Abstract

Experimental demonstration of rotational Doppler-effect due to direct and simultaneous excitation of orthogonal elliptically-polarized fundamental and vortex modes in a two-mode optical fiber is presented here. The rotation frequency and the trajectory of the zero-intensity point in the two-mode fiber output beam measured as a function of analyzer rotation matches with the S-contour of polarization singularity in the beam, identified via Stokes parameter measurement. The characteristics of the S-contour around the C-point in the output beam is also measured as a function of rotating Dove prism and half-wave plate – Dove prism combination to highlight the role of polarization modifying components on the observed rotational Doppler-effect of vector-vortex beams.

© 2011 OSA

1. Introduction

The classical effect of observing the hands of a watch kept at the center of a rotating stage and rotating in the clockwise direction when viewed from the top appears to rotate fast is known for a long time. This simple observation is known as due to the rotational Doppler-effect (RDE). The effect can be extended to the rotation of the electric field vector of a circularly polarized light beam also carrying orbital angular momentum to either speed up or slow down the beam rotation about its propagation axis resulting in the frequency broadening or frequency shift proportional to the rate of rotation of the field vector or the total angular momentum of the beam [1,2]. First demonstrated by Garetz and Arnold, the frequency of circularly polarized light (σ = ± 1) transmitted through a half-wave plate rotating at frequency Ω/2 is measured to be shifted by σΩ [3]. Associated with the angular Doppler-effect, the rotational frequency shift manifests itself whenever light is emitted, absorbed or scattered by rotating bodies [4,5]. Analogously, an atom moving in a light beam with orbital angular momentum (OAM), a Laguerre-Gaussian (LGnl) beam for example, experiences an azimuthal shift in its resonant frequency [6].

In general, for circularly-polarized LG beams, with both spin (σ) and orbital (l) components of the angular momentum, the Doppler frequency shift is proportional to the total angular momentum (σ + l) which has been quantified directly for the rotation of EM beams at millimeter wavelengths [7]. Subsequently, the angular Doppler-effect was made use of in an elegant demonstration of moving interference patterns in the optical domain for controlled rotation of trapped particles [8]. The RDE was also demonstrated for helical wavefront-vortex optical beams generated using a spiral zone plate [9] and in off-axis optical vortex beam generated using a holographic grating [10]. Further, an OAM spectrum analyzer was demonstrated utilizing the rotational Doppler frequency shifts imparted to LG modes with different ‘l’ values [11].

Though different manifestations of the RDE has been demonstrated amply in free-space and in atomic and molecular systems using scalar optical beams with OAM, very few demonstration of such an effect exist in waveguide medium [1215]. In addition to being practically important systems, optical fibers are an interesting class of inhomogeneous medium wherein the guided optical vortices (OVs) can co-exist with the vector eigen modes (with non-uniform polarization across the beam cross-section) of the fiber wherein the OAM of the light is redistributed among the fiber eigen modes [14,15]. As a result the optical vector-vortex beams excited in fibers offers a unique opportunity to study the interplay between the OAM of the light beam redistributed among the spatially (temporally) varying electric field vector of the guided modes. Recently, the possibility of observing RDE in stationary and rotating optical fibers was proposed and subsequently observed in few-mode fibers by launching an externally-generated LG beam into the fiber [1215].

More recently we reported the first experimental demonstration of direct generation of optical vector-vortex beams (OVVBs) using a short-length of two-mode fiber (TMF) excited by an off-axis and tilted linear-polarized Gaussian beam [16]. Based on this we also demonstrated direct transformation of a circular-polarized Gaussian beam into linear-polarized vector-vortex beams, which offers an additional advantage of switchability between the different 0th -order vector-vortex (anti-vortex) modes guided in the two-mode fiber by simply changing the handedness of the input beam polarization from right- to left- circular or vice-versa [17]. These demonstrations emphasize that the inherent vectorial nature of the annular (vortex) waveguide modes of few-mode optical fibers offers an ideal platform for controlled generation and manipulation of vector-vortex beams.

