Four-wave mixing in a ladder configuration of warm <sup>87</sup>Rb atoms: a theoretical study

Heung-Ryoul Noh; Heung-Ryoul Noh; Han Seb Moon; Han Seb Moon

doi:10.1364/OE.416960

1. Introduction

Four-wave mixing (FWM) is a well-known nonlinear effect arising from third-order optical nonlinearity and requires the interaction of a strong optical field with a nonlinear medium. In atomic media with atomic coherence, the nonlinearity arising from multi-photon atomic coherence effects can be enhanced significantly by the interaction of atoms with coherent light [1–11]. More recently, the spontaneous four-wave mixing (SFWM) effect in atomic media has attracted great interest in the field of quantum optics based on atomic-photon interaction [12–14]. The quantum-correlated photon pairs generated in an atomic ensemble via the SFWM process are important for quantum optics, long-distance quantum communication, and for optical quantum-information processing by a linear optics quantum computer [15–17]. Furthermore, classical measurement via stimulated FWM processes is a novel method for characterization of the quantum-correlated photon pairs via the SFWM processes in atoms [18,19]. In particular, collective two-photon coherence effects in a ladder-type Doppler-broadened atomic ensemble have found interest with regard to applications involving polarization entangled photon-pairs, optimal coherent filtering, and low-noise quantum memory [20–27].

Although FWM and SFWM with ladder-type atomic systems have been experimentally demonstrated, the experimental results have been theoretically understood using merely a simple four-level atomic model. Thus far, numerical simulation based on a simple atomic model has been limited to an accurate description of the dynamics of FWM spectra according to the various experimental parameters [6–18]. Real atoms are characterized by many transitions between the Zeeman sublevels of the hyperfine states and multi-photon coherences via the strong interaction of atoms with intense lasers. Recently Parniak and Wasilewski reported a calculating formalism treating the realistic atom as a collection of many four-level atoms to consider the effect of hyperfine structure of the atom [28]. However, the interferences between many four-level subgroups inevitably exist when the laser intensities are not quite weak so that the Zeeman sublevels must be included in the calculation.

Considering the different transition probabilities between the Zeeman sublevels, the FWM process in real atoms is extremely complex, because the occurrence of this process in a ladder-type atomic system is related to the two- and three-photon coherence phenomena. Furthermore, it is more difficult to simulate an FWM signal in a dense Doppler-broadened atomic medium. In this paper, we present the accurate calculation of FWM spectra for a ladder-type energy level system in $^{87}$Rb atoms considering the interactions between all the magnetic sublevels connected via interactions of not greater than three photons. To the best of our knowledge, the accurate calculation of FWM spectra including all the relevant Zeeman sublevels has not been reported to date.

The calculated results are compared with the experimental results presented in the previous publication [9]. In this work, we could understand the formation mechanism of the FWM signals in the strong-field regime using the nonlinear interactions via multi-photon coherence and the interference effect between the subgroups of four-level atoms. Furthermore, we could also understand how the optical depth, laser linewidths, and coupling field power influence on the FWM spectra.

2. Theory

We calculated the FWM signals for the $5S_{1/2}(F=2) - 5P_{3/2}(F'=3) -5D_{5/2}(F''=2,3,4)$ transitions of $^{87}$Rb atoms [29]. The energy level diagram is shown in Fig. 1(a). The resonance frequencies for the $5S_{1/2} - 5P_{3/2}$ and $5P_{3/2} -5D_{5/2}$ transitions are $\lambda _1 =780.2$ nm and $\lambda _2 =775.8$ nm, respectively. The decay rates of the states $5P_{3/2}$ and $5D_{5/2}$ are given by $2 \pi \times 6.065$ MHz [30] and $2 \pi \times 0.6673$ MHz [31], respectively. The pump and driving fields are tuned to the $5S_{1/2}(F=2) - 5P_{3/2}(F'=3)$ resonance line, and the coupling field is scanned around the $5P_{3/2}(F'=3) -5D_{5/2}(F''=2,3,4)$ transitions. The Rabi frequencies (detunings) for the pump, coupling, and driving fields are defined as $\Omega _p$ ($\delta _p$), $\Omega _c$ ($\delta _c$), and $\Omega _d$ ($\delta _d$), respectively. The detailed energy level diagram, which shows the degenerate magnetic sublevels needed in the calculation, is illustrated in Fig. 1(b). To simplify the notation for the matrix elements, $e^{(\mu )}_{m}$ ($\mu =2,3,4$), $r_{m}$, and $g_m$ are defined as the hyperfine states $\left |F''=\mu ,m\right \rangle $, $\left |F'=3,m\right \rangle $, and $\left |F=2,m\right \rangle $, respectively. A simplified scheme of the laser beams that are used to drive the FWM process is depicted in Fig. 1(c). The propagation direction of the coupling field is opposite to that of the pump field, whereas the propagation direction of the driving field intersects that of the pump field at a small angle. All the fields are linearly polarized, and the polarization of the pump field is perpendicular to those of the coupling and driving fields. When the phase matching conditions are satisfied, two FWM fields can be generated in the $5P_{3/2}(F'=3) -5D_{5/2}$ transition and propagate in different directions, as is described below. The quantization axis is selected as the direction of the electric field of the coupling field. Thus the coupling and driving fields excite the $\pi$ transition, whereas the pump field excites both the $\sigma ^{\pm }$ transitions as shown in Fig. 1(b).

Fig. 1. (a) Energy level diagram for the $5S_{1/2}-5P_{3/2}-5D_{5/2}$ transition in $^{87}$Rb atoms. (b) detailed detailed energy level diagram showing the degenerate magnetic sublevels needed in the calculation. The red, blue, and green lines denote the transitions by the pump, driving, and coupling fields, respectively. (c) Simplified schematic diagram of the laser beams for the FWM process.

