ХАРАКТЕРИСТИКИ ПРОПУСКАНИЯ РУБИДИЕВОГО АТОМНОГО ОПТИЧЕСКОГО ФИЛЬТРА НА ДЛИНЕ ВОЛНЫ 780 нм, ИСПОЛЬЗУЮЩЕГО РАМАНОВСКОЕ УСИЛЕНИЕ
ХАРАКТЕРИСТИКИ ПРОПУСКАНИЯ РУБИДИЕВОГО АТОМНОГО ОПТИЧЕСКОГО ФИЛЬТРА НА ДЛИНЕ ВОЛНЫ 780 нм, ИСПОЛЬЗУЮЩЕГО РАМАНОВСКОЕ УСИЛЕНИЕ
TRANSMISSION CHARACTERISTICS OF A RAMAN-AMPLIFIED ATOMIC OPTICAL FILTER IN RUBIDIUM AT 780 nm
© 2014 г. Wenjin Zhang, Yufeng Peng College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, China Е-mail: yufengp@sohu.com
The transmission characteristics of a Raman-amplified atomic filter that can be used to detect fairly weak signals in free-space quantum-key distribution or laser communications are analyzed and discussed in the coherent and incoherent pump fields respectively. The theoretical model for the calculation of the transmission characteristics of a groundstate Raman-amplified Faraday dispersion atomic optical filter based on Raman gain and Faraday rotation is presented. The results show that the filter in a coherent pump field can achieve higher transmission and larger tunability than that in an incoherent pump field due to elimination of pumping detuning. In addition, the filter has a large scale tunability over 3.5 GHz via the Faraday transmission peak adjusted while its bandwidth is only 66 MHz, which is useful for free-space laser communication and lidar systems.
Keywords: atomic optical filter, Raman light amplification, Rubidium, Faraday dispersion, hyperfine structures.
OCIS codes: 120.2440
Submitted 17.10.2013
Introduction
The Faraday anomalous dispersion optical filter (FADOF), which is based on the anomalous dispersion of Faraday rotation near the resonance lines of certain atomic vapours in a longitudinal magnetic field, have many advantages such as high transmission, narrow bandwidth, excellent out-of-band rejection and wide field of view (±π/2) [1–3]. It can be operated either at the line center [4–6] or in the wings [7–9] of the atomic resonance line subjected to given magnetic fields. Therefore, it has been used as a very efficient spectralfiltering device in many signal-detection systems [10–13], recently in the free-space quantum key distribution (QKD) system [14, 15], which could extract very weak signals from strong background noise and operate well in daytime and at night due to adopting the Faraday dispersive atomic filters.
A Faraday dispersive atomic filter was first described and demonstrated by Ohman (1956) [16]. The theoretical framework of a FADOF considered fine structures of atomic system was established by Yeh (1982) [1]. Yin and Shay (1991) [2] (Harrell et al., 2009 [17]) developed the theoretical framework of the FADOF based on the hy-
perfine structures of atomic system and groundstate transitions in the weak (any) magnetic field by using a quantum-mechanical treatment of the resonant Faraday effect. The transmission characteristics of several alkali atomic FADOFs have been observed and discussed by Menders et al. (1991, Cs D2 line) [3], Peng et al. (1993, Rb D2 line) [18], Hu et al. (1998, Na D lines) [5], Zhang et al. (2001, K D lines) [6], and Zielin′ ska et al. (2012, Rb D1 line) [19], etc. Though those filters achieved good performance, e.g., above 70% transmission (Zhang et al., 2001 [6]), and approximately 1 GHz ultra-narrow bandwidth (Zielin′ ska et al., 2012 [19]), they couldn’t have signal gain or amplification ability for the weak light. As for the weak light amplification, since the early 1990s, the mechanisms, which is based on the Raman gain attributed to stimulated Raman scattering and/or the gain from the dressed state inversion, have been extensively studied (Imamoglu et al., 1991 [20], Zibrov et al., 1995, [21], Zhu et al., 1996 [22], Zhu, 1997 [23], Wanare, 2002 [24], Kang et al., 2003 [25], Kilin et al., 2008 [26], etc.).