Another interesting facet of this direct generation of vector-vortex beams using a two-mode optical fiber is presented here – an experimental investigation of the RDE and hence the rotational frequency shift (RFS) in spatially inhomogeneous polarized optical vortex beams. Such beams can also be categorized as beams containing polarization singularities (PS) or beams with well-defined S-line (S-contour) and C-point [18,19]. The beam with required characteristic is generated directly by propagating circularly polarized Gaussian laser beam as an offset-skew ray in a horizontally-held TMF. The input beam launch is adjusted to selectively excite the circular polarized fundamental mode and orthogonal circular polarized vortex mode simultaneously in the fiber resulting in an elliptically polarized output beam with shifted vortex core and well-defined polarization singularities. The coherent coaxial superposition of the reference (Gaussian) and the signal (annular) modes in a single beam offers a unique way of demonstrating the RDE wherein there is no undesirable of displacement of optical elements and resulting possible modal or polarization variations in the beam that will influence the phase of the constituent waves. Rotation frequency of the vortex core or the zero-intensity point (ZIP) or the S-line in the output radiation field from the fiber around an axis, encompassing the C-point is measured as a function the analyzer rotation and for changing handedness of the input circular polarization [20] to demonstrate RDE in the directly excited vector-vortex beams. Stokes parameters of such beams are also measured and the S-contour in the output beam matches well with the ZIP trajectory. Passing the inhomogeneous polarized beam combination through a rotating Dove prism (DP) and a half-wave plate (HWP) – DP combination further enables us to investigate the role of polarization modifying components on the observed RDE of vector-vortex beams in the vicinity of the PS.

2. Theory

We use two-mode optical fiber (TMF) in our experiment to generate circularly polarized vortex beams to demonstrate RDE. The TMF simultaneously guides the fundamental HE11 mode and four 0th – order vector modes: HE21 e, TE01, HE21 o and TM01 [21]. In terms of the conventional linearly-polarized (LP) mode designation of guided modes in optical fibers, the 0th – order vector modes can also be represented as linear combination of LP11 modes which are four-fold degenerate – two in parity and two in polarization [21]. When the fiber is excited by off-axis skew rays of circularly-polarized (σ = ± 1) Gaussian beam, the resulting circularly-polarized (CP) modes excited in the fiber and hence the output beam from the fiber can be represented as a linear combination of the two orthogonally polarized LP11 modes with same parity as given by Eqs. (1)(4) [22] and Fig. 1(a) below:

et(CP11e+)et(LP11ex)+iet(LP11ey)
et(CP11o+)et(LP11ox)iet(LP11oy)
et(CP11e)et(LP11ex)iet(LP11ey)
et(CP11o)et(LP11ox)+iet(LP11oy)
In a circular-core step-index fiber, the inhomogeneity in the core-clad boundary lifts the degeneracy in the propagation constants (β) of the waveguide modes with the addition of polarization corrections βi = β + Δβi (i = 1 – 4) for the four 0th – order vector modes [20]. The propagation constants for the TE01 and TM01 are different (β2 ≠ β4) and the propagation constants for HE21 e and HE21 o modes are same (β1 = β3). This leads to the possibility that the four CP11 modes of the step-index fiber (Eqs. (1)(4)) can form two stable vortex (CV) modes due to the liner combination of HE21 e and HE21 o modes as (HEe 21 ± iHEo 21) and two unstable (within the fiber) vortex (IV) modes due to the linear combination of the TE01 and TM01 modes as (TE01 ± iTM01) as given by Eqs. (5)(8) and Fig. 1(b):
CV+1+CP11e++iCP11o+
CV1CP11eiCP11o
IV1+CP11e+iCP11o+
IV+1CP11e+iCP11o
In terms of the polarization circularity (σ) and the charge (l) of the vortex beams generated at the fiber output, the above mentioned CP modes (Eqs. (5)(8)) can be written as (σ, l) = ( + 1, + 1); (−1, −1); ( + 1, −1) and (−1, + 1).

 

Fig. 1 Linear combination of modes inside the TMF corresponding to (a) Eqs. (1)(4) and (b) Eqs. (5)(8).

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Considering the fiber output as a coherent superposition of the fundamental Gaussian beam HE11σ1and the vortex beam IVσ2σ2transverse electric field distribution at the fiber output (in free space) is written as