Download Full Size | PDF

To calculate the FWM signals, we solve the following density matrix equation considering all the magnetic sublevels in Fig. 1(b):

(1)$$\dot \rho ={-}\frac{i}{\hbar} \left[ H_0 +V ,\rho \right] +\dot \rho_\textrm{relax},$$

where $\rho$ is the density operator, $\dot \rho _\textrm {relax}$ denotes the relaxation terms including spontaneous emissions [29,32], and $H_0$ ($V$) is the atomic (interaction) Hamiltonian, the explicit expression of which is given below. The matrix elements of the operator ${\dot \rho }_\textrm {relax}$ are presented in the Appendix. In the description of the relaxation terms, we consider the transit-time relaxation resulting from the finite interaction times between the laser light and the atoms rather than solving the time-dependent differential equations [33,34]. In our work, the transit-relaxation decay rate ($\Gamma _t = \frac {2 v_\textrm {mp}}{\sqrt {\pi } d}$) is $1.3 \times 10^5$ s$^{-1}$ [33,34] when the laser beam diameter ($d$) is 2.2 mm and the most probable speed of the atom ($v_\textrm {mp}$) is 260 m/s in the cell of temperature $80^{\circ }$C [9]. It is also worth noting that the $5D_{5/2}$ state in Fig. 1(a) decays to the $6P_{3/2}$ as well as $5P_{3/2}$. As described in the previous report [29], we considered this leakage by means of rate equations. The explicit expressions for the rate equations are presented in the Appendix. In Eq. (1), the atomic Hamiltonian $H_0$ is given by

(2)$$\begin{aligned} H_0 &= -\sum_{F^{\prime\prime}=2}^4 \sum_{m={-}F^{\prime\prime}}^{F^{\prime\prime}} \hbar \left( d_c+d_d -\Delta_{4}^{F^{\prime\prime}} \right) \left| F^{\prime\prime},m \right \rangle \left\langle F^{\prime\prime},m \right|\\ &\quad-\hbar d_d \sum_{m={-}3}^{3} \left| F'=3,m \right \rangle \left\langle F'=3,m\right|, \end{aligned}$$

where $d_c =\delta _c -{\vec k}_c \cdot {\vec v}$ and $d_d = \delta _d -{\vec k}_d \cdot {\vec v}$ are the effective detunings experienced by an atom moving with velocity $\vec v$. The magnitudes of the wave vectors ${\vec k}_d$ and ${\vec k}_c$ are given by $2\pi /\lambda _1$ and $k_c = 2\pi /\lambda _2$, respectively. Similarly, the effective detuning for the pump field is defined as $d_p = \delta _p -{\vec k}_p \cdot {\vec v}$ and the magnitude of ${\vec k}_p$ is given by $k_p = 2\pi /\lambda _1$. Because the direction of ${\vec k}_c$ is opposite to the direction of ${\vec k}_p$ and ${\vec k}_d \simeq {\vec k}_p$, we use the following equations in the calculation

\begin{aligned} d_p = \delta_p -k_p v ,\\ d_c = \delta_p +k_c v ,\\ d_d = \delta_d -k_p v . \end{aligned}

In Eq. (2), $\Delta _{\mu }^{\nu }$ is the frequency difference between the hyperfine states $\left |F''=\mu \right \rangle $ and $\left |F''=\nu \right \rangle $. In Eq. (1), the interaction Hamiltonian $V$ is given by

(3)$$\begin{aligned} V &= \sum_{m={-}2}^2 \sum_{q={-}1}^1 \frac{\hbar}{2} \left( a_q \Omega_p e^{{-}i \delta t} +b_q \Omega_d \right)\\ &\quad \times C_{F=2,m}^{F'=3,m+q} \left| F'=3,m+q \right \rangle \left\langle F=2,m \right|\\ &\quad +\sum_{F=2}^4 \sum_{m={-}3}^3 \sum_{q={-}1}^1 \frac{\hbar}{2} \sqrt{\eta_{2a}}c_q \Omega_c\\ &\quad \times C_{F'=3,m}^{F,m+q} \left| F,m+q \right \rangle \left\langle F'=3,m \right|+\textrm{h.c.}, \end{aligned}$$

where ${\rm h}.c.$ implies Hermitian conjugates, $C_{F_g,m_g}^{F_e,m_e}$ is the normalized transition strength between the states $\left |F_g, m_g \right \rangle $ and $\left |F_e, m_e \right \rangle $ [32], and the difference in the effective detunings of the pump and driving field is given by

(4)$$\delta = d_{p}-d_{d}=\delta_{p}-\delta_{d} .$$

In Eq. (3), $\eta _{2a}$ is the branching ratio from the $5D_{5/2}$ state to the $5P_{3/2}$ state (see Appendix) and the coefficients for the pump, driving, and coupling fields are given by

\begin{aligned} \left( a_{{-}1}, a_{0}, a_{+} \right)&=\left(1/\sqrt{2},0,-1/\sqrt{2} \right),\\ \left( b_{{-}1}, b_{0}, b_{+} \right)&= \left(0,1,0 \right),\\ \left( c_{{-}1}, c_{0}, c_{+} \right)&=\left(0,1,0 \right), \end{aligned}

respectively.

Substitution of Eqs. (2) and (3) into Eq. (1) yields the coupled differential equations for the density matrix elements. We use simplified notations for the matrix elements: e.g., $\rho _{e_{i}^{(\mu )},r_{j}}=\left\langle F''=\mu ,i\right |\rho \left |F'=3,j\right \rangle $, and $\rho _{i,j} =\rho ^*_{j,i}$. Because of the existence of various frequency mixing between the laser fields, the density matrix elements are decomposed into various Fourier components. In this calculation, we consider the interactions of three photons for the optical coherences of the $5S_{1/2}-5P_{3/2}$ and $5P_{3/2}-5D_{5/2}$ transitions. In contrast, the interactions of two photons are considered for the optical coherences of the $5S_{1/2}-5D_{3/2}$ transitions and all the Zeeman coherences and populations. Therefore, the matrix elements for the optical coherences are decomposed as:

