A new achromatic method to generate the optical vortex was proposed and supercontinuum optical vortex generation ranging ~500 to ~800 nm was experimentally demonstrated without spatial nor topological-charge dispersions. In addition, polarization evolution in our system using Jones vectors and matrices was discussed and the condition of the polarizer to transfer polarizations was elucidated. This method is useful for the application to time-resolved nonlinear spectroscopy utilizing ultrabroadband optical vortex pulses in topological materials such as ring-shaped crystals or annular materials.
©2009 Optical Society of America
A Laguerre-Gaussian (LG) mode is one of the modes of paraxial solutions to wave equation and has a helical wavefront. The light beam with helical wavefront is called an optical vortex. The optical vortex or LG mode has unique properties. The beam has a phase singularity on its center, which shows a dark part on its center of intensity profile, and carries an orbital angular momentum of light well defined by the topological charge . These characteristics recently attracted much attention because of their increasing applications in many fields, such as the optical trapping [2, 3, 4, 5] (especially, trapping for Bose-Einstein condensates [6, 7, 8]), microstructure rotation in laser tweezers and spanners [9, 10, 11], super-resolution microscopy [12, 13, 14], quantum information using multidimensional entangled states [15, 16, 17], geometrical (or Berry’s) phase observation  and nonlinear spatial-vortex propagation [19, 20]. However, they have not been fully utilized so far for optical spectroscopy, in particular, ultrafast spectroscopy for topological materials such as a ring-shaped crystal or annular-shaped materials [21, 22], in which closed-loop coherence or Aharonov-Bohm effect  is to be investigated by them. It has been due to the fact that generating methods of ultrashort optical vortex pulse is not well established from the view point of spatial or topological charge-chirp free techniques. Thus, for making good use of ultrashort optical vortex pulses, it is a key technique for managing spatial-and/or topological charge-chirps as well as a conventional frequency chirp.
There are two typical methods to generate the optical vortex from the Hermite-Gaussian beams. One uses spiral phase plates ; the other employs holograms  generated by spatial phase modulators. These two methods are not suitable for generating ultrashort [26, 27, 28, 29, 30, 31] or ultrabroadband vortex pulses [32, 33, 34]. Usually spiral plates are designed for a certain wavelength , hence, they induce topological-charge dispersion for ultrashort or ultrabroadband pulses, leading mixture of eigenstates of a vortex. An achromatic lens consisted of two materials was demonstrated as a modified spiral plate, the bandwidth was, however, limited to ≫100 nm . Computer generated holograms composing of diffraction and phase singularity patterns, which are often used for separation of fundamental and transferred vortex beam, bring spatial dispersion without beam position coincidence [30, 32, 34]. Although this spatial chirp can be eliminated by 4f [27, 28, 29], 2f -2f  or prism configuration [31, 34], it is comparably complicated or rather bandwidth-limited.
In the present paper, we propose a new achromatic method without spatial- nor topological charge-dispersions for generation of ultrabroadband optical vortex pulses. In addition, we experimentally demonstrate our new achromatic method to generate a supercontinuum optical vortex pulse with an almost octave-spanning bandwidth. Moreover, we discuss polarization evolution in our system using Jones vectors and matrices.
We explain the principle of our new achromatic method in this section. Figure 1 shows a conceptual scheme of the experiment by our system. It consists of a polarizer (P), an achromatic quarter-wave plate (AQWP1), an axially symmetric polarizer (ASP), another achromatic quarter-wave plate2 (AQWP2), and an analyzer (A). It converts the incident Gaussian beam into the optical vortex or the LG beam with a certain topological charge, which is controllable with the design of the axially symmetric polarizer (ASP). In our case, ASP is designed for generation of an optical vortex with a topological charge ℓ=±2, having a bandwidth of ~500–~850 nm. The AQWPs’ bandwidths range from ~400 to ~800 nm. Our proposal is similar to ref.  in terms of utilizing polarization singularity for generation of an optical vortex. However, it should be emphasized that not a radial half-wave plate for far-infrared radiation (10.6μm)  but ASP and AQWP’s are employed here, enabling us to obtain a supercontinuum optical vortex pulse with an almost octave-spanning bandwidth in the visible and near-infrared region.
