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Free-induction-decay (FID) magnetometers have evolved as simple magnetic sensors for sensitive detection of unknown magnetic fields. However, these magnetometers suffer from a fundamental problem known as a "dead zone," making them insensitive to certain magnetic field directions. Here, we demonstrate a simple experimental scheme for the dead-zone-free operation of a FID atomic magnetometer. Using a single laser beam containing equal strength of linear- and circular-polarization components and amplitude-modulation at a low-duty cycle, we have synchronously pumped the rubidium-87 atoms with both first- and second-order frequency harmonics. Such a novel pumping scheme has enabled us to observe the free Larmor precession of atomic spins at a frequency of $\Omega_L$ (orientation) and/or 2$\Omega_L$ (alignment) in a single FID signal, depending on the direction of the external magnetic field. We observed that the amplitude of the FID signal does not go to zero for any magnetic field direction, proving the absence of dead zones in the magnetometer. The magnetometer has a sensitivity in the range of 3.5 - 7.8 pT/$\sqrt{Hz}$ in all directions. Such a generic experimental scheme can play a crucial role in developing miniaturized atomic magnetometers for various practical applications, including geomagnetic applications., Comment: 5 pages, 4 figures

The spontaneous parametric down-conversion (SPDC), an inherently random quantum process, produces a non-deterministic photon-pair with strong temporal and spatial correlations owing to both energy and momentum conservation. Therefore, the SPDC-based photon pairs are used for quantum random number generation (QRNG). Typically, temporal correlation in association with an ideal unbiased beam splitter is used for QRNG without fully exploring the spatial correction. As a result, SPDC-based QRNG has a low bit rate. On the other hand, due to the spatial correlation, the photon pairs in non-collinear phase-matched geometry are generated randomly in diametrically opposite points over an annular ring spatial distribution. Therefore, exploring the temporal correlation between photon pairs from different sections of the annual ring can lead to multi-bit QRNG at a high rate, avoiding the need for a beam splitter. As a proof-of-concept, we report on high-bit-rate QRNG by using spatial correlation of photon-pairs by sectioning the SPDC ring of a non-collinear, degenerate, high-brightness source and temporal correlation between the diametrically opposite sections. Dividing the annular ring of the high-brightness photon-pair source based on a 20 mm long, type-0 phase-matched, periodically-poled KTP crystal into four sections, recording the timestamp of the coincidences (widow of 1 ns) between photons from diametrically opposite sections and assigning bits (0 and 1), we extracted 90 million raw bits over 27.7 s at a pump power of 17 mW. We determined the extraction ratio using the minimum entropy evaluation of more than 95% in our case. Using Toeplitz matrix-based post-processing, we achieved a QRNG with a bit-rate of 3 Mbps, passing all NIST 800-22 and TestU01 test suites. The generic scheme shows the possibility of further enhancement of the bit rate through more sectioning of the SPDC ring.

Talbot length, the distance between two consecutive self-image planes along the propagation axis for a periodic diffraction object (grating) illuminated by a plane wave, depends on the period of the object and the wavelength of illumination. This property makes the Talbot effect a straightforward technique for measuring the period of a periodic object (grating) by accurately determining the Talbot length for a given illumination wavelength. However, since the Talbot length scale is proportional to the square of the grating period, traditional Talbot techniques face challenges when dealing with smaller grating periods and minor changes in the grating period. Recently, we demonstrated a Fourier transform technique-based Talbot imaging method that allows for controlled Talbot lengths of a periodic object with a constant period and illumination wavelength. Using this method, we successfully measured periods as small as a few micrometers and detected sub-micrometer changes in the periodic object. Furthermore, by measuring the Talbot length of gratings with varying periods imaged through the combination of a thick lens of short focal length and a thin lens of long focal length and large aperture, we determined the effective focal length of the thick lens in close agreement with the theoretical effective focal length of a thick lens in the presence of spherical aberration. These findings establish the Talbot effect as an effective and simple technique for various sensing applications in optics and photonics through the measurement of any physical parameter influencing the Talbot length of a periodic object.

