Frequency-resolved measurement of the orbital angular momentum spectrum of femtosecond ultra-broadband optical-vortex pulses based on field reconstruction

We propose a high-precision method for measuring the orbital angular momentum (OAM) spectrum of ultra-broadband optical-vortex (OV) pulses from fork-like interferograms between OV pulses and a reference plane-wave pulse. It is based on spatial reconstruction of the electric fields of the pulses to be measured from the frequency-resolved interference pattern. Our method is demonstrated experimentally by obtaining the OAM spectra for different spectral components of the OV pulses, enabling us to characterize the frequency dispersion of the topological charge of the OAM spectrum by a simple experimental setup. Retrieval is carried out in quasi-real time, allowing us to investigate OAM spectra dynamically. Furthermore, we determine the relative phases (including the sign) of the topological-charge-resolved electric-field amplitudes, which are significant for evaluating OVs or OV pulses with arbitrarily superposed modes.


Introduction
During the last decade, optical vortices (OVs) have been enthusiastically studied because of interest in the spatial phase of an electromagnetic wave changing linearly, over its cross-section, with the azimuthal angle ϕ around the beam center. This phase dependence leads to a phase singularity in the center, where there is zero field [1]. The phase profile is characterized by a factor of ϕ (distribution of topological charge as a function of frequency) and the measurement of topological-charge dispersion for ultra-broadband OV pulses. Our measurements are composed of two principal steps. The first one is based on reconstruction of the field information (phase and amplitude) of the OV pulses to be measured. After selection of a desired wavelength component sliced from the whole spectral region of the OV pulses, it interferes with the correspondent tilted (quasi-) plane wave, forming an interferogram where the field information of the OV pulses for this spectral component is preserved. The field distribution can be reconstructed by Fouriertransforming and filtering a certain part out of the two-dimensional spatial frequency domain of the interferogram and performing an inverse Fourier transform of this part, as referred to as Takedaʼs method [29]. Field reconstruction for observing the OV structure has been previously done from many interference images with different phase offsets between an OV and a plane wave [30]. In contrast, our electric-field reconstruction using the spatial Fourier transform needs only one interference pattern between an OV and a tilted plane wave.
Next, making use of the Fourier-relationship between the azimuthal angle ϕ and topological charge m [31], we are able to obtain the OAM spectrum of the electric-field amplitude as a function of the radial coordinate r as well as m, by projecting the reconstructed field (in spatial coordinates: r and ϕ) onto the topological charge domain.
Here it is worth mentioning that, in particular, this step of our method enables the evaluation of the relative phases (including the positive or negative sign) of topologicalcharge-resolved electric-field amplitudes, which is difficult by the use of previous methods. After squaring the absolute value of this OAM spectrum of electric-field amplitude for each m, the power spectrum of the OAM for this spectral component of the OV pulses can finally be obtained by integration of the squared absolute value with respect to r. By our simple experimental setup, which is designed to control the selected position and bandwidth in the whole spectral region of the input to form the interferogram, the frequency-resolved OAM spectra of the OV pulses in the wavelength range are acquired as a function of topological charge m and wavelength λ. Thus, the topological-charge dispersion can also be obtained for each m from this three-dimensional spectrum (power spectrum as a function of m and λ).

Proposed method
The complex electric field of the ultra-broadband Laguerre-Gaussian OV pulses, with the superposition of modes of topological charges m and radial indices p (LG p m modes), propagating in z-direction, can be expressed by   The constant w 0 is the beam waist. The parameter Φ ( ) z G denotes the Gouy phase, which is known to be an additional phase shift for a focused and propagated beam, differing from that for a plane wave. , although they are functions of ω. Only for purpose of describing the principle of our proposed method, we use, for convenience, a continuous wave (with a frequency of ω) by omitting the integration with respect to ω in equation (1). So the electric field of the OV then becomes p m    (7), which is tilted by angle θ (angle between k 0 and −y-axis). Here , y ϕ = r sin , α 0 is the constant phase, and B is a constant amplitude. Setting where the symbol * denotes the complex conjugate. By two-dimensional Fourier transformation into the spatial frequency domain of k x and k y , three peaks appear at ( = k 0 ). After filtering out only the peak appearing at ( = k 0 x , = k k y y 0 ), we apply the inverse Fourier transformation to this term and obtain the + AC part of the interferogram ϕ Here β 0 is constant, hence multiplying the value of − ( ) k y exp i y 0 , which is only related to tilted angle θ, on both sides of the equation (11), we find, by changing the dummy index m to ′ m , , proportional to the electric field of the OV to be measured, is finally obtained as a function of r and ϕ.
For the next main step, we first compute a complex inner product  Although the mode decomposition concerning indices p can be performed, we here focus only OAM-resolution, that is a decomposition to m modes. Hence, we examine m-resolved electric fields with the superposition of p modes. Simultaneous m-and p-mode decompositions will be discussed elsewhere.
Next, we carry out the integration of ( ) D r m 2 for each m with respect to r in order to obtain the OAM power spectrum S m for the OV expressed by equation (7), that is, Here, the maximum radius r max is selected to assure that the part containing field information is included, and the minimum radius r min is determined by interpolation. For ultra-broadband pulses, S m can be rewritten as ω ( ) I m as a function of topological charge m and angular frequency ω, since this procedure of continuous OV involves obtaining the OAM spectrum corresponding to a certain frequency in the broadband OV pulses described by equations (1) and (2). By repeating the calculation above for various spectral components, the frequency-or wavelength-resolved OAM spectra of OV pulses composed of the OAM spectra at different angular frequencies or wavelengths are acquired.

