Research of far-field diffraction intensity pattern in hot atomic Rb sample

Ying Zhang; Xuemei Cheng; Xunli Yin; Jintao Bai; Pei Zhao; Zhaoyu Ren

doi:10.1364/OE.23.005468

1. Introduction

A far-field diffraction ring pattern may appear under suitable conditions when a Gaussian beam passes though a nonlinear medium. Because the light energy distribution can be rearranged, optical limiting can be achieved [1]. Moreover, as an important optical Kerr effect, it can be used to measure the nonlinear refractive index. Superior to z-scan which is limited to the measurements of sheet samples with small nonlinear phase shifts, this method can also be applied to those with large nonlinear phase shifts [2].

For its unique properties and potential applications, the far-field diffraction ring pattern has caused much attention since first reported in 1967s [3]. Similar phenomena have been viewed in a variety of materials, including nematic liquid-crystal films [4], photorefractive materials [5], semiconductors [6], atomic vapors [7], carbonaceous materials [8] and other materials [9, 10]. In some of above mentioned materials, a lot of researches have been done. The investigations have indicated that the diffraction ring pattern is intimately related to the incident power. The size [11] and number [12] of the diffraction rings both increase with the incident intensity increasing. It is also found that the sample position has an effect on the diffraction ring pattern [13]. As an optical Kerr effect, the major factor influencing the diffraction ring pattern is the nonlinear refractive index of the material. That is, diffraction ring patterns can be realized easily in the materials of high nonlinear refractive index. For example, Zidan et al [14] observed the laser-induced diffraction ring patterns from single-wall carbon nanotubes (SWNT), not from multi-wall carbon nanotubes (MWNT) because the nonlinear refractive index of SWNT is larger than MWNT. High nonlinear refractive index is also important in quantum-enhanced phase estimation [15], the generation of the dipole-mode solitons in four-wave mixing (FWM) [16] and the spatiotemporal coherent interference between different nonlinear processes [17]. As we know that the nonlinear refractive index coefficient is closely dependent on the frequency of the incident laser [18], which may further influence the far-field diffraction pattern.

In this work, we take Rb atomic vapor as the nonlinear medium and achieve the far-field diffraction ring patterns around D2 lines of both ⁸⁵Rb and ⁸⁷Rb by tuning the laser frequency. The far-field diffraction ring patterns at different frequencies are compared and theoretically studied. The dependence on power of the incident laser, the atomic number density and the position of sample cell are also investigated both experimentally and theoretically.

2. Experiments

The key part of the experimental setup is shown in Fig. 1(a). The output of the laser source (Spectra-Physics, Matisse TR,CW, tunable from 780 to 990 nm, $ω_{T E M_{00}} = 0.7 mm$ ) is focused into a 100 mm long Rb atomic vapor cell. The focal length of the lens is 500 mm. The temperature of the sample cell is controlled by a heater band.

Fig. 1 (a) Scheme of the key part of the experimental setup. (b) (Experiment) and (c) (Simulation) The absorption spectrum of Rb D2 and the corresponding diffraction patterns.

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Firstly, the frequency dependence of far-field diffraction ring pattern was studied. We set the power of the laser at 100 mW, the temperature of the sample cell at $105 {}^{°}C$ and the front-side of sample cell at the focal point of the lens. Then we tuned the laser frequency and received the diffraction ring patterns as shown in Fig. 1(b) which is agree quite well with the simulations in Fig. 1(c). The curve is the absorption spectrum of Rb D2 lines, in which the four peaks are in accordance to the four transitions as:

\begin{array}{l} a : {}^{87}R b | 5 S_{1 / 2}, F = 1 > \to | 5 P_{3 / 2} > 780.2006 nm \\ b : {}^{85}R b | 5 S_{1 / 2}, F = 2 > \to | 5 P_{3 / 2} > 780.2055 nm \\ c : {}^{85}R b | 5 S_{1 / 2}, F = 3 > \to | 5 P_{3 / 2} > 780.2114 nm \\ d : {}^{87}R b | 5 S_{1 / 2}, F = 2 > \to | 5 P_{3 / 2} > 780.2138 nm \end{array}

