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Four-wave mixing with hermite-gauss modes in rubidium vapor

Pedro Henrique Castro Magalhães [UNESP] Ferreira

Published Jul 27, 2023• 0 citations
OpenAlex

Abstract

In this work, we present an experimental study on the spatial profile of coherent light generated in a forward four-wave mixing (FWM) process. We are interested in investigating how the transverse structure of the incident beams, more specifically of the Hermite-Gauss (HG) modes, is transferred to the generated signal in this nonlinear process. For the experiment, we used a continuous wave diode laser to excite a sample of Rubidium atoms, on the ⁸⁵Rb D2 line, tuned to the frequency of the 5²S_(1/2) ( = 3) → 5²P_(3/2) transition. With a Spatial Light Modulator, we were able to split the laser beam in two and structure them in the desired modes. These two beams, A and B, with wave vectors kA and kB, and linear orthogonal polarizations, with quasi-copropagated configuration, were used to induce two FWM signals in the directions 2kA-kB and 2kB-kA. The spatial profile of incident beams and the two nonlinear signals were captured by a CMOS camera, and the regimes of a thin or extensive medium were investigated. We were able to obtain results for a combination of gaussian-HG beams, as well as HG-HG beams. The profiles obtained by computer simulation were compared with the experimental measurements. Our theoretical model relies on solving the paraxial equation of the generated field. It leads to the overlap integrals of the incident beams, which unfold into some interesting selection rules in the transversal plane, for the thin and extensive medium regimes, and in the longitudinal axis, only for the long medium regime. Although the extensive medium regime presents some additional selection rules, the experimental requirements are difficult to achieve in our system, among them, we highlight the Gouy phase matching. On the other hand, a good agreement between theory and experiment was observed in the thin medium regime.

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Four-wave mixing driven by structured light in atomic media : optical mode transfer and spatial correlations

M. R. L. da Motta

In this work we present a series of results on four-wave mixing (FWM) processes induced by structured light in atomic media. We start with a theoretical study focusing on a cascade FWM process in rubidium vapor, generating a blue light field at 420 nm. In this setting, we show that in the extended-medium regime the Hermite-Gaussian basis presents a unique property of optical structure transfer. We then exploit this result to show that, by carefully tailoring the structure of the pump beams, one can obtain desired spatial modes with cylindrical and elliptical symmetries with high fidelity at the blue light output. Following that, we consider an experimental configuration where two degenerate FWM signals are generated in different directions of space by the same pump fields. In this setup, we present results in heated rubidium vapor evidencing the simultaneous conservation of orbital angular momentum (OAM) in the two FWM processes. We show that the two-channel setting allows to encode the OAM content of the input beams onto the OAM carried by the two FWM signals. Next, we explore the transverse mode dynamics in the context of optical modes contained in the so-called orbital angular momentum Poincaré sphere (PS). Defined in analogy with a general polarization state, this family of modes is parametrized in terms of polar and azimuthal angles. We show that the two FWM signals can also be described as belonging to Poincaré spheres, and that the angles on the output PS are related to those on the input PS by well-defined symmetries. The predicted FWM intensity profiles, as well as the consequences of the symmetry properties, are in good agreement with our experimental results. We also explore interesting scenarios, such as combinations of different PS modes and the fulfillment of one of the symmetry relations in each sphere independently, and restrictions that arise in the extended-medium regime. In the last part, we study the correlations between the light fields participating in the two-channel FWM configuration. Following our recent work on cold atoms, we investigate the correlations originating from the conversion of phase-noise to amplitude-noise as a result of the light-atom interaction, and discuss our attempts to discriminate transverse spatial dependencies on these correlations. Finally, we outline the quantum theory of FWM, exploring the multi-spatial-mode nature of the generated light state and the associated spatial correlations.

•Feb 5, 2024•0 citations

Controlled conversion of transverse symmetries in a four-wave mixing process

J. C. A. Rocha, D. G. Pires, M. R. L. da Motta +4 more

Abstract Precise light manipulation in optical media has been important for developing new communication protocols, particle tracking and control, environmental sensing and quantum computing, to cite a few applications. Here, we explore the nonlinear light-matter interactions of Hermite-Gaussian (HG) beams with a rubidium vapor. We analytically and numerically predict the possibility of generating a nearly pure optical HG mode as a result of the four-wave mixing of two different HG modes. In contrast, four-wave mixing of other linear solutions of the paraxial wave equation in a third-order nonlinear medium results in the up-converted beam consisting of a superposition of multiple modes. In addition, we identify a large family of solutions of the paraxial wave equation, including cylindrical and elliptical modes, that can be generated in the process of the four-wave mixing by precisely adjusting the weights of the input HG beams. This study may find applications in the development of new classical and quantum optical sources, with more spatial versatility.