Et=(x^+iσ1y^)aexp(R2)+(x^+iσ2y^)Rexp(R2iσ2ϕ)
where, ‘a’ is relative amplitude, ‘φ’ is the helical phase of the vortex beam, ‘R’ is the radial distance and x^ and y^ are the unit vectors. In our experiment we use left circularly polarized (σ = −1) Gaussian beam (HE11σ1) launched at an angle into the TMF resulting in the generation of an output beam which is a coherent superposition of the input Gaussian beam (HE11σ1) and the circularly-polarized vortex beam (IV1σ2). The transverse electric field of the resulting vortex beam at the fiber output (in free space) given in Eq. (9) can be simplified as
Et=(x^iy^)aexp(R2)+(x^+iy^)Rexp(R2iϕ+iδ)
where ‘δ’ is the phase shift of the vortex beam with respect to the Gaussian beam at the fiber output. Introducing an analyzer in the output beam and rotating it by an angle θ=Ωt where ‘t’ is time and ‘Ω’ is the angular frequency of the analyzer rotation results in the rotation in the position of the Zero Intensity Point (ZIP), - position of the vortex core – obtained by applying the condition Re Et = Im Et = 0 in Eq. (9) resulting in
a+Rexp{i(σ2[θϕ]σ1θ)}=0
Equating the real and imaginary parts in (11) to zero, the ZIP coordinates are
R=a;ϕ=π/σ2+(1σ1σ2)θ
Equation (12) indicates that the ZIP rotates even in the non-rotating coordinate system associated with a fixed screen placed after a rotating analyzer. At the stationary screen the ZIP rotates with an angular frequency
ΔΩ=ϕt=(1σ1σ2)Ω
From Eq. (13), for same circular polarization ofσ1=σ2, the modes corresponding to ZIP do not rotate at the stationary screen whereas for the case of orthogonal circular polarizations of σ1=σ2 the angular frequency of the ZIP is ΔΩ=2Ω at the stationary screen.

3. Experimental details

Schematic of the experimental setup used for the demonstration of RDE due to polarization singularities (PS) in the output of a TMF is as shown in Fig. 2(a) . Partially-polarized Gaussian beam from a He-Ne laser (632.8 nm) passes through a Glan-Thompson polarizer (P) to get linearly polarized light. The linearly polarized light then passes through a quarter wave plate (QWP) oriented at −45° to convert to left circularly polarized (LCP) light. First, the LCP light beam is focused using a 0.40 NA, 20x microscopic objective lens (L1) onto the cleaved end of the circular core step-index TMF. The calculated dimensionless V-number of the TMF is 3.805 implying that the fiber supports a total of six waveguide modes as mentioned earlier [21]. The output from the fiber is collimated using another microscope objective lens (L2) and the output images are collected using a CCD camera connected through IEEE 1394 card to the computer for data acquisition and analysis.

 

Fig. 2 (a) Schematic of the experimental setup. P – polarizer, QWP – quarter wave-plate, BS – beam splitter, L1 and L2 –objective lenses, TMF – two-mode fiber, A – analyzer, CCD – detector, HWP – half wave-plate, DP – Dove prism; (b) output image and (c) interference pattern.

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For the demonstration of RDE due to polarization singularities reported in this paper, we use ~41 cm long TMF held on 3-axis stage (to give tilt and off-axis positioning) and horizontally without bending (to avoid unwanted mode mixing). The fiber launch system is capable of a small offset (r) and tilt (θ) to the fiber input with respect to fiber axis. The tilt angle for the beam used in the experiments reported here is measured to be ~27 μrad from the back reflection of the fiber input through the beam splitter (BS). For tilted off-axis position of the fiber with respect to the input Gaussian beam we launch skew-rays into the fiber which selectively excite elliptically-polarized vortex modes which when coherently superposed with the fundamental Gaussian mode results in an off-centred vortex beam at the fiber output (Fig. 2(b)). The presence of vortex singularity is confirmed by the appearance of forklet in the two-beam interference pattern (Fig. 2(c)) which, in addition to chromascopic technique can be used to diagnose and study phase singularities and field structures in the generated beams [23,24]. The polarization characteristics of the output beam from the TMF for fixed input launch conditions is carried out via Stokes parameter measurements [25]. The polarization singularities (S-line and C-point) in the output beam are also carried out using two-beam interferometric methods outlined in Ref. [26]. The presence of rotational Doppler-effect in the output beam and its characteristics are carried out using a rotating analyzer ‘A’ or a rotating Dove prism (DP) or a rotating DP – analyzer combination or a rotating half-wave plate (HWP) – DP – analyzer combination to investigate a variety of features due to polarization singularity in the output beam generated via selective excitation of vector-vortex field in a two-mode optical fiber.