We take the absorption spectrum as a reference to investigate the frequency dependence of the diffraction ring patterns. It is seen that the diffraction patterns with most rings are obtained at the frequency detuning about

\pm 0.7 GHz

for all the transitions. Taking the transition b as an example, we find the center of the diffraction ring pattern is bright with positive frequency detuning and dark with negative frequency detuning. It is also found that more rings are obtained from ⁸⁵Rb than ⁸⁷Rb. And for the same isotope ⁸⁵Rb, pattern obtained from the transition c with more rings than the transition b.

Secondly, the power dependence is studied. The cell temperature and position are kept the same as the first experiment. The laser wavelength is fixed at 780.2100 nm. The observed diffraction patterns under the incident power of 30 mW, 70 mW, and 100 mW are shown in Fig. 2(a) and the simulations of the x-y plane diffraction patterns and the radial direction normalized intensity are shown as Fig. 2(b). It is obvious that the number of the rings increases with the incident power increasing.

Fig. 2 (a) The experimental graphs of the far-field diffraction ring patterns with three different incident power. (b) The simulation graphs of x-y plane diffraction patterns and the radial direction normalized intensity with three different incident power.

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Thirdly, we investigated the atomic number density dependence. For the atomic number density is determined by temperature as $\lg N = 4.312 - \frac{4040}{T} - \lg (k T)$ , we can change the atom temperature. The cell position is kept unchanged, the incident power is set at $L$ , and the wavelength at $Δ ϕ (r) = k \int_{z_{0}}^{z_{0} + L} Δ n (r, z) d z = k \int_{z_{0}}^{z_{0} + L} n_{2} I (r, z) d z$ . The cell temperature is varied. The experimental and simulation diffraction patterns of $^{85} Rb | 5 S_{1 / 2}, F = 3 > \to | 5 P_{3 / 2} >$ at the cell temperature of $90^{°} C$ , $100^{°} C$ and $105^{°} C$ (the atomic number density are $3.07 \times 10^{13} m^{- 3}$ , $5.93 \times 10^{13} m^{- 3}$ , and $8.17 \times 10^{13} m^{- 3}$ respectively) are shown in Figs. 3(a)-3(b). It can be seen that the number of the far-field diffraction rings increases with the increasing of the cell temperature (the atomic number density). From the study on temperature dependence, we know that diffraction ring patterns can be observed in the temperature range from $90^{°} C$ to $120^{°} C$ . Hence, any temperature in this range can be chosen. In our case we choose the cell temperature of $105^{°} C$ , therefore, we can obtain the most rings. When the temperature is higher than $120^{°} C$ , the scatter spots will appear at the far-field [19].

Fig. 3 (a) The experimental graphs of the far-field diffraction ring patterns with the cell temperature at $90^{°} C$ , $100^{°} C$ and $105^{°} C$ respectively. (b) The simulation graphs of x-y plane diffraction patterns and the radial direction normalized intensity with the cell temperature at $90^{°} C$ , $100^{°} C$ and $105^{°} C$ respectively.

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At last, we studied the influence of the cell position on the diffraction ring pattern. The power of the laser is still set at 100 mW, the temperature at $105 {}^{°}C$ , and the laser wavelength at 780.2100 nm. The experimental and simulation diffraction patterns are shown in Figs. 4(a)-4(b), when we put the front-side of the sample cell at the position of 240 mm away before the focus point of the lens, the center of the ring pattern is dark. And when we put it at the focus of the lens, we obtain the ring pattern with bright center.

Fig. 4 The experimental and simulation graphs of the far-field diffraction ring patterns with the front side to the sample cell set at the positions of f-240 and f, where f, is the focusing length of the lens.

