Demonstration of a mid-infrared NO molecular Faraday optical filter

Kuijun Wu; Yutao Feng; Juan Li; Guangbao Yu; Linmei Liu; Yuanhui Xiong; Faquan Li

doi:10.1364/OE.25.030916

1. Introduction

Faraday anomalous dispersion optical filter (FADOF) plays a key role in the detection of weak signals and is by far better than any commercial filter for its excellent optical properties, such as near-unity transmission, narrow bandwidth, high noise rejecting capability, fast time response and multi-mode imaging property [1]. Therefore, FADOF has been widely applied in lidar [2–4], sun observations [5, 6], atmospheric sensing [7], laser frequency locking [8, 9], optical communications (laser or quantum) [10, 11], and other areas [12, 13].

FADOF is a type of atomic filters and relies on resonant Faraday anomalous dispersion effect to rotate the polarization of signal light. So its working materials are restricted to paramagnetic atoms and its wavelength is restricted by the atomic transition frequencies. A lot of efforts have been made to expand the working wavelength range of FADOF, especially the proposing and realizing of excited state FADOF (ES-FADOF) [14, 15] and isotope FADOF (I-FADOF) [16]. However, the working wavelengths of FADOF are still limited to several certain wavelengths in the visible and near-infrared region, such as 420nm (⁸⁷Rb) [16], 455 nm (Cs) [17], 532 nm (Rb) [18], 589 nm(Na) [3], 770 nm (K) [4], 795 nm (Rb) [19], and 1529 nm(Rb) [14, 15]. Therefore, the FADOF based on hot atomic species’ transitions cannot meet the various needs from the upcoming applications, especially in mid-infrared optical systems, which need transitions between high-energy excited states. As we all know, there are many more energy levels in a molecule than in an atom, including rotational, vibrational, and electronic energy levels. Therefore, molecular spectrum is much richer in lines. Generally, transitions between electronic levels occur mainly in the ultraviolet and visible region, while, vibrational-rotational transitions occur in the near-infrared and mid-infrared region. Thus, using paramagnetic molecules as working materials of FADOF will greatly expand the wavelength range of FADOF.

Although there are no reports on molecular Faraday optical filters (MFOFs), several molecular absorption optical filters (MAOFs) have been discussed and built at different wavelengths. In the visible region, molecular iodine vapor filters (MIVFs) are widely used, because the injection-seeded double Nd:YAG pulsed laser can be readily temperature tuned across a number of I₂ absorption features at 532 nm, and light at this wavelength has good atmospheric and underwater transmission. MIVFs have been commonly employed as optical frequency filter for diagnostic techniques in fluid mechanics [20], combustion [21], and lidar for atmospheric wind measurement [22] and ocean remote sensing [23]. In the mid-infrared region, Gas Filter Correlation Radiometry (GFCR) is a well-known technique, which has been employed successfully on numerous platforms including aircraft [24], satellites [25], and the space shuttle [26]. GFCR, which uses a sample of the gas being measured as a spectral filter to isolate infrared radiation absorbed or emitted by the target gas in the atmosphere, provides a fast response, high reliability, and a simpler, more compact design compared to conventional mechanical gas sensors.

The primary advantage of MFOFs over MAOFs, including MIVF and GFCR, is their ultrahigh-background rejection. The filter transmission of MFOF results from the dispersive polarization rotations (DPRs) caused by the resonant Faraday effect for light propagating along the magnetic field. Large highly DPRs take place only for light in the vicinity of a paramagnetic molecule resonance, while the polarization rotation of light away from the MFOF transmission peaks are vanishingly small. This results in the out-of-band rejection of the MFOF being determined by the extinction ratio of polarizers. Higher-noise rejection optical filters would have more advantages in the detection of weak signals, especially in the infrared region. Recently, broadband and wide angle infrared polarizers with high extinction ratio have been designed, which provides the possibility of improving the performance of MFOFs [27].

The Faraday rotation spectroscopy (FRS) of paramagnetic molecules such as O₂ [28], NO [29], NO₂ [30], ⁷⁹Br₂ [31] and free radical species (OH [32], ·HO₂ [33]) has been widely studied. More recently, the FRS technology has been developed as a zero-background sensor for ultrasensitive detection of paramagnetic trace gases [34–36]. Few studies, however, have focused on applying the resonant Faraday effect of paramagnetic molecules to develop optical filters.

Our motivation for this work is to develop a new improved FADOF which uses NO molecules as working material, because NO molecules play very important roles in many chemical and physical processes ranging from atmospheric science to combustion reaction. There are three major vibrational absorption bands in the mid-infrared part of NO molecule spectrum. Compared with the first and second overtones at approximately 2.7 and 1.8μm, the fundamental band near 5.2 μm is promising for good filter performance due to the fact that the fundamental band has the strongest vibrational absorption spectrum. We study the NO-MFOF’s transmission characteristics in detail, and find that the difference of Landé g-factors in the R, Q and P branches leads to a strong magnetic field dependence of the filter transmission. Large Faraday rotation transmission (FRT) signal of low rotational levels in the Q branch is obtained at both high and low magnetic field. Transitions in the R and P branches are relatively stronger than Q branch, but their FRT signals are vanishingly small when the external magnetic field is low. Large FRT signals of R and P branches take place only for high magnetic field measurements. Here we report, to the best of our knowledge, the first theoretical and experimental demonstration of using resonant Faraday effect of paramagnetic molecules for developing optical filters.

