in Fig.3.5-b.
Actually, the coupling coefficient κ is found to be only slightly dependent on the polarization. This variation depends on the fiber diameter, as visible in the b panel of Fig. 3.7 but leads to a marginal relative change. In the realization of the nanofiber twisted loop below, we use a fiber diameter of 500 nm for which the relative change is estimated to be less than ±3%. This is illustrated in Fig. 3.7-a by the colored region surrounding the mean coupling strength ¯κ, corresponding to the amplitude of the variation with respect to the polarization. Given that, we can reasonably neglect the effect due to polarization and approximate κ≈κ¯.
Figure 3.7: a- Coupling coefficientκ(at 800 nm) between nanofibers with varying diameter and averaged over the polarization degree of freedom. The colored area refers to the amplitude of the variation with the polarization. b- Relative change of the coupling coefficient as a function of the polarization angle with a fiber diameter varying from 500 nm to 900 nm.
In Fig.3.8, we show the directional coupling coefficientκ/kas a map of the polarization.
We assume the mode in both fibers are linearly polarized with polarization angle θ1 and θ2 respectively, where θ1 and θ2 ∈ [0, π]. While the polarization angle θ1 and θ2 changes from 0 to π, the directional coupling coefficient κ/k varies from -0.06 to 0.06.
Therefore, the changing of polarization angle within the fiber should change the value of directional coupling coefficientκ/k. In an optical fiber, the polarization of the propagat-ing light changes when the birefrpropagat-ingence induced by stress or fiber cross-section geometry changes. This is something that we need to consider about when we further analyze the mode coupling of nanofiber structures.
3.3 Nanofiber interferometer
We realized an optical nanofiber by pulling a commercial single-mode fiber to reach a 500 nm diameter over a length of 1 mm following [4]. The transition between the
com-56 POLARIZATION CONTROL OF LINEAR DIPOLE
Figure 3.8: Directional coupling coefficient κ/k as a map of the polarization. θ1 and θ2 are the angle of linearly polarized light in fiber 1 and fiber 2.
mercial single mode fiber and the single mode nanofiber is adiabatic and its transmission is over 95% [21].
To make a twisted loop structure presented in Fig.3.1-a, we first make a ring in the nanofiber region. Then, by fixing one side of the nanofiber, and rotating the other side, we can slowly increase the length of the entwined part. This is a well-known mechanical phenomenon studied in many contexts [50]. Increased torsion will reduce the size of the loop and bend it locally. At some point the bending exceeds the fiber tolerance and it breaks. In the experiment we carefully choose to remain below this threshold.
With this geometry, the system corresponds to a Sagnac interferometer [51]. As rep-resented in Fig.3.1-b, light propagating towards the loop finds two counter-propagating optical paths. After the entwined region, part of the light is transferred into the clockwise path (red arrow) with a coefficient ofr, whereas the remaining propagates along the anti-clockwise path with a coefficient oft. This two beams propagate separately accumulating a phase of eiβLr, where β is the propagation constant, and Lr is the length of the ring.
Then, both paths interfere back into the entwined region, which acts as a beam-splitter, as represented in Fig.3.1.b. The amplitude of output electromagnetic field can be written as:
As mentioned above, we control the length of the coupling region by varying the torsion applied on the nanofiber, which ultimately tunes the reflection and transmission coeffi-cients of our device. Applied mechanical stress will be translated into an optical response, from reflective to transmissive.
To probe the system transmittance we use a fiber-coupled linearly polarized super-continuous white laser (NKT Photonics SuperK COMPACT). The laser beam is coupled into the single mode fiber (SM800) with the twisted nanofiber region in the middle. The output signal is sent to a spectrometer.
3.3. NANOFIBER INTERFEROMETER 57
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Figure 3.9: The transmission spectrum and the corresponding image of twisted loop. a, the map of transmission spectrum while varying the torsion applied on the nanofiber. b, image of nanofiber structure when there is no coupling (measurement No. 324 and plus).
c, images of nanofiber twised loop when the coupling distance is measured to be fixed (measurement No. 1-59, 130-189 and 290-323). d, image of nanofiber twisted loop when it’s rotating and the coupling distance is changing (measurement No. 60-119).
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Figure 3.10: Transmission spectrum through twisted loop structure. a and b- Experi-mental data and numerical simulation of the transmission spectrum of broad light source through the nanofiber interferometer as a function of the wavelength and coupling dis-tance. c and d- Experimental data (blue) and numerical simulation (red) of the transmis-sion spectrum for an entwined length of 75µm (panel c) and 100µm (panel d) respectively.
The experiment data shown are smoothed using rloess methods with a span of 1%.
