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Chiral Fibers

This is a continuation from the previous tutorial - introduction to gradient index optics.


1. Introduction

Specialized fibers are increasingly being used to manipulate light and to couple light of different wavelengths into and out of fibers in telecommunications and sensing applications. The development of new communication modalities, such as cellular, satellite, and cable communications, has only spurred the growth of optical fiber networks.

Wavelength selectivity is achieved by imposing a periodic modulation of the refractive index along the fiber. This is accomplished most often by exposing photosensitive fiber to modulated ultraviolet (UV) light.

In gratings with periods greatly exceeding the wavelength, the refractive index may also be modulated by microbending, such as may be produced by squeezing the fiber between corrugated plates or by local heating with a CO2 laser or with an electric arc.

Chiral fibers employ an alternative means of implementing periodicity into a glass fiber, which allows for polarization and wavelength selectivity. This extends the functionality of optical fibers and is advantageous in a variety of filter, polarizer, sensor, and laser applications. An example of a double-helix chiral fiber is shown in Fig. 12.1.


Figure 12.1. Side-view of a double-helix chiral fiber grating.


Glass fibers with cores that are either concentric and birefringent, or nonconcentric, are twisted at a high rate as they are passed through a miniature oven in a drawing tower such as the one shown in Fig. 12.2.


Figure 12.2. Infrared camera-based twisting tower producing chiral gratings.


The fiber preform, typically from 100 to 200 μm in diameter, is held between a twisting motor affixed to a translation stage on top and a second translation stage on bottom. The speeds of the translation stages, the twisting rate, and the temperature of the short heat zone are varied under computer control to control the diameter and pitch of the fiber along its length.

The fibers are heated as they pass through a microfilament or a mini–torch-based oven in which the temperature is monitored by either an infrared (IR) camera or a thermocouple. A right- or left-handed structure is produced depending on the sense of the twisting motor rotation.

This flexible fabrication approach produces a stable structure that has double-helix symmetry in the case of twisted concentric birefringent fibers or single-helix symmetry in the case of twisted nonconcentric fibers.

High-contrast chiral gratings can be implemented in a broad range of glass materials free of the constraint that these be photosensitive. In double-helix structures, resonance interactions only occur for co-handed circularly polarized light with the same handedness as the structure, whereas in single-helix structures with nonbirefringent cores, resonance interactions are polarization insensitive.

In addition to gratings with constant pitch, gratings with smoothly varying pitch, which naturally arise in twisting fiber preforms to a specific pitch, can be used to produce broadband in-fiber linear polarizers. In addition to a gradual pitch variation, the phase of the helical fiber may be disrupted by an abrupt twist that creates localized states suitable for narrow-band filter and laser applications.


2. Three Types of Chiral Gratings

Chiral fibers have distinct functionalities in each of three ranges of the ratio of the pitch of the chiral fiber to the optical wavelength in the fiber, \(P/\lambda\equiv{Q}\). The three types of chiral fiber gratings are

  1. Resonant chiral short-period gratings (CSPGs) with pitch equal to the optical wavelength of the order of 1 μm which reflect light within the fiber core,
  2. Nonresonant chiral intermediate- period gratings (CIPGs) with pitch on the order of 10 μm, which scatters light out of the core
  3. Chiral long-period gratings (CLPGs) with pitch on the order of 100 μm, which resonantly couples core modes into co-propagating cladding optical modes.

While the single-helix structures are polarization insensitive, the double-helix structures only interact with co-handed circularly or elliptically polarized light, which have the same handedness as the grating and freely transmit cross-handed light of the orthogonal polarization in the CSPG and CLPG, as well as at the edge of the nonresonant scattering band in the CIPG.


Figure 12.3. Three types of double-helix chiral fiber gratings and their potential applications.


Double-helix chiral fiber gratings and their optical interactions are illustrated schematically in Fig. 12.3.

First, CSPGs reflect co-handed light within the fiber core within a stop band corresponding to a range of wavelength within the fiber for which \(Q=1\). Cross-handed light of the orthogonal polarization is freely transmitted. A series of long-lived modes occur at the edges of the stop band for co-handed light.

Such fibers can serve as polarization-selective spectral filters with a bandwidth relative to the band-center wavelength equal to the fractional fiber birefringence. Because the lowest order mode at the two edges of the stop band are significantly longer lived than other modes, these fibers may serve as the basis for lasers when appropriately doped and pumped. It may also be possible to create lasing in spectrally isolated long-lived defect modes in fibers in which an additional twist is introduced.

Second, CIPGs with values of Q over a broad range between those of the CSPG and the CLPG scatter light out of the fiber. Near the edges of this band, only co-handed light is scattered. As a result, these may serve as polarization and wavelength-selective filters over this polarization-selective scattering band.

