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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 49, Iss. 29 — Oct. 10, 2010
  • pp: 5597–5613

Faraday modulation spectrometry of nitric oxide addressing its electronic X 2 Π A 2 Σ + band: I. theory

Lemthong Lathdavong, Jonas Westberg, Jie Shao, Claude M. Dion, Pawel Kluczynski, Stefan Lundqvist, and Ove Axner  »View Author Affiliations

Applied Optics, Vol. 49, Issue 29, pp. 5597-5613 (2010)

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We give a simple two-transition model of Faraday modulation spectrometry (FAMOS) addressing the electronic X 2 Π ( ν = 0 ) A 2 Σ + ( ν = 0 ) band in nitric oxide. The model is given in terms of the integrated line strength, S, and first Fourier coefficients for the magnetic-field-modulated dispersive line shape function. Although the two states addressed respond differently to the magnetic field (they adhere to the dissimilar Hund coupling cases), it is shown that the technique shares some properties with FAMOS when rotational-vibrational Q-transitions are targeted: the line shapes have a similar form and the signal strength has an analogous magnetic field and pressure dependence. The differences are that the maximum signal appears for larger magnetic field amplitudes and pressures, 1500 G and 200 Torr , respectively.

© 2010 Optical Society of America

OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(020.7490) Atomic and molecular physics : Zeeman effect
(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors
(300.6260) Spectroscopy : Spectroscopy, diode lasers
(300.6540) Spectroscopy : Spectroscopy, ultraviolet
(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

ToC Category:
Atmospheric and Oceanic Optics

Original Manuscript: May 27, 2010
Manuscript Accepted: August 12, 2010
Published: October 7, 2010

Lemthong Lathdavong, Jonas Westberg, Jie Shao, Claude M. Dion, Pawel Kluczynski, Stefan Lundqvist, and Ove Axner, "Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+band: I. theory," Appl. Opt. 49, 5597-5613 (2010)

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  24. Although Robinson provided a theoretical description of the MRS from the eight lines originating from a particular rotational state in the ground configuration (the J′′=13/2 state) for a given set of conditions (a given pressure and magnetic field) , the description does not provide any general information useful for predicting or modeling FAMOS signals from other states or under other conditions in any systematic manner.
  25. Although “optimum conditions” often refer to the situations when the signal-to-noise ratio is maximized, we will not do so here. Because we do not consider the noise in the system in this work, “optimum conditions” will refer to the cases for which the signal is maximized.
  26. J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010). [CrossRef]
  27. We have here utilized the same nomenclature as in Ref. , i.e., a tilde sign indicates that the entity is given in units of inverse centimeters, whereas an overbar shows that the entity is dimensionless. The superscript D indicates that it is normalized with respect to δν˜D/ln⁡2.
  28. When the magnetic field changes direction, the propagation of the light alters between being parallel and antiparallel to the magnetic field. If the quantization axis is taken as the direction of the magnetic field, B, as is customary, LHCP light should alter between inducing ΔM=+1 and ΔM=−1 transitions. However, it is mathematically inconvenient to have a quantization axis whose direction is periodically reversed and thereby to periodically alter the transition rules for a given helicity of the light. We have here instead let the quantization axis be fixed along the direction of B0, even though the magnetic field changes direction.
  29. Using ordinary angular momentum coupling, the transition dipole moment squared of a transition between two states expressed in the same basis sets can be expressed in terms of the square of a Clebsch–Gordan coefficient, which in turn can be expressed in terms of the square of a 3-j symbol.
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  31. As has been discussed previously , this expression for the FAMOS signal differs from that of Ganser et al. as well as Herrmann et al. by (at least) a factor of 3/(2J+1) and that of Fritsch et al. by a factor of 3.
  32. There is an inherent property of a 3-j symbol of the type given in Eq. that its square summed over all possible M′′ values (and thereby all possible M′ values) yields a value of 1/3. This implies that the sum of all relative dipole moments squared becomes unity.
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  35. N is the quantum number that corresponds to the rotational energy of a state adhering to the Hund case (b) that lacks orbital angular momentum (Λ=0), given by BN(N+1), where B is the rotational constant and is associated with the operator (J−S)2, where J and S are the operators for the total angular momentum and the electronic spin, respectively.
  36. As a consequence of a weak coupling between the rotation of the nuclei and the orbital angular momentum of the electron, each lower state is additionally split into two states with opposite symmetry (+ and −, respectively) by a so-called Λ doubling. Because this splitting is smaller than the spin splitting as well as the separation between consecutive rotational levels and transitions only are allowed between states of dissimilar symmetry, this splitting does not give rise to any additional transitions; it can therefore be seen as a perturbation that only shifts the transitions slightly in frequency.
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  39. Takazawa et al. investigated the Q11(J′′) and P11(J′′) branches originating from the Π1/22(J′′) state and the Q12(J′′) and P12Q(J′′) branches from the Π3/22(J′′) state (referred to as Q21(J′′) and P21(J′′) by the authors) and their analysis dealt with light inducing ΔM=0 transitions .
  40. Because the total line strength of a transition is not altered by a splitting of the level, it is possible to conclude that S¯Π,MJ,Σ,1/2L+S¯Π,MJ,Σ,−1/2L=S¯Π,ΣL=1 and SΠ,MJ,Σ,1/2R+SΠ,MJ,Σ,−1/2R=S¯Π,ΣR=1, which, in turn, leads to S¯Π,MJ,Σ,1/2L−S¯Π,MJ,Σ,1/2R=S¯Π,MJ,Σ,−1/2R−S¯Π,MJ,Σ,−1/2L, which is Eq. .
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  42. SΠ,Σ and SΠ,Σ′ are related to each other through the relation SΠ,ΣNNO=SΠ,Σ′pNO.
  43. With a splitting of the transition of gSμBB0, and a value of the Bohr magneton of 4.6710−5cm−1/G, a splitting of 25cm−1 for a magnetic field of 25×104G provides a value of the gS factor of 2.1.
  44. C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).
  45. The model is appropriate for the cases when the magnetic splitting of the upper level significantly exceeds that of the lower state, which it does for all states except for those with the lowest rotational quantum number (it is valid primarily for states with J>6.5). However, because the states with lowest rotational quantum number in general also have lower line strengths than those with larger J, this is not a severe limitation.
  46. Because the final detectability of a technique is given by a number of entities, including the available laser power at the transition used, properties of the polarizers as well as the detectors, including noise and disturbances, this analysis cannot yet fully assess the detectability and thereby the true applicability of the FAMOS technique when electronic transitions are addressed. Despite this, the present work, with its characterization of the optimum conditions for FAMOS addressing electronic transitions has provided a first step toward such an assessment.

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