2. Theory

The Zeeman and Faraday effect of paramagnetic molecules play key roles in MFOFs. The principle scheme of a MFOF is shown in Fig. 1. Generally, a MFOF consists of a molecular cell sandwiched between two crossed linear polarizers and immersed in a longitudinal magnetic field. The output light of the first polarizer is linearly polarized, and resolved into equal amplitude left-hand and right-hand circularly polarized components (LHCP and RHCP). The Zeeman splitting of the energy levels leads to a different index of refraction between LHCP and RHCP for light frequencies in the immediate neighborhood of the sharp molecular resonance lines. This then results in a net rotation of the linear polarized light, which provide an opportunity for the second, initially crossed, polarizer to transmit the light. Simultaneously, the Zeeman absorption of the rotation light in the molecular cell is expected to be weak. The ideal transmission spectrum of MFOF will be attainable when the absorption is minimal and the polarization direction rotates exactly π/2. Therefore, the FRT calculation of MFOF shall take into account both Zeeman spectra and resonant Faraday effect.

Fig. 1 Schematic drawing of the discussed MFOF, consisting of a gas cell in a homogeneous, constant magnetic field B between two crossed polarizers, P1 and P2. The Faraday polarization rotation (represented by red arrows) results in the transmission through the second polarizer.

Download Full Size | PDF

2.1 Zeeman Spectra

The MFOF characteristics relies on the Zeeman splitting of the energy levels in the paramagnetic molecules caused by an external magnetic field. The magnetically affected energy shift of a level is given by

Δ E = - M_{J} g_{J} μ_{B} B

where

M_{J}

is the magnetic quantum numbers,

g_{J}

is the Landé g-factor,

μ_{B}

is the Bohr magneton,

B

is the magnetic-field strength.

The intermediate coupling case between Hund’s coupling case(a) and case (b) is a better approximation for NO molecule either at higher rotational quantum numbers or at lower rotational levels, for both ${}^{2}Π_{\frac{1}{2}}$ and ${}^{2}Π_{\frac{3}{2}}$ subsystems. The values of $g_{J}^{}$ in Hund’s coupling case (a-b) are given by

g_{J}^{} = \frac{3 / 2 \pm [2 (J - 1 / 2) (J + 3 / 2) - 3 Y / 2 + 3] / X}{J (J + 1)}

where

X = \sqrt{4 (J -1/2) (J + 3 / 2) + {(Y - 2)}^{2}}

, and

Y = A_{V} / B_{V}

;

A_{V}

and

B_{V}

are spin-orbit coupling and rotational constants at vibrational level

V

, respectively.

The line intensities for each Zeeman sub-transition, $S_{J^{'} M^{'} {J^{'}}^{'} M^{″}}$ , are proportional to the transition strengths $S_{Ω J^{'} J^{″}}$

S_{J^{'} M^{'} {J^{'}}^{'} M^{″}} = {\bar{S}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}} \cdot S_{Ω J^{'} J^{″}}

where

M^{'}

,

M^{″}

and

J^{'}

,

J^{″}

are the

M

and

J

quantum numbers for the upper and lower states, respectively.

The line-strength factor, ${\bar{S}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}$ , can be described by the corresponding 3J symbol

{\bar{S}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}} = {(\begin{matrix} \begin{matrix} J^{'} \\ - M^{'} \end{matrix} & \begin{matrix} 1 \\ Δ M \end{matrix} & \begin{matrix} J^{″} \\ M^{″} \end{matrix} \end{matrix})}^{2}

where

Δ M = M^{'} - M^{″}

gives the selection rule on

M

. For the Faraday experimental configuration (magnetic field parallel to optical axis), this is

Δ M = \pm 1

.

Δ M =+ 1

transitions are excited by RHCP (

σ^{+}

), while,

Δ M = - 1

transitions are excited by LHCP (

σ^{-}

).

One inherent property of this dimensionless entity is that the sum of all split transitions must equal unity, i.e.,

\sum_{M^{'} M^{″}} {\bar{S}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}} = 1

For the Faraday experimental configuration, the line-strength factors of $σ_{\pm}$ components need to carry a factor of 3/2 to meet the normalization requirement given in Eq. (5).

The schematic illustration of the magnetically induced vibration-rotation transition $v^{'}, J^{'}, M^{'} \leftarrow v^{″}, J^{″}, M^{″}$ is shown in Fig. 2, exemplified for the P(2.5) transition of the ${}^{2}Π_{3 / 2}$ state. The energy level with a given total angular momentum J is split into several sub-states due to the lifting of the degeneracy caused by an external magnetic field B. The number of sub-states depends on the magnetic quantum number M of the rotational level, and Each sub-state split from the degeneracy level has an energy separation of $M_{J} g_{J} μ_{B} B$ , which is schematically shown in Fig. 2(a). Therefore, the frequency of each magnetically induced sub-state transitions $M^{'} \leftarrow M^{″}$ , can be expressed as $v_{J^{'} M^{'} {J^{'}}^{'} M^{″}} = v_{0} + ({M^{'}}_{J} {g^{'}}_{J} - {M^{″}}_{J} {g^{″}}_{J}) μ B$ , which indicates that each sub-state transition’s frequency varies linearly with B (in weak field), as seen from Fig. 2(b). The line-strength factor for each individual Zeeman sub-transition is calculated according to Eq. (4).

Fig. 2 Zeeman energy level pattern for the P(2.5) line of the fundamental vibrational transition $X {}^{2}Π_{\frac{3}{2}} (v = 1 \leftarrow 0)$ . (a) The Zeeman splitting of the lower level $J^{″} = 2.5$ and the upper one $J^{'} = 1.5$ . (b) line-strength factor versus relative frequency shift. (c) Zeeman sub-levels of P(2.5) line versus the magnetic field

Download Full Size | PDF

2.2 Resonant Faraday Effect

The application of a longitudinal magnetic field on a paramagnetic molecule induces both magnetic circular dichroism (MCD) and magnetic circular birefringence (MCB) via the Zeeman effect. MCD and MCB are two main contributors to the change in polarization state of the light and play key role in the transmission characteristics of MFOF.