\begin{aligned} \label{} &\rho_{e_{m}^{(\mu)},r_{m\pm 1}} = \rho_{e_{m}^{(\mu)},r_{m\pm1}}^{(1)} e^{{-}i \delta t}+\rho_{e_{m}^{(\mu)},r_{m\pm1}}^{(2)} e^{i \delta t},\\ &\rho_{e_{m}^{(\mu)},r_{m}} = \rho_{e_{m}^{(\mu)},r_{m}}^{(1)}, \quad \rho_{e_{m}^{(\mu)},r_{m\pm 2}} = \rho_{e_{m}^{(\mu)},r_{m\pm 2}}^{(1)},\\ &\rho_{r_{m},g_{m}} = \rho_{r_{m},g_{m}}^{(1)} +\rho_{r_{m},g_{m}}^{(2)} e^{{-}2 i \delta t},\\ &\rho_{r_{m},g_{m\pm 1}} = \rho_{r_{m},g_{m\pm1}}^{(1)} e^{{-}i \delta t}+\rho_{r_{m},g_{m\pm1}}^{(2)} e^{i \delta t},\\ &\rho_{r_{m},g_{m\pm 2}} = \rho_{r_{m},g_{m\pm 2}}^{(1)}+\rho_{r_{m},g_{m\pm 2}}^{(2)} e^{{-}2 i \delta t},\\ &\rho_{r_{m},g_{m\pm 3}} = \rho_{r_{m},g_{m\pm 3}}^{(1)} e^{{-}i \delta t},\\ &\rho_{e_{m}^{(\mu)},g_{m\pm 1}} = \rho_{e_{m}^{(\mu)},g_{m\pm1}}^{(1)} e^{{-}i \delta t},\\ &\rho_{e_{m}^{(\mu)},g_{m}} = \rho_{e_{m}^{(\mu)},g_{m}}^{(1)},\\ &\rho_{e_{m}^{(\mu)},e_{m}^{(\mu')}} = \rho_{e_{m}^{(\mu)},e_{m}^{(\mu')}}^{(1)}, \end{aligned}

where $\mu ,\mu '=2,3,4$. The decompositions of the Zeeman coherences and populations are given by

\begin{aligned} \label{} &\rho_{r_{m},r_{m+1}} = \rho_{r_{m},r_{m+1}}^{(1)} e^{{-}i \delta t}+\rho_{r_{m},r_{m+1}}^{(2)} e^{i \delta t},\\ &\rho_{r_{m},r_{m+2}} = \rho_{r_{m},r_{m+2}}^{(1)},\\ &\rho_{g_{m},g_{m+1}} = \rho_{g_{m},g_{m+1}}^{(1)} e^{{-}i \delta t}+\rho_{g_{m},g_{m+1}}^{(2)} e^{i \delta t},\\ &\rho_{g_{m},g_{m+2}} = \rho_{g_{m},g_{m+2}}^{(1)},\\ &\rho_{e_{m}^{(\mu)},e_{m}^{(\mu)}} = \rho_{e_{m}^{(\mu)},e_{m}^{(\mu)}}^{(1)},\,\, \rho_{r_{m},r_{m}} = \rho_{r_{m},r_{m}}^{(1)},\,\, \rho_{g_{m},g_{m}} = \rho_{g_{m},g_{m}}^{(1)}. \end{aligned}

All the higher interactions are neglected in the calculation.

Finally, we obtain the coupled differential equations for the Fourier components of the density matrix elements, which are then solved in the steady-state regime as a function of $\delta _p$, $\delta _c$, $\delta _d$, and $v$. The density matrix elements that are responsible for the FWM signal are the optical coherences $\rho _{e_{m}^{(\mu )},r_{m\pm 1}}$. Owing to the symmetry, the coherences for the $\sigma ^{\pm }$ transitions are identical to each other: here, the $\sigma ^{-}$ component is chosen for analysis of the FWM signal. Because two oscillation frequencies exist for those optical coherences, we have two FWM signals, associated with $\rho _{e^{(\mu )}_{m-1},r_{m}}^{(1)}$ and $\rho _{e^{(\mu )}_{m-1},r_{m}}^{(2)}$, which are approximately proportional to the following:

(5)$$\left| \sum_{m={-}3}^3 \sum_{\mu=2}^{4} C_{F'=3,m}^{F=\mu,m-1} \int_{-\infty}^{\infty} \frac{d v}{\sqrt{\pi}u} e^{-(v/u)^2} \rho_{e^{(\mu)}_{m-1},r_{m}}^{(i)} \right|^2,$$

where $i=1,2$. In Eq. (5), the FWM signals are averaged over the Maxwell-Boltzmann velocity distribution where $u$ is the most probable speed. In Eq. (5), the integration is performed by assuming that the intensities of the laser fields are constant. We note that because the decomposed components of the density matrix elements for the optical coherences are responsible for the optical signals such as probe absorption, FWM1, and FWM2, the decomposition of the density matrix elements is still valid even when $\delta =0$. Furthermore, the calculated results are still located within three photon approximation when $\delta =0$ because the higher order terms neglected in our calculation have nonzero value even in this case.

In the configuration of the laser fields shown in Fig. 1(c), there exist two possible FWM signals, i.e., FWM1 and FWM2, which are associated with the density matrix elements $\rho _{e^{(\mu )}_{m-1},r_{m}}^{(1)}$ and $\rho _{e^{(\mu )}_{m-1},r_{m}}^{(2)}$ in Eq. (5), respectively, satisfying the phase matching conditions. As shown in Fig. 2, the laser fields for FWM1 and FWM2 have different configurations. To explain the principle for generating two FWM signals, we consider a subset of the energy levels in Fig. 1(b): $e_{m-1}^{(4)}$ and $r_{m}$. In the FWM1 scheme in Fig. 2(a), the emission of coupling and pump photons followed by absorption of a driving photon creates a FWM1 photon. The conservation of momentum vectors results in the wave vector of FWM1, as given by

(6)$${\vec k}_\textrm{FWM1} = {\vec k}_p +{\vec k}_c -{\vec k}_d .$$

The configuration of the laser wave vectors is shown in Fig. 2(a). Thus, the direction of the generated FWM1 field is exactly opposite to that of the driving field, and FWM1 is experimentally measurable. In contrast, in the FWM2 scheme in Fig. 2(b), an FWM2 photon is generated as a result of the emission of the coupling and driving photons followed by the absorption of a pump photon. From the conservation of the momentum vectors, the wave vector of FWM2 is given by

(7)$${\vec k}_\textrm{FWM2} ={-}{\vec k}_p +{\vec k}_c +{\vec k}_d ,$$

and the configuration of the laser wave vectors is shown in Fig. 2(b). As shown, the direction of the FWM2 signal is not along the propagation direction of the driving field. Thus, the FWM2 signal is not detected in the experiment [9].