3.3. NANOFIBER INTERFEROMETER 59 To show the relationship between the transmission spectrum and the movement of the nanofiber twisted loop, we measured the transmission spectrum and its corresponding image of twisted loop in camera each time we rotate one side of the fiber 5 degrees. We plot the transmission spectrum as a function of wavelength in Fig.3.9-a and the ordinate is the number of measurements. The length of entwined part is measured by analyzing the image taken with a calibrated camera, corresponding to the coupling length. The camera is calibrated with a resolution test target (Thorlabs) at the focus plane. By measuring the coupling length in each photo taken by camera, we find the coupling length changes periodically instead of continuously when we rotate one side of the twisted loop. When the coupling length is fixed, the rotating of fiber end lower the tension in the system.
The coupling length starts to decrease when the loop begins to rotate, and the coupling length stops decreasing after the loop rotates 180 degrees and released one crossing in the entwined part. In Fig.3.9-c, we show the image of twisted loop with 3, 4 and 5 crossing.
At the alternation of these two states, the state of the fiber is unstable. To study the influence of coupling length on the transmission spectrum, we focus on the part when the coupling length is changing or the part when loop is rotating.
Fig. 3.10-a shows a map of the transmission spectrum through the nanofiber twisted loop structure as a function of the entwined region size.
To understand the spectrum obtained for a given entwined region length, also pre-sented in Figs. 3.10-c and d in blue, one has to note that when the fiber diameter is fixed, the extension of the evanescent part of the field increases with the wavelength.
Accordingly the coupling strength κ and so the effective reflection coefficient r change too as shown by the red curves in Fig. 3.10-c and d. In consequence, the spectrum for a fixed coupling length leads to the interference pattern visible in Fig. 3.10 and agrees with our description of the device as an interferometer. Moreover, increasing the cou-pling length leads to a shift of the interference to smaller wavelengths. In Fig. 3.10-b we represent the same map as in Fig. 3.10-a, calculated from Eq. 3.17. Despite the variabil-ity of many experimental parameters, our theoretical model shows good agreement with the experimental data. This agreement can be verified quantitatively in panel c and d, where we show the measured (blue) and calculated (red) spectrum for a coupling length of respectively 75 µm and 100 µm with no adjustable parameters.
However, the contrast of the observed spectra is relatively low (0.2 to 0.6). Besides, the contrast seems to be different between nearby peaks, which is different from the numerical simulation. It is more obvious in Fig.3.10-d, the experimentally measured transmission spectrum shows four peaks. The peaks at 700 nm and and 850 nm have higher contrast while the peaks at 790 nm and 920 nm have lower contrast.To understand the limiting factor of this phenomenon, we did numerical simulations of transmission spectrum to study the effect of loss during the mode coupling. In our two fibers coupling model, the energy transfer between the two fibers were regarded as lossless. In reality, the surface of tapered nanofiber is not purely smooth. There are a lot of small defects which scatter the light into free space as we can see in the camera after a nanofiber is fabricated. For twisted nanofiber structure, the losses come from both the coupling between two fibers at the twisted part and the light propagating in a bending fiber. Therefore, we add parameters representing coupling loss and propagation loss based on the equation.3.18.
We have:
Aout = [((1−ρr)r)2+ ((1−ρt)t)2](1−ρ)eiβLrA0, (3.20) where ρr is the coupling loss for reflection coefficient r, ρt is the coupling loss for trans-mission coefficient t, ρ is the propagation loss for ring part.
60 POLARIZATION CONTROL OF LINEAR DIPOLE
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Figure 3.11: a, Optical microscope image of a nanofiber twisted loop. b, Schematic of introducing losses in the Sagnac interferometer equivalent optical setup. The transmission coefficient with loss is labeled as t0. The reflection coefficient with loss is labeled as r0.
When we take the coupling loss into consideration, we found that the loss is higher for longer wavelength. In figure 3.12-a, we plot the directional coupling coefficient κ as a function of wavelength. The fiber diameter is fixed at 500 nm. The longer wave-length gives higher coupling coefficient. For the same coupling wave-length, we can imagine the longer wavelength bouncing back and forth more times than the shorter wavelength in the entwined region. Thus for the longer wavelength, there will be more loss.
Besides, there is unbalanced loss between reflection coefficient r and transmission coefficient t, which caused the contrast difference between nearby peaks. In Fig.3.12-b,c, we plot the transmission spectrum when coupling lengthz = 100µmand z = 75µmwith 10 percent loss for reflection coefficient r and 2 percent loss for transmission coefficient t. We take the one with 100 µm coupling length as an example, the peaks at 700 nm and and 850 nm have higher contrast while the peaks at 790 nm and 920 nm have lower contrast, which is similar as the measurement. Therefore, we think the contrast difference between nearby peaks comes from the unbalanced loss between reflection coefficientrand transmission coefficient t. In reality, this can be due to the geometric asymmetry of nanofiber structure.
By tuning the coupling length, we can adjust the high contrast region to the wavelength we need, as we can see in the Fig.3.10-a, the high contrast region shifts from 700 nm to 750 nm when we change the coupling length from 100 µm to 75 µm.