Third, CLPGs with ratio \(Q\approx100\) resonantly couple core modes into co-propagating cladding optical modes when the difference between the propagation constants of the core and various cladding modes is compensated by the grating constant. The sensitivity of these resonances to the optical characteristics of the cladding and its surroundings makes the fibers ideal for sensing pressure, temperature, and liquid fluid level.


Figure 12.4. Overview of performance of a double-helix chiral fiber grating giving the ratio of right-to-left circularly polarized transmission versus \(Q=P/\lambda\). Measurements for small values of Q covering the resonant band-gap and the long-wavelength side of the nonresonant scattering gap were obtained for microwave transmission through a chiral rod, while measurements at the short-wavelength side of the nonresonant band and at larger values of Q were obtained from optical transmission. The dashed line connects the spectral ranges in which experimental results were obtained.


An overview of the performance of chiral fiber gratings with double-helix symmetry is presented in Fig. 12.4. The figure gives the variation of the ratio of right-to-left circularly polarized transmission with Q using data that are shown on the right side of Fig. 12.3, which is discussed in detail later in this tutorial.

Measurements for small values of Q covering the resonant gap and the long-wavelength side of the nonresonant scattering gap were obtained for microwave transmission through a chiral rod, whereas measurements at the short-wavelength side of the intermediate nonresonant band were obtained for IR radiation in the telecommunications range.

Because neither measurements nor calculations were carried out in the central portion of the nonresonant scattering gap, this region is indicated by the dashed curve. This curve reflects our belief that scattering will persist, and polarization selectivity will be lost in this region in which light is scattered at a substantial angle from the fiber axis.

A similar study was conducted for single-helix chiral fibers in two ranges of the parameter Q corresponding to the CIPG and CLPG in the optical spectral range and for \(Q\sim1\) corresponding to the CSPG in the microwave range. The results for single-helix chiral fibers show that transmission is polarization insensitive with an overall performance for any polarization, which is similar to that for RCP waves used in Fig. 12.4. Thus, the performance of single-helix gratings is similar to that of isotropic FBGs and long-period gratings (LPGs).


3. Chiral Short-Period Grating: In-Fiber Analog of CLC

3.1. Fabrication Challenges

CIPG and CLPG structures are produced with sufficient precision that highquality optical polarizers, filters, and sensors have been manufactured. The precision of short-pitch optical CSPGs that have been produced so far is lower, reflecting the greater fabrications challenge.

Although optical CSPGs appear uniform under microscopic inspection, polarization-selective stop bands with spectral shape appropriate for filter applications have not been produced. The properties of CSPGs will, therefore, be illustrated via microwave propagation in scaled up versions of the structure shown in Fig. 12.1.


3.2. Analogy to 1D Chiral Planar Structure

The polarization and wavelength-selective properties of chiral fibers rely on the symmetry of the structure. Because specific features of optical interactions arise as the symmetry of structures is lowered, an appreciation for the operation of chiral gratings can be obtained by comparing the optical interactions in chiral fibers to those in periodic structures with higher symmetry.

The highest symmetry periodic structure with refractive index modulation in a single direction consists of alternating isotropic layers. Propagation of waves directed along the normal to the layers may be described using a one-dimensional (1D) model.

The symmetry is lower in anisotropic planar chiral structures such as cholesteric liquid crystals (CLCs) and in periodic isotropic fibers such as fiber Bragg gratings (FBGs). Chiral short-period fiber gratings are obtained by lowering the symmetry in either of these structures; they lack the transverse translation symmetry of CLCs and the axial rotation symmetry of FBGs.


3.3. Comparison of 1D Chiral to 1D Isotropic Layered Structures

In the simplest periodic dielectric structure, composed of alternating layers with different refractive indices, orthogonal modes in different polarization states propagating normal to the structure are degenerate and polarization is maintained throughout the structure.

Light may, therefore, be taken to be linearly polarized without loss of generality. As a result of interference in the periodic structure, light cannot propagate normal to the planes for wavelengths within certain bands. Light in these forbidden bands or band gaps penetrates the sample as a standing exponentially decaying evanescent wave. Because there are no propagating optical modes within the band gap, the density of photon states vanishes as the sample size increases.

The energy density of the standing wave component of the light in the first mode at the high-frequency band edge has maxima in the low-index layers and nodes in the high-index layers. This leads to a concentration of energy in regions with low refractive index, which is consequently referred to as the air band.

The opposite situation, in which nodes fall within the low-index layer while maxima coincide with the high-index layers, prevails at the low-frequency edge. This leads to a concentration of energy in the region with high refractive index. This spectral range is, therefore, called the dielectric band.