In our proposed system, the polarization transformation is a key technique. Here, we explain the polarization evolution of the beam in our system, using Jones vectors and matrices . An incident optical beam with a spatial Gaussian profile is set to be linearly-polarized with the Jones vector of , by the horizontal linear polarizer P whose Jones matrix is expressed
by . It is then converted into left (counterclockwise) circularly-polarized beam giving the vector , by AQWP1 whose matrix is . This beam is entered into the axially symmetric polarizer ASP. It transmits only the radially-polarized component and its matrix is yielded by , where ϕ is the azimuthal angle in the beam cross section. This polarizer is made of a photonic crystal and purchased from Photonic Lattice Inc. After passing through ASP, the beam is turned to be radially polarized with the vector . The polarization is locally linear. However, the polarization directions depends on the azimuthal coordinates ϕ. It should be noted that not only their amplitude but their phase depends on ϕ in the form of exp(iϕ). It is essential for generating an optical vortex. The obtained radially polarized beam is guided into the second achromatic quarter-wave plate AQWP2, whose fast axis is perpendicular to that of AQWP1. The Jones matrix of AQWP2 is described by . The polarizations of the passed beam depend on the azimuthal angle ϕ, giving the Jones vector . Finally, only the vertical component is extracted by the vertical linear analyzer A with the matrix of . Consequently, the Jones vector of the beam comes to be . Its polarization is linearly vertical. However, unlike usual uniform linear polarization, its phase depends on ϕ as indicated by the factor exp(i2ϕ), expressing an ℓ=2 optical vortex. Thus we obtain the optical vortex beam. For ℓ=-2 optical vortex generation with another sign, AQWP’s should be interchanged. In addition, other designs of ASP supply other ℓ-value vortices.
The polarizations represented by Jones vectors at points after passing through the components are summarized in Table 1. It should be emphasized that Jones matrices of components used in our system are independent of wavelength λ as well as the radial coordinate ρ in the beam cross section, resulting in capability of spatial and topological-charge free optical vortex generation even with an ultrabroadband pulse.
3. Experimental setup and results
In this section, we describe experimental demonstration of ultrabroadband optical vortex pulse generation without spatial nor topological-charge dispersions. Figure 2 shows our experimental setup. The light source that we used was a Ti:sapphire laser regenerative amplifier with a center wavelength of 795 nm and a repetition rate of 1 kHz. The pulse from the regenerative amplifier was focused into a 3.5 mm-thick sapphire crystal by a plano-convex lens L1 with a focal length of 150mm for supercontinuum or white light continuum generation. The generated white light pulse, spectrally ranging from ~450 to ~900 nm with a full-width at one-thousandth maximum as depicted in Fig. 3, was roughly collimated by a plano-convex lens L2 with a focal length of 70 mm. The supercontinuum pulse with a Gaussain spatial profile was guided into our achromatic vortex generation system.
After passing through the achromatic vortex generation system, a supercontinuum vortex pulse was obtained. It was spectrally filtered by bandpass filters, and its beam profile was monitored at the point Q by a charge-coupled device (CCD) or directed into the Mach-Zehnder-type interference system to investigate topological charge. By changing bandpass filters, the supercontinuum vortex pulse was spectrally resolved.
First, we investigate position of the dark part of the spectral-resolved beams to examine the spatial dispersion. Beam intensity profiles and line profiles at the point Q are shown in Fig. 4 for the spectrally-resolved vortex pulses with (a) a center wavelength λ0=800 nm and a bandwidth Δλ=11 nm, (b) λ0=680 nm and Δλ=11 nm, and (c) λ0=500 nm and Δλ=65 nm from a generated supercontinuum. Line profiles show horizontal and vertical intensities along the lines including the beam center. They all have dark spots on the center, reflecting a character of optical vortices. Figure 4(d) shows the superposition of their beam intensity profiles. Their dark spots well coincide at the center position with one another, giving no spatial dispersion, at least in the wavelength range of 500–800 nm.
Second, we examine interference patterns between spectrally-resolved vortex pulses. The vortex pulse propagating along the one arm in our Mach-Zehnder-type interferometer is reflected three times, while the vortex pulse is reflected six times along the other arm (three-reflection retroreflector RR was used for the delay line). Hence, interference patterns between vortex pulses with the same |ℓ| value but different signs were observed, since a reflection changes the sign of topological charge. Observed interference patterns for (a) λ0=800 nm and Δλ=11 nm, (b) λ0=680 nm and Δλ=11 nm, and (c) λ0=500 nm and Δλ=65 nm, which are spectrally-resolved from a generated supecontinuum, are depicted in Fig. 5(a), (b) and (c), respectively. They all yield clear four-pronged fork patterns, indicating that spectrally-resolved vortex pulses before entering the interferometer definitely possess the same topological charge of ℓ=2 as designed. These results show generation of topological charge-free optical vortex pulse with a broadband width, at least in the wavelength range of 500–800 nm. The conversion efficiency from spatially-Gaussian to vortex with an ultrabroadband was evaluated to be ~18%, which was comparable with the maximum efficiency in principle described in the discussion section. Somewhat of discrepancy between the experimental and theoretical conversion efficiencies is attributed to the imperfectness of polarization transformation of ASP and AQWP’s for ultrabroadband pulses as well as their transmittivities (~90% for ASP and ~99% for AQWP’s). Although not fully investigated, the maximum input fluence of the supercontinuum pulse available to our achromatic vortex generation system is limited to several µJ/cm2 mainly by the damage threshold of ASP.