Journal of Optics•Feb 22, 2022•3 citations

Forward four-wave mixing experiments in hot and cold atomic systems

Alexandre Mendes de [UNESP] Almeida Junior

In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.

•Feb 28, 2023•0 citations

Analysis of the transverse spatial image of beams transmitted and generated through a non-linear process in rubidium vapor

Patrick Alyson Gouveia TORRES

In this work we describe an experimental investigation on the spatial intensity distribution of the two transmitted beams and the two beams generated in a four-wave mixing (FWM) process, using rubidium vapor as a nonlinear medium. The two FWM signals are detected in the 2⃗kF − ⃗kP and 2⃗kP − ⃗kF directions, where ⃗kF and ⃗kP are the incident beam directions. A diode laser, responsible by the two Ąelds that driven the FWM process, is tuned to the 5S1⇑2(F ≙ 3) Ð→ 5P3⇑2 of the 85Rb. The images of the four beams are obtained using a CMOS camera for a Ąxed atomic density and a Ąxed frequency of the input beams. From each image we can determine the spatial distribution of the intensity of the respective beam. We analyzed the spatial intensity distribution of the signals, as well as the spatial intensity Ćuctuation. The spatial intensity distribution of the transmitted beams shows a decrease of the beam waist for intensities close to the saturation intensity, while for high intensities the spatial proĄle is almost constant. This behavior indicates an autofocusing effect, related to the variation of the refractive index of the medium. The spatial distribution of the intensity Ćuctuation is obtained considering an average surface for each image, and is used to calculate the autocorrelation of the transmitted beams and the two generated FWM beams. Working with the intensity Ćuctuation, we can deĄne a transverse spatial correlation length, which is used to compare the behavior of the four beams with that of a free beam. In this sense, the autocorrelations of intensity Ćuctuations of the transmitted beams show a decrease in the correlation length in the same region of intensities where we observe the self-focusing effect, and as we increase the intensity, the correlation length approaches the value obtained for the free beam. On the other hand, the correlation length for the autocorrelations of the signals generated by FWM increases as the intensity of the incident beams increases.

•Jan 17, 2023•0 citations

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M. R. L. da Motta

We present a theoretical analysis of the spatial shape (transverse and longitudinal) of two degenerate four-wave mixing (FWM) signals induced by fields Ea and Eb, with nonuniform transverse profiles, and wave-vectors ka and kb. The two signals are generated in a cold sample of two-level atoms, in the forward directions 2ka − kb and 2kb − ka. In all cases, the transverse shapes of the generated signals (both near- and far-field) are dictated by the projection of the nonlinear coherence onto the Laguerre-Gaussian (LG) function space. Our calculations show that the longitudinal profile of each generated beam is affected by the spectral characteristics of the medium, described by the third order nonlinear susceptibility. For Gaussian incident fields with equal detunings from resonance, the FWM beam shapes are completely symmetric, with an asymmetry induced by different detunings of incident fields. By considering incident beams with LG distributions, the topological charge conservation determines the orbital angular momentum of each signal and the longitudinal profiles show similar behaviors to the case with Gaussian beams. The phase discontinuities of LG modes reveal a rotation and distortion of the phase distribution of the FWM beams caused by the detunings. We discuss for selected configurations, the relations between both generated beams and their propagation properties. We also present preliminary experimental results of the power spectra of the two FWM signals generated in our magneto-optical trap of rubidium atoms.

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Efficient transfer of spatial intensity and phase information of arbitrary modes via four-wave mixing in an atomic vapor

Onkar N. Verma, Ravi Kamal Pandey, Raja Ram Yadav +1 more

We propose a scheme to transfer the spatial intensity as well as phase information encoded initially in the spatial profile of a weak probe field onto a newly generated Stokes field in a nonlinear process of four-wave mixing (FWM). The FWM process is explored in a gaseous medium consisting of atoms modeled as a three-level system in the $\mathrm{\ensuremath{\Lambda}}$ configuration. We found that different orders of Hermite-Gaussian and Laguerre-Gaussian modes carried by a probe beam are effectively transferred to a Stokes beam. Interestingly, the transferred intensity of the Stokes beam is a clone of the probe beam, whereas the phase profile is conjugate to the probe. The phase distribution of the transmitted modes at the exit of the medium is explored by superimposing them on a copropagating plane wave. Moreover, the parametric amplification due to the nonlinear process of FWM prevents any information loss due to linear absorption and permits us to work at high optical depths. We found a structural similarity between transferred images of about $99%$, which provides clear evidence for the successful transfer of information in the FWM process.