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3. Theoretical discussions

Now we turn to the theoretical analysis of the experimental phenomena. Due to third-order nonlinearity of the medium, the refractive index is dependent on the light intensity as: $n = n_{0} + n_{2} I$ , where $n_{0}$ is the linear refractive index, $I$ is the incidence intensity, and $n_{2}$ is the nonlinear refractive coefficient, expressed as [20]:

n_{2} = \frac{12 π^{2}}{n_{0}^{2} c} Re χ^{(3)} = \frac{12 π^{2}}{n_{0}^{2} c} \times (- \frac{4}{3} N {(ρ_{b b} - ρ_{a a})}^{e q} {| μ_{b a} |}^{4} \frac{T_{1} T_{2}^{2}}{ε_{0} ℏ^{3}} \frac{Δ T_{2}}{{(1 + Δ^{2} T_{2}^{2})}^{2}})

where

N

is the atomic number density,

{(ρ_{b b} - ρ_{a a})}^{e q}

is the population difference between the excited and the ground states in thermal equilibrium,

T_{1}

is the excited state lifetime,

T_{2}

is the dipole dephasing time,

Δ

is the frequency detuning, and

μ_{a b}

is the electric dipole moment. The variation of

n_{2}

is plotted with respect to

Δ T_{2}

by setting

- \frac{4}{3} N {(ρ_{b b} - ρ_{a a})}^{e q} | μ_{b a} |^{4} \frac{T_{1} T_{2}^{2}}{ε_{0} ћ^{3}} = 1

as shown in Fig. 5. It is seen in Fig. 5 and Eq. (2) that

n_{2} > 0

at the positive detunings and

n_{2} < 0

at the negative detunings, which is a typical self-focusing phenomenon. For a self-defocusing phenomenon obtained by cross-phase modulation (XPM) [21, 22], contrary results is obtained. That is

n_{2} < 0

at the positive detunings and

n_{2} > 0

at the negative detunings.

Fig. 5 The real part of the third-order nonlinear susceptibility $χ^{(3)}$ versus $Δ T_{2}$ .

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The nonlinear refractive index will cause a lateral additional phase shift when a laser beam transmits through a nonlinear medium. If we define the light transmitting direction as $z$ axis with the coordinate origin at the beam waist, the lateral additional after the light passing though the sample cell with a length of $L$ will be [20]:

Δ ϕ (r) = k \int_{z_{0}}^{z_{0} + L} Δ n (r, z) d z = k \int_{z_{0}}^{z_{0} + L} n_{2} I (r, z) d z

in which

z_{0}

is the coordinate of the front-side of the sample cell. Considering the linear absorption coefficient

α

, the lateral additional phase shift of a Gaussian TEM₀₀ beam is then expressed as:

Δ ϕ (r) = k \int_{z_{0}}^{z_{0} + L} n_{2} I (0, 0) \frac{r_{0}^{2}}{r_{p}^{2} (z)} e x p (- α (z - z_{0})) \exp (- 2 r^{2} / r_{p}^{2} (z)) d z

where

r

is the radial coordinate,

r_{0}

is the waist radius,

I (0, 0)

is the central light intensity,

z_{R} = π r_{0}^{2} / λ

is the Rayleigh length of the beam,

r_{p} (z) = r_{0} {[1 + {(z / z_{R})}^{2}]}^{1 / 2}

is the beam radius,

R (z) = z + z_{R}^{2} / z

is the radius of curvature of the wave face of the Gaussian beam.

Based on the Fraunhofer approximation of the Fresnel-Kirchhoff diffraction formula [23], the far-field diffraction intensity is:

I = {| \frac{1}{i λ D} |}^{2} {| \int_{0}^{\infty} \int_{0}^{2 π} E (r, z_{0} + L) \exp (- i k r θ \cos ϕ) \exp [- i (\frac{k r^{2}}{2 R (z_{0})} + Δ ϕ (r)] r d r d ϕ |}^{2}

in which

D

is the distance between the backside of the sample cell and the receiving screen, and

θ

and

φ

are the far-field diffraction angle and the angular coordinates of the exit surface, respectively. If we denote the radical coordinate of the far-field to be

ρ

, and

θ

equals to

ρ / D

.