4. Results and discussion

4.1. Measurement of Stokes parameters of vector-vortex beams

The polarization singularities in the output beam from the TMF for a fixed input condition are determined via point-wise Stokes parameters measurement directly from the CCD images measured [25,27]. From these images the polarization singularity positions are determined from the zeros of the three Stokes parameters: S-line by S3 = 0 and C-point by the intersection of the surfaces S1 = 0 and S2 = 0 [28]. The Stokes parameters are measured experimentally from the six intensity measurements: I(0; 0), I(π/2; 0), I(π/4; 0) and I(3π/4; 0) measured by orienting the polarizer in the output beam at 0°,90°,45° and 135° angles and I(π/4; π/2) and I(3π/4; π/2) are measured by introducing a QWP before the polarizer, with its fast axis oriented at 90°, and recording intensities at polarizer angles 45° and 135° respectively. From the six intensity images measured using the CCD, the Stokes parameters are obtained for each point using the intensity value at the same coordinate from the images. This is done using a MATLAB code written for the purpose, which loads the grayscale bitmap images from the CCD as matrices and processes them. After the point-wise Stokes parameters are calculated they are displayed and point-wise calculations of the polarization parameters are performed as in the case of normal Stokes parameters. Using this method, the four Stokes parameters are obtained for each pixel in the image. This enables the construction of the polarization ellipse individually at each point in the cross section of the image. The important point is that the images used in the Stokes parameter calculation are to be captured with the CCD in the same position, and care must be taken that the output QWP does not cause any shift in the beam when introduced in its path. Small spill over in the S3 image in Fig. 3 is due to this misalignment of optical components.

 

Fig. 3 Stokes parameters of the output beam from the TMF for input QWP oriented at −45°.

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Figure 3 shows the Stokes parameters (S0 – S3) of the output beam from the TMF for the input QWP orientated at −45° launching left-circular polarized light. Here, S0 is the total field intensity, S1, S2 gives the linear components and S3 the ellipticity of the output beam. Also, S3 is positive for right-handed elliptical polarization and negative for left-handed elliptical polarization. The region where S3 = 0 the polarization is linear and the zero line corresponds to the S-line. The region where S1 = S2 = 0, the angle of the polarization ellipse major axis is undefined, giving the condition for circular polarization or the C-point. We also confirmed these polarization singularities measurements by carrying out the interferometric measurements as discussed below, which also helps us to demonstrate the RDE in these beams.

4.2 Interferometric identification of polarization singularities

With the QWP oriented at −45° the left-circularly polarized Gaussian beam coupled into the TMF results in nested elliptically polarized vortex output – region of left-handed elliptical polarization surrounded by the region of right-handed polarization as shown in Fig. 3. These two beam polarizations are respectively due to the on-axis and off-axis waves guided through the short length of the TMF. The two-beam interferometer constructed (Fig. 2(a)) in parallel allows us to characterize the phase properties of the output beam from the TMF. As mentioned before, the appearance of a forklet pattern centered at the zero-intensity point (ZIP) of the output beam clearly indicates the presence of helical phase structure in the output beam (Fig. 2(b)). Now, rotating the analyzer A, kept after the L2 lens in the output beam, in clockwise (CW) direction results in the ZIP rotation within the beam in the same direction as the analyzer rotation without appreciable change in the beam intensity. This clearly indicates that the output beam is a superposition of the left-circularly polarized fundamental mode and the right-hand elliptically polarized vortex mode due to the TE01 + iTM01 fiber mode combination excited simultaneously inside the fiber (Eq. (7)).

From the CCD images of the output beam intensity measured for every 5° rotation of analyzer a movie made is shown in Fig. 4(a) . From the images we calculate the X and Y coordinates of the geometric centre and that of the ZIP denoted respectively as Xc, Yc, Xmin and Ymin using MATLAB program. From these measurements we calculate the ZIP rotation with respect to the beam centre. The X-Y plot of the ZIP rotation as a function of the analyzer rotation is shown in Fig. 4(b). It is seen from the figure that the ZIP trajectory for a rotating analyzer with respect to stationary screen makes two (4π) rotations (cycles of maxima and minima) for one (2π) rotation of the analyzer implying that the angular frequency (ΔΩ) of the ZIP is twice the angular frequency of the analyzer (Ω) i.e., ΔΩ = 2Ω. It is important to note that this trajectory of the ZIP exactly corresponds to the S-line trajectory measured using the Stokes parameters in Sect. 4.1. To identify the C-point in the output beam, we introduced a QWP in the output beam (as shown in Fig. 2(a)) at fixed QWP angles of 8° and 33°. The beam passes through the rotating analyzer ‘A’ gives us the azimuth curves a1 and a2 respectively as shown in Fig. 4(b). The intersection of the two curves within the S-contour is the C-point. The boundary between the fundamental and the vortex modes in the output beam corresponds to the S-line and within the S-contour is the C-point.