The band structure in the vicinity of the stop band and the energy density near the center of the sample for light resonant with the first modes at the high- and low-frequency edges of the stop band are illustrated schematically in Fig. 12.5.

The wavelength of the evanescent wave within the band gap and at the first modes at the band edges is twice the period, \(\lambda=2a\), as illustrated in Fig. 12.5b. Higher order band gaps occur at values of \(k=2\pi/\lambda\), which are multiples of \(\pi/a\).


Figure 12.5. (a) Photonic band structure of a layered dielectric system with period a. (b) Dark and light layers correspond to high and low refractive indices, respectively. The electric field (E) and intensity (I) near the center of the sample are shown.


We next consider optical propagation in samples in which full-rotation symmetry is replaced by double-helix symmetry. This occurs in anisotropic-layered CLCs and in structured thin films (STFs). CLCs occur naturally in certain beetles and can be synthesized at different pitches in mixtures containing chiral moieties between glass plates in which the molecules are oriented and anchored on the inside surfaces of the plates.

STFs are produced by oblique deposition of dielectric materials on a rotating substrate. The molecular ordering for CLCs is illustrated in Fig. 12.6b. Rod-shaped molecules, represented by short bars, are partially ordered in each layer with an average polarization along a direction called the molecular director.

The director rotates with displacement perpendicular to the birefringent planes to form a periodic helical macrostructure, which can be either right- or left-handed with period a and pitch \(P=2a\). The pitch of the helix can be as small as 100 nm.


Figure 12.6. (a) Photonic band structure of a CLC with period a and pitch \(P=2a\). (b) Arrows indicate the electric field direction being aligned along or perpendicular to the director.


For sufficiently thick films, normally incident co-handed circularly-polarized light is nearly totally reflected within a band centered at vacuum wavelength \(\lambda_c\) such that the wavelength within the medium equals the pitch, \(\lambda_c/n=P\), where \(n\) is the average of the extraordinary and ordinary refractive indices of the medium, \(n_e\) and \(n_o\), respectively.

The bandwidth corresponding to the difference in vacuum wavelength between the first modes on the long- and short-wavelength sides of the band is \(\Delta\lambda=\lambda_c\Delta{n}/n\), where \(\Delta{n}=n_e-n\), and \(n=(n_e+n_o)/2\).

Over this wavelength range, the standing evanescent wave is composed of counter-propagating waves of the same sense of circular polarization with equal amplitudes at any depth within the sample. In any plane of the structure, the electric field oscillates in a line with a fixed orientation relative to the director (Fig. 12.6b).

At the first mode at the long-wavelength edge of the band, the polarization is parallel to the ordinary axis, while at the other extreme of the stop band, it is aligned along the extraordinary axis. As the wavelength is tuned across the stop band, the polarization rotates by 90 degrees in every plane.

The reflection spectrum of co-handed light is qualitatively similar to that in an isotropic-layered sample. In contrast, however, cross-handed radiation is freely transmitted by the chiral structure.

Propagation near the stop band of isotropic binary samples and in anisotropic chiral structures may be compared using a scattering matrix computer simulation. In the simulation, the CLC is treated as a set of equal-thickness anisotropic layers with thickness significantly less than the wavelength of light with the direction of the axes of the optical indicatrix in successive layers rotated by the same small angle within the plane of the layer. The simulation gives the transmittance and reflectance spectra of the CLC and the layered dielectric structures, as well as the intensity distribution inside the sample.

The results of the computer simulation of electromagnetic energy density within the sample for linearly polarized light in a layered dielectric sample and for co-handed circularly polarized light in a CLC structure are compared at the first and second modes at the band edge in Fig. 12.7 and Fig. 12.8.


Figure 12.7. Distribution of the energy density of the electromagnetic field inside a one-dimensional periodic sample at the wavelength of the \(n=1\) mode for layered (a) and CLC samples (b). The refractive indices are 1.47 and 1.63 for the layers of the layered dielectric structure and for the ordinary and extraordinary indices of the CLC. The sample thickness is 16 μm and the period is 0.2 μm. The energy density of the incident wave is unity.


Figure 12.8. Distribution of the energy density of the electromagnetic field inside a one-dimensional periodic sample at the wavelength of the \(n=2\) mode for layered (a) and CLC samples (b). The parameters of the samples are the same as in Fig. 12.6.


Results are for structures with indices of refraction 1.47 and 1.63, period \(a=0.2\) μm, and total thickness 16 mm. In the layered binary material, the indices correspond to those of two equal-thickness layers, whereas they correspond to ordinary and extraordinary indices in the CLCs.

Interference in layered structures leads to strong modulation of light in layered structures on the scale of a wavelength. In contrast, wavelength-scale oscillations of the electromagnetic energy within the CLC sample are nearly absent within the stop band.

