Righting the handedness’ dichotomy: schemes of Sagnac interferometer containing chiral elements for suppression of dependence on circular birefringence and circular dichroism

Hourieh Exir; Ilya Golub

doi:10.1364/OE.21.024112

1. Introduction

Sagnac interferometer/Sagnac loop mirror has many applications in gyroscopy, sensing, communications etc due to their robustness and easy integration in photonic devices. In many uses of SI it is advantageous to amplify the measured properties e.g. by employing a ring resonator placed in the Sagnac loop to magnify birefringence [1,2]. The opposite, i.e. suppression of device under test (DUT) certain properties, may be of benefit too: for example, the dependence on the DUT orientation, polarization dependent loss (PDL) and LB were suppressed using a π-shift/placing a half-wave plate (HWP) in the SI at different, case specific angles [3–8]. All this research was done for linear birefringence devices. With the emergence of devices such as switches/modulators and sensors based on controllable chiral waveguides /fibers [9–12], and since CB has properties very different from those of LB, the questions “Is it possible to immunize the properties of SI from dependence on CB or CD of a chiral optical element placed in the loop, as it was done for LB and PDL placed in a SI?”, and thus “Is it possible to cancel/compensate for CB or CD?” are pertinent from both theoretical and practical points of view.

Circular dichroism is an optical property of chiral materials leading to differential absorption of the left and right circularly polarized light. Measurements of CD provide valuable information on molecular structure [13,14] and it is used in optical components such as notch filters, sensors and polarizers made from liquid crystals [15] or twisted/helix fibers [16–18]. However, there are cases where CD is large [19], detrimental [20] or when one needs to cancel it while performing zero adjustment/calibrating CD measurement or when measuring optical properties of materials like dyes embedded in chiral media.

In this work we consider two configurations of SI containing a chiral optical element and show SI reflection independence on CB or cancelation of the polarization dependence/CD of this element. The CD/polarization independence in transmission of chiral element placed in a π-shifted SI is verified experimentally. We also compare/show the differences between SI containing either circular birefringence or linear birefringence devices. Only cases of devices which have dominant either CB or LB are considered; when both are present like in a single mode fiber by itself, without any other devices, the situation becomes more complicated and is not studied here.

2. Jones matrix analysis

The light propagation through the chiral/birefringent element defined as DUT is represented by Jones matrix, _{$[J_{D U T}]$}[21]

[J_{D U T}] = \exp (- \frac{α L}{2}) [\begin{matrix} \cosh (\frac{η}{4} + i \frac{Δ}{2}) & - i \sinh (\frac{η}{4} + i \frac{Δ}{2}) \\ i \sinh (\frac{η}{4} + i \frac{Δ}{2}) & \cosh (\frac{η}{4} + i \frac{Δ}{2}) \end{matrix}],

In Eq. (1),

Δ = 2 π (n_{L} - n_{R}) L / λ

where

n_{L}

and

n_{R}

are the refractive indices for the left and right circular polarizations,

λ

is the wavelength of light, L is the chiral sample thickness, and is the CD, defined as

η = (α_{L} - α_{R}) L

where

α_{L}

and

α_{R}

are the absorption coefficients for the left and right circular polarizations, respectively, and is the mean absorption coefficient (in intensity),

α = (α_{L} + α_{R}) / 2

Using Jones matrix analysis, we calculate the reflectance and transmittance of a SI containing a chiral element, with and without a HWP placed in the loop. Figure 1 shows a Sagnac interferometer [22,23] formed from a 50:50, 2 × 2 fiber coupler containing a chiral element/DUT and an optional HWP which introduces a π-shift between the clockwise and counterclockwise propagating beams. R, denoted as a dashed line in Fig. 1, represents the effect of the folded loop (coordinate-conversion) inside the SI [4].

Fig. 1 Configuration of a Sagnac interferometer with a DUT and an optional HWP in the loop.