Experimental setup
In our experiment, a mode-locked Ti:sapphire laser oscillator is used as a light source (center wavelength: ∼800 nm, repetition rate: ∼80 MHz). Femtosecond pulses from the oscillator are focused into a photonic crystal fiber (PCF; core diameter μ 2.3 m, length: ∼30 mm, zerodispersion wavelength: 790 nm) to broaden their spectra (bandwidth: ∼600-∼950 nm). As shown in figure 1, a linearly-polarized ultra-broadband pulse from the PCF is split into two pulses in separate arms by a beam splitter (BS1). One beam passes through a pair of achromatic convex lenses with focal lengths of = f 1 200 and = f 2 100 mm, and its beam size shrinks by a factor of 1/2. Then this beam is converted into ultra-broadband OV pulses, with nominal topological charge of = m 2 and made up of a superposition of radial modes with indices p, by the use of an ultra-broadband OV converter (composed of an axially-symmetric wave plate ASWP and two quarter-wave plates AQWP1 and 2), in a way identical to our previous work [20]. The other beam is magnified by a pair of achromatic convex lenses (focal lengths = f 3 100 and = f 4 200 mm) by a factor of 2, for use as a reference quasi-plane wave in comparison with the shrunk beam above. The two beams are interferometrically recombined by another beam splitter (BS2), which can also control the interference angle (set to be ∼°0.25 in the present experiment) between these two arms.
Instead of using many band-pass filters to obtain interferograms at different wavelength components, which is not practical for ultra-broadband pulses, here we introduce a reflective grating G (groove density = N 235 mm −1 ) to convert different wavelengths to correspondent spatial frequencies. We filter out a certain wavelength component by putting a width-adjusted slit at the focal image plane of a concave mirror (radius of curvature = − R 500 mm) and by rotating the grating G. By this experimental setup, only the desired component in the whole spectrum is reflected back by a slightly-tilted mirror positioned accurately behind the slit to form an interferogram which is detected by a charge-coupled device (CCD) camera. This experimental setup realizes full control of the position and the bandwidth of the wavelength range by allowing the adjustment of the slit width and rotation of the grating, which is essential for frequency-resolved OAM measurements for ultra-broadband OV pulses. Moreover, we incorporate a newly-built software that enables quasi-real-time measurement (acquisition and processing rate is higher than − 2 s 1 even for a high resolution camera with 2560 × 1920 pixels). This enables the inspection of a calculated OAM spectrum dynamically and more reliably in real-time, as shown below.