This difference between the intensity distribution within the sample in layered and CLC structures is displayed for the first and second modes at the stop-band edge. The low-frequency modulation of intensity in Fig. 12.7 and Fig. 12.8 reflects the Bloch wave vector of these modes. Resonances at a particular thickness occur at multiples of half of the Bloch wavelength. The number of peaks inside the medium gives the mode number from the band edge.

The differences in the nature of light propagation through layered and chiral structures lead to quantitative changes in transmission and reflection spectra. Figure 12.9 shows transmittance spectra calculated for layered and CLC structures with the same parameters as in Fig. 12.7 and Fig. 12.8.


Figure 12.9. Comparison of transmittance for binary layered and CLC structures in a large spectral range (a) and at the band edge (b). The spectra are shifted in (b) so that the peaks of the first mode at the band edge coincide.


The stop band is wider for co-handed light in the CLC than in the binary layered structure because the field for the modes at the two band edges in CLC is aligned along or perpendicular to the director throughout the structure, as seen in Fig. 12.6, so the full index difference is experienced.

In contrast, the field experiences an index of refraction in binary samples at the band edges, which lies between values in each of the layers because the intensity is not strictly confined to one or another of the dielectric layers.

At the same time, the integrated intensity of the wave on resonance with band-edge modes is seen in Fig. 12.7 and Fig. 12.8 to be larger in the CLC than in the binary structure.

First, the intensity is not modulated in the chiral structure, which contributes a factor of 2 in integrated intensity relative to that in the binary sample, and second, the peak intensity is somewhat greater in the chiral sample.

The enhanced integrated intensity reflects the lengthened dwell time for light within the CLC structure. The lengthened photon dwell time at the band edge corresponds to a narrower line width for corresponding modes. Because the photon dwell time in the structure varies as \(1/n^2\), for the \(n\)th mode from the band edge, the lifetimes of the first band-edge mode are substantially longer than those for other modes, lasing is initiated in the first band-edge mode in presence of gain. Low-threshold lasing has been observed for the first mode at the band edge of CLCs.


3.4. Microwave Experiments

The principles of operation of a CSPG are most readily demonstrated using a scaled-up model that functions in microwave frequency range in which perfectly periodic structures are readily produced. Plastic rods with dimensions comparable to the wavelength of the microwave radiation were either milled or twisted as the rod was passed through a heat zone.

A band gap was not seen for \(\lambda\sim{P}\) when a single thin plastic rod was wound around the thicker rod, as shown in Fig. 12.10a. But when a second helix displaced by one-half the pitch was wound around the thicker rod so that the chiral structure had the symmetry of a double helix, as shown schematically in Fig. 12.10b, a stop band was observed.


Figure 12.10. Examples of fibers possessing (a) single- and (b) double-helix symmetry.


In analogy with planar chiral structures, a reflection band for co-handed radiation is formed over the range of vacuum wavelengths over which the wavelength in the fiber is equal to the pitch, \(\lambda_{vac}/n_\text{eff}=P\), where \(n_\text{eff}\) is the effective index that reflects the distribution of propagating energy between the core and cladding of the fiber.

The width of the gap in the vacuum wavelength is then given by \(\Delta\lambda_{vac}=P\Delta{n}/\langle{n}\rangle\) where \(\Delta{n}\) is the birefringence, which is the difference in effective index of refraction for linearly polarized light aligned along the slow and fast axes of the fiber, and \(\langle{n}\rangle\) is the average of these indices. Because the birefringence in these structures is high, the bandwidth is wide and the attenuation coefficient for co-handed light is large.

Measurements of microwave complex microwave field transmission coefficient, \(E\exp(i\phi)\) were carried out using a vector network analyzer. Here \(E\) is the amplitude and \(\phi\) the phase, so that the real and imaginary parts of \(E\exp(i\phi)\) are the in- and out-of-phase components of the field transmission coefficient.

Measurements of transmission of co- and counter-handed radiation within the 1.5- to 2.3-cm wavelength range through a rod of a length equal to 78P placed between a series of Teflon rods, which couple the grating shown in Fig. 12.11 to a source and detector.


Figure 12.11. Double-helix polymeric rod with a rectangular cross-section. Arrow in the middle indicates a place at which the fiber was cut and twisted around its axis to produce a chiral twist, which gives rise to a defect state in the center of the stop band.


Horns with polarization states set to produce either right or left circularly polarized (RCP or LCP) radiation were used to launch and detect the microwave radiation. A stop band is observed in transmission with \(\Delta\lambda\sim0.01\lambda_c\) with a reflection peak over the same range centered at \(\lambda_c=2.09\) cm.