Hong-Ou-Mandel (HOM) interferometry has emerged as a valuable tool for quantum sensing applications, particularly in measuring physical parameters that influence the relative optical delay between pair photons. Unlike classical techniques, HOM-based quantum sensors offer higher resolution due to their intrinsic dispersion cancellation property. Despite this advantage, achieving precise measurements of optical delay crucial for practical applications often involves time-consuming integration and post-processing with traditional statistical methods. To address this challenge, our recent work focused on optimizing optical delay measurements in a time-efficient manner. By carefully selecting the length of a 1 mm periodically-poled KTP (PPKTP) crystal for pair photon generation, we achieved a remarkable group index measurement precision of $\sim 6.75\times 10^{-6}$ per centimeter of sample length, surpassing the previous maximum precision by over 400$\%$. These current measurements maintain fast detection and high photon counts, which are essential for practical quantum sensing applications. The HOM-based method, while limiting the measurement range, can be extended by compensating for photon delay using an optical delay stage. As a proof-of-principle, we measured the group index variation of PPKTP over a temperature range up to 200$^{\circ}$C with a precision in the range of one part per million ($\sim$10$^{-6}$). This advancement not only contributes to quantum sensing but also holds promising implications for high-precision and long-range measurements in quantum optical coherence tomography.

As a vector version of scalar Bessel beams, Poincar\'{e}-Bessel beams (PBBs) have attracted a great deal of attention due to the presence of polarization singularities and their nondiffraction and self-healing properties. Previous studies of PBBs have been restricted primarily to understanding the disinclination patterns in the spatially variable polarization, and many of the properties of PBBs remain unexplored. Here, we present a theoretical and experimental study of the polarization characteristics of PBBs, investigating a variety of their features. Using a mode transformation of a full Poincar\'{e} (FP) beam in a rectangular basis, ideally carrying 100$\%$ polarization coverage of polarization states represented on the surface of the Poincar\'{e} sphere, we observe the PBB as the superposition of an infinite number of FP beams, as each ring of PBB has polarization coverage >75$\%$. We also observe the resilience of a PBB's degree of polarization to perturbation. The polarization-ellipse orientation map of PBBs shows the presence of infinite series of C-point singularity pairs. The number of such series pairs is decided by the number of C-point singularity pairs of the FP beam. The dynamics of C-point singularity pairs in the self-healing process show a non-trivial creation of new singularities and recombination of existing singularities. Such dynamics provide insight into ``Hilbert Hotel'' style evolution of singularities in light beams. The present study can be useful for imaging in the presence of depolarizing surroundings, studying turbulent atmospheric channels, and exploring the rich mathematical concepts of transfinite numbers., Comment: 9 pages,11 figures

Hong-Ou-Mandel (HOM) interference, the bunching of two indistinguishable photons on a balanced beam-splitter, has emerged as a promising tool for quantum sensing. There is a need for wide spectral-bandwidth photon pairs (for high-resolution sensing) with high brightness (for fast sensing). Here we show the generation of photon-pairs with flexible spectral-bandwidth even using single-frequency, continuous-wave diode laser enabling high-precision, real-time sensing. Using 1-mm-long periodically-poled KTP crystal, we produced degenerate, photon-pairs with spectral-bandwidth of 163.42$\pm$1.68 nm resulting in a HOM-dip width of 4.01$\pm$0.04 $\mu$m to measure a displacement of 60 nm, and sufficiently high brightness to enable the measurement of vibrations with amplitude of $205\pm0.75$ nm and frequency of 8 Hz. Fisher-information and maximum likelihood estimation enables optical delay measurements as small as 4.97 nm with precision (Cram\'er-Rao bound) and accuracy of 0.89 and 0.54 nm, respectively, therefore showing HOM sensing capability for real-time, precision-augmented, in-field quantum sensing applications.

Historically, infinity was long considered a vague concept - boundless, endless, larger than the largest - without any quantifiable mathematical foundation. This view changed in the 1800s through the pioneering work of Georg Cantor showing that infinite sets follow their own seemingly paradoxical mathematical rules. In 1924, David Hilbert highlighted the strangeness of infinity through a thought experiment now referred to as the Hilbert Hotel paradox, or simply Hilbert's Hotel. The paradox describes an "fully" occupied imaginary hotel having infinite number of single-occupancy rooms, the manager can always find a room for new guest by simply shifting current guests to the next highest room, leaving first room vacant. The investigation of wavefield singularities has uncovered the existence of a direct optical analogy to Hilbert's thought experiment. Since then, efforts have been made to investigate the properties of Hilbert's Hotel by controlling the dynamics of phase singularities in``fractional'' order optical vortex beams. Here, we have taken such proposals to the next level and experimentally demonstrated Hilbert's Hotel using both phase and polarization singularities of optical fields. Using a multi-ramped spiral-phase-plate and a supercontinuum source, we generated and controlled fractional order vortex beams for the practical implementation of Hilbert's Hotel in scalar and vector vortex beams. Using a multi-ramped spiral-phase-plate, we show the possibility for complicated transitions of the generalized Hilbert's Hotel. The generic experimental scheme illustrates the usefulness of structured beams in visualizing unusual mathematical concepts and also for fractional vector beams driven fundamental and applied research., Comment: 9 pages, 5 figures

Full Poincar\'e (FP) beams, a special class of fully correlated beams generated through the coaxial superposition of a Laguerre-Gauss and fundamental Gaussian modes of orthogonal polarizations, contain all possible polarization states on the surface of the Poincar\'e sphere in a single beam. While the presence of all unconventional polarization states makes the FP beams useful for various applications, the dependence of the refractive index on the polarization restricts the efficient generation of FP beams across the electromagnetic spectrum through nonlinear frequency processes. To avoid such difficulty, we use two contiguous BIBO crystals with orthogonal optic axes and generate an ultrafast FP beam at 405 nm with average power as high as 18.3 mW at a single-pass conversion efficiency of 2.19%. Using Stokes parameters and Stokes phases, we observed the doubling of C-points and L-lines singularities and the orbital angular momentum in the SHG process. We have also devised a new technique to estimate the polarization coverage of the pump and SHG FP beams and observed the variation in polarization coverage with the intensity weightage of the constituting beams. Interestingly, the SHG beam has the highest polarization coverage for the FP beam of equal intensity weightage of the superposed beams. We validated our experimental results in close agreement with the theoretical results.

We report on an imaging-inspired measurement of orbital angular momentum (OAM) using only a simple tilted lens and an Intensified Charged Coupled Device (ICCD) camera, allowing us to monitor the propagation of OAM structured photons over distance, crucial for free-space quantum communication networks. We demonstrate measurement of OAM orders as high as 14 in a heralded single-photon source (HSPS) and show, for the first time, the imaged self-interference of photons carrying OAM in a modified Mach-Zehnder Interferometer (MZI). The described methods reveal both the charge and order of a photons OAM, and provide a proof of concept for the interference of a single OAM photon with itself. Using these tools, we are able to study the propagation characteristics of OAM photons over distance, important for estimating transport in free-space quantum links. By translating these classical tools into the quantum domain, we offer a robust and direct approach for the complete characterization of a twisted single-photon source, an important building block of a quantum network.

We report on a simple and compact experimental scheme to generate high power, ultrafast, higher order vortex array beams. Simply by using a dielectric microlens array (MLA) and a plano-convex lens we have generated array beams carrying the spatial property of the input beam. Considering the MLA as a 2D sinusoidal phase grating, we have numerically calculated the intensity pattern of the array beams in close agreement with the experimental results. Using Gaussian embedded vortex beams of order as high as l= 6, we have generated vortex array beams with individual vortices of order as high as l= 6. We have also theoretically derived the parameters controlling the intensity pattern, size and the pitch of the array and verified experimentally. The single-pass frequency-doubling of the vortex array at 1064 nm in a 1.2 mm long BiBO crystal produced green vortex array of order, l_sh= 12, twice the order of the pump beam. Using lenses of different focal lengths, we have observed the vortex array of all orders to follow the focusing dependent conversion similar to the Gaussian beam. The maximum power of the green vortex array is measured to be 138 mW at a single-pass efficiency as high as ~3.65%. This generic experimental scheme can be used to generate array beams of desired spatial intensity profile across wide wavelength range by simply changing the spatial profile of the input beam., Comment: 4 pages, 5 figures

