Frequency modulation absorption spectroscopy: Theory

Chapter VI: Frequency Modulation Spectroscopy of YbOH

6.1 Frequency modulation absorption spectroscopy: Theory

Frequency modulation absorption spectroscopy (FM spectroscopy) allows the sen- sitive measurement of the absorption (or dispersion) signals of very weak spectral features [148–150]. This is accomplished by pushing the absorption and dispersion signals into the radio frequency (rf) regime. Dealing with rf signals reduces the 1/𝑓 noise in the system while additional noise rejection is provided via phase-sensitive detection of the rf signals. Working in the rf regime is especially beneficial in our CBGB source as many of the noise sources are in the Hz-kHz range (the pulse tube operates at 4 Hz and the laser locks at kHz frequencies) and FM spectroscopy enables significant suppression of these noise sources.

A brief description of the theory behind FM spectroscopy is given below; more thorough descriptions can be found in Ref. [148, 151, 152]. The electric field of a

single-frequency laser is given by [148, 151]

𝐸₀(𝑡)= 𝐸₀exp𝑖𝜔₀𝑡+𝑐 .𝑐 ., (6.1) where𝜔₀ is the carrier frequency of the unmodulated laser light. The laser beam is then passed through a phase modulator that is driven by a sinusoidal rf field of frequency 𝜔_𝑚, the modulation frequency. Following the phase modulator, the electric field of the laser is given by

𝐸_{𝑃 𝑀} =𝐸₀exp[𝑖(𝜔₀𝑡+𝑀sin𝜔_𝑚𝑡)] +𝑐 .𝑐 .

=𝐸₀exp(𝑖𝜔₀𝑡)

∞

∑︁

𝑛=−∞

𝐽_𝑛(𝑀)exp(𝑖(𝜔₀+𝑛𝜔_𝑚)𝑡) +𝑐 .𝑐 .,

(6.2)

where 𝐽_𝑛(𝑀) is the order 𝑛Bessel function of argument 𝑀. 𝑀 is the modulation depth which is dependent on the amplitude of the rf field driving the phase modulator.

The frequency spectrum of the phase modulated laser will contain a component at the carrier frequency,𝜔₀, and sideband components located at integer multiples of

±𝜔_𝑚 from the carrier.

Now consider a sample with a frequency-dependent absorption coefficient, 𝛼(𝜔), and frequency-dependent index of refraction, 𝜂(𝜔). The amplitude attenuation (absorption1) is given by 𝛿(𝜔) = 𝛼(𝜔)𝑙/2 and the phase shift is given by 𝜙(𝜔) = 𝜂(𝜔)𝑙 𝜔/𝑐 (dispersion)2. The frequency dependence of 𝛿(𝜔) and 𝜙(𝜔) are the absorption and dispersion lineshapes, respectively. Then the transmission function of the sample is [151]

𝑇(𝜔) =exp[−𝛿(𝜔) −𝑖 𝜙(𝜔)]. (6.3) If the frequency modulated laser is passed through the sample the electric field of the resulting, transmitted laser beam is

𝐸_𝑇 =𝐸₀exp(𝑖𝜔₀𝑡)

∞

∑︁

𝑛=−∞

𝑇(𝜔₀+𝑛𝜔_𝑚)𝐽_𝑛(𝑀)exp(𝑖(𝜔₀+𝑛𝜔_𝑚𝑡)) +𝑐 .𝑐 . (6.4) If the transmitted laser is detected with a fast photodiode, the signal will be pro- portional to the intensity of the laser, 𝐼_𝑇(𝜔) =

𝐸¯_𝑇

2. The signal will then contain frequency components at dc and at integer multiples of𝜔_𝑚, due to the beating of any pair of the laser frequency components separated by integer multiples of 𝜔_𝑚.

1If this expression is compared to Eq. 3.8 in Ch. 3, we can see that 2𝛿(𝜔) =𝜎(𝜔)𝑛(𝑡)𝑙and therefore,𝛼(𝜔)=𝜎(𝜔)𝑛(𝑡).

2𝑐is the speed of light and𝑙is path length.

If phase sensitive detection is used, the terms oscillating at a frequency of𝜔_𝑚 can be isolated. Expanding Eq. 6.4 in the limit of weak absorption and dispersion and only keeping the terms with a frequency of𝜔_𝑚 gives

𝐸¯_𝑇

𝜔𝑚

=𝐸²

0exp[−2𝛿(𝜔₀)]

×

"

2 cos𝜔_𝑚𝑡

∞

∑︁

𝑛=0

𝐽_𝑛(𝑀)𝐽_𝑛+1(𝑀) [𝛿(𝜔₀− (𝑛+1)𝜔_𝑚) −𝛿(𝜔₀+ (𝑛+1)𝜔_𝑚) +𝛿(𝜔₀−𝑛𝜔_𝑚) −𝛿(𝜔₀+𝑛𝜔_𝑚)]

+2 sin𝜔_𝑚𝑡

∞

∑︁

𝑛=0

𝐽_𝑛(𝑀)𝐽_𝑛+1(𝑀) [𝜙(𝜔₀− (𝑛+1)𝜔_𝑚) −𝜙(𝜔₀−𝑛𝜔_𝑚) +𝜙(𝜔₀+ (𝑛+1)𝜔_𝑚) −𝜙(𝜔₀+𝑛𝜔_𝑚)]].