Fig. 2. Simple four-level diagrams and configurations of the wave vectors for (a) component 1 and (b) component 2 of the FWM signals.

Download Full Size | PDF

3. Results

Figure 3 depicts the typical results for the experimental (upper) and calculated (lower) spectra of the FWM1 and FWM2 signals. All the experimental results presented in this paper were obtained from the previous publication [9]. The beam diameters of the three laser fields are 2.2 mm and the temperature of the atomic cell is $80{^\circ }$C. The intensity of the pump and driving fields are 0.13 and 0.26 mW/mm$^2$, respectively, and the intensity of the coupling field is 2.6 mW/mm$^2$ in Fig. 3(a) and 26 mW/mm$^2$ in Fig. 3(b). The frequencies of the pump and driving fields are fixed at the $5S_{1/2}(F =2) - 5P_{3/2}(F'=3)$ transition, whereas the coupling field is scanned around the $5P_{3/2}(F'=3) - 5D_{5/2}(F''=2,3,4)$ transitions. In the calculation, the powers of both the pump and driving fields are decreased intentionally by a factor of $10^{-3}$, which corresponds to an optical depth (OD) of 6.9, whereas the power of the coupling field is decreased to half of that used in the experiment. This decrease in the power is attributed to the absorption of the laser power in the atomic cell and the inaccurate measurement of the laser beam diameter, and is discussed in detail below. Because the power of the coupling field is reduced only in the presence of the pump field, a small absorption occurs in contrast to the case of the pump and driving fields.

Fig. 3. Experimental and calculated FWM spectra (FWM1 and FWM2) with the intensity of the coupling field in the experiments (calculations) being (a) 2.6 mW/mm$^2$ (1.3 mW/mm$^2$) and (b) 26 mW/mm$^2$ (13 mW/mm$^2$). The value of the OD for the pump and driving fields in the calculations is 6.9.

Download Full Size | PDF

In Fig. 3, the prominently intense FWM signal near $\delta _c =0$ corresponds to the $5P_{3/2}(F'=3) - 5D_{5/2}(F''=4)$ transition and the weak dispersive-like signal near $\delta _c = 30$ MHz corresponds to the $5P_{3/2}(F'=3) - 5D_{5/2}(F''=3)$ transition. The extremely weak signal for the $5P_{3/2}(F'=3) - 5D_{5/2}(F''=2)$ transition, found in the calculated results, was not detected experimentally because of the weak strength of this transition line. In Fig. 3, the FWM signals for the $5P_{3/2}(F'=3) - 5D_{5/2}(F''=2, 3$, and $4)$ transitions are labeled as the FWM signals $F''=2$, $F''=3$, and $F''=4$, respectively. A further detailed study for the dispersive-like FWM signal $F''=3$ will be given below. A comparison of the calculated results for FWM1 with the experimental results indicates that they are in good agreement. Note that a FWM2 signal, whose theoretical prediction is shown also on the same plot, has not been experimentally measured [9]. Thus, we conclude that the FWM signals that were detected experimentally are attributable to the matrix elements $\rho _{e^{(\mu )}_{m-1},r_{m}}^{(1)}$ in Eq. (5).

Figure 4 shows the variation in the calculated FWM spectra for different values of the OD for the power of the pump and driving fields. The experimental results are presented for comparison purposes. The experimental traces were recorded by setting the intensity of the pump and driving fields to 0.13 and 0.26 mW/mm$^2$, respectively, whereas the coupling power was 2.6 and 26 mW/mm$^2$, as shown in Fig. 4(a) and 4(b), respectively. As in Fig. 3, the coupling power for the calculated FWM spectra was set to half the values used in the experiments. The values specified for the OD are 0, 2.3, 4.6, 6.9, and 9.2 (from top to bottom) in Fig. 4. As expected, the linewidth becomes narrower as the value of the OD increases. The actual OD that was used in the experiment was approximately 6.9, and is supported by the calculated results; the experimental and calculated results are in good agreement when the OD is approximately $6.9 \sim 9.2$. Therefore, in this study, we adopted a value of 6.9 for the OD.

Fig. 4. Variation of calculated FWM spectra for different OD values together with the experimental results. The power of the pump, coupling, and driving fields for the experimental traces are (a) 0.13, 0.26, and 2.6 mW/mm$^2$ and (b) 0.13, 0.26, and 26 mW/mm$^2$, respectively.

Download Full Size | PDF

We investigated the effect of the linewidth of the laser on the calculated FWM spectra. The results are shown in Fig. 5. The power of the laser is the same as that used to obtain the results in Fig. 4, and the value of the OD for the pump and driving fields was set at 6.9 for the calculation. In Fig. 5, the linewidths for the calculated FWM spectra, displayed from top to bottom, are 0, 0.5, 1, 1.5, and 2.0 MHz, respectively. As the linewidth increases, the narrow dips near $\delta _c =0$ in Fig. 5 gradually diminish, and the overall width of the spectrum increases. A comparison of the calculated results with the experimental results (shown at the bottom in Fig. 5) indicates that agreement between the calculated and experimental results is achieved when the linewidth is approximately 0.5 MHz.

Fig. 5. Changes in the calculated FWM spectra varying the linewidth of the lasers shown together with the experimental results. The intensity of the pump, coupling, and driving fields for the experimental traces are (a) 0.13, 0.26, and 2.6 mW/mm$^2$ and (b) 0.13, 0.26, and 26 mW/mm$^2$, respectively. The OD is 6.9.