In 2008, Shan et al. [27] demonstrated experimentally an ultranarrow-bandwidth Ramanamplified atomic filter (RAAF) that was
“Оптический журнал”, 81, 4, 2014
11
a composite of a Raman light amplifier (RLA) and a dispersive atomic optical filter (i.e. FADOF), which achieved a weak light amplification of about 55 times, a bandwidth of ~ 60 MHz, and frequency tunability of ~ 1 GHz in an incoherent pump field. Compared with a conventional FADOF, the RAAF could be more suitable for operating in the free-space QKD or laser-communication systems; however, the detailed theoretical model and the analyses of transmission characteristics for this category of filter in the coherent (incoherent) pump field have not been reported.
In this paper, taking into account Raman gain mechanism in the Doppler-broadened atomic system, the theoretical model for the transmission characteristics of a Raman-amplified Faraday dispersion atomic filter is presented by basing on Raman gain involving the coupling field and pump field and Faraday rotation in the magnetic field. The gain properties, transmission and passband width of the filter subjected to the coherent and incoherent pump fields are analyzed and discussed respectively. We obtain not only the filter frequency tunability of over 3.5 GHz, which is important for free-space optical communication and lidar systems subjected to large Doppler shift. In addition, the other characteristics such as the transmission, gain, and bandwidth of the filter under the action of the coherent and incoherent pump fields are discussed.
1. Theory
The schematic diagram of the RAAF apparatus is shown in Fig. 1a. The coupling, probe (i.e. signal) and pump lasers are emitted from three different lasers, respectively. The coupling laser and probe laser have linear orthogonal polarizations and propagate in the same direction, opposite to that of the pump laser. After the three laser beams enter into the cell 1, the signal gain occurs under the combined action of coupling laser and pump laser; the probe field is amplified. The amplified probe light passes through the P4 and P5 to the Faraday cell 2, while the coupling light is distinguish by the P4. Due to the Faraday rotation effect in an axial magnetic field, the amplified signal light which frequency lies only in the transmission passband of the dispersive atomic filter, can experience a 90° rotation of the polarization and emerge apparently low absorption and high transmission.
The three-level Λ-type system for Rubidium
atoms with the laser-coupled D2 transitions is depicted in Fig. 2. Rubidium has only one stable isotope, 85Rb, with the isotope 87Rb, which com-
poses almost 28% of naturally occurring Rubidium (72.17% 85Rb and 27.83% 87Rb). The
levels |1ñ, |2ñ and |3ñ represent the ground states |5S1/2, F = 2ñ, |5S1/2, F = 3ñ and the exited state 5P3/2 of 85Rb atom, and like wise, the ground
Pump Laser
(а) M
Coupling Laser Signal Laser
P1 M
HW P2
P5 B P6 D
A P3M Rb cell1 M P4 Rb cell2
Signal Laser
x
y
(b)
z2 = L2 z z1 = 0, z1 = L1 , z2 = 0
Fig. 1. Schematic diagram of the RAAF apparatus (a). P1−4 stand for polarizing beam splitters (PBS), M for a mirror, P5−6 for Glan− Thompson prism polarizers, HW for a halfwave plate, A for an attenuator, and D for a photodiode detector. Schematic of the rotation of the signal-light plane of polarization (b).
F = 1,2,3,4
5P3/2
∆
3 ∆1
F= 3 5S1/2
Pump Laser
Signal Laser
3.036 GHz
2 1
Coupling Laser
F= 2 5S1/2
Fig. 2. 85Rb far-detuned three-level Λ system. Δ1(Δ) is the coupling (probe) detuning. For 87Rb atoms, the ground hyperfine states are 5S1/2, F = 2 and 5S1/2, F = 1, and the excited hyperfine states are 5P3/2, F = 3, 2, 1, 0.