Results and discussion
Figure 2(a) shows the interferogram centered at 700 nm (bandwidth: ∼10 nm) between a generated nominally-pure (m = 2) ultra-broadband OV pulse and a tilted quasi-plane-wave pulse. The intensity and phase profiles, reconstructed by the procedure described in section 2, are respectively shown in figures 2(b) and (c). We find that both the intensity and phase profiles are clearly reconstructed, the former of which closely resembles the directly measured intensity profile shown in figure 2(f). The retrieved intensity profile was compared with the directly measured intensity profile by a CCD camera, giving a mean-squared error G of ∼ − 10 4 , as listed in table 1. This indicates that our reconstruction is excellent. The mean-squared error (per-pixel error) G is defined [32] by where I ij retr and I ij meas are the retrieved and directly-measured intensity profiles (normalized to have unity peak) at ( ) 1920), respectively. The error is considered to be mainly due to the speckle in the directly measured intensity profile, which exhibits a more sensitive dependence in this profile than in the interferogram. The reconstructed phase profile shows the azimuthal phase dependence clearly with topological charge of = m 2. From the reconstructed field, the OAM spectrum of the electric-field amplitude and displayed in our software in real time. This figure shows that the OV pulses to be measured possess a topological charge of m = 2 with very high contrast. Even though the optical power corresponding to contamination of m is below 1% compared with that of the dominant value of m, it can still be measured by our method. By rotating the grating G, the OAM spectra at other wavelengths were also measured. Figures 3(a) and (e) show typical examples of interferograms of other spectral components (centered at 800 and 900 nm, respectively; bandwidths ∼10 nm). As is often the cases with interference patterns between OVs ( ⩾ m 1) and plane waves, we find that a singularity with = m 2 breaks into two singularities with m = 1. Even though a singularity breaks, electric-field reconstruction is well performed, as indicated in figures 3(b), (c), (f) and (g). It should be noted that the phase profiles reconstructed by our method clearly resolve the splitting of the singularities, as shown in figures 3(c) and (g). The OAM spectra at 800 and 900 nm are obtained as shown in figures 3(d) and (h), respectively, which are evaluated to be similar to that at 700 nm in our case. Whereas they show that the = m 2 modes are dominant with small side modes below 1%, the power ratios of the = m 0 modes to the = m 2 modes at 800 and 900 nm are higher than that at 700 nm. These higher power ratios cause the splitting of the singularities. The mean-squared errors G at 800 and 900 nm were evaluated to be ∼ − 10 3 and ∼ − 10 4 , respectively, as listed in table 1. These results confirm that our generated ultra-broadband OV pulses possess a topological charge of = m 2 with considerably high contrast throughout their whole spectral region.
In order to further assess and evaluate our method, we introduce mixed-type OV pulses as inputs. The method we use here is spatially blocking half of the input OV pulses by a knife edge lying exactly along a diameter of the OV pulses (nominally = m 2), generated as in the previous step. Only the part with the π-range azimuthal angle of the pulse beam passes through and forms the interferogram with a reference quasi-plane wave pulse. The half-blocked OV pulse is no longer an eigenmode of the paraxial equation and the spatial window of azimuthal angle broadens the OAM spectrum of the original OV pulses [31,33].  case, which is reasonable. This is because almost null intensity was recorded in the half-portion of the measured region in the half-blocked case. Figure 4(d) shows the measured OAM power spectrum, together with the theoretically calculated result. The theoretical results take into consideration OAM spectral broadening (equivalently the diffraction effect) by half-blocking.  1, 1, 2, 3 and 5, respectively). To our knowledge, this represents the first determination of the relative phases of OAM-resolved electric fields. This capability is important for characterizing OVs or OV pulses with arbitrarily superposed modes, for example fractional vortices [26,27].
To analyze the frequency dispersion of OAM spectrum or topological-charge dispersion of ultra-broadband OV pulses, which is crucial for the previously-mentioned applications using OV pulses, we performed measurements of the frequency-resolved OAM spectrum ranging from 650 to 900 nm. By continuously rotating the grating G (using a spectrometer in front of the camera to simultaneously monitor the spectral information of an input), a series of normalized OAM spectra for consecutive spectral components were attained from the correspondent input interferograms. Figure 5(a) shows the measured frequency-resolved OAM spectrum in this wavelength range (sampling wavelength-spacing is 50 nm) as a function of topological charge m and wavelength λ of the OV pulses to be measured (also plotted in linear and log scales with interpolation for wavelength, as shown in figures 5(b) and (c), respectively). By slicing this spectrum with a m-constant plane, we are able to obtain the topological-charge dispersion, that is, the topological charge m as a function of wavelength or frequency. From figure 5, while topological-charge dispersion is sufficiently small in this case, our method, in general, enables one to characterize topological-charge dispersion even with such small dispersion. In addition, our results support the fact that our previously-reported methods generating ultrashort or ultra-broadband OV pulses by using axially-symmetric polarizers or waveplates [19,20] are free from topological-charge dispersion.

Conclusion
In conclusion, we have proposed and demonstrated a high-precision interferometric electricfield reconstruction method in the spatial domain, combined with frequency slicing, to obtain the frequency-resolved OAM power spectrum (topological charge distribution) of femtosecond ultra-broadband OV pulses (∼600 − ∼950 nm). Our method is based on the spatial reconstruction of the electric-fields of the pulses to be measured from the frequency-resolved interference patterns. We experimentally applied this method to ultra-broadband OV pulses with nominally-pure topological charge. It is found that the electric fields in the spatial domain were reconstructed well, giving mean-squared errors of below ∼ − 10 3 between the retrieved and directly-measured beam intensity profiles. The obtained OAM spectra gave high precision results, enabling us to even detect below 1% contamination. In our method, the retrieval proceeds in quasi-real time, allowing us to investigate the OAM spectra dynamically. Furthermore, the comparison with experimental and theoretical results (intensity/phase profiles and OAM spectra) for a half-blocked OV pulse confirms the reliability of our method. To our knowledge, this is the first demonstration of the determination of the relative phases of OAMresolved electric fields. This capability of relative-phase determination is important for characterizing OVs or OV pulses with arbitrarily superposed modes. In addition, our method for providing the OAM-and frequency-resolved spectra enables us to access the frequency dispersion of the topological charge of the ultra-broadband OV pulses to be measured. Our measurement method should also prove practically applicable in related fields of research and application, such as ultrafast nonlinear spectroscopy and quantum information processing by ultra-broadband OV pulses.