Download Full Size | PDF

We additionally investigated the effect on the FWM spectra by varying the power of the coupling field. The experimental results are shown in Fig. 6(a), and the calculated results for laser linewidths of 0 and 0.5 MHz are shown in Fig. 6(b) and 6(c), respectively. The intensities of the pump and driving fields were 0.13 and 0.26 mW/mm$^2$, respectively, and an OD of 6.9 was used for the calculations. The traces in Fig. 6(a) represent the FWM spectra with the coupling field intensities of 2.6, 5.2, 7.9, 13, and 26 mW/mm$^2$ (bottom to top), and 1.3, 2.6, 3.9, 6.6, and 13 mW/mm$^2$ (bottom to top) in Fig. 6(b) and 6(c), respectively.

Fig. 6. Dependence of the FWM spectra on the power of the coupling field: (a) experimental and (b) calculated results with a laser linewidth of 0 MHz, and (c) calculated results with a laser linewidth of 0.5 MHz. The intensity of the coupling field is 2.6, 5.2, 7.9, 13, and 26 mW/mm$^2$ (bottom to top) in (a), and the intensity in (b) and (c) is half that in (a).

Download Full Size | PDF

As observed in Fig. 6, good agreement exists between the calculated and experimental results. Although the effect of the laser linewidth is not prominent, consideration of a laser linewidth of 0.5 MHz may improve the agreement between the calculated and experimental results. A comparison of the calculated results with the experimental results indicates that the effective power of the coupling laser is reduced to approximately half the value used in the experiment as mentioned in the discussion accompanying Fig. 3. As can be seen in Fig. 6, as the power of the coupling field increases, the overall FWHM of the spectrum centered at the resonance ($\delta _c =0$) becomes wider with two peaks emerging when the power of the coupling field approximates 20 mW. The separation between these peaks increases monotonously with increasing power, which results from Autler-Townes splitting [35] due to dressing of $5P_{3/2} \rightarrow 5D_{5/2}$ transition by coupling laser.

As can be seen in Fig. 3, the FWM signal exhibits a dispersive-like lineshape near the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=3)$ transition (FWM signal $F''=3$). To investigate the formation mechanism for this dispersive-like signal, we calculated FWM spectra by varying the intrinsic properties of the atoms intentionally. Figure 7(a) shows the FWM spectra according to the variation of the oscillator strength for the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=4)$ cycling transition line and Fig. 7(b) shows those for various hyperfine energy spacings of the $5D_{5/2}$ state. In Fig. 7, the powers of the pump, coupling, and driving fields are 0.13 $\mu$W/mm$^2$, 13 mW/mm$^2$, and 0.26 $\mu$W/mm$^2$, respectively, that are the same values as in the results in Fig. 3(b).

Fig. 7. Calculated FWM spectra according to (a) the artificial oscillation strengths of the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=4)$ transition line and (b) the artificial energy spacings of the $5D_{5/2}$ state.

Download Full Size | PDF

In Fig. 7(a), the OS is defined as the ratio of the artificial oscillator strength to the original value for the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=4)$ cycling transition line. This calculation is performed because strong transition strength for the cycling transition line might influence the FWM signal $F''=3$ in Fig. 7(a). In Fig. 7(a), the values of the OS are 1.0, 0.8, 0.6, 0.4, 0.2, and 0 (from top to bottom), and lower three spectra are enlarged by a factor of three or two to provide a clear view. The spectrum at the OS of 1.0 is identical to FWM1 in Fig. 3(b) exhibiting a Fano-resonance-like dispersive lineshape for the FWM signal $F''=3$ [36]. As the OS is reduced intentionally, the dispersive-like signal monotonically changes to Lorentzian-like signal. When the OS is zero, the FWM signal $F''=3$ exhibits a symmetric lineshape because the influence of the strong cycling transition line is now excluded. In Fig. 7(a), we can see that the narrow dip near $\delta _c = 30$ MHz originates from the dip at the center of the FWM signal $F''=3$. We note that the weak FWM signal $F''=2$ shows still a dispersion-like lineshape, which results from the influence of the relatively strong $5P_{3/2}(F'=3) - 5D_{5/2} (F''=3)$ transition. In sum, the dispersive-like signal for the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=3)$ transition results from the influence of the strong cycling transition line.

The calculated FWM spectra depending on the variation of the hyperfine energy spacings of the $5D_{5/2}$ state are shown in Fig. 7(b). The trace labeled as $n \Delta _4^3$ represents the FWM spectrum calculated with the hyperfine energy spacings enlarged intensionally by a factor of $n$ ($n=1,2,3,5,10$, and 15). As $n$ increases the FWM signal $F''=3$ changes to a Lorentzian-like lineshape, because the influence of the strong $5P_{3/2}(F'=3) - 5D_{5/2} (F''=4)$ cycling transition line diminishes. At the same time the FWM signal $F''=4$ also becomes symmetric, as the influence of the FWM signal $F''=3$ is reduced in turn.

4. Conclusion

We presented a theoretical study of the FWM spectra of the $5S_{1/2}-5P_{3/2}-5D_{5/2}$ transition of $^{87}$Rb atoms. The density matrix elements responsible for the FWM spectra were calculated as a function of various detuning values and the velocity from the density matrix equations considering all the relevant magnetic sublevels, and subsequently integrated over a Maxwell-Boltzmann velocity distribution. We could discriminate two types of FWM signals in the calculation, and compared one of them, propagating in the direction exactly opposite to that of the driving field, with the experimental results. We also investigated the effects of the OD, laser linewidths, and the power of the coupling field on the FWM signal. These investigations revealed that appropriate values for the OD for the pump (and driving) field and linewidth were 6.9 and 0.5 MHz, respectively. The dispersive-like FWM signal at the $5P_{3/2}(F'=3) - 5D_{5/2} (F''=3)$ transition line is found to be resulted from the influence of the strong $5P_{3/2}(F'=3) - 5D_{5/2} (F''=4)$ cycling transition line.