Considering Eq. (4) cannot be integrated if $r_{p} (z)$ is variable, and therefore $r_{p} (z)$ is always taken to be fixed as $r_{p} (z_{0})$ in numerical simulation for thin samples [24]. As for our case, the distance from the lens to the position of the beam waist (coordinate origin of the in this paper) and the waist radius $r_{0}$ are calculated to be 470 mm and 172 μm though the lens transformation. And the Rayleigh length $z_{R}$ is then obtained to be 119 mm by inserting the value of $r_{0}$ into the equation $z_{R} = π r_{0}^{2} / λ$ . It is seen that our sample is a thick sample because the Rayleigh length $z_{R}$ in our experiments is close to the length of the sample cell (100 mm). The thick sample can be regarded as a stack of thin samples. Since the light is strongest at the entrance side of the sample cell, the nonlinear refraction modification at this place plays the key role. Therefore, we also set $r_{p} (z) = r_{p} (z_{0})$ in the numerical simulations. Under this approximation, $Δ ϕ (r)$ can be written as $Δ ϕ (0) \exp (- 2 r^{2} / r_{p}^{2} (z_{0}))$ . Figures 6(a)-6(c) show the theoretical far-field diffraction intensity patterns for $Δ φ (0)$ equal to $π$ , $2 π$ , and $3 π$ respectively by setting $λ = 780.2100 nm$ , $r_{p} (z_{0}) = 0.2 mm$ , $D = 3.5 m$ , $R (z_{0}) = 502 mm$ in Eq. (5). It is seen that one, two and three rings appear in far-field diffraction patterns respectively. That is, the ring number increase with the increasing of the nonlinear lateral phase shift $Δ φ (0)$ .

Fig. 6 The simulations of the far-field diffraction intensity patterns for (a) $Δ φ (0) = π$ , (b) $R (z_{0})$ , and (c) $Δ φ (0) = 3 π$ .

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With the same parameters, the far-field diffraction patterns at different experimental conditions are also simulated, as shown in Fig. 1(b), Fig. 2(b), Fig. 3(b) and Fig. 4(b). The theoretical pictures are in good agreement with the experimental results, proving the thin sample approximation also stands in our case.

From Eq. (4), it is obvious that $Δ φ (0)$ is proportional to $I (0, 0)$ and n₂. And n₂ is further proportional to the atomic number density N. That is, larger $Δ φ (0)$ is achieved with higher laser power or larger atomic number density (higher cell temperature), and therefore, more rings are obtained as shown in Fig. 2(b) and Fig. 3(b).

In Figs. 4(a) and 4(b), it is shown that the center of the pattern is dark and is surrounded by concentric bright rings when we put the sample cell at the position of 240 mm away before the focus point of the lens in which $Δ = 0.7 GHz$ , $R (z_{0}) = z_{0} + z_{R}^{2} / z_{0} = - 277 mm < 0$ and bright at the focus since $Δ = 0.7 GHz$ , $R (z_{0}) = 502 mm > 0$ . This phenomenon is in agreement with those studies reported before [25]. The center of the far-field diffraction pattern is dark when $n_{2}$ and $R (z_{0})$ have the opposite signs, and a bright center appears when $n_{2}$ and $R (z_{0})$ have the same sign.