At shorter wavelengths, the beginning of a broad dip is observed for co-handed radiation. The drop in transmission is not associated with a corresponding increase in reflection. The broad peak in LCP radiation reflects the frequency dependence of coupling of the horns to the Teflon rod and does not change appreciably as the length of the Teflon rods in the system increases.

The transit time of the wave passing through the Teflon rods sandwiching the central chiral rod is given by the spectral derivative of the phase, \(d\phi/d\omega\), shown in Fig. 12.12b.

The high value of the delay time at the band edge is associated with a resonance with a spectrally narrow long-lived state at the band edge. In this case, \(\phi\) increases by \(\pi\) as the wavelength is tuned thorough a single narrow mode. The broad gap in transmission below 1.9 cm is due to scattering out of the fiber and is discussed later in the section on the CIPG.


Figure 12.12. Broadband polarized spectra of microwave radiation transmitted through the polymeric rod, shown in Fig. 12.8. (a) Transmission spectrum on semilog scale. (b) Delay time obtained from the phase derivative of the transmitted RCP and LCP fields and the ratio of the transmission shown on linear scale.


3.5. Optical Measurements

Despite that CSPG with uniformity sufficient for filter applications has not been made yet, CSPGs with substantial reflectivity were fabricated. The presently achieved level of the pitch variation is near 1%.

This level of uniformity is apparently already sufficient to obtain lasing from highly birefringent-doped fibers but is not high enough for filter applications. The quality of optical CSPGs has been improving rapidly and we anticipate it will shortly be the basis of useful devices.


4. Chiral Intermediate-Period Grating

4.1. Symmetry of CIPG Structures

We next consider the differences between propagation in a planar and in a fiber geometry. Because the sample is no longer uniform in the transverse direction, core modes may be scattered into modes with propagation vector pointing away from the fiber axis when the difference between the propagation constant of the core mode and the scattered wave is compensated by the grating constant associated with the periodic structure.

In FBGs, the forward and backwards propagating core modes are coupled via the grating constant, which is \(2\pi/a\), where \(a\) is the grating period. In chiral fibers, core modes are similarly coupled via the grating condition above. In single-helix chiral gratings \(a=P\), whereas in double-helix gratings, \(a=P/2\), giving grating constants of \(2\pi/P\) and \(4\pi/P\), respectively. We focus here on double-helix gratings because these are polarization selective.

In short-period gratings, the gratings constant is so large that the forward propagating core mode can only couple into a single other mode, the backwards-scattered core mode.

As a result, reflection is analogous to that of chiral planar materials such as cholesteric liquid crystals or STFs. Propagation near the stop band of CSPGs is, therefore, essentially 1D and closely mimics the propagation in planar chiral structures. Similarly, propagation in isotropic FBGs and in single-helix chiral fibers is similar to that in binary-layered structures.

In weakly twisted fibers, the grating constant is too small to couple the core mode to the backward propagating core modes because the propagation constant of these modes differs by more than \(2\pi/a\).

For many years, twisted fibers with pitch of tens of centimeters have been studied because of their potential for reducing polarization mode dispersion, which limits the bandwidth of transmission.

Optical fibers with similar pitch were also used either to create circularly birefringent fibers for maintaining circularly polarized light or for converting the state of polarization.

In all of these applications, the goal is to have light of different polarization interact with the fiber in a similar manner to reduce the polarization sensitivity of the fiber. It was shown that reducing the pitch leads to further reduction of the polarization sensitivity. In contrast, fibers that are twisted to a much shorter pitch are highly polarization selective.


4.2. Microwave Experiments

A broad gap in the transmission spectrum can be clearly seen at the short-wavelength side of Fig. 12.12. This gap exists only for co-handed polarization and corresponds to the double-helix CIPG.

The transit time shown in Fig. 12.12b is found to be the same for left- and right-handed circular polarization in a right-handed structure. This is in contrast to a drop in the transit time within the stop band and an increase at band-edge modes for co-handed relative to cross-handed radiation.

The transit time in the intermediate band is consistent with the linear dispersion at lower frequencies. These results confirm that the dip in the CIPG is not the result of coherent resonant reflection within the chiral fiber but is the result of scattering out of the fiber.


4.3. Optical Measurements

High-quality CIPGs have been fabricated for applications from 980 to 2000 nm. This range may be expanded to shorter or longer wavelength by scaling the structure. These structures are of particular interest because of the multiplicity of applications for which they would be well suited.

In CIPGs with pitches of order 10, or 100 microns co-handed circularly polarized light can be selectively scattered from the fiber, whereas the orthogonal polarization is freely transmitted over a wide range of wavelengths near the edges of this band.

For a specific wavelength, the resonant pitch for short- or long-period chiral gratings is defined by the corresponding phase-matching condition. In contrast, light is scattered out of the fiber by CIPGs for a broad range of pitch. Over this range, the direction of the scattered light changes from the near-backward to the near-forward direction.