Because the SFWM process is a highly important phenomenon for photon pair generation in the quantum optics and quantum information fields, a detailed and precise understanding of FWM spectra is of tremendous importance. A theoretical calculation of the SFWM process in real atoms is currently in progress. We believe that our results can be applied to the characterization and engineering of narrowband quantum sources based on atom-photon interfaces.

Appendix

The matrix elements of the operator $\dot \rho _\textrm {relax}$ for the transitions 5$S_{1/2}$–5$P_{3/2}$–5$D_{5/2}$ are presented in the Appendix. Detailed energy level diagram for the 5$S_{1/2}$–5$P_{3/2}$–$5D_{5/2}$ transition and relaxations in $^{87}$Rb atoms is shown in Fig. 8. For simplicity, the angular momenta $F$, $F'$, and $F''$ represent the states 5$S_{1/2}$, 5$P_{3/2}$, and 5$D_{5/2}$ states, respectively. The populations at the states $5S_{1/2}$, $5P_{3/2}$, and $5D_{5/2}$ are defined as $P_{F}^m$, $Q_{F'}^{m'}$, and $T_{F''}^{m''}$, respectively. The populations at the $6S_{1/2}$, $4D_{3/2}$, $4D_{5/2}$, $5P_{1/2}$, and $6P_{3/2}$ states are defined as $P_a{_{F}^m}$, $P_b{_{F}^m}$, $P_c{_{F}^m}$, $P_d{_{F}^m}$, and $P_e{_{F}^m}$, respectively. The decay rates of the $5P_{3/2}$, $5P_{1/2}$, $5D_{5/2}$, $6P_{3/2}$, $6S_{1/2}$, $4D_{5/2}$, and $4D_{3/2}$ states are given by $\Gamma _1$($=2 \pi \times 6.065$ MHz), $\Gamma _{1a}$($=2 \pi \times 5.746$ MHz), $\Gamma _2$($=2 \pi \times 0.6673$ MHz), $\Gamma _3$($=2 \pi \times 1.3$ MHz), $\Gamma _4$($=2 \pi \times 3.492$ MHz), $\Gamma _5$($=2 \pi \times 1.7$ MHz), and $\Gamma _{6}$($=2 \pi \times 1.8$ MHz), respectively [29]. The branching ratios in Fig. 8 are given as follows: $\eta _{2a}=0.74$, $\eta _{2b}=0.26$, $\eta _{3a}=0.23$, $\eta _{3b}=0.55$, and $\eta _{3c}=0.22$ [29]. It should be noted that the transition rate ($R$) of each transition line is equal to the square of $C_{F_g,m_g}^{F_e,m_e}$ times the branching ratio.

Fig. 8. Detailed energy level diagram for the 5$S_{1/2}$–5$P_{3/2}$–$5D_{5/2}$ transition and relaxations in $^{87}$Rb atoms.

