift acquired from the point of modulation.
Consequently, the optical field at the position of a subsequent detector is given by
EH(t) =E0· ∞
k=−∞
Jk(β)H(ωL+kωm) exp[−i(ωL+kωm)t] .
The corresponding photocurrent I(t) is in turn correlated with the local power on the detector:
The factor I0 can be expressed in terms of the photodiode responsivity η and its active area A:
I0=ηA·1
2cε0|E0|2
DC contributions have k=k, so that the overall DC current IDC=I0
∞ k=−∞
|Jk(β) H(ωL+kωm)|2 (8.1) simply is proportional to the sum of the intensities contained in the individual sidebands. Additionally, the photodiode current has components at all integer multiples of the modulation frequency, which may be extracted by according demodulation and subsequent averaging over a certain time interval. According to the sampling theorem, this always has to be longer than the inverse Nyquist frequency 2νm, which ultimately limits the attainable signal recovery band-width. In practice, as indicated in Figure 8.1, demodulation is accomplished by mixingI(t) with the output of a phase-locked oscillator at the desired frequency componentnνm and relative phaseϕm. Subsequent low-pass filtering then only leaves contributions with k−k−n= 0, so that the resulting current is
Inνm =gI0 g represents the overall electronic gain in these processes. Because Jk(β) van-ishes for|k| → ∞, suitable estimates for any given value ofβ can be found with a finite summation by substituting
∞ k=−∞
−→
q−n
k=−q
124 NICE-OHMS OF OXYGEN AT CRYOGENIC TEMPERATURES
-3 -2 -1 0
0
0
0
0
0
0
0
0
1 2 3 -3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
-30 -20 -10 0 10 20 30
-300 -200 -100 0 100 200 300
(ºL-ºh)/¢ºh
-300 -200 -100 0 100 200 300
(ºL-ºh)/¢ºh
-30 -20 -10 0 10 20 30
Absorption ('m= 0) Dispersion ('m=¼/2)
ºm=¢ºh/10
ºm= 10¢ºh ºm=¢ºh
ºm= 100¢ºh
Figure 8.2: Gallery of typical heterodyne lineshapes. Pure absorption signals are obtained forϕm= 0, whileϕm=π/2 yields pure dispersion signals. Both are plotted to scale at each individual modulation frequency. At higher modulation frequencies, one can clearly distinguish the independent interaction of the individual sidebands with the sample. The lineshapes of the absorption coefficient and refractive index are assumed to be homogeneously broadened and added in gray for comparison.
OPTICAL FREQUENCY MODULATION 125 in (8.2), whereq≥nwould be the order of approximation. Some representative optical heterodyne signals calculated from (8.2) with n = 1 and q = 5 are illustrated in Figure 8.2 for the case of an absorbing gas. The optical transfer function is thus given by equation (4.13), where the absorption coefficient α is assumed to have a Lorentzian, homogeneously broadened lineshape with∆νh = 1 MHz. The modulation index is taken to beβ= 1. Depending onϕm, both pure absorption as well as pure dispersion signals can be recovered. This is possible because FM spectroscopy simultaneously compares both the amplitude and the phase of all transmitted sidebands.
8.1.2 Dual-frequency modulation
Dual-frequency modulation (DFM) has originally been considered by DeVoe et al. as a means of comparing a laser frequency to a radio-frequency standard us-ing an optical resonator [115]. This is achieved by lockus-ing the laser to the cavity and in turn the cavity’s free spectral range to an rf standard. Alternatively, the same principle can also be applied vice versa to stabilize a frequency source on the FSR of a given resonator, as required for the NICE-OHMS technique.
Instead of using two independent modulations, a modulation of the modu-lation frequencyωm itself is considered here. If the latter is varied at frequency Ωm, the resulting optical field becomes
E(t) =E0exp[−iωLt]·exp[−iαsin (ωmt+βsin(Ωmt))]
=E0· ∞ k,l=−∞
Jk(α)Jl(kβ) exp[−i(ωL+kωm+lΩm)t] ,
where α and β are the respective modulations indices. Any subsequent optical elements shall again be characterized by the overall transfer functionH(ω). The detector’s photocurrent then is Demodulation at Ωm with relative phase Φm and suitable low-pass filtering finally gives In analogy to the preceding section, the sum can again be approximated by limiting it to finite values with the substitution
126 NICE-OHMS OF OXYGEN AT CRYOGENIC TEMPERATURES
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0
ErrorSignal
(!m- 2¼FSR)/ m
50 kHz 100 kHz 150 kHz Detuning
Figure 8.3: Dual-frequency modulation error signal. It is calculated from(8.3) with Φm =π/2 under the assumption that the cavity is on resonance with the laser light.
The additional curves show the effect of increasing detuning ofωLaway from the cavity resonance, in steps of the cavity’s HWHM of about 50 kHz.
As before, minimum suitable values for qk and ql depend on the actual mod-ulation indices α and β. For qk = ql = 5 and α =β = 1, Figure 8.3 shows a calculated error signal for a cavity withF = 10 000 and detection in reflection.
The corresponding optical transfer function is thus given by equation (7.7). The free spectral range is assumed to be 1 GHz and Ωm= 1 MHz.
The error signal is obtained at constant laser frequency and can obviously serve to determine the deviation of ωm from the cavity’s FSR in a feedback loop. In contradiction to previous statements [112], however, small changes of the laser frequency away from resonance are found to significantly affect the quality of the signal. In particular, its derivative changes sign when these are on the order of the cavity linewidth or larger, making any further stabilization efforts futile. The maximum tolerable deviation, defined as the point where the signal derivative at ωm = 2πFSR reaches zero, is depicted in Figure 8.4 as a function of the modulation frequency Ωm. It demonstrates that the laser frequency has to be actively held on resonance with the cavity to obtain a useful error signal.
As a concluding remark it should be pointed out that the DFM technique might also serve to measure the cavity length to high precision, since the latter is directly connected with the FSR value. One would therefore have to simply count ωm while it is locked to the free spectral range.