MCB refers to the difference in phase shifts between the two circular components of the linear polarized light, LHCP and RHCP. The phase shifts for LHCP and RHCP, induced by each $J^{'} M^{'} \leftarrow J^{″} M^{″}$ transition from a lower state to an upper state, can be written as

ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{L, R} (\bar{v}) = \frac{S_{J^{'} M^{'} {J^{'}}^{'} M^{″}} N_{x} L}{2} χ_{d i s p}^{L, R} (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D}, δ {\bar{v}}_{L}^{D})

where

N_{x}

is the density of absorbers, and L is the interaction length.

$χ_{d i s p}^{L, R}$ are the area-normalized dispersive lineshape functions for LHCP and RHCP, and can be expressed as

χ_{d i s p}^{L, R} (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D}, δ {\bar{v}}_{L}^{D}) = - {\hat{χ}}_{0} Im [w (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D} + i δ {\bar{v}}_{L}^{D})]

where

{\hat{χ}}_{0}

is the peak value of the area-normalized absorption Gaussian lineshape function defined as

{\hat{χ}}_{0} = \sqrt{\ln 2} / (\sqrt{π} δ v_{D})

,

ω

is the complex error function given by

ω (z) = ω \frac{i}{π} \int_{- \infty}^{\infty} \frac{e^{- t^{2}}}{z - t} d t

, and

δ {\bar{v}}_{L}^{D}

is the Doppler-width-normalized collisional broadening defined as

δ {\bar{v}}_{L}^{D} = δ v_{L} \sqrt{\ln 2} / δ v_{D}

.

δ v_{L}

and

δ v_{D}

are the collisional and Doppler broadening, respectively.

Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D}

represents the Doppler-width-normalized frequency detuning from the center of the magnetically induced vibration-rotation transition

v^{'}, J^{'}, M^{'} \leftarrow v^{″}, J^{″}, M^{″}

, given by

Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D} = (Δ v_{J^{'} {J^{'}}^{'}}^{0} - Δ v_{J^{'} M^{'} {J^{'}}^{'} M^{″}}) \sqrt{\ln 2} / δ v_{D}

, where

Δ v_{J^{'} {J^{'}}^{'}}^{0}

and

Δ v_{J^{'} M^{'} {J^{'}}^{'} M^{″}}

represents the frequency detuning from the center of the unsplit and magnetically induced shift of the transition, given by

Δ v_{J^{'} {J^{'}}^{'}}^{0} = v - v_{J^{'} {J^{'}}^{'}}^{0}

and

Δ v_{J^{'} M^{'} {J^{'}}^{'} M^{″}} = ({M^{'}}_{J} {g^{'}}_{J} - {M^{″}}_{J} {g^{″}}_{J}) μ B

respectively.

Therefore, the difference in phase shifts between LHCP and RHCP can be expressed as

Δ ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}} (\bar{v}) = ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{L} (\bar{v}) - ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{R} (\bar{v})

Since all transitions between the magnetically shifted sub-states contribute to the polarization rotation of the incident light at the same time, although possibly with different degrees, the total phase shift of the light for the vibration-rotation transition $v^{'}, J^{'} \leftarrow v^{″}, J^{″}$ needs to be written as a sum including all possible magnetically split transitions

Δ ϕ_{J^{'} {J^{'}}^{'}} (\bar{v}) = \sum_{M^{'} M^{″}} [ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{L} (\bar{v}) - ϕ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{R} (\bar{v})]

MCD can be qualitatively explained by similar arguments. The attenuation difference $Δ δ_{J^{'} M^{'} {J^{'}}^{'} M^{″}} (\bar{v})$ resulted from split transitions due to Zeeman effect can thereby be written as

Δ δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}} (\bar{v}) = δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{L} (\bar{v}) - δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{R} (\bar{v})

where

δ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{L, R} (\bar{v})

are the attenuations for LHCP and RHCP of each magnetically induced

J^{'} M^{'} \leftarrow J^{″} M^{″}

transition

δ_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{L, R} (\bar{v}) = \frac{S_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}} N_{x} L}{2} χ_{a b s}^{L, R} (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{D}, δ {\bar{v}}_{L}^{D})

$χ_{d i s p}^{L, R}$ are the area-normalized absorption lineshape functions for LHCP and RHCP, and can be expressed as

χ_{a b s}^{L, R} (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D}, δ {\bar{v}}_{L}^{D}) = {\hat{χ}}_{0} Re [w (Δ {\bar{v}}_{J^{'} M^{'} {J^{'}}^{'} M^{″}}^{D} + i δ {\bar{v}}_{L}^{D})]

The sum of attenuation differences of all possible magnetically split transitions is given by

Δ δ_{J^{'} {J^{'}}^{'}} (\bar{v}) = \sum_{M^{'} {M^{'}}^{'}} [δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{L} (\bar{v}) - δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{R} (\bar{v})]

The average attenuation of LHCP and RHCP is given by

{\bar{δ}}_{J^{'} {J^{'}}^{'}} (\bar{v}) = \frac{1}{2} \sum_{M^{'} {M^{'}}^{'}} [δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{L} (\bar{v}) + δ_{J^{'} M^{'} {J^{'}}^{'} {M^{'}}^{'}}^{R} (\bar{v})]

The phase shift and attenuation difference of the 8 individual $J^{'} M^{'} \leftarrow J^{″} M^{″}$ transitions depicted in Fig. 3, as well as their sums for $Δ M = 1$ and $Δ M = - 1$ transitions, under the influence of an external magnetic field. The phase shifts and attenuations corresponding to RHCP and LHCP light are given in blue and olive respectively, while, ones for the difference in phase shifts and attenuations between the two circular components are given in red. The lineshapes of both phase shift and attenuation for different Zeeman sub-transitions are symmetric around the absorption line center, because there is a transition $- M^{″} \to - M^{″} - 1$ with equal transition probability with opposite sign in Zeeman shift for every transition $M^{″} \to M^{″} + 1$ in the Faraday experimental configuration. This implies that as the frequency of the linearly polarized light is swept across the transition, its two circularly polarized light components, LHCP and RHCP, will experience dissimilar phase shifts but opposite magnetic tuning behaviors, which in turn leads to a rotation of the direction of the polarization of the linearly polarized light. The magnetic field also gives rise to the difference in attenuations between the LHCP and RHCP for light frequencies in the immediate neighborhood of the molecular resonance lines, as illustrated in Fig. 3.