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The normalized Jones vectors for the right and left circularly polarized inputs are represented as $E_{R}^{i n} = (1 / \sqrt{2}) {[1 - i]}^{T}$ and $E_{L}^{i n} = (1 / \sqrt{2}) {[1 i]}^{T}$ . The effect of the optical elements on the input light of a given polarization state can be described by their corresponding Jones matrices. Following a procedure similar to Ref [4], the SI reflection and transmission matrices $[R (λ)]$ and $[T (λ)]$ can be expressed as

[R (λ)] = [K_{\times}] [R] [J_{D U T}] [J_{H W P}] [K_{∥}] + {[K_{∥}]}^{T} {[J_{H W P}]}^{T} {[J_{D U T}]}^{T} {[R]}^{T} {[K_{\times}]}^{T},

[T (λ)] = [K_{∥}] [R] [J_{D U T}] [J_{H W P}] [K_{∥}] + {[K_{\times}]}^{T} {[J_{H W P}]}^{T} {[J_{D U T}]}^{T} {[R]}^{T} {[K_{\times}]}^{T},

where

[K_{∥}]

and

[K_{\times}]

are the 2 × 2 coupler’s parallel-coupling and cross-coupling matrices, [R], [J_DUT], [J_HWP] are the Jones matrices of R, HWP, and DUT, all for the light traversing the SI clockwise. The corresponding transposed matrices represent the effect of these elements on the counterclockwise light in the loop. The matrices in Eqs. (2) and (3) are

\begin{array}{l} [R] = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}], [J_{H W P 4}] = [\begin{matrix} \cos 2 ρ & \sin 2 ρ \\ \sin 2 ρ & - \cos 2 ρ \end{matrix}], \\ [K_{∥}] = [\begin{matrix} {(1 - k_{x})}^{1 / 2} & 0 \\ 0 & {(1 - k_{y})}^{1 / 2} \end{matrix}], [K_{\times}] = [\begin{matrix} i {(k_{x})}^{1 / 2} & 0 \\ 0 & i {(k_{y})}^{1 / 2} \end{matrix}], \end{array}

In Eq. (4), ρ is the orientation of the axes of the HWP with respect to the laboratory coordinate system defined by the plane of SI and the coupler,

k_{x}

and

k_{y}

are the power–coupling coefficients for the x and y polarizations. Assuming that the coupling ratio of the lossless 50:50 coupler is polarization independent, we have

k_{x} = k_{y} = 0.5

.

First, we consider a SI with a chiral element but without a HWP. In this case, [J_HWP] and [J_HWP]^T are omitted in Eqs. (2) and (3). Then we obtain T = 0, while the reflections for right or left circularly polarized input fields are, respectively:

R_{r} = \exp (- α L) {| \cos h (A) - \sin h (A) |}^{2} \neq R_{l} = e x p (- α L) {| \cos h (A) + \sin h (A) |}^{2},

where

A = η / 4 + i Δ / 2

Eq. (5) shows that for SI containing chiral element but not HWP the reflection is polarization dependent since it is distinct for right or left circularly polarized input fields. In case of negligible loss, α_L = α_R = 0, R_out = 1. This is in contrast to such an SI containing a lossless linear birefringent DUT, since it’s reflection depends on both LB orientation and magnitude [3,8]. Thus, a SI containing a lossless chiral element as e.g. a spectral filter can be used as a reflector for a laser.

The transmission and reflection of a SI containing a chiral element and HWP are calculated from Eqs. (2) and (3) for either right or left circularly polarized inputs (preliminary results for T were obtained in [24]):

T_{r} = T_{l} = e x p (- α L) {| \cos h (A) s i n (2 ρ) + i \sin h (A) \cos (2 ρ) |}^{2} .

R_{r} = R_{l} = e x p (- α L) {| \sin h (A) s i n (2 ρ) + i \cos h (A) \cos (2 ρ) |}^{2} .