 

Fig. 4 (a) (Media 1) Rotating image at the TMF output as a function of the analyzer rotation (b) Polarization singularities – S-lines and C-point – in the output from the TMF excited by left-circularly polarized Gaussian beam.

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4.3 Vector-vortex beam through rotating Dove prism - half-wave plate

It has been reported that for linearly polarized Gaussian light beam the Dove prism behaves as a polarizer with poor extinction ratio and that this ratio decreases with the increasing prism base angle [29]. The deviation in the linear polarization state to slightly elliptical state of the output Gaussian beam depends on the angle of rotation of the DP [30]. Instead of linearly polarized Gaussian we used elliptically polarized vortex beams containing PS through a rotating DP and report the corrections of the field in the vicinity of the PS. Our measurements are carried out using a DP whose base angle is 45°. For this base angle we first measure the polarization leakage of the DP by passing linearly and circularly polarized Gaussian input beams through the DP as baseline. For the linearly polarized Gaussian input beam, for the DP placed between parallel polarisers the intensity transmitted dropped upto 30% of its peak value implying that the output beam becomes significantly elliptically polarized. For circularly polarized Gaussian input beam by keeping the DP at every 22.5° and rotating the analyzer we measure the deviation in the output polarization from input circular polarization with respect to the DP orientation angle. We observe that the deviation from the circular polarization of the input beam is a maximum for 22.5° and a minimum for 157.5° DP orientations. Further experiments are in progress to understand the behaviour of the DP for circularly polarized input beams. Independent of the polarization of the output beam from the TMF, passing the vortex beam with shifted core through an image-rotating DP inverts the vortex location by 180°. For such a beam, one complete rotation of the DP in the CW direction with respect to the beam axis results in the CCW rotation of the ZIP at twice the angular rotation frequency due to the image rotation property of the DP [31]. The X-Y plot of the ZIP rotation (Fig. 5 ) shows a symmetric circular trajectory for the rotation around the beam center at twice the rotation frequency of the DP.

 

Fig. 5 Rotation of ZIP due to rotating DP (black) and analyzer rotation for different fixed orientation of DP (colours).

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To study the effect of DP on the polarization of the vortex beam passing through it, we fix the DP at every 22.5° and rotate the analyzer kept after the DP in the CW direction. Figure 5 also shows the X-Y plot of the ZIP rotating in the CW direction at twice the rotation frequency for every 45° orientation of the DP as a function of the analyzer rotation. In our case, upon passing the elliptically polarized, shifted-core vortex beam through the DP we see a clear departure from the circularity of the ZIP rotation for different orientations of the DP. For the DP oriented at 22.5°, we see that the ZIP trajectory is more elliptical and is less elliptical for DP orientation of 157.5°. These significant departures from circularity of the ZIP trajectory for different DP orientations clearly indicate the polarization leakage from the DP [29,30] and confirm our earlier measurement using circularly polarized Gaussian beams. Also, for all orientations of the DP, the ZIP rotates at twice the angular frequency of the analyzer rotation in the same direction as the analyzer rotation direction. As mentioned earlier, the ZIP trajectory coincides with the S-contour of the polarization singularity in the output beam for the rotation of the DP and the analyzer. The trajectory of the S-contour is measured by keeping the DP at every 45° and rotating the analyzer CW. For one complete rotation of the analyzer, the S-contour rotates twice in the same clockwise direction for all the positions of the Dove prism and these trajectories going around the periphery of the DP rotation trajectory as shown in Fig. 5. The elliptical shape in the S-contour trajectories is due to the polarization leakage from the DP [29,30].

Instead of adjusting ZIP position in the beam by passing it through the DP we also manipulate the polarization character and hence the PS of the output beam by passing it through a rotating HWP. The combination of half-wave plate and Dove prism introduced in the output beam given by Eq. (10) rotates both the polarization and the image at twice the angular frequency of analyzer rotation. One complete CW rotation of these two elements together rotates the ZIP and the PS twice in the CCW direction as shown by continuous line central circle in Fig. 6 . As our output beam from the TMF is a combination of both left and right circular polarized light, introduction of the rotating HWP changes the phase difference between field components and hence the polarization of the beam passing through it by equal and opposite amount to both the circular components and hence the resulting behaviour of ΔΩ = 2Ω of the ZIP. Next, we introduced a rotating analyzer after the HWP-DP combination positioned at different fixed angles to study the behaviour of the polarization singularities (S-lines and C-points) in the output beam. The trajectory of the S-line for every 45° orientation of the HWP-DP combination also rotates at twice the angular frequency of the analyzer rotation but now rotating in CCW direction. For a complete rotation of analyzer these trajectories go along the periphery of the circle measured for the HWP-DP ZIP trajectory as shown in Fig. 6. The change in the direction of rotation of the ZIP trajectory as compared to that in Fig. 5 is due to the introduction of the HWP which changes the rotation direction of the circular polarization of the output beam. Also, due to the introduction of additional phase to the output beam polarization by the HWP, the starting point of the ZIP due to polarization singularity in the output beam changes its starting position. From the above sets of measurements, it is clear that with the DP, the S-line rotates in the CW direction whereas with the HWP-DP combination, the S-line rotates in the CCW direction due to the rotation of circular polarization by the HWP.