The polarization selectivity of the optical CIPG is most readily demonstrated near the short-wavelength edge of the scattering band. Polarization selectivity was described earlier for microwave radiation near the long-wavelength edge of the scattering band.


Figure 12.13. Ratio of right-to-left circularly polarized transmission through a chiral intermediate period grating. Inset shows broadband in-fiber polarizer based on an intermediate chiral grating with nonuniform pitch.


Figure 12.13 shows the spectrum of the extinction ratio between co- and cross-handed circularly polarized components at the short-wavelength edge of the nonresonant scattering band.

The structure has a pitch of 45 μm and a length of 11 mm. Appreciable scattering of cross-handed radiation occurred in the middle of the band, leaving regions near the long- and short-wavelength band edges in which scattering of cross-handed radiation was negligible. Polarizers may, therefore, be produced at the band edges of the CIPG.

Chiral optical fibers at the long-wavelength edge of the scattering band may also be produced. A portion of the fiber with a pitch of 0.95 μm and a length of 500 μm has an extinction ratio close to 9 dB at 1460 nm.


4.4. Synchronization of Optical Polarization Conversion and Scattering

In the process of producing stretches of fiber with uniform pitch from an untwisted preform, which can be spliced to standard fibers, the preform twist must be accelerated to reach the desired pitch and finally decelerated. This process introduces new functionality to chiral fibers.

We find that it is possible to create an optical fiber with variable pitch in which linearly polarized light along the slow axis is transmitted without attenuation while orthogonally polarized incident light is strongly attenuated as it is scattered out of the fiber.

This indicates that the two processes of polarization conversion and scattering are synchronized. Thus, an elliptically polarized mode deriving from the slow linearly polarized mode of the untwisted fiber is not scattered at any point along the fiber but is freely transmitted as its state of polarization is continuously transformed.

At the same time, light with the orthogonal polarization is strongly scattered from the sample. As a result, when the final pitch of the right-handed sample falls in the range of the CIPG, linearly polarized light incident along the slow axis is efficiently converted to left circularly polarized light while light polarized along the fast axis is scattered out of the fiber.

Because the reverse process converts one component of elliptically polarized light to linear polarized light, decelerating the twist will restore the light to its initial state of linear polarization along the slow axis. This structure serves as an efficient linear polarizer with low insertion loss.

We first analyze the polarization evolution along the adiabatically twisted optical fiber with birefringent core using a 4 x 4–transfer matrix approach, designed for planar chiral media. This calculation can account for the longitudinal propagation, but not for scattering out of the core mode, which occurs in samples with transverse structural variation.

In this approach, the grating period is not constrained to be constant. We confirmed that the transfer matrix method converges as the thickness of individual anisotropic layers with ordinary and extraordinary refractive indices, \(n_f\) and \(n_s\), along the fast and slow axes, respectively, and the angle between consecutive layers are proportionately reduced and the number of layers correspondingly increases. Measurements of polarized transmission demonstrate the synchronization of conversion and scattering in twisted fibers, such as the fiber shown in Fig. 12.14.


Figure 12.14. Side-view of an adiabatically twisted optical fiber with rectangular core.


We calculate the conversion of linearly polarized 1.5-μm-wavelength radiation for different values of the birefringence in a 16-mm long stack of 1-μm thick anisotropic layers structure twisted to a final right-handed pitch of 110 μm. The linear twist acceleration is close to that of the fibers in which measurements are presented later in this tutorial.


Figure 12.15. Calculation of conversion of linear polarization to right circular polarization. The residual left circularly polarized light is shown in the graph for different values of the birefringence of the untwisted fiber.


The incoming linearly polarized wave is oriented along the fast fiber axis. The conversion into a right circularly polarized wave is displayed in the plot in Fig. 12.15 by the residual transmission of left-circularly polarized light. The wave is linearly polarized as it enters the sample at \(z=0\), so that half of the energy is left-circularly polarized, corresponding to the \(-3\) dB level on the graph.

For small values of birefringence \(\Delta{n}=n_s-n_f\) conversion is negligible. For higher values of \(\Delta{n}=0.004\), the residual left-circularly polarized light falls as P decreases, with a conversion into the right-circularly polarized wave of more than 99% conversion at \(z=16\) mm.

For larger values of \(\Delta{n}\), the conversion is again not optimal. At \(\Delta{n}=0.013\), the substantial intensity of left-circularly polarized light indicates that the resulting polarization is far from circular. This can also be seen from the variation of the effective refractive indices in Fig. 12.16.