(6.5) Phase sensitive detection is accomplished by mixing the photodiode signal with a pickoff of the original rf drive via an rf mixer. The resulting dc signal will depend on the phase angle,𝜃, the phase difference between the two paths from the rf oscillator to the rf mixer. At an arbitrary value of𝜃, the mixer output will be a sine and cosine weighted mixture of the absorption and dispersion signals [151]

𝐼_{𝐹 𝑀}(𝜔) =cos𝜃 𝐴_{𝐹 𝑀}(𝜔) +sin𝜃 𝐷_{𝐹 𝑀}(𝜔), (6.6) where

𝐴_{𝐹 𝑀}(𝜔) =𝐸²

0exp[−2𝛿(𝜔)]

∞

∑︁

𝑛=0

𝐽_𝑛(𝑀)𝐽_𝑛+1(𝑀)

× [𝛿(𝜔− (𝑛+1)𝜔_𝑚) −𝛿(𝜔+ (𝑛+1)𝜔_𝑚) +𝛿(𝜔−𝑛𝜔_𝑚) −𝛿(𝜔+𝑛𝜔_𝑚)]

(6.7)

is the FM absorption signal and 𝐷_{𝐹 𝑀}(𝜔)=𝐸²

0exp[−2𝛿(𝜔)]

∞

∑︁

𝑛=0

𝐽_𝑛(𝑀)𝐽_𝑛+1(𝑀)

× [𝜙(𝜔− (𝑛+1)𝜔_𝑚) −𝜙(𝜔−𝑛𝜔_𝑚) +𝜙(𝜔+ (𝑛+1)𝜔_𝑚) −𝜙(𝜔+𝑛𝜔_𝑚)]

(6.8)

is the FM dispersion signal.

FM spectroscopy is often conducted in the low modulation depth limit,𝑀 ≲ 1, where the frequency spectrum of the laser consists of the carrier and only the two first order

sidebands. In this limit 𝐽₀(𝑀) ≈1, 𝐽_±1(𝑀) ≈ 𝑀/2, and𝐽_|𝑛|≥2(𝑀) ∼ O (𝑀²) ≈ 0 and Eq. 6.7 and 6.8 reduce to [151]

𝐴_{𝐹 𝑀}(𝜔) [𝑀 ≤ 1] = 𝐸²

0

2 exp[−2𝛿(𝜔)]𝑀[𝛿(𝜔−𝜔_𝑚) −𝛿(𝜔+𝜔_𝑚)] (6.9) and

𝐷_{𝐹 𝑀}(𝜔) [𝑀 ≤ 1] = 𝐸²

0

2 exp[−2𝛿(𝜔)]𝑀[𝜙(𝜔−𝜔_𝑚) +𝜙(𝜔+𝜔_𝑚) −2𝜙(𝜔)]. (6.10) Note that in the above equations the amplitude of both 𝐴_{𝐹 𝑀} and 𝐷_{𝐹 𝑀} are pro- portional to 𝑀, the modulation depth. Therefore, operating at higher modulation depths will increase the FM signal size. However, at high enough modulation depths, higher-order sidebands will no longer be negligible and will need to be accounted for. Eq. 6.9 and 6.10 provide a good intuitive understanding of how FM absorption works. The FM absorption signal is the result of the differential absorption between the two sidebands, while the FM dispersion signal is the difference between the average phase shift of the sidebands and twice the phase shift of the carrier.

The enhanced sensitivity provided by FM absorption is often used to measure weak absorption features. In the case of weak absorption, the amplitude attenuation of the carrier due to absorption of the sample will be negligible. In this limit (absorption of ≲ 1%), the factor of exp[−2𝛿(𝜔)] ≈ 1 in Eq. 6.7, 6.8, 6.9, and 6.10. If the absorption is ≳ 1%, then the carrier will be attenuated and this effect will need to be accounted for in the lineshape.

𝐼_{𝐹 𝑀} given in Eq. 6.6 is refered to as the in phase signal, the demodulated signal that is in phase with the rf reference at the mixer. Experimentally, the phase angle 𝜃 describing the mixing of the absorption and dispersion components in𝐼_{𝐹 𝑀} is not easily determined. Therefore, it is also convenient to measure the in-quadrature signal as well. This is accomplished with an I and Q demodulator, which is a dual output mixer, mixing the input signal with the rf reference (I channel) and the input signal with the rf reference after a 90^◦ phase shift is applied (Q channel). The quadrature FM signal is given by [151]

𝑄_{𝐹 𝑀}(𝑤)=sin𝜃 𝐴_{𝐹 𝑀}(𝜔) −cos𝜃 𝐷_{𝐹 𝑀}(𝜔). (6.11) With a simultaneous measurement of both𝐼_{𝐹 𝑀} and𝑄_{𝐹 𝑀} the phase angle,𝜃 can be determined. Examples of absorption, dispersion, in-phase and in-quadrature FM signals are shown in Section 6.2 and 6.3 below. A discussion of the lineshapes themselves is also given.