 

Fig. 6 Rotation of ZIP due to rotation of HWP-DP combination and rotation of analyzer for different fixed orientation of the HWP-DP combination.

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5. Conclusions

By constructing a simple one-beam interferometer within a horizontally held TMF we achieve coherent superposition of the fundamental mode with CP11 vortex modes. Using such a mode combination we demonstrated RDE which appears for rotating beams with nonzero and different OAM: for a 2π rotation of the analyzer with respect to the stationary screen the ZIP trajectory rotates twice (4π) for left circularly polarized input light. The S-line of the polarization singularity identified in the output beam from the TMF rotates around the C-point in the same manner thereby establishing the link between the ZIP and the PS in the TMF output. Similar vector singularity effects can also be anticipated in partially coherent orthogonally polarized beam combinations [32,33].

Acknowledgements

The authors acknowledge the Department of Science and Technology (DST), India for financial support for the work and Y.V. Jayasurya (KVPY student) for developing Stokes parameter measurement methodology. V.V.G.K.I acknowledges CAS, UoH for research fellowship.

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31. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

32. Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008). [CrossRef]  

33. A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009). [CrossRef]  

References

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  1. S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
    [CrossRef] [PubMed]
  2. M. Padgett, “Electromagnetism: like a speeding watch,” Nature 443(7114), 924–925 (2006).
    [CrossRef] [PubMed]
  3. B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half wave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
    [CrossRef]
  4. B. A. Garetz, “Angular Doppler effect,” J. Opt. Soc. Am. 71(5), 609–611 (1981).
    [CrossRef]
  5. I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78(13), 2539–2542 (1997).
    [CrossRef]
  6. L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
    [CrossRef]
  7. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
    [CrossRef]
  8. J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
    [PubMed]
  9. I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
    [CrossRef]
  10. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
    [CrossRef] [PubMed]
  11. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
    [CrossRef] [PubMed]
  12. C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
    [CrossRef]
  13. C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
    [CrossRef]
  14. T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).
  15. A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
    [CrossRef]
  16. N. K. Viswanathan and V. V. G. K. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
    [CrossRef] [PubMed]
  17. V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
    [CrossRef]
  18. J. F. Nye, “Natural focusing and fine structure of light: Caustics and wave dislocations,” Institute of Physics Publishing, Bristol (1999).
  19. For a complete list of published references in the field of singular optics refer to http://www.cis.rit.edu/~grovers/SO/so.html
  20. N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
    [CrossRef]
  21. A.W. Snyder and Love, Optical waveguide theory, (Chapman and Hall, 1983)
  22. A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: Optical vortices,” Opt. Spectrosc. 85, 272–280 (1998).
  23. O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
    [CrossRef] [PubMed]
  24. O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
    [CrossRef] [PubMed]
  25. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and polarized light, North-Holland, 1977.
  26. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
    [CrossRef] [PubMed]
  27. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
    [CrossRef]
  28. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
    [CrossRef] [PubMed]
  29. M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).
  30. I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
    [CrossRef]
  31. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).
  32. Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
    [CrossRef]
  33. A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
    [CrossRef]

2010

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[CrossRef]

N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
[CrossRef]

2009

N. K. Viswanathan and V. V. G. K. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
[CrossRef] [PubMed]

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

2008

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

2007

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

2006

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

M. Padgett, “Electromagnetism: like a speeding watch,” Nature 443(7114), 924–925 (2006).
[CrossRef] [PubMed]

2005

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
[CrossRef]

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

2004

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
[CrossRef]

2003

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
[CrossRef]

2002

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

1999

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

1998

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: Optical vortices,” Opt. Spectrosc. 85, 272–280 (1998).