Physical Review A•Nov 22, 2022•12 citations
In this work, we present an experimental study on the spatial profile of coherent light generated in a forward four-wave mixing (FWM) process. We are interested in investigating how the transverse structure of the incident beams, more specifically of the Hermite-Gauss (HG) modes, is transferred to the generated signal in this nonlinear process. For the experiment, we used a continuous wave diode laser to excite a sample of Rubidium atoms, on the ⁸⁵Rb D2 line, tuned to the frequency of the 5²S_(1/2) ( = 3) → 5²P_(3/2) transition. With a Spatial Light Modulator, we were able to split the laser beam in two and structure them in the desired modes. These two beams, A and B, with wave vectors kA and kB, and linear orthogonal polarizations, with quasi-copropagated configuration, were used to induce two FWM signals in the directions 2kA-kB and 2kB-kA. The spatial profile of incident beams and the two nonlinear signals were captured by a CMOS camera, and the regimes of a thin or extensive medium were investigated. We were able to obtain results for a combination of gaussian-HG beams, as well as HG-HG beams. The profiles obtained by computer simulation were compared with the experimental measurements. Our theoretical model relies on solving the paraxial equation of the generated field. It leads to the overlap integrals of the incident beams, which unfold into some interesting selection rules in the transversal plane, for the thin and extensive medium regimes, and in the longitudinal axis, only for the long medium regime. Although the extensive medium regime presents some additional selection rules, the experimental requirements are difficult to achieve in our system, among them, we highlight the Gouy phase matching. On the other hand, a good agreement between theory and experiment was observed in the thin medium regime.
In this work we present a series of results on four-wave mixing (FWM) processes induced by structured light in atomic media. We start with a theoretical study focusing on a cascade FWM process in rubidium vapor, generating a blue light field at 420 nm. In this setting, we show that in the extended-medium regime the Hermite-Gaussian basis presents a unique property of optical structure transfer. We then exploit this result to show that, by carefully tailoring the structure of the pump beams, one can obtain desired spatial modes with cylindrical and elliptical symmetries with high fidelity at the blue light output. Following that, we consider an experimental configuration where two degenerate FWM signals are generated in different directions of space by the same pump fields. In this setup, we present results in heated rubidium vapor evidencing the simultaneous conservation of orbital angular momentum (OAM) in the two FWM processes. We show that the two-channel setting allows to encode the OAM content of the input beams onto the OAM carried by the two FWM signals. Next, we explore the transverse mode dynamics in the context of optical modes contained in the so-called orbital angular momentum Poincaré sphere (PS). Defined in analogy with a general polarization state, this family of modes is parametrized in terms of polar and azimuthal angles. We show that the two FWM signals can also be described as belonging to Poincaré spheres, and that the angles on the output PS are related to those on the input PS by well-defined symmetries. The predicted FWM intensity profiles, as well as the consequences of the symmetry properties, are in good agreement with our experimental results. We also explore interesting scenarios, such as combinations of different PS modes and the fulfillment of one of the symmetry relations in each sphere independently, and restrictions that arise in the extended-medium regime. In the last part, we study the correlations between the light fields participating in the two-channel FWM configuration. Following our recent work on cold atoms, we investigate the correlations originating from the conversion of phase-noise to amplitude-noise as a result of the light-atom interaction, and discuss our attempts to discriminate transverse spatial dependencies on these correlations. Finally, we outline the quantum theory of FWM, exploring the multi-spatial-mode nature of the generated light state and the associated spatial correlations.
Abstract Precise light manipulation in optical media has been important for developing new communication protocols, particle tracking and control, environmental sensing and quantum computing, to cite a few applications. Here, we explore the nonlinear light-matter interactions of Hermite-Gaussian (HG) beams with a rubidium vapor. We analytically and numerically predict the possibility of generating a nearly pure optical HG mode as a result of the four-wave mixing of two different HG modes. In contrast, four-wave mixing of other linear solutions of the paraxial wave equation in a third-order nonlinear medium results in the up-converted beam consisting of a superposition of multiple modes. In addition, we identify a large family of solutions of the paraxial wave equation, including cylindrical and elliptical modes, that can be generated in the process of the four-wave mixing by precisely adjusting the weights of the input HG beams. This study may find applications in the development of new classical and quantum optical sources, with more spatial versatility.