12 “Оптический журнал”, 81, 4, 2014
states |5S1/2, F = 1ñ, |5S1/2, F = 2ñ and the exited state 5P3/2 for 87Rb atom, respectively. A coupling laser is used to connect the ground state 5S1/2 (F = 2 for 85Rb or F = 1 for 87Rb) ↔ the excited
state 5P3/2 transition, and a probe laser couples the 5P3/2 ↔ 5S1/2 (F = 3 for 85Rb or F = 2 for 87Rb) transition. A pump laser with a rate Г is applied to the Rb 5S1/2 (F = 3 for 85Rb or F = 2 for 87Rb) ↔ 5P3/2 transition and excites the atoms to the Rb 5P3/2 excited state.
According to above three-level system, and the
electric-dipole and the rotating-wave approxima-
tions, the Hamiltonian in the interaction picture
can be expressed as
H1
=
-
2
êêëéêêΩ00
0
-2(∆1 - ∆) g
Ω g -2∆1
úúûúúù
.
(1)
Here Δ1 is the detuning of the coupling beam from the atomic transition ω31 and Δ is the detuning of the probe field from the atomic transition
ω32; Ω = μ31E1/ћ and g = μ32E2/ћ are the Rabi frequencies of the coupling laser (with amplitude
E1) and probe laser (with amplitude E2); and μ31 and μ32 are the relevant dipole moments. For convenience’s sake, the Rabi frequencies are taken
as real parameters, and the quantum dynamics of
the three-level atomic system can be described by
the Liouville equation
¶ρ ¶t
=
1 i
[H,ρ]-
1 2
{γ
,ρ}.
(2)
The full statements of Equation (2) are expressed as follows:
ρ11 = γ 31ρ33 + iΩ(ρ31 -ρ13) 2,
ρ22 = -Γρ22 +(Γ + γ 32 )ρ33 + ig(ρ32 -ρ23) 2,
ρ33 = Γρ22 -(Γ + γ 32 + γ 31)ρ33 + +iΩ(ρ13 -ρ31) 2 + ig(ρ23 -ρ32) 2,
ρ12 = -[Γ 2- i(∆1 - ∆)]ρ12 - igρ13 2 + iΩρ32 2, ρ13 = -[Γ 2 + γ 32 2 + γ 31 2- i∆1 ]ρ13 -
- igρ12 2 + iΩ(ρ33 -ρ11) 2,
ρ23 = -[Γ + γ 32 2 + γ 31 2- i∆]ρ23 -
- iΩρ21 2 + ig(ρ33 -ρ22) 2,
ρ11 + ρ22 + ρ33 = 1.
(3)
Here γij (i, j = 1–3) are the radiative decay rates from state |iñ to state |jñ. Since the atomic
filter is used to detect a weak optical signal, we
neglect the second order in g and solve Equation
(3) in the steady state to get
ρ32 =
ig{(ρ22 -ρ33)(Γ + γ 31 + γ 32 -2i∆1)
(Γ + γ 31 + γ 32 -2i∆1)
´
{ } }´
éëΓ -2i(∆1 - ∆)ùû + Ω2(ρ33 -ρ11) (2Γ + γ 31 + γ 32 + 2i∆)éëΓ -2i(∆1 - ∆)ùû + Ω2
,
(4)
ρ11
=
Ω2
Ω2Γ(Γ (Γ + γ 31
+ +
γ 31 γ 32
+ γ 32) )(3Γ +
+ γ 32
)
+
+ Γγ 31 +Γγ 31
ëêé4∆12 ëéê4∆12
+ (Γ + (Γ
+ +
γ 31 γ 31
+ +
γ 32)2 ûùú γ 32)2 úùû
,
ρ22
=
Ω2
(Γ
Ω2 (Γ + γ 31 + γ 32)´ + γ 31 + γ 32)(3Γ + γ 32
)
+
+
Γγ
31
´ éëê4∆12
(Γ +
+ (Γ
γ 32 +γ
)
31
+
γ
32
)2
ùúû
,
ρ33
=
Ω2
(Γ
+
γ
31
Ω2Γ ´ + γ 32)(3Γ
+
γ
32 )
+
+Γγ
31
´(Γ + γ 31 + γ 32) êéë4∆12 + (Γ + γ 31 +
γ
32 )2
ûúù
.
(5) (6) (7)
Considering the atoms interacting with the
optical field, the complex susceptibility χ± at the probe field frequency can be obtained from the
polarization
P±
(z,t)
=
1 2
ε0χ±
E±
(z,t)eˆ±
exp
éëi(kz
-
ωt)ûù
+
c.c.