We report a novel experimental scheme for single-pass second harmonic generation (SHG) of vector vortex beam in the blue. Using an ultrafast Ti:Sapphire laser of pulse width ~17 fs and a set of spiral phase plates in polarization based Mach-Zehnder interferometer (MZI) we have generated vector vortex beams of order as high as lp = 12 at an average power of 860 mW. Given the space-variant polarization of the vector vortex beam, and the dependence of nonlinear frequency conversion processes on the polarization of the interacting beams, using two contiguous bismuth borate crystals with optic axis orthogonal to each other, we have frequency-doubled the near-IR vector vortex beam into visible vector vortex beams with order as high as lsh=24. The maximum output power of the vector vortex beam of order, lsh =2 is measured be as high as 20.5 mW at a single-pass SHG efficiency of 2.4 %. Controlling the temporal delay in the MZI, we have preserved the vector vortex nature of the beams at both pump and frequency-doubled beams at ultrafast timescales. The measurement on mode purity confirms the generation of high quality vector vortex beams at pump and SHG wavelengths. The generic experimental scheme can be used to generate vector vortex beams across the electromagnetic spectrum.

We report on a simple experimental scheme based on a pair of cylindrical lenses (convex and concave) of same focal length and common optical elements producing high power optical beams in 1D and/or 2D Airy intensity profiles with laser polarization as control parameter. Using an ultrafast Yb-fiber laser at 1064 nm of average power of 5 W in Gaussian spatial profile and pulse-width of ~180 fs, we have generated 1D and 2D Airy beams at an efficiency of 80% and 70%, respectively, and pulse width of ~188 fs and ~190 fs, respectively. We have measured the transverse deflection rate of 1D and 2D beams to be ~5.0 (10^-5) 1/mm and ~2.0 (10^-5) 1/mm, respectively. Simply rotating the polarization state of the 1D cubic phase modulated beam in the experiment we can produce 1D and 2D Airy beams on demand. Using a 5 mm long bismuth borate (BiB3O6) we have also studied frequency-doubling characteristics of both 1D and 2D Airy beams. Like 2D Airy beam, the 1D Airy beam also produce frequency-doubled 1D Airy and an additional 1D spatial cubic structure. Like the Gaussian beams, we have observed the focusing dependent conversion efficiency for both 1D and 2D Airy beams producing green 1D and 2D Airy beams of output powers in excess of 110 mW and 150 mW for 3.4 W and 2.8 W of fundamental power respectively.

We propose and experimentally demonstrate a novel experimental scheme to generate high peak power, segmented, smooth, zero-order Bessel beams with tunable range. Illuminating the axicon with hollow Gaussian beams (HGBs) of different orders we have generated Bessel beams of varying range at different positions away from the axicon. The presence of dark core at the center of the HGBs removes the effect of imperfection in the axicon tip. As a result, the entire power of the input beam is transformed into zero-order Bessel beam without any on-axis intensity modulation. We observe the decrease in range and increase in peak power of the zero-order Bessel beam with the order of HGBs. Controlling the superposition of the HGBs of different orders to the axicon we have demonstrated the increase in the range of the Bessel beam. The current technique can also produce Bessel beams of different intensity distribution including single peak or multiple peak Bessel beams. Using single-pass second harmonic generation in nonlinear crystals of different lengths we have further verified the increase of peak power of the Bessel beam with the order of the HGBs and also the increase in the range of the Bessel beam due to the superposed HGBs. The generic experimental scheme can be used at different wavelengths and timescales (continuous-wave to ultrafast)., Comment: Four pages, Five figures

We report on controlling the bi-photon orbital angular momentum (OAM) eigenmodes in the spontaneous parametric down conversion process by simply adjusting the asymmetry of the pump vortex beam. Adjusting the optic axis of the spiral phase plate (SPP) of phase winding corresponding to OAM mode, l, with respect to the beam propagation axis, we have transformed a Gaussian beam into an asymmetric vortex beam with OAM modes, l, l-1, l-2, 0 with different weightages. Pumping the nonlinear crystal with such asymmetric vortices and controlling their asymmetry we have tailored the spiral spectrum of the biphoton OAM eigenmodes. Calculating the Schmidt number of the biphotons we observe the increase in the spiral bandwidth of the OAM eigenmodes and hence the dimensionality of the system. Although we have restricted our study to show the increase in spiral bandwidth of the biphotons by simply controlling the asymmetry of the pump vortices, we can, in principle, further enhance the dimensionality of the entangled states by manipulating the pump beam size and crystal length.