Equations (6) and (7) show that for a chiral element, placed in a π-shifted SI, both the reflection and transmission are CD/polarization independent as they are identical for right and left circularly polarized inputs. Physically, this is the result of the HWP forcing the DUT to be probed by both left and right circular polarizations counter-propagating in SI. In practice, the degree of CD compensation depends on factors such as how close to 50:50 the splitting ratio of the coupler is etc.

In contrast, as shown in Eq. (8) below, for light transmitted directly by the chiral element one obtains:

T_{r} = e x p (- α L) {| \cos h (A) - \sin h (A) |}^{2} \neq T_{l} = e x p (- α L) {| \cos h (A) + \sin h (A) |}^{2} .

From Eqs. (6) and (7) it follows that controlling the chiral element parameter A either thermally [9], electrically [10], optically using band-edge shift [11] or changing chiral media concentration [12], polarization/CD independent switching/modulation or sensing can be achieved. Thus, similar to polarization independent measurements [3,4], switching [5] and sensing [6,7] with non-chiral/LB components inside a π-shifted SI, chiral devices placed in such an interferometer provide a new way/degree of freedom to manipulate light.

It is instructive to compare properties of SI containing either circular or linear birefringence by summarizing the present results and those of Refs. 3, 4 and 8 – Tables 1 and 2.

Table 1. Transmission and reflection of SI containing a device with a CB where Δ is the optical rotation and ρ is the orientation of the axes of the HWP.

View Table | View all tables in this article

Table 2. Transmission and reflection of SI containing a device with LB where α is the orientation angle of the birefringence axes of the DUT, and δ is the birefringence of the DUT.

View Table | View all tables in this article

For simplicity, the equations for lossy elements are not shown. When only one expression/value is given for reflection or transmission, obviously, they are polarization independent. One concludes that 1) for lossless elements, all schemes are polarization independent 2) In the presence of loss, introduction of HWP makes SI with either CB or LB polarization independent, resulting in cancellation of CD or PDL, respectively. 3) SI with lossless CB behaves as a 100% mirror, while in order to achieve R = 1 for a SI containing a lossless LB device, an insertion of a HWP at ρ = 0 is required [8].

3. Experimental demonstration and results

To demonstrate the CD/polarization independence in transmission of a chiral element placed in a SLI containing a HWP, we used a Left Handed Structure (LHS) manufactured by Chiral Photonics consisting of a linear polarizer sandwiched between two circular polarizers. The twisted fiber based polarizers are spliced with their slow/fast axis oriented in such a way as to make LHS transmit right circularly polarized light while heavily attenuating/scattering left circular polarization [18,25]. Figure 1 shows the experimental configuration for the π-shifted fiber Sagnac interferometer with a DUT and an optional HWP in the loop. Using a tunable laser, we carry out the CD compensating experiment by measuring the spectral transmission of the LHS chiral element placed inside (Fig. 1) and outside of the SI. The quarter-wave plate (QWP) orientation could be changed to generate left or right circularly polarized light while a HWP introduces a π-shift between the clockwise and counterclockwise propagating beams, and LHS is used as DUT.

The direct transmission of the LHS without SI exhibits a large difference for the left and right circularly polarized light - Fig. 2. The spectral variation in transmission is attributed to the transformation of the circular polarization to elliptical in the SMF fiber connectors and non-ideal character of the custom-made LHS.

Fig. 2 Transmission spectra of the chiral element for the right and left circularly polarized light without the SI.

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The transmission of the chiral element placed inside the π–shifted SI for the right and left circularly polarized light versus wavelength is shown in Fig. 3.

Fig. 3 Transmission spectra of the π–shifted SI with a chiral element for the right and left circularly polarized light.

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It can be seen from Fig. 3 that the transmission of SI with LHS and HWP is practically identical for right and left circularly polarized inputs (the residual CD at ~1535 nm can be caused by finite bandwidth of the 1550 nm centered coupler and other optical elements). By comparing the CD of the LHS placed inside (solid curve) and outside (dashed curve) π-shifted SLI, the near cancellation of the CD in transmission of the chiral element is obvious - Fig. 4.