Download Full Size | PDF

(i) The off-diagonal matrix elements are given by $\left\langle \mu ,m_{\mu} \left| {\dot \rho}_\textrm{relax} \right| \nu,m_{\nu} \right \rangle {=} -S_\textrm{rate} \left\langle \mu ,m_{\mu} \left| \rho \right| \nu,m_{\nu} \right \rangle ,$ where $S_\textrm{rate} =\left\{ \begin{array}{ll} \Gamma_2 +\Gamma_t \, , & \; (\mu,\nu)\in (F,F) \, , \\ \left(\Gamma_1 + \Gamma_2 +\gamma_c \right)/2 \, , & (\mu,\nu) \in (F,F') \, , \\ \left( \Gamma_2 +\gamma_p +\gamma_c \right) /2 \, , & (\mu,\nu) \in (F,F) \, , \\ \left( \Gamma_1 +\gamma_p \right) /2 \, , & (\mu, \nu ) \in (F',F) \, , \end{array} \right.$ where $\gamma _p$ and $\gamma _c$ are the liewidths of the pump and coupling lasers, respectively.
(ii) The matrix elements for the state $5P_{3/2}(F'=3)$ are given by $\begin{aligned} & \left\langle F'=3 ,m_1 \left| {\dot \rho}_\textrm{relax} \right| F'=3,m_{2} \right \rangle {=} - \left( \Gamma_1 +\Gamma_t \right) \left\langle F'=3 ,m_{1} \left| \rho \right| F'=3,m_{2} \right \rangle \\ &+\eta_{2a} \Gamma_2 \sum_{F=2}^{4} \sum_{q={-}1}^{1} C_{3,m_1}^{F\prime\prime,m_1+q} C_{3,m_2}^{F\prime\prime,m_2+q} \left\langle F^{\prime\prime},m_1 +q \left| \rho \right| F^{\prime\prime} , m_2 +q \right \rangle ,\\ &+ \delta_{m_1 ,m_2} \sum_{F\prime\prime = 2}^{3} \sum_{m\prime\prime =m_1 -1}^{m_1 +1} \Gamma_{6} {R_{6a}}_{3 , m_1}^{F\prime\prime , m^{\prime\prime}} {P_b}_{F\prime\prime}^{m\prime\prime} + \delta_{m_1 ,m_2} \sum_{F^{\prime\prime} = 2}^{4} \sum_{m^{\prime\prime} =m_1 -1}^{m_1 +1} \Gamma_5 {R_{5}}_{3 , m_1}^{F^{\prime\prime} , m^{\prime\prime}} {P_c}_{F^{\prime\prime}}^{m^{\prime\prime}}\\ &+ \delta_{m_1 ,m_2} \sum_{F^{\prime\prime} = 2}^{2} \sum_{m^{\prime\prime} =m_1 -1}^{m_1 +1} \Gamma_4 {R_{4a}}_{3 , m_1}^{F^{\prime\prime} , m} {P_a}_{F^{\prime\prime}}^{m^{\prime\prime}} . \end{aligned}$
(iii) The matrix elements for the state $5S_{1/2}(F=2)$ are given by $\begin{aligned} & \left\langle F=2 ,m_{1} \left| {\dot \rho}_\textrm{relax} \right| F=2,m_{2} \right \rangle \\ & =\Gamma_{1} \sum_{F'=1}^{3} \sum_{q={-}1}^{1} C_{2,m_1}^{F',m_{1} +q} C_{2,m_2}^{F',m_{2} +q} \left\langle F',m_{1} +q \left| \rho \right| F' , m_{2} +q \right \rangle \\ &+ \delta_{m_{1},m_{2}}\sum_{F' = 1}^{3} \sum_{m'=m_{1} -1}^{m_{1} +1} \Gamma_{3} {R_{3a}}_{F , m_{1}}^{F' , m'} {P_{e}}_{F'}^{m'} ,\\ &+ \delta_{m_{1},m_{2}}\sum_{F' = 1}^{2} \sum_{m'=m_{1} -1}^{m_{1} +1} \Gamma_{1a} {R_{1a}}_{F' , m_{1}}^{F' , m'} {P_{d}}_{F'}^{m'} ,\\ & -\Gamma_{t} \left( \left\langle F=2,m_{1} \right| \rho \left| F=2, m_{2} \right \rangle {-}\frac{1}{8}\right) \delta_{m_{1},m_{2}} , \end{aligned}$ and $\left({\dot \rho}_\textrm{relax} \right)_{ij} = \left({\dot \rho}_\textrm{relax}\right)_{ji}^* , \quad \textrm{for} \quad i \neq j .$
The relaxation terms for other states are given by the following rate equations.
(iv) The rate equations for the states $5P_{3/2}(F' =0,1,2)$ are given by $\begin{aligned} {\dot Q}_{F'}^{m} &= -\left( \Gamma_1 +\Gamma_t \right) Q_{F'}^{m} + \sum_{F = F' -1}^{F' +1} \sum_{m =m-1}^{m+1} \left( \Gamma_2 {R_{2a}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}}{T}_{F^{\prime\prime}}^{m^{\prime\prime}} \right.\\ &\quad \left. + \Gamma_{6} {R_{6a}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}} {P_b}_{F^{\prime\prime}}^{m^{\prime\prime}} + \Gamma_5 {R_{5}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}} {P_c}_{F^{\prime\prime}}^{m^{\prime\prime}} + \Gamma_4 {R_{4a}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}} {P_a}_{F^{\prime\prime}}^{m^{\prime\prime}} \right). \end{aligned}$
(v) The rate equations for the states $6P_{3/2}(F'=1,2,3)$ are given by $\dot{P}_e{_{F'}^{m}} ={-}\left( \Gamma_3 +\Gamma_t \right) P_e{_{F'}^{m}} + \sum_{F^{\prime\prime} = F' -1}^{F' +1} \sum_{m^{\prime\prime} =m-1}^{m+1} \Gamma_2 {R_{2b}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}} {T}_{F^{\prime\prime}}^{m^{\prime\prime}} .$
(vi) The rate equations for the states $6S_{1/2}(F'=1,2)$ are given by ${\dot P}_a{_{F'}^{m}} ={-}\left( \Gamma_4 +\Gamma_t \right) P_a{_{F'}^{m}} + \sum_{F^{\prime\prime} = F' -1}^{F' +1} \sum_{m^{\prime\prime} =m-1}^{m+1} \Gamma_3 {R_{3b}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}}{P_e}_{F^{\prime\prime}}^{m^{\prime\prime}} .$
(vii) The rate equations for the states $5P_{1/2}(F'=1,2)$ are given by $\dot{P}_d{_{F'}^{m}} ={-}\left( \Gamma_{1a} +\Gamma_t \right) P_d{_{F'}^{m}} + \sum_{F^{\prime\prime} = F' -1}^{F' +1} \sum_{m^{\prime\prime} =m-1}^{m+1} \left( \Gamma_4 {R_{4b}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}}{P_a}_{F^{\prime\prime}}^{m^{\prime\prime}} + \Gamma_{6} {R_{6b}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}} {P_b}_{F^{\prime\prime}}^{m^{\prime\prime}} \right) .$
(viii) The rate equations for the states $4D_{3/2}(F'=0,1,2,3)$ are given by $\dot{P}_b{_{F'}^{m}} ={-}\left( \Gamma_{6}+\Gamma_t \right) P_b{_{F'}^{m}} + \sum_{F = F' -1}^{F' +1} \sum_{m =m-1}^{m+1} \Gamma_3 {R_{3c}}_{F' , m}^{F , m}{P_e}_{F}^{m} .$
(ix) The rate equations for the states $4D_{5/2}(F'=1,2,3,4)$ are given by $\dot{P}_c{_{F'}^{m}} ={-}\left( \Gamma_5 +\Gamma_t \right) P_c{_{F'}^{m}} + \sum_{F^{\prime\prime} = F' -1}^{F' +1} \sum_{m^{\prime\prime} =m-1}^{m+1} \Gamma_3 {R_{3d}}_{F' , m}^{F^{\prime\prime} , m^{\prime\prime}}{P_e}_{F^{\prime\prime}}^{m^{\prime\prime}} .$

Funding

National Research Foundation of Korea (2020M3E4A1080030, 2020R1A2C1005499).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. J. C. Petch, C. H. Keitel, P. L. Knight, and J. P. Marangos, “Role of electromagnetically induced transparency in resonant four-wave-mixing schemes,” Phys. Rev. A 53(1), 543–561 (1996). [CrossRef]

2. Y.-Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21(14), 1064–1066 (1996). [CrossRef]

3. A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83(20), 4049–4052 (1999). [CrossRef]

4. Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91(22), 221108 (2007). [CrossRef]

5. F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78(1), 013834 (2008). [CrossRef]

6. R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79(3), 033814 (2009). [CrossRef]

7. F. Wen, H. Zheng, X. Xue, H. Chen, J. Song, and Y. Zhang, “Electromagnetically induced transparency-assisted four-wave mixing process in the diamond-type four-level atomic system,” Opt. Mater. 37, 724–726 (2014). [CrossRef]

