# Spectral characteristics of a modified inverted-Y system beyond rotating wave approximation

###### Abstract

The spectral properties of a multilevel atomic system interacting with multiple electromagnetic fields, a modified inverted-Y system, have been theoretically investigated. In this study, a numerical matrix propagation method has been employed to study the spectral characteristics beyond the validity regime of the rotating wave approximations. The studied atomic system, comprising of several basic sub-systems i.e. lambda, ladder, vee, N and inverted-Y, is useful to study the interdependence among these basic sub-systems. The key features of the obtained probe spectra as a function of coupling strength and detuning of the associated electromagnetic fields show inter-conversion, splitting and shifting of the transparency and absorption peaks. The dressed and doubly dressed state formalism have been utilized to explain the numerically obtained results. This study has application in design of novel optical devices capable of multi-channel optical communication along with switching.

## I Introduction

Studies on atomic systems interacting with electromagnetic fields has a long and prosperous history. However, in the recent past, multi-level atomic systems have been investigated rigorously either to gain an insight into the fundamental features of the atom-field interactions or to implement the physical behaviors in optical devices. A plethora of cutting edge technologies has already been attributed to the optical phenomena discovered due to the advancement of spectroscopy techniques. The three level atomic systems which facilitated the first observation of quantum interference effects in atomic systems are one of the most basic building blocks for the optical switching devices using Electromagnetically induced transparency (EIT) Harris et al. (1990); Boller et al. (1991) and electromagnetically induced absorption (EIA) Akulshin et al. (1998). Apart from these three level systems, namely Li and Xiao (1995a, b); Mishra et al. (2018), ladder Moseley et al. (1995); Kumar and Singh (2009); Khoa et al. (2016) and vee Zhao et al. (2002); Ying et al. (2014) atomic systems, there are atomic systems consisting higher number of atomic levels which have already been proven to display wonderful physical properties which not only have enriched physics but the application value is enormous too. Two promising four level atomic systems, i.e. inverted Y Gao et al. (2000); Yan et al. (2001a); Qi (2010a); Yan et al. (2012); Ali, Sabir et al. (2016); Yadav and Wasan (2017) and N configuration Goren et al. (2004); Kong et al. (2007); Abi-Salloum et al. (2011); Abd-Elnabi and Osman (2013); Phillips et al. (2013); Ying et al. (2015); Islam et al. (2017); Tuan et al. (2018), have already been earmarked for their application in non-linear spectroscopy for producing large Kerr non-linearities in an optical medium Kou et al. (2010); Yang et al. (2015). These third order non-linear effects relies on large field strengths to obtain considerably enhanced third order optical susceptibility. Historically, increasing the number of atomic levels interacting with increased number of externally applied electromagnetic fields increases the diversity of observed physical phenomena but also the complexity to a many fold both theoretically and experimentally.

In this article, a modified inverted-Y system, hereafter denoted as system, is considered which comprises of all the three basic three-level sub-systems (, ladder and Vee) and two basic four-level sub-systems (inverted-Y and N). This system is investigated with variation in external field parameters like field strength and detuning. The aim of this study is to investigate this atomic system for cold atoms to control the probe absorption in a desired manner for device applications. Here, we have used a numerical matrix propagation method in which a complete density matrix has been propagated through time in order to obtain the transient characteristics as well as the steady state condition for a probe field absorption. The rotating wave approximations (RWA) are routinely employed in the spectroscopy for finding pertinent informations regarding the interaction of atomic systems with electromagnetic fields in steady-state. Though extremely successful, RWA suffers severely when the atom-field interactions are far from the resonance condition and the applied electromagnetic fields are strong enough to be treated perturbatively. The numerical approach utilized in this article, to solve the Liouville equation is described in detail in Nakano and Yamaguchi (1994, 1995) and a complex algebraic form of this method is numerically implemented in this article. The obtained results employing this approach, have also been explained using dressed and doubly dressed state formalism.