In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.
In this work we describe an experimental investigation on the spatial intensity distribution of the two transmitted beams and the two beams generated in a four-wave mixing (FWM) process, using rubidium vapor as a nonlinear medium. The two FWM signals are detected in the 2⃗kF − ⃗kP and 2⃗kP − ⃗kF directions, where ⃗kF and ⃗kP are the incident beam directions. A diode laser, responsible by the two Ąelds that driven the FWM process, is tuned to the 5S1⇑2(F ≙ 3) Ð→ 5P3⇑2 of the 85Rb. The images of the four beams are obtained using a CMOS camera for a Ąxed atomic density and a Ąxed frequency of the input beams. From each image we can determine the spatial distribution of the intensity of the respective beam. We analyzed the spatial intensity distribution of the signals, as well as the spatial intensity Ćuctuation. The spatial intensity distribution of the transmitted beams shows a decrease of the beam waist for intensities close to the saturation intensity, while for high intensities the spatial proĄle is almost constant. This behavior indicates an autofocusing effect, related to the variation of the refractive index of the medium. The spatial distribution of the intensity Ćuctuation is obtained considering an average surface for each image, and is used to calculate the autocorrelation of the transmitted beams and the two generated FWM beams. Working with the intensity Ćuctuation, we can deĄne a transverse spatial correlation length, which is used to compare the behavior of the four beams with that of a free beam. In this sense, the autocorrelations of intensity Ćuctuations of the transmitted beams show a decrease in the correlation length in the same region of intensities where we observe the self-focusing effect, and as we increase the intensity, the correlation length approaches the value obtained for the free beam. On the other hand, the correlation length for the autocorrelations of the signals generated by FWM increases as the intensity of the incident beams increases.
We present a theoretical analysis of the spatial shape (transverse and longitudinal) of two degenerate four-wave mixing (FWM) signals induced by fields Ea and Eb, with nonuniform transverse profiles, and wave-vectors ka and kb. The two signals are generated in a cold sample of two-level atoms, in the forward directions 2ka − kb and 2kb − ka. In all cases, the transverse shapes of the generated signals (both near- and far-field) are dictated by the projection of the nonlinear coherence onto the Laguerre-Gaussian (LG) function space. Our calculations show that the longitudinal profile of each generated beam is affected by the spectral characteristics of the medium, described by the third order nonlinear susceptibility. For Gaussian incident fields with equal detunings from resonance, the FWM beam shapes are completely symmetric, with an asymmetry induced by different detunings of incident fields. By considering incident beams with LG distributions, the topological charge conservation determines the orbital angular momentum of each signal and the longitudinal profiles show similar behaviors to the case with Gaussian beams. The phase discontinuities of LG modes reveal a rotation and distortion of the phase distribution of the FWM beams caused by the detunings. We discuss for selected configurations, the relations between both generated beams and their propagation properties. We also present preliminary experimental results of the power spectra of the two FWM signals generated in our magneto-optical trap of rubidium atoms.
We propose a scheme to transfer the spatial intensity as well as phase information encoded initially in the spatial profile of a weak probe field onto a newly generated Stokes field in a nonlinear process of four-wave mixing (FWM). The FWM process is explored in a gaseous medium consisting of atoms modeled as a three-level system in the $\mathrm{\ensuremath{\Lambda}}$ configuration. We found that different orders of Hermite-Gaussian and Laguerre-Gaussian modes carried by a probe beam are effectively transferred to a Stokes beam. Interestingly, the transferred intensity of the Stokes beam is a clone of the probe beam, whereas the phase profile is conjugate to the probe. The phase distribution of the transmitted modes at the exit of the medium is explored by superimposing them on a copropagating plane wave. Moreover, the parametric amplification due to the nonlinear process of FWM prevents any information loss due to linear absorption and permits us to work at high optical depths. We found a structural similarity between transferred images of about 9999%99, which provides clear evidence for the successful transfer of information in the FWM process.