In the case of a complete conversion to the circularly polarized wave, both curves would converge to \(n_\text{av}=(n_s+n_f)/2\). The measured birefringence in the fiber used was about 0.013. In accord with results in Fig. 12.15, the polarization state of the wave at \(z=16\) mm at a final pitch of 110 μm is not circular.


Figure 12.16. Variations in the effective refractive indices for orthogonally polarized eigenmodes in adiabatically twisted birefringent optical fiber. Linear polarizations along the fast and slow axes and right and left circularly polarized radiation are intrinsic modes of the untwisted fiber and of the fiber twisted with short constant pitch. While the cross-handed circular polarization is not scattered, the co-handed circular polarization is scattered out of the fiber.


We studied the polarization evolution for this value of \(\Delta{n}\) for a fiber in which the pitch is accelerated in the first half and decelerated in the second half of the structure.

Figure 12.17 illustrates the polarization evolution in a right-handed 32-mm structure with \(\Delta{n}=0.013\), with accelerating and then decelerating twist. Figure 12.17a shows the evolution of the wave initially polarized along the fast fiber axis on the Poincare´ sphere in the laboratory frame. The surface of the Poincare´ sphere represents all possible states of polarization. Points on the equator correspond to linearly polarized waves, whereas north and south poles represent right- and left-circularly polarized waves, respectively.

It can be seen that an initially linearly polarized wave moves towards the north pole, as illustrated by the arrow originating from the point F. The polarization trajectory rotates around the north pole without actually reaching it, in agreement with Fig. 12.15.

Then, after passing the fiber midpoint, the polarization state moves back to the equator, reaching the point indicated by the second arrow. This point on the sphere depends on the precise angle of rotation of the fiber modulo \(2\pi\) and represents a linearly polarized wave. The evolution of the linearly polarized wave initially oriented along the slow fiber axis is a symmetrical curve originating from the point S and moving towards the south pole.


Figure 12.17. Calculation of conversion of linear (along fast axis) polarization in a right-handed optical fiber with \(\Delta{n}=0.013\). (a) Evolution of the polarization in the laboratory’s coordinate system shown on the Poincare´ sphere. (b) Evolution of the polarization in the structure’s coordinate system.


To get a better physical picture of the polarization evolution, we recalculated the path on the Poincare´ sphere in a coordinate system, which rotates with the fiber, in which the X and Y-axes coincide with the slow and fast fiber axes at any cross-section. The results of the calculations are shown in Fig. 12.17b.

The initially linearly polarized wave represented by the point F evolves towards the north pole along a sphere meridian, stops at the same distance from the pole as in Fig. 12.17a, and then returns to the initial point. Again, scattering is not taken into account and the evolution of the linearly polarized wave oriented along the slow axis is symmetrical with respect to the center of the sphere.

Measurements were made on a twisted fiber prepared from a custom preform with refractive indices in the visible spectral range for the core and cladding of 1.69 and 1.52, respectively. The fiber preform was drawn down to a 125-μm diameter with a 10- x 5-μm rectangular core.

This preform was further drawn and twisted as it passed through a miniature oven. The control of the twisting tower was integrated into a single computer program in which the precise translation, drawing and twisting of the fiber, and its temperature were coordinated.

A side image of the twisted fiber is shown in Fig. 12.14. We compare the performance of three samples prepared with linear twist acceleration and deceleration, with shortest pitch of 110 μm.

These samples are (1) final cladding diameter of 14.5 μm and total length of 32 mm; (2) final cladding diameter of 14 μm and total length of 44 mm, and (3) final cladding diameter of 14 μm and total length of 51 mm. At a wavelength of 1.5 μm, the untwisted core supports two modes with orthogonal polarizations with a difference in effective refractive index difference of approximately 0.013.

An Agilent 83437A broadband unpolarized light source was used for spectral measurements between 1450 and 1600 nm. The incident light was polarized along either the slow or the fast fiber axis using a fiber-connected walk-off linear polarizer. The output light was guided to a fiber connected to an Agilent 86145B optical spectrum analyzer without having its polarization analyzed.


Figure 12.18. Measurements of wavelength dependence of light transmission through optical fibers twisted with different pitch profile.


Measurements of optical transmission through the three twisted fiber samples are shown in Fig. 12.18. Curves of transmission for linearly polarized light initially oriented along the slow fiber axis, for which scattering is minimal, are shown for samples 1, 2, and 3. The corresponding extinction ratios for these samples, which are the ratios of transmission for waves polarized along the fast and slow fiber axes, are also shown.

The splice-induced insertion losses were near 1.5 dB and may be seen in transmission at a wavelength of 1450 nm for all three samples in Fig. 12.18. At longer wavelengths, there are relatively small twist-induced insertion losses, which are approximately 0.2 dB for sample 1 and 0.5 dB for samples 2 and 3 at \(\lambda\) = 1550 nm.