The article is organized as follows. In section II, the numerical matrix propagation (NMP) technique is discussed. In section III, the study of transient characteristics of inverted-Y system and equivalence of NMP method with the conventional RWA method is established. In section IV, the probe absorption in system is explored in steady state regime and the obtained results are discussed. The conclusion of work is presented in section V.

## Ii Numerical Matrix Propagation

An in depth description of the numerical solution technique employed in this article has been provided in Nakano and Yamaguchi (1994, 1995). A brief outline of the theoretical background and numerical implementation is provided here for completeness. The generalized electromagnetic radiation composed of M individual field components can be written as,

(1) |

where and represent the field strength and angular frequency for field component. The Hamiltonian of the composite atom-field system can be written as,

(2) |

where is the atomic Hamiltonian described in the atomic basis states as,

(3) |

The electromagnetic interaction Hamiltonian can not be written exactly in the atomic basis states due to the infinite sequence of multipole interactions that can originate from the interaction of the atom with the electromagnetic fields. Under the long wavelength approximation (i.e. dipole approximation), the interaction can be written as,

(4) |

The evolution of the density matrix () for all the atomic levels can be described by the Liouville equation as,

(5) |

The first part in right-hand side of the Liouville equation describes the unitary evolution whereas the second part describes the decay of the composite atom-field system. The second part can be written explicitly in terms of the individual decay rates between different states as,

(6) |

and,

(7) |

The decay rates can also be written as,

(8) |

with the property

(9) |

and

(10) |

where are the feeding parameters and is dephasing factor due to phase changing collisions which is considered zero in this study. To solve Liouville equation 5, density matrix elements are divided into the diagonal and off-diagonal elements denoted by complex variables and .

The evolution of population of state can be obtained by solving following equation,

(11) |

and the equations from which time dependent off-diagonal density matrix elements can be evaluated, are written as,

(12) |

and

(13) |

The resulting time series of the propagated matrix can be utilized to obtain the populations of individual states or the coherence of individual transitions which gives absorption between states and .

## Iii Transient characteristics and Comparison with RWA

In order to study the numerical stability of the utilized algorithm, we characterize the transient response of an inverted-Y (IY) system consisting four levels along with three electromagnetic fields with coupling strengths connecting states and with , as shown in figure 1(a). The population of the ground state and the coherence of the transition of the inverted-Y system as a function of scaled time is shown in figure 2 (a) and (b) respectively. As expected, after initial rapid variations, the steady state is achieved at and without any external perturbation the atomic system continues to be in the steady state.

In order to establish the equivalence of the numerical matrix propagation (NMP) method with the RWA method in the low field strength and small detuning regime, the evolution of the density matrix equations has been studied using both the formalisms in the atomic system described above. The density matrix equations of an inverted-Y system (figure. 1(a)) can be explicitly written in the rotating wave approximations as,

and

(14) |

where is detuning of the applied electromagnetic fields from the corresponding atomic transitions between states and . , and are the decay rates of the excited states , and respectively. The total population is conserved by . The non-radiative relaxation rate of the ground state, i.e. , is zero for this closed system. The steady-state of the above set of equation can be obtained using the matrix method described as,

(15) |

The eigenvector corresponding to the ‘zero’ eigenvalue of the Liouvillian super-operator () correspond to the steady-state value of the density matrix .

In order to perform the comparison, the steady state value of the density matrix is obtained from both the method for two different coupling strength and detuning regime and the results are shown in figure 3. Figure 3 (a) correspond to the low coupling strength and low detuning regime with Rabi frequencies , and with detuning values and . The additional parameters utilized in the numerical matrix propagation method is propagated upto . The equivalence of the RWA (dashed line) and NMP (solid line) methods in low field strength is clearly visible in the graph. The slight shift in the peak of the value can be attributed to the counter rotating terms neglected in the RWA method.

However, in the high field strength and large detuning regime, though the spectral features remains same, the difference between the outcomes of both the methods grow further. In this regime, the field strengths are , and with detuning values and . The time sequence parameters of the NMP method were kept same.