Fig. 3 Individual contributions to the phase shift and attenuation from the $X {}^{2}Π_{\frac{3}{2}} P (2.5)$ transition. (a) The phase shifts from the individual $M^{″} \to M^{'}$ transitions for the magnetically split $X {}^{2}Π_{\frac{3}{2}} P (2.5)$ transition, where the phase shifts corresponding to LHCP light are given in green whereas the ones for RHCP are given in blue. (b) The sum phase shift over all $M^{″} \to M^{'}$ transitions for ∆M = ± 1. The red curve shows the total phase shift, given by $Δ ϕ = ϕ_{L}^{} - ϕ_{R}^{}$ . (c) The attenuation from the individual $M^{″} \to M^{'}$ transitions. (d) The sum attenuation over all $M^{″} \to M^{'}$ transitions for ∆M = ± 1. The red curve shows the total attenuation, given by $Δ δ = δ_{L}^{} - δ_{R}^{}$ .

Download Full Size | PDF

The FRT calculation of MFOF takes into account both Zeeman absorption and Faraday rotation. FRT is strongly dependent on the frequency of the incident light, as well as the field strength B, the interaction length L and the density of the paramagnetic molecules. The FRT for the magnetically induced vibration-rotation transition $v^{'}, J^{'} \leftarrow v^{″}, J^{″}$ can be given by

T = \frac{1}{2} \exp (- {\bar{δ}}_{J^{'} {J^{'}}^{'}}) [\cos h (Δ δ_{J^{'} {J^{'}}^{'}}) - \cos (2 Δ ϕ_{J^{'} {J^{'}}^{'}})]

Figure 4 shows the schematic illustration of the principle for ideal transmission spectrum of MFOF. Zero detuning refers to the absorption line center of the unperturbed transition. When the linearly polarized light passes through the magnetically induced paramagnetic molecules, its polarization is rotated, as given in blue dots, and at the same time, the modified absorption arises. The Doppler spectrum is given in olive and shows two bands which are split from the absorption line center due to the Zeeman effect. The curve given in red shows the transmission spectrum of MFOF, depending on both absorption strength and rotation angle of the linearly polarized light. The linearly polarized light will pass through the analyzer if it is outside the absorption bands and its polarization rotation angle approximates to π/2.

Fig. 4 Transmission spectrum (red, solid), Optical rotation (blue, short dash) and Doppler spectrum (olive, dash dot) of the $X {}^{2}Π_{\frac{3}{2}} Q (1.5)$ transition of NO.

Download Full Size | PDF

3. Experiment

The schematic of the experimental set-up used for measuring the FRT signal of NO MFOF is shown in Fig. 5. A commercial CW external-cavity QCL (Daylight Solutions 41052-MHF-012-KD0357) with higher-solution mode-hop free wavelength tuning capability was used as the spectroscopic source. In this work, the laser was operated in CW mode at a thermoelectrically-cooled chip temperature of 18 °C and working current of 460 mA. The QCL source can be tuned over the range of 1,820 cm⁻¹ to 1,974 cm⁻¹ (5.07 μm-5.49 μm), with a line-width less than 0.001 cm⁻¹ and a scanning rate of about 10 cm⁻¹/s. The wavelength scanning range of this QCL covers the R, Q and P branch in the fundamental vibrational transition of NO near 5.2 μm.

Fig. 5 The schematic of the experimental set-up used for measuring the FRT signal of NO MFOF

Download Full Size | PDF

The collimated output laser beam (with a spot size of about 2 mm in diameter) is split into two independent pathways by a wedged CaF₂ window BS₁ (Thorlabs BSW520) in order to obtain a sample channel and a calibration one. The calibration beam is transmitted through a 101.6 mm long solid germanium etalon (LightMachinery OP-5483-101.6) in the ambient air after reflected by a gold-plated reflection mirror (Thorlabs PF10-03-M03), and detected by a thermoelectrically cooled mercury cadmium telluride (MCT) detector (VIGO PVI-3-TE-6). The etalon with a free spectral range (FSR) of 0.012 cm⁻¹ enables the conversion of scan time to relative wavelength. The sample beam passes through an absorption gas cell in a uniform axial magnetic field after polarize by a ProFlux infrared polarizer (MOXTEK SIR3-5), and then split into two pathways by another wedged window BS₂ same with BS₁. One beam is detected directly by a photodetector which is the same with the calibration channel for Zeeman absorption signal. The other beam is transmitted through an infrared analyzer of same parameters with the polarizer, and then detected by a detector same with the Zeeman channel for the FRT signal of NO MFOF. The transmission performance and extinction ratio of the infrared polarizer and analyzer are about 90% and 20,000:1 respectively within the tuning range of the QCL. The three photodetectors’ built-in preamplifiers (VIGO MIPDC-F-20) have a bandwidth of 20 MHz, and their outputs are sent to a 1 G sample/s, 100MHz bandwidth digital oscilloscope (Tektronix DPO 2014).