The calculation from Eq. (2) as shown in Fig. 5 indicates the value of $Re χ^{(3)}$ is positive at positive detunings and negative at negative detunings. Furthermore, in Eq. (2) we know $n_{2} \propto Re χ^{(3)}$ , so the nonlinear refractive coefficient $n_{2}$ is positive at positive detunings and negative at negative detunings. From Eq. (4) we can obtain the relationship that $Δ φ (0) \propto n_{2} \propto Re χ^{(3)}$ , so $Δ φ (0)$ and $n_{2}$ reach the maximum at $Δ = \pm 0.7 GHz$ . Together with the simulation shown in Fig. 2(b) and Fig. 3(b), which shows the ring number increases with the incident power and atom temperature increasing, we can explain why the most far-field diffraction rings are obtained at $Δ = \pm 0.7 GHz$ in Fig. 1(b). The two experimental pictures of ${}^{85}R b | 5 S_{1 / 2}, F = 2 > \to | 5 P_{3 / 2} >$ shown in Fig. 1(b) from Eq. (2), we see the nonlinear refractive coefficient is in proportion to the atomic number density. The isotope ratio of our sample was measured to be ${}^{85}N / {}^{87}N = 2.590 \pm 0.001$ by the means of DFWM [26], thus the nonlinear refractive coefficient of ⁸⁵Rb is larger than that of ⁸⁷Rb. So more far-field diffraction rings are obtained from ⁸⁵Rb than ⁸⁷Rb. And for the same isotope ⁸⁵Rb, pattern are obtained from the transition b ( $| 5 S_{1 / 2}, F = 3 > \to | 5 P_{3 / 2} >$ ) with more rings than the transition c ( $| 5 S_{1 / 2}, F = 2 > \to | 5 P_{3 / 2} >$ ) because the transition dipole moment of the former transition is larger than the latter one [20].

In our work, the value of $Δ ϕ (0)$ can be determined experimentally according to the fact that the diffraction rings number equals to $| Δ ϕ (0) / π |$ [27] (as indicated in Fig. 6). That is, experimentally, we can count the rings and determine the value of $Δ ϕ (0)$ , then $n_{2}$ can be calculated from the mentioned equation to be $1.97 \times 10^{- 16} {cm}^{2} / W$ , $1.99 \times 10^{- 16} {cm}^{2} / W$ , and $2.02 \times 10^{- 16} {cm}^{2} / W$ at the cell temperature of $105^{°} C$ and the laser power of $30 mW$ , $70 mW$ and $100 mW$ respectively. From the value of $n_{2}$ for different laser power, we see that it is independent on the laser power. In order to verify the results, we directly calculated the value of $n_{2}$ by setting $Δ = 0.7 GHz$ and the temperature at $105^{°} C$ ,so the atomic number density atomic $N$ is $8.17 \times 10^{13} m^{- 3}$ ( $\lg N = 4.312 - \frac{4040}{T} - \lg (k T)$ ). When we insert ${(ρ_{b b} - ρ_{a a})}^{e q}$ , $T_{1}$ , $T_{2}$ , $ε_{0}$ , $ℏ$ and $μ_{a b}$ to Eq. (2) we can calculate the $n_{2}$ is equal to $2.0 \times 10^{- 16} {cm}^{2} / W$ , which is consistent to the experimental data calculated above. And from Eq. (2) we can also observe the nonlinear refractive index $n_{2}$ has no relation with the incident power.

4. Conclusions

In this paper, we studied the laser frequency, laser power, atomic number density and cell position dependence of the far-field diffraction ring patterns in a hot atomic Rb medium. The experimental results are in good agreement with the theoretical simulations with Fresnel-Kirchhoff diffraction considering the third-order nonlinear optical effect. The information obtained suggest that the far-field diffraction cannot only be an effective means of measuring the nonlinearity of a medium, but also a way to determine the parameters, such as atomic number density, resonant frequency and transition dipole.

Acknowledgments

This work was supported by the National“973 Plan” Preliminary Study Special of China, (No. 2012CB723407), the National Science Foundation of China (NSFC) (No. 61475125), Northwest University Science Foundation (No. 13NW12), the Shaanxi Provincial Department of education project (No. 12JK0599) and the Innovation experiment project of Northwest University (No. 2014049).

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Research of far-field diffraction intensity pattern in hot atomic Rb sample

Abstract

1. Introduction

2. Experiments

3. Theoretical discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (5)

Optics Express