## Iv Results and discussions

In the previous section, the stability of the numerical implementation has been presented along with the comparison with the rotating wave approximations. The established validity of the numerical matrix propagation method enables one to employ the same in field strength and frequency regimes commonly not considered in case of RWA.

In this section, a modified inverted-Y system (figure. 1 (b)) has been studied after incorporating an additional coupling field connecting the ground state to another excited state. This particular choice of an atomic system serves the purpose of integrating all the primitive three level atomic systems namely , ladder and vee along with an option to study the effect of the merger of two of the basic four level atomic system, i.e. the inverted-Y system and N system. The level diagram of the proposed atomic system is shown in figure. 1 (b) where the unperturbed atomic states are denoted as , . The applied electromagnetic fields are described by coupling strength , and detuning , where subscript represents the state and through which electromagnetic field couples. The atomic system consist a sub-system , a ladder sub-system and a vee system . The inverted-Y sub-system is formed by the transition paths and . Finally the N sub-system is formed via the transitions . Assuming the field and detuning for the transition as a probe field, response of all the above atomic sub-systems can be simultaneously probed. In the proceeding section, the coherence of the density matrix element (i.e. ) is analyzed to identify the key spectral features of the system. If not explicitly described, the field values are the following: , , and . The probe beam detuning has been varied in a wide range and the relevant part of the spectral response are depicted in the various plots.

To begin with, the detuning of the coupling field has been varied from to for the inverted-Y system as well as the system and the obtained results are shown in figure 4 . Figure 4 (a) and (b) show the amalgamated spectrum for inverted-Y and system respectively and the individual spectrum corresponding to the white lines drawn in plots (a) and (b) are presented in plots (c) and (d) respectively. For inverted-Y system (figure 1 (a)), as reported in the earlier studies Qi (2010b), two sharp transparencies (i.e EIT) have been obtained at the specific detuning positions where the two photon resonance condition in the system and ladder system are satisfied. For both the resonant coupling fields, two EIT dips merge to give a single sharp EIT at zero probe field detuning. The coupling of ground state to excited state via field of strength in the system (figure 1 (b)) has shown different spectral feature than that of inverted-Y system. At zero detuning of , figure 4 (b)) shows higher at resonant probe (encoded by red color) implying the existence of an absorption peak rather than a transparency as obtained for the case of inverted-Y system.

A qualitative understanding of such change of spectral behavior from inverted-Y to system can be obtained using a semi-classical dressed state formalism as explained in the following discussion. In this formalism, the bare states coupled with strong field are transformed into a new atom-photon basis state which is known as dressed state. In the case of resonant driving fields, the dressed states can be determined by diagonalizing the interaction Hamiltonian. The interaction Hamiltonian of the inverted-Y system, after RWA and dipole approximation, can be written as,

(16) |

As the strength of the probe field is weaker than the strength of all other coupling fields, the probe field can be neglected while evaluating the eigen dressed states corresponding to the Hamiltonian in equation 16. The resulting three eigen-frequencies and their corresponding eigen dressed states are

.

The absorption process in this formalism can be understood by determining the transition probabilities between bare state and upper dressed states, which can be expressed as,

(17) |

where is ground state, and .

The transition between and corresponds to the transition at line center (i.e. ) does not exist due to its zero transition probability (i.e. ). The other transitions (i.e. between and ) exhibit non-zero transition probability. As a consequence, the inverted-Y system exhibits EIT at line center along with the two absorption peaks surrounding the EIT. This is schematically shown in figure 5 (a) and (b).

In the system, the ground state also couples with a state via a strong field which results in conversion of the bare ground state into dressed state. Thus, the dressed states for system include three upper dressed states due to bare states , , as evaluated for the case of inverted-Y system, and two lower dressed states due to coupling of states and through strong field . The lower dressed states can be evaluated by diagonalizing two-level interaction Hamiltonian after RWA and dipole approximation,

and the eigen-energies and corresponding eigen dressed states are given as,

.

.