We report on a novel experimental scheme to generate continuous-wave (cw), high power, and higher-order optical vortices tunable across mid-IR wavelength range. Using cw, two-crystal, singly resonant optical parametric oscillator (T-SRO) and pumping one of the crystals with Gaussian beam and the other crystal with optical vortices of orders, lp = 1 to 6, we have directly transferred the vortices at near-IR to the mid-IR wavelength range. The idler vortices of orders, li = 1 to 6, are tunable across 2276-3576 nm with a maximum output power of 6.8 W at order of, li = 1, for the pump power of 25 W corresponding to a near-IR vortex to mid-IR vortex conversion efficiency as high as 27.2%. Unlike the SROs generating optical vortices restricted to lower orders due to the elevated operation threshold with pump vortex orders, here, the coherent energy coupling between the resonant signals of the crystals of T-SRO facilitates the transfer of pump vortex of any order to the idler wavelength without stringent operation threshold condition. The generic experimental scheme can be used in any wavelength range across the electromagnetic spectrum and in all time scales from cw to ultrafast regime., Comment: 4 pages, 5 figures

We report on generation and control of tunable, broad orbital angular momentum (OAM) spectrum of a vortex beam. Using two spiral phase plates (SPPs), we have converted the Gaussian beam of Yb-fiber femtosecond laser at 1064 nm into optical vortices of orders, l=1-6, with conversion efficiency >95%. By adjusting the transverse shift of the SPPs with respect to the incident Gaussian beam axis, we have transformed the symmetric (intensity distribution) optical vortex of order, l, into an asymmetric vortex beam of broad OAM spectrum of orders, l, l-1, l-2,..,0. While the position of the SPPs determines the weightage of the OAM modes, however, the weightage of the higher OAM modes transfer to lower OAM modes with the shift of SPPs and finally resulting a Gaussian beam (l=0). Using single pass frequency doubling of asymmetric vortices in a 5-mm-long bismuth triborate crystal, we have transferred the pump OAM spectra, l, l-1, l-2, ..., 0, into the broad spectra of higher order OAM modes, 2l, 2l-1, 2l-2, ...,0 at green wavelength, owing to OAM conservation in nonlinear processes. We have also observed that the conversion efficiency of the frequency-doubled vortices of all orders increases with the increase in the asymmetry of the pump vortices. Using pump power of 4.6 W, we have generated symmetric vortices at green wavelength of output powers, 1.2, 0.76, 0.47, 0.4, 0.32, and 0.3 W and orders of ls=2, 4, 6, 8, 10, and 12, respectively., Comment: Four pages, four figures

We report on experimental demonstration of high power, ultrafast, high repetition rate vortex beam source tunable in the near-IR wavelength range. Based on single-pass optical parametric generation of Yb-fiber laser of vortex order lp=1 in a 50 mm long MgO doped periodically poled LiNbO3 crystal, the source produces signal beam in vortex profile of order ls=1 across 1433-1553 nm. Additionally, the source produces broadband idler radiation tunable across 3379-4132 nm in the Gaussian beam profile. We observed that the vortex profile of the pump beam is always transferred to the signal beam due to the highest overlapping integral among the interacting beams and the idler maintains a Gaussian spatial profile owing to conservation of orbital angular momentum in optical parametric processes. For pump power of 4.72 W, the signal and idler beams have maximum power of 1.7 W at 1509 nm and 0.48 W at 3625 nm respectively. The signal vortex beam has output pulses of width 637 fs at a repetition rate of 78 MHz. The signal (idler) has spectral width of 4.3 nm (129.5 nm) and passive peak-to-peak power fluctuation better than 3% (1.1%) over 30 minutes, respectively.

In this paper an explicit cosmological model which allows cosmological singularities are discussed. The generalized power-law and asymptotic expansions of the baro-tropic fluid index $\omega$ and equivalently the deceleration parameter $q$, in terms of cosmic time $'t'$ are considered. Finally, the strength of the found singularities is discussed., Comment: 12 pages, 3 figures, references added and text revised

We report on a compact, simple and robust high brightness entangled photon source at room temperature. Based on a 30 mm long periodically poled potassium titanyl phosphate (PPKTP), the source produces non-collinear, type0 phase matched, degenerate photons at 810 nm with pair production rate as high 39.13 MHz per mW at room temperature. To the best of our knowledge, this is the highest photon pair rate generated using bulk crystals pump with continuous-wave laser. Combined with the inherently stable polarization Sagnac interferometer, the source produces entangled state violating the Bells inequality by nearly 10 standard deviations and a Bell state fidelity of 0.96. The compact footprint, simple and robust experimental design and room temperature operation, make our source ideal for various quantum communication experiments including long distance free space and satellite communications.

The authors considered the bulk viscous fluid in $f(R, T)$ gravity within the framework of Kaluza-Klein space time. The bulk viscous coefficient $(\xi)$ expressed as $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, where $\xi_0$, $\xi_1$ and $\xi_2$ are positive constants. We take $p=(\gamma-1)\rho$, where $0\le\gamma\le2$ as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of $f(R, T)=R+2f(T)$, where $f(T)=\lambda T$, $\lambda$ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient $(\xi)$. The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to $\xi_2\frac{\ddot{a}}{\dot{a}}$, hence the second law of thermodynamics is not valid, however, it is valid with the generalized second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when, the bulk viscous coefficient is proportional to $\xi=\xi_1\frac{\dot{a}}{a}$, $\xi=\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$ and $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, so the second law of thermodynamics and the generalized second law of thermodynamics is satisfied throughout the evolution. We calculate statefinder parameters of the model and observed that, it is different from the $\wedge$CDM model. Finally, some physical and geometrical properties of the models are discussed., Comment: 20 pages, 10 figures. reference added, text clean, abstract modified

Hybrid entangled states, having entanglement between different degrees-of-freedom (DoF) of a particle pair, are of great interest for quantum information science and communication protocols. Among different DoFs, the hybrid entangled states encoded with polarization and orbital angular momentum (OAM) allow the generation of qubit-qudit entangled states, macroscopic entanglement with very high quanta of OAM and improvement in angular resolution in remote sensing. Till date, such hybrid entangled states are generated by using a high-fidelity polarization entangled state and subsequent imprinting of chosen amount of OAM using suitable mode converters such as spatial light modulator in complicated experimental schemes. Given that the entangled sources have feeble number of photons, loss of photons during imprinting of OAM using diffractive optical elements limits the use of such hybrid state for practical applications. Here we report, on a simple experimental scheme to generate hybrid entangled state in polarization and OAM through direct transfer of classical non-separable state of the pump beam in parametric down conversion process. As a proof of principle, using local non-separable pump state of OAM mode l=3, we have produced quantum hybrid entangled state with entanglement witness parameter of W-1.25 violating by 8 standard deviation. The generic scheme can be used to produce hybrid entangled state between two photons differing by any quantum number through proper choice of non-separable state of the pump beam.

In this article, we considered the bulk viscous fluid in the formalism of modified gravity in which the general form of a gravitational action is $f(R, T)$ function, where $R$ is the curvature scalar and $T$ is the trace of the energy momentum tensor within the frame of flat FRW space time. The cosmological model dominated by bulk viscous matter with total bulk viscous coefficient expressed as a linear combination of the velocity and acceleration of the expansion of the universe in such a way that $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, where $\xi_0$, $\xi_1$ and $\xi_2$ are constants. We take $p=(\gamma-1)\rho$, where $0\le\gamma\le2$ as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are obtained by assuming a particular model of the form of $f(R, T)=R+2f(T)$, where $f(T)=\lambda T$, $\lambda$ is constant. We studied the four possible scenarios for different values of $\gamma$, such as $\gamma=0$, $\gamma=\frac{2}{3}$, $\gamma=1$ and $\gamma=\frac{4}{3}$ with the possible positive and negative ranges of $\lambda$ to observe the accelerated expansion history of the universe. Finally, a big-rip singularity is observed., Comment: 19 pages, references added, conclusion section is rewritten and possible future work is given