In our experiment, a pair of Nd-Fe-B permanent magnets and a Helmholtz coil are used interchangeably to produce uniform axial magnetic field. The permanent magnets are for high field, while the Helmholtz coil is for low field. The magnetic field intensity between the Nd-Fe-B magnets could be changed from 40 mT to 320 mT, by varying the distance of those two magnets. The Helmholtz coil is air cooled and could reach a maximum magnetic field of only 88 mT, limited by the current supply. With the interchangeable using of this two magnets, a tunable magnetic field from zero to 320 mT is available. The magnetic field intensity is measured by a 3-axis Hall Effect Gaussmeter (Coliy Technology G93).

4. Results and Discussion

We recorded the FRT signal of NO MFOF, as well as its Zeeman absorption spectrum with the mid-infrared QCL. Figure 6 shows the typical raw traces of the light intensity recorded by the sample and calibration channels over the range of 1,820 cm⁻¹ to 1922. cm⁻¹, taking around 20 s. Several absorption lines of H₂O are also clearly observed over the tuning range of the QCL, due to the aerial water vapor in the laboratory air. The top trace and the bottom one show the voltage signals of the two channels, at the experimental conditions of B = 9 mT and B = 78 mT, respectively. The output wavelength and intensity of the QCL are monitored by the solid germanium etalon, for its tuning rate and light energy are not linear in certain spectrum regions. The relative variation of laser frequency is calibrated by tracking the peak positions of the interference fringes of the solid germanium etalon, while, the output power is monitored by recording the peak values.

Fig. 6 The recorded output traces over the range of 1,820 cm⁻¹ to 1,922 cm⁻¹ for FRT spectrum at the experimental conditions of B = 9 mT and B = 78 mT, and the etalon spectrum used to determine the tuning rate of the laser.

Download Full Size | PDF

As can been seen from Fig. 6, there is a strong magnetic field dependence for the filter transmission spectrum of NO MFOF. Single branch transmission spectrum takes place at low magnetic field, while comb-like optical transmission is obtained at high field. As discussed in Zeeman Spectra, this is because of the total angular momentum J dependence of the Landé g-factor.

The summation of all allowed transitions weighted by their transition probabilities can be defined as effective $g_{J}$ value, g_eff, which determines the Faraday rotation angle.

\begin{matrix} g_{e f f}^{R} = g_{J^{'}}^{v^{'}} + (g_{J^{'}}^{v^{'}} - g_{{J^{'}}^{'}}^{{v^{'}}^{'}}) \cdot \frac{J}{2} \\ g_{e f f}^{Q} = g_{J^{'}}^{v^{'}} + (g_{{J^{'}}^{'}}^{{v^{'}}^{'}} - g_{J^{'}}^{v^{'}}) \cdot \frac{1}{2} \\ g_{e f f}^{P} = g_{J^{'}}^{v^{'}} + (g_{{J^{'}}^{'}}^{{v^{'}}^{'}} - g_{J^{'}}^{v^{'}}) \cdot \frac{J + 1}{2} \end{matrix}

A comparison of calculated g_eff values, as well as integrated line intensity, for P, Q and R branches in the ²Π_3/2 and ²Π_1/2 (v = 1←0) states, is shown in Fig. 7. The values of effective $g_{J}$ factor is positive in the ²Π_1/2 subsystem, but negative in the ²Π_3/2 subsystem, for P and R branch. The g_eff values of the Q branch in the ²Π_3/2 subsystem give positive signals for transitions with J ≤ 9.5, and give negative signals for transitions with J > 9.5. In the Q branch of the Ω = 1/2 subsystem, all values are positive. The g_eff value, as well as the integrated line intensity, are of similar magnitude for P and R branches in the ²Π_1/2 and ²Π_3/2 subsystems, but the g_eff value of Q(1.5) in the ²Π_3/2 subsystem is about thirty times larger than that of P and R branches in the ²Π_1/2 and ²Π_3/2 subsystems. The integrated line intensity for the P and R branches of the ²Π_1/2 subsystem is about twice large than that of ²Π_3/2 subsystem. For Q branch, the reverse is true.

Fig. 7 The calculated g_eff values and integrated line intensity for the ²Π_3/2 and ²Π_1/2 (v = 1←0) states.

Download Full Size | PDF

Low J lines in Q branch of ²Π_3/2 state have relatively large g_eff values, therefore exhibit the largest magnetic tuning rate dν/dB and Faraday rotation angle. Thus, large FRT signals of low rotational levels in Q-transitions, especially for Q(1.5), are obtained at both high and low magnetic field. For higher J values, the FRT signals are vanishingly small when the external magnetic field is low. Large FRT signals of R and P branches take place only for high magnetic field measurements, though their integrated line intensity is relatively stronger than Q branch.

We measure the FRT signals of NO MFOF for different pressure and B field values, and record the Zeeman absorption spectrum. Figure 8 shows Faraday transmission characteristics and Zeeman absorption behavior of the Q branch near 1875.7 cm⁻¹ at a pressure of 80 mbar. For the particular case of the Q-transitions, $J^{″} = J^{'} = J$ and $g_{{J^{'}}^{'}} \approx g_{J^{'}}$ . This implies that all the Zeeman sub-transitions will experience the same magnetically induced frequency detuning shift, given by $\pm g_{J} μ_{B} B$ . With increasing B field, the two absorption peaks of the Zeeman sub-transitions with $Δ M = + 1$ and $Δ M = - 1$ will separate from each other linearly in frequency, therefore the absorption strength at the absorption line center of the unperturbed transition decrease and the Faraday rotation transmission increase.

Fig. 8 The Faraday transmission characteristics and Zeeman absorption behavior of the Q branch at different B field values.