As the upper dressed state has zero component of bare state , the transition probabilities and are zero, while the rest of the transitions have non-zero transition probabilities. This is depicted in figure 5 (d). The calculated location of each transitions and hence the existence of absorption peaks in frequency space are at -6.0, -1.0, 1.0, 6.0.

The absorption peaks at -1 and 1 merge with each other and result in a broad single peak. Thus, there are total three peaks observable. The non-zero detuning leads to a detuned system where the off-resonant two-photon Raman resonance gives rise to an additional absorption peak near zero probe detuning along with the EIT dip at position where two-photon resonance condition of detuned system is satisfies Hemmer et al. (1989). Consequently, for non-zero , four absorption peaks at resonance and a EIT at off resonant probe detuning exist. Also, since the eigenvalues and dressed states are actually function of detuning and strengths of all coupling fields, their variations can result in a change in location of spectral features and their strength as well.

The obtained spectral features in above study can further be tailored by varying the detuning of another coupling field. For the rest of the studies, the system was made far detuned so that its effect can be separated while studying the effect of other systems. The spectra of the probe absorption as a function of detuning for and are shown in left and right column of figure 6 respectively. Since the electromagnetic field with strength directly couples the ground state and an excited state , its detuning may affect the coherence between the two ground states and . This results in shift in the EIT feature of far-detuned system as shown in the figure 6 (as indicated by an arrow in figure 6 (c)). Similarly, the resonant spectral features in this figure also show the dependence on the field detuning . It can be noted here that, in contrast to resonant case, where four absorption peaks were observed near the probe resonance frequency, the resonance features get modified for non-zero and only show three absorption peaks with sharp transparencies between the peaks. The effect of detuned system can also be noted by comparing figures 6 (a) and (b) or (c) and (d). The detuned system affected the spectrum in terms of strength only. This study summarizes that by varying , the absorption features of probe can be considerably tailored and transparency can be obtained.

In order to investigate the dependence of the detuned EIT and the resonant spectral features further on other parameters of the externally applied electromagnetic fields, the detuning of another coupling field is varied in three different conditions and the obtained results are shown in figure 7. The three different conditions chosen are and , as contrasting spectral features have been obtained for these detuning values in previous study. In all the configurations of the detuning , the detuned EIT shows negligible change and hence is not shown in figure 7, whereas the resonant spectra is modified in all the three conditions. This negligible change of EIT may be because of absence of direct decay channel from excited state to ground state resulting in no change in the coherence between ground states and .

Figure 7 (a), (b) and (c) correspond to values of and . Previously it has been observed that non-zero gives rise to three absorption peaks. Among these three absorption peaks, two of the peaks show variation in its strength with the variation in , while the third peak is independent of . The position of these peaks depend on positive or negative values of . This is clearly visible from figure 7 (a) and the position of these peaks are inter changed in figure 7 (c). Along with this, for the case of far detuned values i.e. (figure 7 (a) and (c)), a single transparency exist at the same probe field frequency for all the values of detuning . For the resonant case i.e. (figure 7 (b)), there exist four absorption peaks in which the central peak shows a shift in its position with the detuning . Another central peak shows the change in its strength as well as position with the variation in detuning (figure 7 (b)). In addition to this, a sharp transparency window begins to appear as the detuning value is increased in either positive or negative side. Thus, the strength of the absorption peaks can be tuned by applying appropriate detuning and to attain a large transparency, should be kept non-zero. These spectral characteristics of the system can be useful for developing optical switching devices.

Subsequent to the studies on effect of detuning, the spectral features of the probe field have also been studied by varying the strength of all the coupling fields. The variation in probe absorption with the variation in coupling strength is shown in figure 8. For lower field strength , the EIT feature (at ) corresponding to the detuned system () is not observable. It begins to appear as the field strength is increased. The resonant spectral feature also shows dependence on the coupling field strength . With increase in value, the small central absorption peak (shown by an arrow in figure) that appear due to the detuned system, merges with the other peaks as shown in figure 8 (b) and (d).