As mentioned above, our new achromatic method enables us to generate supercontinuum optical vortices without spatial-chirp nor topological-charge dispersions. In this experiment, although active compensation of frequency chirp was not carried out, a Fourier-transform limited pulse can be obtained by using a 4f-system with a spatial phase modulator [37, 38], even for a pulse with an over-octave-spanning bandwidth.
We discuss our system from the view point of circular polarization decomposition. Arbitrary polarization Ẽ is decomposed into a linear combination of left circular polarization and right circular polarization , as
Here the bracket (a,b) represents the inner product as (a,b)=a † b, where a † is the adjoint vector of a (transposed complex conjugate vector of a). For the set of P and AQWP1, its joint Jones matrix F 1 is , and . Thus, F 1 converts arbitrary polarizations to the left circular polarization ẼL with amplitude reduction of factor 1=√2. Similarly, for the set of AQWP2 and A, its Jones matrix F 2 is , where 0 T is the transposed vector of . Since
F 2 extracts the ẼR-component of arbitrary polarization Ẽ, as a horizontal polarization.
Radial polarization Ẽrad just after passing through ASP can be decomposed into
This indicates that Ẽrad is the superposition of constant-amplitude left circular polarization ẼL and right circular polarization ẼR, as shown in Fig. 6. It should be noted here that only the right circular polarization component has azimuthal angle ϕ-dependent phase 2ϕ while the left circular polarization component has the uniform phase. Hence, it enables us to generate a linear polarized optical vortex with topological charge ℓ=2. Regarding the conversion efficiency from the initial linear polarization to final linear polarized optical vortex , the amplitude is reduced by a factor 1/2. Thus, maximum conversion efficiency in intensity in our system is 25 %.
Next, we discuss the necessary and sufficient conditions of the optical component at the position of ASP in our system to generate an ultrabroadband optical vortex pulse without spatial nor topological-charge dispersions. We put the Jones matrix of the component as
and the input constant polarization to the achromatic vortex pulse generation system as . Hence, the output polarization Ẽout from the system is expressed by
where the rotation matrix is used and it is assumed that the optical component is rotated by angle θ. Therefore, necessary and sufficient conditions of the component for generation an ultrabroadband optical vortex pulse without spatial nor topologicalcharge dispersions are:
1) rotation angle θ is nϕ=2(n=±1,±2,±3, ⋯)
2) “P 12+P 21 is nonzero and wavelength-independent” or “P 11-P 22 is nonzero and wavelength-independent”.
The ASP in our system is one of the simplest examples (n=±2 and P 11=1,P 12=P 21=P 22=0). However, it should be noted that, only in even n cases, the component is axially symmetric (n/2-fold axially symmetry).
We proposed a new achromatic method to generate the optical vortex without spatial- nor topological charge-dispersions and demonstrated the dispersion free supercontinuum optical vortex in the simple setup. Our experiment showed that our method generate the supercontinuum optical vortex without spatial nor topological charge dispersion (ℓ=±2) with an almost octave-spanning bandwidth in the wavelength of 500 to 800 nm. In addition, we discuss polarization evolution in our system using Jones vectors and matrices, clarifying the condition of the polarizer to transfer polarizations. Our method is useful and powerful for the application to time-resolved nonlinear spectroscopy employing ultrabroadband optical vortex pulses in topological materials such as ring-shaped crystals and annular-shaped materials, in order to investigate closed loop coherence.
The authors would like to thank S. Tanda and Y. Toda for their useful discussion and encouragement. This work was partially supported by Grant-in-Aid for the 21st Century COE program on “Topological Science and Technology” from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and Grant-in-Aid for Scientific Research (B), 2008-2010, No. 20360025 from Japan Society for the Promotion of Science (JSPS).
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