=
(8)
( )= N0 µ±ρ3±2eˆ± + c.c. .
Consequently, the atomic susceptibility χj± of the jth σ± probe transition in the cell 1 can be
written as
χj±
=
µ2j± N0 ε0 g±
ρ±32,
(9)
where +(–) denotes the left(right)-hand circular component, ћ is the Plank’s constant divided by 2p, ε0 is the permittivity of free space, μj± are the relevant dipole moments, and N0 (with unit atom/cm3) is the total atomic number density, given by [28]
lg(N0) = -4529.6 T -3.991lgT + + 0.00059T + 34.8325.
(10)
“Оптический журнал”, 81, 4, 2014
13
We consider the coupling laser and probe laser
passing through a Doppler-broadened atomic cell
1 in the same direction. An atom moving towards
the probe beam (with frequency ωp) with velocity ν is affected by the probe frequency detuning
upshifted to (Δ + ωpν/c) and the frequency detuning of the coupling beam (with frequency ωc) upshifted to Δ1 + ωcν/c. Considering that atoms are in classical thermal equilibrium with the one-
dimensional Maxwellian velocity distribution of N(ν)dν = N0exp(–ν2/u2)/(u√`p)dν (u is the most probable velocity and is defined as u = (2kBT/M)1/2), the final value of χ± is obtained by integrating over the atomic velocity ν and summing over all
the components of Equation (9)
å òχ± =
2N0 πuε0
j
µ2j± g±
+¥ -¥
ρ3±2e-ν2
u2 dν.
(11)
1.1. RAAF gain
We assume that an x-direction linearly polarized probe beam enters the atomic gas cell 1 of length L1 (Fig. 1b) at z1 = 0 and travels along the positive z-direction. The electric radiation field of the probe beam can be expressed in terms of its right- and left-circularly polarized component as
E (z)
=
E0 2
{ëéexp(ik+z)
+
exp(ik-z)ûù
xˆ
+
+i ëéexp(ik+z) - exp(ik-z)ùû yˆ} exp(-iωt),
(11)
where k± are the circular wave-number components, given by
k±
=
ω c
n±
+
i 2
α±.
(12)
By virtue of Equation (11), at z1 = 0 and L1, the x components of this probe beam may be writ-
ten as
E1x (0) = E0xˆ exp(-iωt),
(13)
=
E0 2
E1x (L1) = ëéexp(ik+L1) + exp(ik-L1)ùû xˆ exp(-iωt),
(14)
respectively. The transmission coefficient of the
RLA, defined by TrRLA = I1х(L1)/I1х(0), is then
given by
TrRLA
=
E1x (L1) 2 E1x (0) 2
=
=
exp(-
α+
+ 2
α-
L1)
=
exp(GL1),
(15)
where G is the mean gain coefficient of the RAAF. It can be written as
G
= - α+
+ α2
=
-
ωp 2c
Im(χ+
+
χ- ),
where c is the speed of light in free space.
(16)
1.2. RAAF transmission
When the amplified probe beam goes through
the cell 1, polarizing beam splitter P4 and Glan–
Thompson prism polarizer P5 to the Faraday cell 2,
the
x
component
of the polarized E2x (0) = E1x (L1
wave ).
at
z2
=
0 is (17)
Owing to Faraday magneto-optical effect, the
plane of polarization will rotate when the probe
beam propagates through the cell 2 of length L2. By virtue of Equations (11), (14) and (17), at
z2 = L2, the transmitted wave which is parallel to the y-direction can be written as
E2y
(L2 )
=
i
E0 4
éëexp(ik+L1)
+
exp(ik-L1)ûù
´
(18)
´ ëéexp(ik+¢ L2) - exp(ik-¢ L2)ûù yˆ exp(-iωt).
The transmission Tr of the RAAF (z1 = 0 →
→ z2 = L2), defined by Tr = I2y(L2)/I1x(0), is then
given by
Tr =
E2y (L2) 2 E1x (0) 2
=
(19)
=
1 2
exp(GL1
-
αL2
)[cosh(∆αL2
)
-
cos(2ρL2
)].