These obtained results from the NMP method (as discussed above) can be explained using the doubly dressed approach Yan et al. (2001b), for the case of where . In this approach, initially, the bare atomic states coupled with stronger coupling field form the primary dressed states. One of these primary dressed states again gets dressed due to its coupling with another bare state. We considered a particular case for (figure 8 (d)). The states and coupled with the strong field of strength form primary dressed states and with energies and respectively, where . The dressed state is again coupled with the bare state through the field which creates doubly dressed states and with energy and respectively, where . The formation of these dressed states are depicted in figure 9. The transition of lower dressed states with three upper dressed states via probe field give rise to six possible transitions. The calculated location of all these possible transitions in frequency domain are -23.4, -18.4, -3.7, 1.3, 1.4 and 6.4. These calculated locations of transition peaks are in agreement with the obtained results through NMP method as shown in figure 8 (d).

The study on the effect of variation in coupling strength on probe absorption characteristics has also been performed and the corresponding spectrum is plotted in figure 10. For a lower value of the coupling strength, i.e. , one large dispersive EIT feature appears at the detuning position fixed by the value. As is increased, this large EIT feature splits into two smaller EIT features. With further increase in , the separation between two EIT peaks increases (figure 10 (a) and (c)). This shows that the strength of the coupling field modifies the coherence created between two ground states and due to its direct coupling with the state to . Using the dressed state approach, this can be attributed to shift in location of the allowed transitions due to increase in the coupling strength . When we look into resonance spectra of the probe absorption, it is observed that these are also considerably modified as the strength of the coupling field is varied. A large transparency at lower coupling strength gets converted into the absorption peak at higher coupling strength . The observed broad transparency, within which this absorption peak appears, could be a result of depletion in the population from the ground state due to strong coupling .

The effect of variation in coupling strength on the probe absorption has been studied for resonant and off resonant detuning values of . The obtained results are shown in figure 11. For the case of resonant condition i.e. , one can obtain a coupling strength dependent disappearance and reappearance of the absorption peaks in the central region. For a weak coupling strength , the upper transition between states and acts as a perturbation and the system can be considered as a perturbed N-system. The black continuous curve in figure 11 (c) shows the corresponding spectrum which is similar to the earlier reported spectrum for the case of N-system Abi-Salloum et al. (2009). From the figure 11 (a), it is clear that as strength becomes comparable to other coupling strengths, this perturbed N-system becomes equivalent to system resulting in four absorption peaks as observed earlier (shown by blue curve in figure 4 (d)).

For greater than the strength of the other coupling fields, the obtained results can be again explained by using the doubly dressed state formalism. The method is same as described earlier. Here, the primary dressed states are created by field coupling the bare states and . The newly formed dressed states have eigen-energies . The one of the resonantly close primary dressed state (one with energy ) gets doubly dressed due to its interaction with the field . The energy of these doubly dressed states are (see figure 12). The location of all allowed transitions between two lower dressed states and three upper dressed states are expected to be at -22.8, -17.8, -6.6, -1.5, 2.0, 7.0, which is consistent with the results obtained using NMP method (figure 11 (c)). In case of a far off resonant detuning condition , the spectral structure of the central region remains almost independent of the coupling strength , while amplitude of absorption depends on .

## V Conclusion

A five-level modified inverted-Y system, i.e. system, comprising of basic three-level sub-systems, i.e. , ladder and vee systems, and basic four-level sub-systems, i.e. N and inverted-Y, has been investigated for probe absorption characteristics using a numerical matrix propagation method. The superiority of this method over the well known RWA method has been established by investigating an inverted-Y system within and beyond the validity regime of RWA method. The presence of a strong coupling field connecting the ground state to another state in the atomic system leads to conversion of resonant probe transparency (obtained in the inverted-Y system) into absorption. The transparency in system is recovered when the aforementioned coupling field is kept off-resonant. Apart from this, the coupling field detuning dependent splitting of the transparency and coupling field strength dependent shifting of the transparency and absorption have also been obtained for this system. The numerically obtained results are also found consistent with the dressed and doubly dressed state formalism. This study shows that the system can be used to design optical devices for switching and multi-channel optical communication.

## Vi Acknowledgments

Charu Mishra is grateful for financial support from RRCAT, Indore under HBNI, Mumbai program.

## References

- Harris et al. (1990) S. E. Harris, J. E. Field, and A. Imamoğlu, Phys. Rev. Lett. 64, 1107 (1990), URL http://link.aps.org/doi/10.1103/PhysRevLett.64.1107.
- Boller et al. (1991) K.-J. Boller, A. Imamoğlu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991), URL http://link.aps.org/doi/10.1103/PhysRevLett.66.2593.
- Akulshin et al. (1998) A. Akulshin, S. Barreiro, and A. Lezama, Physical Review A 57, 2996 (1998).
- Li and Xiao (1995a) Y.-q. Li and M. Xiao, Phys. Rev. A 51, R2703 (1995a), URL http://link.aps.org/doi/10.1103/PhysRevA.51.R2703.
- Li and Xiao (1995b) Y.-q. Li and M. Xiao, Phys. Rev. A 51, 4959 (1995b), URL https://link.aps.org/doi/10.1103/PhysRevA.51.4959.
- Mishra et al. (2018) C. Mishra, A. Chakraborty, A. Srivastava, S. K. Tiwari, S. P. Ram, V. B. Tiwari, and S. R. Mishra, Journal of Modern Optics 65, 2184 (2018), eprint https://doi.org/10.1080/09500340.2018.1502824, URL https://doi.org/10.1080/09500340.2018.1502824.
- Moseley et al. (1995) R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, Optics Communications 119, 61 (1995), ISSN 0030-4018, URL http://www.sciencedirect.com/science/article/pii/003040189500316Z.
- Kumar and Singh (2009) M. A. Kumar and S. Singh, Phys. Rev. A 79, 063821 (2009), URL https://link.aps.org/doi/10.1103/PhysRevA.79.063821.
- Khoa et al. (2016) D. X. Khoa, P. V. Trong, L. V. Doai, and N. H. Bang, Physica Scripta 91, 035401 (2016), URL http://stacks.iop.org/1402-4896/91/i=3/a=035401.
- Zhao et al. (2002) J. Zhao, L. Wang, L. Xiao, Y. Zhao, W. Yin, and S. Jia, Optics Communications 206, 341 (2002), ISSN 0030-4018, URL http://www.sciencedirect.com/science/article/pii/S0030401802014165.
- Ying et al. (2014) K. Ying, Y. Niu, D. Chen, H. Cai, R. Qu, and S. Gong, Journal of Modern Optics 61, 631 (2014), eprint http://dx.doi.org/10.1080/09500340.2014.904019, URL http://dx.doi.org/10.1080/09500340.2014.904019.
- Gao et al. (2000) J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, Phys. Rev. A 61, 023401 (2000), URL https://link.aps.org/doi/10.1103/PhysRevA.61.023401.
- Yan et al. (2001a) M. Yan, E. G. Rickey, and Y. Zhu, Phys. Rev. A 64, 043807 (2001a), URL https://link.aps.org/doi/10.1103/PhysRevA.64.043807.
- Qi (2010a) J. Qi, Physica Scripta 81, 015402 (2010a), URL http://stacks.iop.org/1402-4896/81/i=1/a=015402.
- Yan et al. (2012) D. Yan, Y.-M. Liu, Q.-Q. Bao, C.-B. Fu, and J.-H. Wu, Phys. Rev. A 86, 023828 (2012), URL https://link.aps.org/doi/10.1103/PhysRevA.86.023828.
- Ali, Sabir et al. (2016) Ali, Sabir, Ray, Ayan, and Chakrabarti, Alok, Eur. Phys. J. D 70, 27 (2016), URL https://doi.org/10.1140/epjd/e2015-60533-5.
- Yadav and Wasan (2017) K. Yadav and A. Wasan, Physics Letters A 381, 3246 (2017), ISSN 0375-9601, URL http://www.sciencedirect.com/science/article/pii/S037596011730734X.