Download Full Size | PDF

A number of measurements for the peak transmission with respect to the magnetic field and pressure was carried out in detail for Q(1.5) and R(6.5) in the ${}^{2}Π_{3 / 2}$ state, which is shown in Fig. 9. The peak transmission of Q(1.5) rises with the magnetic field increasing and approaches a maximum signal until the slope decreases. For each pressure there is a magnetic field amplitude that maximizes the FRT signal and the optimum magnetic fields depend on the value of pressure. The optimum magnetic field strength increases with rising pressure due to pressure broadening. Two opposing effects will affect the FRT signal as the pressure increases: the increasing number of molecules enlarges the rotation angle which enhances the signal, while, the increasing pressure broadening results in a less effective absorption which attenuates the signal. The B field strength are at optimum when the corresponding Zeeman shift is in the same order of magnitude as the spectral line width of the zero-field transition. The g_eff value of R(6.5) is barely one third of that of Q(1.5), as shown in Fig. 7. Therefore, the peak transmission of R(6.5) rises with the increasing magnetic field from 5 to 320 mT.

Fig. 9 The transmittance of the peak transmission changing with the magnetic field and pressure for Q(1.5) and R(6.5) in the ${}^{2}Π_{3 / 2}$ state

Download Full Size | PDF

In order to verify the accuracy of the theoretical model constructed in Section 2, a series of comparisons between the model predictions and the experimental results are made and analyzed for the P (from P_1.5 to P_14.5), Q (from Q_0.5 to Q_6.5) and R (from R_0.5 to R_22.5)branches of the fundamental vibrational transition $X {}^{2}Π_{\frac{1}{2}}, X {}^{2}Π_{\frac{3}{2}} (v = 1 \leftarrow 0)$ of NO, under different experimental conditions. Figure 10 shows the measured and simulated lineshape of the MFOF transmission spectrum for the Q branch of NO at the experimental conditions of T = 300 K, P = 80 mbar and B = 120 mT. Also shown in Fig. 10 is the residual of the experimental data and simulation result. The values for the line center frequency $v_{0}$ and the transition strengths $S_{Ω J^{'} J^{″}}$ are taken from the HITRAN2016 database [37]. The good agreement between the measured MFOF transmission spectrum and the calculated line shapes indicates that the theoretical model can accurately predict the behavior of MFOF.

Fig. 10 Experimental and simulated lineshapes of the NO MFOF transmission spectrum for the Q branch at the experimental conditions of T = 300 K, P = 80 mbar and B = 120 mT.

Download Full Size | PDF

Whereas the simulation result still has a maximum residual of ± 8% at about 1875.8 cm⁻¹ (the Q_1.5 transition) compared with the experimental data. Two possible reasons are considered for this discrepancy: the inhomogeneity of the magnetic field inside molecular cell and the uncertainty of the theoretical approximation of Landé g-factor. In the first case, larger difference between the MFOF model and measurement values is expected for the Q(1.5) transition, because the Q(1.5) transition has the maximum g_eff value as shown in Fig. 7, and therefore the most sensitively response to the non-uniformity of the magnetic field. The magnetic field distribution measurement revealed that the magnetic field in the cell center was about 2.8 mT lower than at the end position. The MFOF simulation assumed a constant value of B = 120 mT for Fig. 10. The non-uniformitiy in the magnetic field is likely to produce substantial discrepancy between the measurement and the model, because the Zeeman splitting rate for the Q(1.5) line is 3.7 × 10⁻⁴ cm⁻¹/mT. Another possible effect is the uncertainty of the theoretical approximation of Landé g-factor. The values of $g_{J}^{}$ used in the MFOF model are obtained through Eq. (2), which is derived from the approximate Zeeman operator and wave functions that account for S uncoupling but not the smaller effects of L uncoupling. In fact, there are two degenerate states for each rotational transition, which are split by the rotation of the nuclear framework, an effect known as Λ-doubling. According to Radford’s theory [38], g-factors for two degenerate states are written as a sum of small corrections of $g_{J}^{o}$

\begin{matrix} g_{J}^{e} = g_{J}^{0} + {(δ g_{J})}_{S} + {(δ g_{J})}_{N} + {(δ g_{J})}_{L}^{+} \\ g_{J}^{e} = g_{J}^{0} + {(δ g_{J})}_{S} + {(δ g_{J})}_{N} + {(δ g_{J})}_{L}^{-} \end{matrix}

where

g_{J}^{e}

and

g_{J}^{f}

are the g-factors for two degenerate states,

{(δ g_{J})}_{S}

and

{(δ g_{J})}_{N}

are the theoretical g-factor corrections for the anomalous spin magnetic moment of the electron and the rotation of the nuclei, and

{(δ g_{J})}_{L}

is a correction for L uncoupling. Based on Radford’s theory. The g-factors decreases by no more than 0.1% without theoretical corrections of

{(δ g_{J})}_{S}

{(δ g_{J})}_{N}

and

{(δ g_{J})}_{L}

, which causes a negligible effect on the FRS signal. However, higher order Zeeman effects, which originate in the mixing of the zero-order levels by the applied magnetic field should be taken into account for improving the calculation accuracy of g-factors. According to the experimental result of Liu et al [39], the second order Zeeman effect should be considered for fields higher than 250 mT but lower than 1 T, while the third or higher order terms should be included for field beyond 1 T.