Here α , Δα and ρ are the mean absorption coefficient, circular dichroism, and rotatory power, respectively. They can be written as
α=
α+¢
+ α-¢ 2
=
ωp 2c
Im(χ+¢
+
χ-¢ ),
∆α =
α+¢
- α-¢ 2
=
ωp 2c
Im(χ+¢
- χ-¢ ),
ρ=
ωp 2c
(n+¢
-n-¢ )
=
ωp 4c
Re(χ+¢
-
χ-¢ ).
Here χ¢± are the dielectric susceptibilities seen by the left and right circularly polarized com-
ponents of the probe light in the magnetic field.
It can be given by our previous work [28]
χ±¢
=
i
3 8π
ln2e2 πmε0 ν21
´
å´
n,j
NeF f ∆νDn
CF
C3j
C6j
W
(δνn
± δνnsj
+ ia).
(20)
14 “Оптический журнал”, 81, 4, 2014
where
CF = (2J1 +1)(2F1 +1)(2F2 +1),
C3j = çèçæç-Fm2F
1 ±1
F1 mF
1ø÷ö÷÷÷2
,
C6j = ìïíïïîïJF21
1 I
F1 J2
üþïïïýï2
,
òW(δνn
±
δνnsj
+
ia)
=
i π
+¥ -¥
δνn
exp(-t2 ) ± δνnsj + ia
-
tdt,
δνn = 2
ln2 ν- νoF ∆νDn
,
δνnsj = 2
ln2 ∆νnsj , a = ∆νDn
ln2 ∆νL . ∆νDn
Here e and m are the charge and mass of an
electron, respectively, Δνnsj is the Zeeman hyperfine splitting depending on the external magnetic
field strength, NeF is the population density of the ground-state hyperfine level F of Rb atoms,
f is the absorption oscillator strength of Rb D2 line, C3j is 3j coefficient, C6j is the 6j coefficient, ΔνDn is the Doppler width, and ΔνL is the Lorentz width.
laser which is ~ 1GHz detuned from the |5S1/2, F = 3ñ → |5P3/2, F = 3ñ pump transition of 85Rb atom. The beam diameter dcoupling of the coupling light is 1.5×10–3 m. The gain cell length L1 is 0.1 m and its temperature T1 = 376 K. The Faraday cell length L2 is 0.04 m and its temperature T2 = 372 K. The axial magnetic strength B in the Faraday cell is 0.0205 T. According
to Equation (19), the peak transmission of the
Raman-amplified atomic filter as a function of
probe detuning from a resonance transition for 5S1/2(F = 3) → 5P3/2 of 85Rb in an incoherently pumped field (pump detuning Δp ≈ –1 GHz ) is shown in Fig. 3. It can be found that the RAAF
displays a single dominant peak with transmission of 18.4 (corresponding to G = 29) at 4.0 GHz relative to the absorption peak 1a with FWHM
(full width at half maximum) of 66 MHz. The an-
alytical results correspond greatly to the experi-
mental results carried out by Shan [27].
Fig. 4a, b show an atomic absorption spectrum
in cell 1 and a typical transmission spectrum
Transmission
Results and discussion
It is assumed that the coupling light power Pcoupling (Pcoupling ≈ áE1〉2) is 236 mW, and the detuned value ∆1 of its central frequency is 4 GHz red detuned from the transition |1ñ = |5S1/2, F = 2ñ → |3ñ = |5P3/2, F = 3ñ of 85Rb D2 line. The coupling light also acts as the pumping
20
Tr1
16
12
8
4
Tr1a
Tr3 Tr4
0
−8 −4
0
4
Probe Detuning, GHz
Fig. 3. The transmission of the Raman-amplified atomic filter versus the probe detuning. Pcoupling = 236 mW, dcoupling = 1.5×10−3 m, T1 = = 376 K, T2 = 372 K, L1 = 0.1 m, L2 = 0.04 m, B = 0.0205 T and Δ1 = –4.0 GHz, Δp = –0.964 GHz.