1997

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78(13), 2539–2542 (1997).
[CrossRef]

1994

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
[CrossRef]

1981

B. A. Garetz, “Angular Doppler effect,” J. Opt. Soc. Am. 71(5), 609–611 (1981).
[CrossRef]

1979

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half wave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[CrossRef]

Alexeyev, A. N.

T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
[CrossRef]

Alexeyev, C. N.

T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
[CrossRef]

Allen, L.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

Arlt, J.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Arnold, S.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half wave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[CrossRef]

Babiker, M.

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
[CrossRef]

Barreiro, S.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

Bekshaev, A. Y.

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

Bialynicka-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78(13), 2539–2542 (1997).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78(13), 2539–2542 (1997).
[CrossRef]

Bogatyryova, G. V.

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

Bogatyryova, H. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

Chernyshov, A. A.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Courtial, J.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

Dennis, M. R.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

Dholakia, K.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: Optical vortices,” Opt. Spectrosc. 85, 272–280 (1998).

Fadeyeva, T. A.

T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

Failache, H.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

Felde, Ch. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

Flossmann, F.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Garetz, B. A.

B. A. Garetz, “Angular Doppler effect,” J. Opt. Soc. Am. 71(5), 609–611 (1981).
[CrossRef]

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half wave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[CrossRef]

Hanson, S. G.

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

Inavalli, V. V. G. K.

N. K. Viswanathan and V. V. G. K. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
[CrossRef] [PubMed]

Jayasurya, Y. V.

N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
[CrossRef]

Krishna Inavalli, V. V. G.

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[CrossRef]

Lesso, J. P.

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

Lezama, A.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

Macdonald, M.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

Maksimyak, A. P.

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

Maksimyak, P. P.

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

Mokhun, A. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

Moreno, I.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
[CrossRef]

Padgett, M.

M. Padgett, “Electromagnetism: like a speeding watch,” Nature 443(7114), 924–925 (2006).
[CrossRef] [PubMed]

Padgett, M. J.

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

Paez, G.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
[CrossRef]

Paterson, L.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Petrov, D. V.

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

Philip, G. M.

N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
[CrossRef]

Polyanskii, P. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

Power, W. L.

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
[CrossRef]

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

Sibbett, W.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Slyusar, V. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Soskin, M. S.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

Strojnik, M.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
[CrossRef]

Tabosa, J. W. R.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

Torner, L.

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

Torres, J. P.

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

Ushenko, Y. A.

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

Viswanathan, N. K.

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[CrossRef]

N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
[CrossRef]

N. K. Viswanathan and V. V. G. K. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
[CrossRef] [PubMed]

Volke-Sepulveda, K.

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Volyar, A. V.

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: Optical vortices,” Opt. Spectrosc. 85, 272–280 (1998).

Yavorsky, M. A.

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
[CrossRef]

Am. J. Phys.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75(2), 163–168 (2007).
[CrossRef]

Appl. Opt.

O. V. Angelsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and Y. A. Ushenko, “Role of caustics in the formation of networks of amplitude zeros for partially developed speckle fields,” Appl. Opt. 43(31), 5744–5753 (2004).
[CrossRef] [PubMed]

O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Feasibilities of interferometric and chromascopic techniques in study of phase singularities,” Appl. Opt. 44(24), 5091–5100 (2005).
[CrossRef] [PubMed]

J. Mod. Opt.

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

J. Opt. A, Pure Appl. Opt.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11(9), 094010 (2009).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Optical vortices in rotating weakly guiding ideal optical fibers,” J. Opt. A, Pure Appl. Opt. 6(8), 762–768 (2004).
[CrossRef]

C. N. Alexeyev, A. N. Alexeyev, and M. A. Yavorsky, “Effect of one-axis anisotropy on the propagation of optical vortices in rotating optical fibers,” J. Opt. A, Pure Appl. Opt. 7(1), 63–72 (2005).
[CrossRef]

J. Opt. Soc. Am.

B. A. Garetz, “Angular Doppler effect,” J. Opt. Soc. Am. 71(5), 609–611 (1981).
[CrossRef]

JETP Lett.

I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76(8), 486–489 (2002).
[CrossRef]

Ch. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88(7), 418–422 (2008).
[CrossRef]

Nature

M. Padgett, “Electromagnetism: like a speeding watch,” Nature 443(7114), 924–925 (2006).
[CrossRef] [PubMed]

Opt. Commun.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half wave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[CrossRef]

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112(3-4), 141–144 (1994).
[CrossRef]

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[CrossRef]

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4-6), 257–268 (2003).
[CrossRef]

Opt. Express

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[CrossRef] [PubMed]

J. Arlt, M. Macdonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, “Moving interference patterns created using the angular Doppler-effect,” Opt. Express 10(16), 844–852 (2002).
[PubMed]

Opt. Lett.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28(14), 1185–1187 (2003).
[CrossRef] [PubMed]

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003).
[CrossRef] [PubMed]

N. K. Viswanathan and V. V. G. K. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
[CrossRef] [PubMed]

Opt. Spectrosc.