Hollow Gaussian beams (HGB) are a special class of doughnut shaped beams that do not carry orbital angular momentum (OAM). Such beams have a wide range of applications in many fields including atomic optics, bio-photonics, atmospheric science, and plasma physics. Till date, these beams have been generated using linear optical elements. Here, we show a new way of generating HGBs by three-wave mixing in a nonlinear crystal. Based on nonlinear interaction of photons having OAM and conservation of OAM in nonlinear processes, we experimentally generated ultrafast HGBs of order as high as 6 and power >180 mW at 355 nm. This generic concept can be extended to any wavelength, timescales (continuous-wave and ultrafast) and any orders. We show that the removal of azimuthal phase of vortices does not produce Gaussian beam. We also propose a new and only method to characterize the order of the HGBs.

The perfect vortex is a new class of optical vortex beam having ring radius independent of its topological charge (order). One of the simplest techniques to generate such beams is the Fourier transformation of the Bessel-Gauss beam. The variation in ring radius of such vortices require Fourier lenses of different focal lengths and or complicated imaging setup. Here we report a novel experimental scheme to generate perfect vortex of any ring radius using a convex lens and an axicon. As a proof of principle, using a lens of focal length f=200mm, we have varied the radius of the vortex beam across 0.3-1.18mm simply by adjusting the separation between the lens and axicon. This is also a simple scheme to measure the apex angle of an axicon with ease. Using such vortices we have studied non-collinear interaction of photons having orbital angular momentum (OAM) in spontaneous parametric down-conversion (SPDC) process and observed that the angular spectrum of the SPDC photons are independent of OAM of the pump photons rather depends on spatial profile of the pump beam. In the presence of spatial walk-off effect in nonlinear crystals, the SPDC photons have asymmetric angular spectrum with reducing asymmetry at increasing vortex radius.

We report on efficient nonlinear generation of ultrafast, higher order "perfect" vortices at the green wavelength. Based on Fourier transformation of the higher order Bessel-Gauss beam generated through the combination of spiral phase plate and axicon we have transformed the Gaussian beam of the ultrafast Yb-fiber laser at 1060 nm into perfect vortices of power 4.4 W and order up to 6. Using single-pass second harmonic generation (SHG) of such vortices in 5-mm long chirped MgO-doped, periodically poled congruent LiNbO$_3$ crystal we have generated perfect vortices at green wavelength with output power of 1.2 W and vortex order up to 12 at single-pass conversion efficiency of 27% independent of its order. This is the highest single-pass SHG efficiency of any optical beams other than Gaussian beams. Unlike the disintegration of higher order vortices in birefringent crystals, here, the use of quasi-phase matching process enables generation of high quality vortices even at higher orders. The green perfect vortices of all orders have temporal and spectral width of 507 fs and 1.9 nm, respectively corresponding to a time-bandwidth product of 1.02.

We have considered optical beams with phase singularity and experimentally verified that these beams, although being classical, have properties of two mode entanglement in quantum states. We have observed the violation of Bell's inequality for continuous variables using the Wigner distribution function (WDF) proposed by Chowdhury et al. [Phys. Rev. A \textbf{88}, 013830 (2013)]. Our experiment establishes a new form of Bell's inequality in terms of the WDF which can be used for classical as well as quantum systems., Comment: 7 pages, 9 figures and 1 table

We make a source of entangled photons (SEP) using spontaneous parametric down-conversion (SPDC) in a non-linear crystal and study the spatial distribution of photon pairs obtained through the down-conversion of different modes of light including higher order vortices. We observe that for the Gaussian pump, the thickness of the SPDC ring varies linearly with the radius of pump beam. However, in case of vortex carrying beams, two concentric SPDC rings are formed for beams above a critical radius. The full width at half maximum (FWHM) of SPDC rings increase with increase in the order of optical vortex beams. The presence of a critical beam width for the vortices as well as the observed FWHM of the SPDC rings are supported with our numerical results., Comment: 6 pages, 8 figures