8. U. Khadka, H. Zheng, and M. Xiao, “Four-wave-mixing between the upper excited states in a ladder-type atomic configuration,” Opt. Express 20(6), 6204–6214 (2012). [CrossRef]

9. Y.-S. Lee and H. S. Moon, “Atomic coherence effects in four-wave mixing process of a ladder-type atomic system,” Opt. Express 24(10), 10723–10732 (2016). [CrossRef]

10. Y.-S. Lee and H. S. Moon, “Doppler-free three-photon coherence in Doppler-broadened diamond-type atomic system,” Opt. Express 25(5), 5316–5326 (2017). [CrossRef]

11. A. Leszczyński, M. Parniak, and W. Wasilewski, “Phase matching alters spatial multiphoton processes in dense atomic ensembles,” Opt. Express 25(1), 284–295 (2017). [CrossRef]

12. Y.-S. Lee, S. M. Lee, H. Kim, and H. S. Moon, “Single-Photon Superradiant Beating from Doppler-Broadened Ladder-Type Atomic Ensemble,” Phys. Rev. A 96(6), 063832 (2017). [CrossRef]

13. J. Park, T. Jeong, H. Kim, and H. S. Moon, “Time-Energy Entangled Photon Pairs from Doppler-Broadened Atomic Ensemble via Collective Two-Photon Coherence,” Phys. Rev. Lett. 121(26), 263601 (2018). [CrossRef]

14. A. Cerè, B. Srivathsan, G. K. Gulati, B. Chng, and C. Kurtsiefer, “Characterization of a photon-pair source based on a cold atomic ensemble using a cascade-level scheme,” Phys. Rev. A 98(2), 023835 (2018). [CrossRef]

15. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001). [CrossRef]

16. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008). [CrossRef]

17. N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeater based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83(1), 33–80 (2011). [CrossRef]

18. T. Jeong, J. Park, and H. S. Moon, “Stimulated measurement of spontaneous four-wave mixing from warm atomic ensemble,” Phys. Rev. A 100(3), 033818 (2019). [CrossRef]

19. T. Jeong and H. S. Moon, “Joint spectral intensity of entangled photon pair from warm atomic ensemble via stimulated emission beat interferometry,” Opt. Lett. 45(9), 2668–2671 (2020). [CrossRef]

20. B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111(12), 123602 (2013). [CrossRef]

21. Y.-S. Lee, S. M. Lee, H. Kim, and H. S. Moon, “Highly bright photon-pair generation in Doppler-broadened ladder-type atomic system,” Opt. Express 24(24), 28083–28091 (2016). [CrossRef]

22. T. Jeong, Y.-S. Lee, J. Park, H. Kim, and H. S. Moon, “Quantum interference between autonomous single-photon sources from Doppler-broadened atomic ensemble,” Optica 4(10), 1167–1170 (2017). [CrossRef]

23. K. T. Kaczmarek, P. M. Ledingham, B. Brecht, S. E. Thomas, G. S. Thekkadath, O. Lazo-Arjona, J. H. D. Munns, E. Poem, A. Feizpour, D. J. Saunders, J. Nunn, and I. A. Walmsley, “High-speed noise-free optical quantum memory,” Phys. Rev. A 97(4), 042316 (2018). [CrossRef]

24. R. Finkelstein, E. Poem, O. Michel, O. Lahad, and O. Firstenberg, “Fast, noise-free memory for photon synchronization at room temperature,” Sci. Adv. 4(1), eaap8598 (2018). [CrossRef]

25. D. J. Whiting, N. Šibalić, J. Keaveney, C. S. Adams, and I. G. Hughes, “Single-photon interference due to motion in an atomic collective excitation,” Phys. Rev. Lett. 118(25), 253601 (2017). [CrossRef]

26. J. Park, H. Kim, and H. S. Moon, “Polarization-Entangled Photons from a Warm Atomic Ensemble Using a Sagnac Interferometer,” Phys. Rev. Lett. 122(14), 143601 (2019). [CrossRef]

27. S. Gao, O. Lazo-Arjona, B. Brecht, K. T. Kaczmarek, S. E. Thomas, J. Nunn, P. M. Ledingham, D. J. Saunders, and I. A. Walmsley, “Optimal Coherent Filtering for Single Noisy Photons,” Phys. Rev. Lett. 123(21), 213604 (2019). [CrossRef]

28. M. Parniak and W. Wasilewski, “Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification,” Phys. Rev. A 91(2), 023418 (2015). [CrossRef]

29. H. R. Noh and H. S. Moon, “Transmittance signal in real ladder-type atoms,” Phys. Rev. A 85(3), 033817 (2012). [CrossRef]

30. P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B: At., Mol. Opt. Phys. 41(15), 155004 (2008). [CrossRef]

31. D. Sheng, A. Perez Galvan, and L. A. Orozco, “Lifetime measurements of the 5d states of rubidium,” Phys. Rev. A 78(6), 062506 (2008). [CrossRef]

32. H. R. Noh and H. S. Moon, “Calculation of line shapes in double-resonance optical pumping,” Phys. Rev. A 80(2), 022509 (2009). [CrossRef]

33. J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D₁ line,” J. Phys. B: At., Mol. Opt. Phys. 29(12), 2629–2643 (1996). [CrossRef]

34. G. W. Choi and H. R. Noh, “Line shapes in sub-Doppler DAVLL in the ⁸7Rb-D2 line,” Opt. Commun. 367, 312–315 (2016). [CrossRef]

35. S. H. Autler and C. H. Townes, “Stark Effect in Rapidly Varying Fields,” Phys. Rev. 100(2), 703–722 (1955). [CrossRef]

36. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

Four-wave mixing in a ladder configuration of warm ⁸⁷Rb atoms: a theoretical study

Abstract

1. Introduction

2. Theory

3. Results

4. Conclusion

Appendix

Funding

Disclosures

References

Cited By

Figures (8)

Equations (22)

Optics Express