- Goren et al. (2004) C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, Phys. Rev. A 69, 053818 (2004), URL https://link.aps.org/doi/10.1103/PhysRevA.69.053818.
- Kong et al. (2007) L. Kong, X. Tu, J. Wang, Y. Zhu, and M. Zhan, Optics Communications 269, 362 (2007), ISSN 0030-4018, URL http://www.sciencedirect.com/science/article/pii/S0030401806008522.
- Abi-Salloum et al. (2011) T. Abi-Salloum, S. Snell, J. Davis, and F. Narducci, Journal of Modern Optics 58, 2008 (2011), eprint https://doi.org/10.1080/09500340.2011.603441, URL https://doi.org/10.1080/09500340.2011.603441.
- Abd-Elnabi and Osman (2013) S. Abd-Elnabi and K. Osman, International Journal of Modern Physics B 27, 1350037 (2013).
- Phillips et al. (2013) N. B. Phillips, I. Novikova, E. E. Mikhailov, D. Budker, and S. Rochester, Journal of Modern Optics 60, 64 (2013), eprint https://doi.org/10.1080/09500340.2012.733433, URL https://doi.org/10.1080/09500340.2012.733433.
- Ying et al. (2015) K. Ying, Y. Niu, D. Chen, H. Cai, R. Qu, and S. Gong, Optics Communications 342, 189 (2015), ISSN 0030-4018, URL http://www.sciencedirect.com/science/article/pii/S0030401814012449.
- Islam et al. (2017) K. Islam, D. Bhattacharyya, A. Ghosh, D. Biswas, and A. Bandyopadhyay, Journal of Physics B: Atomic, Molecular and Optical Physics 50, 215401 (2017), URL http://stacks.iop.org/0953-4075/50/i=21/a=215401.
- Tuan et al. (2018) A. N. Tuan, D. L. Van, and B. N. Huy, J. Opt. Soc. Am. B 35, 1233 (2018), URL http://josab.osa.org/abstract.cfm?URI=josab-35-6-1233.
- Kou et al. (2010) J. Kou, R. G. Wan, Z. H. Kang, H. H. Wang, L. Jiang, X. J. Zhang, Y. Jiang, and J. Y. Gao, J. Opt. Soc. Am. B 27, 2035 (2010), URL http://josab.osa.org/abstract.cfm?URI=josab-27-10-2035.
- Yang et al. (2015) X. Yang, K. Ying, Y. Niu, and S. Gong, Journal of Optics 17, 045505 (2015), URL http://stacks.iop.org/2040-8986/17/i=4/a=045505.
- Nakano and Yamaguchi (1994) M. Nakano and K. Yamaguchi, Phys. Rev. A 50, 2989 (1994), URL https://link.aps.org/doi/10.1103/PhysRevA.50.2989.
- Nakano and Yamaguchi (1995) M. Nakano and K. Yamaguchi, Chemical Physics Letters 234, 323 (1995), ISSN 0009-2614, URL http://www.sciencedirect.com/science/article/pii/000926149500058C.
- Qi (2010b) J. Qi, Physica Scripta 81, 015402 (2010b), URL http://stacks.iop.org/1402-4896/81/i=1/a=015402.
- Hemmer et al. (1989) P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel, J. Opt. Soc. Am. B 6, 1519 (1989), URL http://josab.osa.org/abstract.cfm?URI=josab-6-8-1519.
- Yan et al. (2001b) M. Yan, E. G. Rickey, and Y. Zhu, Phys. Rev. A 64, 013412 (2001b), URL https://link.aps.org/doi/10.1103/PhysRevA.64.013412.
- Abi-Salloum et al. (2009) T. Abi-Salloum, S. Meiselman, J. Davis, and F. Narducci, Journal of Modern Optics 56, 1926 (2009), eprint https://doi.org/10.1080/09500340903199939, URL https://doi.org/10.1080/09500340903199939.