This indicates that the wave polarized along the slow axis very closely follows the evolution path that originates at the point S and is symmetrical to the trajectory shown in Fig. 12.17.

In contrast, the wave initially polarized along the fast axis is strongly suppressed at a wavelength of 1550 nmin all three samples. This indicates that the wave at 1550 nm initially polarized along the fast axis does not follow the trajectory shown in Fig. 12.17 but approaches the center of the Poincare´ sphere as it is scattered.

This behavior is possible only if scattering from the fiber is fully synchronized with polarization conversion. This indicates that at any point along the trajectory shown in Fig. 12.17, one elliptically polarized wave is scattered and an orthogonal wave is not. This conclusion is bolstered by the comparison of these measurements and transfer matrix calculations.

The high extinction ratio over a 100-nm range, seen from Fig. 12.18, indicates that synchronization takes place over a broad spectral range. This range is independent of the twist acceleration or deceleration, as can be seen from the comparison of results for samples 2 and 3, but the range shifts with fiber diameter, as can be seen from the measurements in sample 1.

This shift is expected because the product of the shortest pitch and the effective refractive index of the core mode determines the spectral position of the scattering band in CIPGs.

Increasing the core diameter results in a larger effective refractive index. This, in turn, shifts the position of the intermediate scattering gap to longer wavelengths. The synchronization of polarization conversion and wave scattering along a chiral optical fiber with varying pitch makes possible a broad-band, low-loss, and high extinction ratio linear and circular in-fiber polarizers.


5. Chiral Long-Period Grating

In both isotropic LPGs and CLPGs, the core mode is resonantly coupled to co-propagating cladding modes. This is the basis of their use as sensors.

The double-helix CLPG may have a particular advantage because the cross-handed wave, which is not scattered by the CLPG, is available for monitoring the integrity of the fiber system.

In addition, the sinusoidal modulation of the dielectric constant gives rise to a single set of dips without harmonics for each cladding mode.


5.1. Optical Measurements

5.1.1. Double-Helix CLPG

A double-helix CLPG was made with pitch of 78 μm and was 55 mm long. Sharp dips in transmission were observed at wavelengths associated with coupling of the core mode to distinct co-propagating cladding modes.

In conventional fiber LPGs, the wavelength of the resonant dips is determined by the condition, \(k_\text{core}-k_\text{clad}=2\pi/1\), where \(k_\text{core}\) is the propagation constant of the core mode, \(k_\text{clad}\) is the propagation constant of a particular cladding mode, and \(a\) is the period of the grating.

A similar phase-matching condition applies to CLPGs. The wavelength of strongest extinction of the CLPG can be readily adjusted by changing the chiral pitch. The sensitivity of these resonances to the optical characteristics of the cladding and its proximate environment makes the fibers ideal for ambient sensing.

The ratio of right- to left-circularly polarized transmission through the sample is shown in Fig. 12.19. The results show that while LCP light is not affected by the right-handed structure, RCP light displays dips in transmission. The inset shows the shift of the transmission dip when the CLPG is surrounded by gasoline.


Figure 12.19. Ratio of right-to-left circularly polarized transmission through a double-helix chiral long-period grating.


5.1.2. Single-Helix CLPG

Single-helix CLPGs are also capable of coupling core modes with the copropagating cladding modes. The single-helix structure does not match the geometry of lightwave of any particular polarization. This leads to polarization insensitive interaction of the lightwave with the structure. The preform for single-helix CLPG could be made of low numerical aperture (NA) fiber with a nonconcentric core.

The performance of these structures is similar to that of regular fiber LPGs produced by microbending. As a result, the index contrast achievable cannot be larger than the difference in index of the core and cladding, which is smaller than the grating strength available in double-helix CLPGs.

The performance of the single-helix CLPG made of a custom silica fiber with nonconcentric core is shown in Fig. 12.20. Both single- and double-helix CLPGs have their advantages for sensor applications. The main similarities and differences between single- and double-helix CLPGs are summarized in Table 12.1.


Figure 12.20. Polarization insensitive transmission through a single-helix chiral long-period grating.


Table 12.1. Key parameters of single- and double-helix CLPGs


6. Conclusion

In conclusion, we have demonstrated the distinctive polarization and wavelength-selective properties of optical CLPGs and intermediate-period gratings that make them useful as sensors, filters, and polarizers. We have also discussed the desirability of chiral short-period gratings as lasing sources and discussed progress towards this goal.

Because of the flexible method of fabrication, which does not require photosensitive glass, chiral fiber gratings can be produced to perform a wide array of functions over a broad wavelength range including IR and THz.


The next tutorial introduces in detail about laser mirrors and regenerative feedback.


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