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: Optical vortices,” Opt. Spectrosc. 85, 272–280 (1998).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light Beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[CrossRef]

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78(13), 2539–2542 (1997).
[CrossRef]

Proc. SPIE

T. A. Fadeyeva, A. N. Alexeyev, and C. N. Alexeyev, “Rotational Doppler effect in weakly guiding optical fibres,’,” Proc. SPIE 6254, 625401 (2006).

N. K. Viswanathan, G. M. Philip, and Y. V. Jayasurya, “Rotational frequency shift in cylindrical vector beam due to skew rays in few-mode optical fibers,” Proc. SPIE 7613, 761307, 761307-9 (2010).
[CrossRef]

Ukr. J. Phys.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Rotation of arbitrary optical image and the rotational Doppler effect,” Ukr. J. Phys. 49, 490–495 (2004).

A. N. Alexeyev, C. N. Alexeyev, T. A. Fadeyeva, and A. V. Volyar, “Analysis of singularity properties of the radiation field in low-mode optical fibers,” Ukr. J. Phys. 7(1), 11–17 (2006).
[CrossRef]

Other

J. F. Nye, “Natural focusing and fine structure of light: Caustics and wave dislocations,” Institute of Physics Publishing, Bristol (1999).

For a complete list of published references in the field of singular optics refer to http://www.cis.rit.edu/~grovers/SO/so.html

A.W. Snyder and Love, Optical waveguide theory, (Chapman and Hall, 1983)

R. M. A. Azzam, and N. M. Bashara, Ellipsometry and polarized light, North-Holland, 1977.

Supplementary Material (1)

» Media 1: AVI (86 KB)     

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Figures (6)

Fig. 1
Fig. 1

Linear combination of modes inside the TMF corresponding to (a) Eqs. (1)(4) and (b) Eqs. (5)(8).

Fig. 2
Fig. 2

(a) Schematic of the experimental setup. P – polarizer, QWP – quarter wave-plate, BS – beam splitter, L1 and L2 –objective lenses, TMF – two-mode fiber, A – analyzer, CCD – detector, HWP – half wave-plate, DP – Dove prism; (b) output image and (c) interference pattern.

Fig. 3
Fig. 3

Stokes parameters of the output beam from the TMF for input QWP oriented at −45°.

Fig. 4
Fig. 4

(a) (Media 1) Rotating image at the TMF output as a function of the analyzer rotation (b) Polarization singularities – S-lines and C-point – in the output from the TMF excited by left-circularly polarized Gaussian beam.

Fig. 5
Fig. 5

Rotation of ZIP due to rotating DP (black) and analyzer rotation for different fixed orientation of DP (colours).

Fig. 6
Fig. 6

Rotation of ZIP due to rotation of HWP-DP combination and rotation of analyzer for different fixed orientation of the HWP-DP combination.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

e t ( C P 11 e + ) e t ( L P 11 e x ) + i e t ( L P 11 e y )
e t ( C P 11 o + ) e t ( L P 11 o x ) i e t ( L P 11 o y )
e t ( C P 11 e ) e t ( L P 11 e x ) i e t ( L P 11 e y )
e t ( C P 11 o ) e t ( L P 11 o x ) + i e t ( L P 11 o y )
C V + 1 + C P 11 e + + i C P 11 o +
C V 1 C P 11 e i C P 11 o
I V 1 + C P 11 e + i C P 11 o +
I V + 1 C P 11 e + i C P 11 o
E t = ( x ^ + i σ 1 y ^ ) a exp ( R 2 ) + ( x ^ + i σ 2 y ^ ) Re x p ( R 2 i σ 2 ϕ )
E t = ( x ^ i y ^ ) a exp ( R 2 ) + ( x ^ + i y ^ ) Re x p ( R 2 i ϕ + i δ )
a + Re x p { i ( σ 2 [ θ ϕ ] σ 1 θ ) } = 0
R = a ; ϕ = π / σ 2 + ( 1 σ 1 σ 2 ) θ
Δ Ω = ϕ t = ( 1 σ 1 σ 2 ) Ω

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