5. Conclusions

We have demonstrated what is to our knowledge the first MFOF. In this work, we focus on mid-infrared NO-MFOF and study its transmission as a function of magnetic field and pressure in detail. We developed a complete theory to describe the performance of NO-MFOF and simulated its transmission characteristics through our theoretical model based on the elementary quantum theory of optical dispersion and the interference of the two circularly polarized beams. We also measured the FRT signals of NO-MFOF for different pressure and B field values over the range of 1,820 cm⁻¹ to 1922. cm⁻¹ using a mid-infrared QCL and found that single branch transmission spectrum took place at low magnetic field, while comb-like optical transmission was obtained at high field. A number of measurements for the transmittance of the peak transmission changing with the magnetic field and pressure was carried out in detail. And good agreement between the simulated MFOF model and measurement values was achieved, which indicated that the theoretical model could accurately predict the behavior of MFOF. The MFOF transmission spectrum varies distinctly with the magnetic field. The NO-MFOF with comb-like optical transmission spectrum obtained at high magnetic field is able to provide more efficient filtering in many applications, such as atmospheric remote sensing and combustion diagnosis. NO-MFOF with single branch transmission spectrum taking place at low magnetic field may play a potential role in the coming mid-infrared free-space optical communication with QCL. This filtering method can also be extended to the lines of other paramagnetic molecules. MFOFs based on resonant Faraday anomalous dispersion effect of paramagnetic molecules are able to provide more efficient filtering in many applications.

Funding

National Key R&D Program of China (2017YFC0211900); National Natural Science Foundation of China (NSFC) (61705253); Open Research Fund of Key Laboratory of Spectral Imaging Technology, Chinese Academy of Sciences (LSIT201701D).

Acknowledgments

The authors gratefully acknowledge helpful and informative discussions with Gong Shunsheng on experimental details of this work.

References and links

1. D. J. Dick and T. M. Shay, “Ultrahigh-noise rejection optical filter,” Opt. Lett. 16(11), 867–869 (1991). [PubMed]

2. Y. Yang, X. Cheng, F. Li, X. Hu, X. Lin, and S. Gong, “A flat spectral Faraday filter for sodium lidar,” Opt. Lett. 36(7), 1302–1304 (2011). [PubMed]

3. H. Chen, M. A. White, D. A. Krueger, and C. Y. She, “Daytime mesopause temperature measurements with a sodium-vapor dispersive Faraday filter in a lidar receiver,” Opt. Lett. 21(15), 1093–1095 (1996). [PubMed]

4. C. Fricke-Begemann, M. Alpers, and J. Höffner, “Daylight rejection with a new receiver for potassium resonance temperature lidars,” Opt. Lett. 27(21), 1932–1934 (2002). [PubMed]

5. M. Oliviero, G. Severino, and G. Esposito, “Planning magneto-optical filters for the study of magnetic oscillations of the Sun,” Astrophys. Space Sci. 328(1–2), 325–329 (2010).

6. P. F. Moretti, F. Berrilli, A. Bigazzi, S. M. Jefferies, N. Murphy, L. Roselli, and M. P. Di Mauro, “Future instrumentation for solar physics: a double channel MOF imager on board ASI Space Mission ADAHELI,” Astrophys. Space Sci. 328(1–2), 313–318 (2010).

7. S. D. Harrell, C. Y. She, T. Yuan, D. A. Krueger, J. M. C. Plane, and T. Slanger, “The Faraday filter-based spectrometer for observing sodium nightglow and studying atomic and molecular oxygen associated with the sodium chemistry in the mesopause region,” J. Atmos. Sol-Terr. Phy. 72(17), 1260–1269 (2010).

8. X. Miao, L. Yin, W. Zhuang, B. Luo, A. Dang, J. Chen, and H. Guo, “Note: Demonstration of an external-cavity diode laser system immune to current and temperature fluctuations,” Rev. Sci. Instrum. 82(8), 086106 (2011). [PubMed]

9. Q. Q. Sun, Y. L. Hong, W. Zhuang, Z. W. Liu, and J. B. Chen, “Demonstration of an excited-state Faraday anomalous dispersion optical filter at 1529 nm by use of an electrodeless discharge rubidium vapor lamp,” Appl. Phys. Lett. 101(21), 211102 (2012).

10. X. Shan, X. P. Sun, J. Luo, Z. Tan, and M. D. Zhan, “Free-space quantum key distribution with Rb vapor filters,” Appl. Phys. Lett. 89(19), 191121 (2006).

11. J. A. Zielińska, F. A. Beduini, V. G. Lucivero, and M. W. Mitchell, “Atomic filtering for hybrid continuous-variable/discrete-variable quantum optics,” Opt. Express 22(21), 25307–25317 (2014). [PubMed]

12. J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes, and E. S. Polzik, “High purity bright single photon source,” Opt. Express 15(13), 7940–7949 (2007). [PubMed]

13. F. Wolfgramm, X. Xing, A. Cerè, A. Predojević, A. M. Steinberg, and M. W. Mitchell, “Bright filter-free source of indistinguishable photon pairs,” Opt. Express 16(22), 18145–18151 (2008). [PubMed]

14. L. Yin, B. Luo, A. Dang, and H. Guo, “An atomic optical filter working at 1.5 μm based on internal frequency stabilized laser pumping,” Opt. Express 22(7), 7416–7421 (2014). [PubMed]

15. L. Yin, B. Luo, Z. Chen, L. Zhong, and H. Guo, “Excited state Faraday anomalous dispersion optical filters based on indirect laser pumping,” Opt. Lett. 39(4), 842–844 (2014). [PubMed]

16. L. Ling and G. Bi, “Isotope Rb-87 Faraday anomalous dispersion optical filter at 420 nm,” Opt. Lett. 39(11), 3324–3327 (2014). [PubMed]

17. Y. Wang, X. Zhang, D. Wang, Z. Tao, W. Zhuang, and J. Chen, “Cs Faraday optical filter with a single transmission peak resonant with the atomic transition at 455 nm,” Opt. Express 20(23), 25817–25825 (2012). [PubMed]

18. R. I. Billmers, S. K. Gayen, M. F. Squicciarini, V. M. Contarino, W. J. Scharpf, and D. M. Allocca, “Experimental demonstration of an excited-state Faraday filter operating at 532 nm,” Opt. Lett. 20(1), 106–108 (1995). [PubMed]

19. J. A. Zielińska, F. A. Beduini, N. Godbout, and M. W. Mitchell, “Ultranarrow Faraday rotation filter at the Rb D₁ line,” Opt. Lett. 37(4), 524–526 (2012). [PubMed]

20. J. N. Forkey, N. D. Finkelstein, W. R. Lempert, and R. B. Miles, “Demonstration and characterization of filtered Rayleigh scattering for planar velocity measurements,” AIAA J. 34(3), 442–448 (1996).

21. D. Müller, R. Pagel, A. Burkert, V. Wagner, and W. Paa, “Two-dimensional temperature measurements in particle loaded technical flames by filtered Rayleigh scattering,” Appl. Opt. 53(9), 1750–1758 (2014). [PubMed]

22. Z. S. Liu, D. C. Bi, X. Q. Song, J. B. Xia, R. Z. Li, Z. J. Wang, and C. Y. She, “Iodine-filter-based high spectral resolution lidar for atmospheric temperature measurements,” Opt. Lett. 34(18), 2712–2714 (2009). [PubMed]

23. R. Dai, W. Gong, J. Xu, X. Ren, and D. Liu, “The edge technique as used in Brillouin lidar for remote sensing of the ocean,” Appl. Phys. B-Lasers O 79(2), 245–248 (2004).

24. J. Niu, M. N. Deeter, J. C. Gille, D. P. Edwards, D. C. Ziskin, G. L. Francis, A. J. Hills, and M. W. Smith, “Carbon monoxide total column retrievals by use of the measurements of pollution in the troposphere airborne test radiometer,” Appl. Opt. 43(24), 4685–4696 (2004). [PubMed]

25. J. R. Drummond, J. S. Zou, F. Nichitiu, J. Kar, R. Deschambaut, and J. Hackett, “A review of 9-year performance and operation of the MOPITT instrument,” Adv. Space Res. 45(6), 760–774 (2010).

26. H. G. Reichle, B. E. Anderson, V. S. Connors, T. C. Denkins, D. A. Forbes, B. B. Gormsen, R. L. Langenfelds, D. O. Neil, S. R. Nolf, P. C. Novelli, N. S. Pougatchev, M. M. Roell, and L. P. Steele, “Space shuttle based global CO measurements during April and October 1994, MAPS instrument, data reduction, and data validation,” J. Geophys. Res. Atmos. 104(D17), 21443–21454 (1999).

27. M. Dai, W. Wan, X. Zhu, B. Song, X. Liu, M. Lu, B. Cui, and Y. Chen, “Broadband and wide angle infrared wire-grid polarizer,” Opt. Express 23(12), 15390–15397 (2015). [PubMed]

28. E. J. Zhang, B. Brumfield, and G. Wysocki, “Hybrid Faraday rotation spectrometer for sub-ppm detection of atmospheric O_2.,” Opt. Express 22(13), 15957–15968 (2014). [PubMed]

29. T. A. Blake, C. Chackerian Jr, and J. R. Podolske, “Prognosis for a mid-infrared magnetic rotation spectrometer for the in situ detection of atmospheric free radicals,” Appl. Opt. 35(6), 973–985 (1996). [PubMed]

30. X. Yang, Y. Chen, P. Cai, H. Wang, J. Chen, and C. Xia, “Application of Zeeman modulation Faraday spectroscopy to the measurement of a magnetic field,” Appl. Opt. 37(21), 4806–4809 (1998). [PubMed]

31. C. D. Boone, F. W. Dalby, and I. Ozier, “Magnetic rotation molecular spectroscopy using an oscillating field,” J. Chem. Phys. 113(19), 8594–8607 (2000).

32. W. Zhao, G. Wysocki, W. Chen, E. Fertein, D. Le Coq, D. Petitprez, and W. Zhang, “Sensitive and selective detection of OH radicals using Faraday rotation spectroscopy at 2.8 µm,” Opt. Express 19(3), 2493–2501 (2011). [PubMed]

33. B. Brumfield, W. Sun, Y. Ju, and G. Wysocki, “Direct in situ quantification of HO₂ from a flow reactor,” J. Phys. Chem. Lett. 4(6), 872–876 (2013). [PubMed]

34. R. Lewicki, J. H. Doty 3rd, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33 µm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 106(31), 12587–12592 (2009). [PubMed]

35. Y. Wang, M. Nikodem, E. Zhang, F. Cikach, J. Barnes, S. Comhair, R. A. Dweik, C. Kao, and G. Wysocki, “Shot-noise limited Faraday rotation spectroscopy for detection of Nitric Oxide isotopes in breath, urine, and blood,” Sci. Rep. 5, 9096 (2015). [PubMed]

36. Y. Qian and H. Sun, “Improved approach for ultra-sensitive detection of NO,” Opt. Express 19(2), 739–747 (2011). [PubMed]

37. L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J. M. Flaud, R. R. Gamache, J. J. Harrison, J. M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Muller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Ra. 130, 4–50 (2013).

38. H. E. Radford, “Microwave Zeeman effect of free hydroxyl radicals,” Phys. Rev. 122(1), 114–130 (1961).

39. Y. Y. Liu, Y. Q. Guo, H. P. Liu, J. L. Lin, X. Y. Liu, G. M. Huang, F. Y. Li, and J. R. Li, “On the nonlinearity of the Zeeman effect of NO in the moderate field by intracavity laser magnetic resonance at 1842 cm⁻¹,” Phys. Lett. A 272(1–2), 80–85 (2000).

Demonstration of a mid-infrared NO molecular Faraday optical filter

Abstract

1. Introduction

2. Theory

2.1 Zeeman Spectra

2.2 Resonant Faraday Effect

3. Experiment

4. Results and Discussion

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Equations (17)

Optics Express