Fig. 4 CD of the chiral element when it is placed inside (solid curve) and outside (dashed curve) of the π-shifted SI.

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Note that the LHS has an extreme case of CD - one of the circular polarizations simultaneously propagating in the SI is almost totally absorbed resulting in 3dB loss. The losses of the 50:50 coupler, U-bracket for the HWP and the HWP itself, all add up to 5 dB.

4. Conclusion

In conclusion, we have shown that the reflectance of a Sagnac interferometer containing a lossless chiral element is circular birefringence independent and we introduced a method, using a π-shift, to remedy the polarization dependence/CD impairment in transmission and reflection of SI containing lossy chiral devices. These schemes allow use of chiral photonic devices placed in Sagnac loop mirrors for switching/modulation, filtering and sensing and avoiding CB and CD-imposed limitations on the laser loop mirror performance.

Acknowledgment

This work was partially supported by a grant from National Science and Engineering Research Council of Canada (NSERC).

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Mode	Without HWP	With HWP
Transmission, loss ≠ 0	Polarization dependent	Polarization independent
Transmission, loss = 0	0	$T = \cos^{2} (\frac{Δ}{2}) \sin^{2} (2 ρ) + \sin^{2} (\frac{Δ}{2}) \cos^{2} (2 ρ)$
Reflection, loss ≠ 0	Polarization dependent	Polarization independent
Reflection, loss = 0	1	$R = \sin^{2} (\frac{Δ}{2}) \sin^{2} (2 ρ) + \cos^{2} (\frac{Δ}{2}) \cos^{2} (2 ρ)$

Mode	Without HWP	With HWP
Transmission, loss ≠ 0	Polarization dependent	Polarization independent
Transmission, loss = 0	$\sin^{2} \frac{δ (λ)}{2} \sin^{2} 2 α$	$\cos^{2} \frac{δ (λ)}{2} \sin^{2} 2 ρ$ , For ρ = 0, T = 0
Reflection, loss ≠ 0	Polarization dependent	Polarization independent
Reflection, loss = 0	$1 - \sin^{2} \frac{δ (λ)}{2} \sin^{2} 2 α$	$1 - \cos^{2} \frac{δ (λ)}{2} \sin^{2} 2 ρ$ , For ρ = 0, R = 1

Mode	Without HWP	With HWP
Transmission, loss ≠ 0	Polarization dependent	Polarization independent
Transmission, loss = 0	0	$T = \cos^{2} (\frac{Δ}{2}) \sin^{2} (2 ρ) + \sin^{2} (\frac{Δ}{2}) \cos^{2} (2 ρ)$
Reflection, loss ≠ 0	Polarization dependent	Polarization independent
Reflection, loss = 0	1	$R = \sin^{2} (\frac{Δ}{2}) \sin^{2} (2 ρ) + \cos^{2} (\frac{Δ}{2}) \cos^{2} (2 ρ)$

Mode	Without HWP	With HWP
Transmission, loss ≠ 0	Polarization dependent	Polarization independent
Transmission, loss = 0	$\sin^{2} \frac{δ (λ)}{2} \sin^{2} 2 α$	$\cos^{2} \frac{δ (λ)}{2} \sin^{2} 2 ρ$ , For ρ = 0, T = 0
Reflection, loss ≠ 0	Polarization dependent	Polarization independent
Reflection, loss = 0	$1 - \sin^{2} \frac{δ (λ)}{2} \sin^{2} 2 α$	$1 - \cos^{2} \frac{δ (λ)}{2} \sin^{2} 2 ρ$ , For ρ = 0, R = 1

Righting the handedness’ dichotomy: schemes of Sagnac interferometer containing chiral elements for suppression of dependence on circular birefringence and circular dichroism

Abstract

1. Introduction

2. Jones matrix analysis

3. Experimental demonstration and results

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Tables (2)

Equations (8)

Optics Express