Transmission
Absorption
(a)
1
0.9 1 0.8 3 0.7 2
0.6 −8 −6 −4 −2 0 2 4
4
6
8 10
1 0.8 0.6 0.4 0.2
0 −10
(b)
Tr1 FWHM
Tr1a 1.98 GHz
Tr3 Tr4
−8 −6 −4 −2 0 2 4 6
Frequency Detuning, GHz
Tr2
8
10
Fig. 4. The atomic absorption spectrum in cell
1 without coupling light at room temperature
(293 K) (a). This spectrum is used to calibrate
the transmission spectrum of the filters. The
absorption peaks 1, 2, 3, and 4 correspond to
the F= and
87Rb 3ñ → 87Rb
|5|5SP13/2/2,ñ,F85=R2bñ|5→S1/|52,P3F/2=ñ, |5S1/2, F = 1ñ → |5P3/2ñ
85Rb |5S1/2, 2ñ → |5P3/2ñ transitions,
respectively. The position of absorption peak 2
is set to zero detuning point. The transmission
spectrum of the dispersive atomic filter without
the coupling light (b). T1 = 376 K, T2 = 372 K, L1 = 0.1 m, L2 = 0.04 m and B = 0.0205 T.
“Оптический журнал”, 81, 4, 2014
15
of the FADOF in cell 2 without coupling light, respectively. The FADOF displays a 1.98 GHz bandwidth (FWHM) at a maximum peak (Tr1) transmittance of 94.6%. It should be noted that the Rb-FADOF possesses a secondary maximum peak Tr2 in the strong magnetic field [9], while it doesn’t emerge in reference [27] mainly because the transmission frequency of peak Tr2 is beyond the scan range of signal-light frequency (≈15 GHz).
Fig. 5a, b represent the gain coefficient, transmission and FWHM of the RAAF in the incoherent pump field [27] as a function of coupling-light power, respectively. It can be seen that with the coupling-light power (Pcoupling) increasing from 60 to 95 mW, the RAAF can achieve a larger gain, higher transmission and a narrower bandwidth. When Pcoupling varies from 95 to 500 mW, the filter can obtain an ultra-narrow bandwidth of 66 MHz and the transmission coefficient of over 2.5. It is worth mentioning that the saturation of transmission coefficient appears when Pcoupling attains to approximatelly 472 mW, as shown in Fig. 5a, which means that too larger coupling-laser power is unnecessary to get a higher transmission.
The outstanding characteristic of the RAAF is the light amplification by comparison with traditional FADOFs. According to Equations (6) and (7), we can get
ρ33 ρ22
=
Γ
Γ + γ 32
TRANSMISSION CHARACTERISTICS OF A RAMAN-AMPLIFIED ATOMIC OPTICAL FILTER IN RUBIDIUM AT 780 nm
© 2014 г. Wenjin Zhang, Yufeng Peng College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, China Е-mail: yufengp@sohu.com
The transmission characteristics of a Raman-amplified atomic filter that can be used to detect fairly weak signals in free-space quantum-key distribution or laser communications are analyzed and discussed in the coherent and incoherent pump fields respectively. The theoretical model for the calculation of the transmission characteristics of a groundstate Raman-amplified Faraday dispersion atomic optical filter based on Raman gain and Faraday rotation is presented. The results show that the filter in a coherent pump field can achieve higher transmission and larger tunability than that in an incoherent pump field due to elimination of pumping detuning. In addition, the filter has a large scale tunability over 3.5 GHz via the Faraday transmission peak adjusted while its bandwidth is only 66 MHz, which is useful for free-space laser communication and lidar systems.
Keywords: atomic optical filter, Raman light amplification, Rubidium, Faraday dispersion, hyperfine structures.
OCIS codes: 120.2440
Submitted 17.10.2013
Introduction
The Faraday anomalous dispersion optical filter (FADOF), which is based on the anomalous dispersion of Faraday rotation near the resonance lines of certain atomic vapours in a longitudinal magnetic field, have many advantages such as high transmission, narrow bandwidth, excellent out-of-band rejection and wide field of view (±π/2) [1–3]. It can be operated either at the line center [4–6] or in the wings [7–9] of the atomic resonance line subjected to given magnetic fields. