Sensitive measurement of the magneto-optic effect in the near infrared wavelength region with weak alternating magnetic fields

Jae Young Jo; Yudeuk Kim; Chang Kwon Hwangbo; Kyong Hon Kim

doi:10.1364/OME.8.002636

1. Introduction

Faraday effect refers to a physical phenomenon which is related to rotation of a light beam’s polarization direction when the beam passes through a magneto-optic (MO) material along the direction of the induced magnetic fields. The angle of rotation $Δ θ$ can be expressed by $Δ θ = V B L$ , where $L$ is the length of the MO material through which the light is transmitted, $B$ is the strength of the magnetic fields along the direction of the light propagation. The proportional constant, $V$ , which represents the magnitude of the faraday effect, is called the Verdet constant. It is a unique property of a MO material which has a specific value for its own [1].

The Faraday effect of the MO materials is very important in applications to optical isolators and circulators. The optical isolators and circulators allow light transmission from light sources only in a forward direction and prevent unwanted back reflections. Bulk-type optical isolators and circulators are widely used in various applications to lasers, optical communications, and optical signal processing. However, there are no practical devices of integrated-type optic isolators and circulators yet, although many research groups keep working actively on those devices [2]. For such integrated-type optic isolators and circulators, MO materials of a large Verdet constant are very essential. A sensitive and accurate measurement method of the Verdet constant of the MO materials is required for development of an optimized MO material.

Many studies have been conducted to measure the Verdet constant using a constant magnetic field and using an alternating current (AC) magnetic field. The measurement method based on the constant direct current (DC) magnetic field requires a strong electromagnet which can be achieved by a large driving current to generate a large magnetic field and to obtain a sufficient measurement sensitivity [3]. However, the measurement method based on the AC magnetic field requires a relatively small magnetic field which is synchronized with the applied AC current for a proper measurement sensitivity [4–8]. It is also well known that the Verdet constant of the MO materials is relatively one order lower in the near infrared (NIR) 1,550 nm wavelength than that in the visible wavelength region [4,5,7]. A very sensitive and accurate measurement scheme of the Verdet constant of the MO materials is needed for the NIR wavelength region.

Recently polymer-based MO materials are demonstrated for high magneto-optic properties [5,9–12]. Conjugated polymers, such as regioregular polythiophenes, polyalkoxythiophenes, and poly(arylene ethynylene), and chiral polymers doped with stable organic biradicals are reported to have large Verdet constants in the visible wavelength region. However, there is no detailed investigation reported for the MO effect of such materials at various sample preparation conditions especially in the NIR wavelength region.

In this report, we describe a sensitive measurement method of the Verdet constant of MO materials at NIR wavelengths with an auto-balanced photoreceiver (ABPR) and a lock-in amplifier (LIA) under relatively weak AC magnetic fields. We develop the theoretical formula for the measurement method by accounting the characteristics of the ABPR and the LIA. The accuracy of this method was verified by measuring the Verdet constant of a well-known material of BK7, and used to measure the Verdet constant of the regioregular poly(3-alkyl) thiophenes (P3HT) prepared at various conditions of thermal baking and annealing processes.

2. Theory and experiment

The experimental setup measuring the Verdet constant is schematically shown in Fig. 1. The Verdet constant of a sample is determined by measuring the relative intensity variation between two orthogonally polarized components of a linearly polarized laser beam after passing through it. A laser diode (LD) is used as a light source, and its polarization is aligned to a linear polarizer whose polarization direction is adjusted to about 45° with respect to a polarization beam splitter (PBS) at the output side. When the linearly polarized laser beam passes through a MO sample placed in the middle of a homemade solenoid, its polarization direction rotates in accordance with the Faraday effect caused by the alternating magnetic field under an AC input. Then, the laser beam is divided into vertical and horizontal components by the PBS, and each of the separated beams is detected by each of the signal and reference port photodiodes of an ABPR as signal and reference beam powers, P_sig and P_ref, respectively. The ABPR measures the difference between the signal and reference port powers with an internal balance circuit, and delivers an output current for AC optical beam inputs according to the following equation under the auto-balanced mode operation [13]:

I_{A u t o - b a l} = 100 \cdot R \cdot [P_{sig} (f) - g \cdot P_{ref} (f)] .

where

R

is the responsivity of the photodiodes of the ABPR and

g

is the gain factor that makes the DC component of the output signal to zero. The output signal from the ABPR is transmitted into the input signal port of a LIA. The LIA amplifies only the signal embedded in noises by synchronizing it to the reference input coming from the alternating current driving the solenoid regardless of the phase difference between the signal and the reference inputs. Since the filtered output of a dual LIA is in a form of root-mean-square (RMS) value, the actual peak values should be taken by multiplying them with

\sqrt{2}

.

Fig. 1 Schematic diagram of the experimental setup measuring the Verdet constant with an auto-balanced photoreceiver.

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The Verdet constant of the MO sample is determined based on the measured parameters according to the following theoretical analysis. In Fig. 1, the input polarized laser beam with its polarization direction of an angle $θ$ with respect to the PBS suffers an additional polarization rotation of $Δ θ$ due to the Faraday effect when it passes through the sample which is placed in the solenoid driven by an AC current. The laser powers detected by the signal and reference photodiodes of the ABPR are related to (i) the polarization-dependent transmittances $T^{∥}$ and $T^{⊥}$ through the sample, (ii) the beam coupling efficiencies $T_{sig}$ and $T_{ref}$ to each of the photodiodes including the transmittance and reflectance through the PBS, and (iii) the reciprocal polarization rotation (RPR) $θ_{r}$ within the sample. Thus, the laser beam powers, $P_{sig}$ and $P_{ref}$ , coupled into the signal and reference photodiodes of the ABPR are expressed as follows:

P_{sig} = T_{sig} T^{∥} P_{L D} \cos^{2} (θ + Δ θ + θ_{r}),

P_{ref} = T_{ref} T^{⊥} P_{L D} \sin^{2} (θ + Δ θ + θ_{r}),

where

P_{L D}

is the beam power of the laser diode right after the linear polarizer. Substitution of Eqs. (2) and (3) into Eq. (1) provides the output from the ABPR as [13]

P_{sig} - g P_{ref} = T_{sig} T^{∥} P_{L D} \cos^{2} (θ + Δ θ + θ_{r}) - g T_{ref} T^{⊥} P_{L D} \sin^{2} (θ + Δ θ + θ_{r}) .

In this equation, both the reciprocal polarization rotation angle

θ_{r}

and the polarization-dependent transmittances

T^{∥}

and

T^{⊥}

through the sample can affect the output of the ABPR, and cannot be estimated separately. By considering the samples of isotropic materials, we can ignore the RPR, i.e.,

θ_{r}

≈ 0. However, we cannot ignore the polarization-dependent transmittances which may be affected by the refractive index difference between the sample surface and the air, the beam incident angle to the sample, the PBS’s transmittance and reflectance, and so on. Thus, using the simple trigonometric relationships, the right-side terms of Eq. (4) can be written as, when θ = 45°,

P_{sig} - g P_{ref} = \frac{1}{2} P_{LD} [T_{sig} T^{∥} {1 - \sin (2 Δ θ)} - g T_{ref} T^{⊥} {1 + \sin (2 Δ θ)}] .

The polarization angle θ of the polarizer is fixed at 45° for the experimental measurements in this paper. Since the gain factor

g

eliminates the DC components of current outputs of the auto-balanced detectors, unrelated terms to Δθ in Eq. (5) can be ignored. Then, the output current of the ABPR can be written, from Eq. (1), as below:

I_{Auto - bal} = - 50 R P_{LD} \sin (2 Δ θ) [T_{sig} T^{∥} + g T_{ref} T^{⊥}],

and the gain factor can be derived, from the DC mode operation condition of the ABPR, as

g = T_{sig} T^{∥} / T_{ref} T^{⊥} .

The parameters

T^{∥}

,

T^{}

,

T_{sig}

and

T_{ref}

are obtained by utilizing the different operation modes of the ABPR. Its balance mode output

I_{Bal}

provides an output current without applying the gain factor

g

while the signal monitor mode output

I_{SM}

provides only the current output of the signal port. These mode outputs are expressed as [13],

I_{Bal} = 20 R [P_{s i g} - P_{r e f}],

I_{SM} = - 10 R P_{s i g} .

We measure and compare these mode outputs for two different cases, one when

T^{∥} \neq T^{⊥}

with a sample and the other when

T^{∥} = T^{⊥} = 1

with no sample between the solenoid coils. In addition, when no sample is placed in the setup, we adjust the input polarization direction to make

I_{Bal}

= zero so that

T_{sig}

and

T_{ref}

are the same. From these processes, we can obtain

T^{∥}

,

T^{⊥}

,

T_{sig}

and

T_{ref}

as follows:

T^{∥} = I_{SM}^{m} / I_{S M}^{0},

T^{⊥} = (I_{SM}^{m} + I_{Bal}^{m}) / I_{S M}^{0},

T_{sig} = T_{ref} = - I_{S M}^{0} / (5 P_{LD}),

where

I_{S M}^{m}

and

I_{S M}^{0}

are the signal monitor mode output currents for the sample present and absent cases, respectively. Finally from substitution of the relations of Eqs. (10) – (12) and of

Δ θ = V B L

into Eq. (6), the Verdet constant of the sample can be expressed as

V = \frac{1}{2 L B_{R M S}} \sin^{- 1} [I_{A u t o - b a l} / (20 I_{S M}^{m})] .

Since the LIA amplifies the output current of the ABPR

I_{Auto - bal}

by a gain G, its output provides a calibrated RMS voltage

R_{L I A} (= I_{A u t o - b a l} Z_{0})

for the impedance

Z_{0}

of a voltage detector. Then, Eq. (13) can be rewritten as

V = \frac{1}{2 L B_{R M S}} \sin^{- 1} [R_{L I A} / (20 V_{S M}^{m})],

where

B_{RMS}

and

V_{SM}^{m}

are the RMS alternating magnetic field and the output voltage measured at the signal monitoring mode of the ABPR, respectively. The length L of the MO sample and the strength of the magnetic field B are measured with an electronic micrometer (resolution <1

μm

) and a gauss meter, respectively.

3. Result

To verify the validity of our measurement method and theoretical analysis, we have measured the Verdet constant of the well-known BK7 sample. The Verdet constants of the BK7 samples of various thicknesses were measured to be 44.07 $° / T m$ with a standard deviation of 0.14 $° / T m$ at 1,550 nm wavelength and 63.59 $° / T m$ with a standard deviation of 0.15 $° / T m$ at 1,310 nm wavelength as shown in Fig. 2(a). The Verdet constants of the 3-mm-long BK7 sample of 1,550 nm and 1,310 nm wavelengths were also measured at various magnetic flux densities as shown in Fig. 2(b). The measured values were almost a constant value around 42.84 $° / T m$ with a standard deviation of 0.46 $° / T m$ at 1,550 nm wavelength and a constant value around 62.27 $° / T m$ with a standard deviation of 1.05 $° / T m$ at 1,310 nm wavelength.

Fig. 2 (a) The measured Verdet constants of the BK7 samples of three different thicknesses, and (b) measured Verdet constants of the 3-mm-long BK7 sample as a function of the applied magnetic field, both at 1,550 and 1,310 nm wavelengths.

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The Verdet constant of the P3HT samples was measured in the same way as that used for the BK7 sample. Cholorobenzene was used as a solvent for the P3HT with a concentration of 30 ml/mg, and the P3HT solution was placed on a thin glass substrate. Many samples having a thin layer of the P3HT solution covering the glass substrate were prepared by varying annealing conditions and baking times. The Verdet constant of the glass substrate was first measured by placing it at the sample position in Fig. 1. The total Verdet constant value of the P3HT sample on the glass substrate was measured by placing it the sample position. Then, the Verdet constants of the P3HT materials prepared at various conditions were taken by subtracting the Verdet constant of the glass substrate from the total Verdet constant values. The measured results are plotted in Fig. 3 for the applied magnetic field of 100 Gauss. The highest measured value of the Verdet constant of the P3HT sample is 14,132 and 11,017 $° / T m$ at the wavelengths of 1,310 and 1,550 nm, respectively, when the sample was prepared at a baking condition of 80 °C during 30 minutes and a annealing condition of −1°C/min. However, measured Verdet constants fluctuate as the baking period changes. This is due to non-uniformly crystallized lamellae structures over the P3HT films. Nevertheless, for optimum baking conditions, crystallization takes place over larger areas which is observed by a strong X-ray diffraction intensity (inset on Fig. 3(b)) than non-optimum baking conditions and no baking condition.

Fig. 3 Measured Verdet constants of the P3HT samples prepared at a thermal annealing condition of -1 °C /min. as functions of the baking time at two different baking temperatures of 80 (a) and 150 °C (b) for two wavelengths of 1,310 and 1,550 nm. The inset shows the measured X-ray diffraction data for the samples annealed at various baking times.

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The optical transmittance properties of the P3HT samples prepared at various thermal treatment conditions were measured with a tungsten lamp and an optical spectrum analyzer by placing each of the samples between them. As the measured Verdet constant value of the P3HT samples at 1,550 nm wavelength increases, the absorption resonance condition starts to appear around 1,200 nm wavelength as shown in Fig. 4. This may be attributed to the two-photon absorption process between the highest molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) whose energy gap is 1.94 eV (λ = 640 nm) [16]. Thus, as the wavelength approaches to the resonance region, the Verdet constant expects to increase according to Ref [14]. The wavelength dependency of the MO properties of paramagnetic rare-earth ion containing mediums is known, for a case of single transition frequency, as

V = \frac{E}{λ^{2} - λ_{0}^{2}},

Here λ₀ indicates the resonant transition wavelength of the paramagnetic ions, and E represents a proportional constant depending on the concentration of magnetic ions per volume, the Lande splitting factor, and the transition probability [15]. From our measurement it was found that the optimum condition for the large Verdet constant case is very sensitive to the P3HT sample preparation conditions and environment effects.

Fig. 4 The measured Verdet constants of the BK7 samples of three different thicknesses at 1,550 and 1,310 nm wavelength.

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4. Conclusion

We have demonstrated a sensitive measurement method of the Verdet constant of magneto-optic samples with weak AC magnetic fields using an ABPR and LIA. We have also measured the Verdet constant of P3HT samples prepared at various baking and annealing conditions, and obtained the highest values of 14,132 and 11,017 $° / T m$ at the wavelengths of 1,310 and 1,550 nm, respectively, for a sample prepared at a baking condition of 80°C during 30 minutes and a following annealing condition of −1°C/min.

Funding

Future Semiconductor Device Technology Development Program (project # 10044735) funded by the Ministry of Trade, Industry & Energy of the Korean government and the Korea Semiconductor Research Consortium, and by the Basic Science Research Programs through the National Research Foundation of Korea funded by the Korean Ministry of Education (2017R1D1A1B03035790, 2018R1D1A1B07051001).

References and links

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Sensitive measurement of the magneto-optic effect in the near infrared wavelength region with weak alternating magnetic fields

Abstract

1. Introduction

2. Theory and experiment

3. Result

4. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (15)

Optical Materials Express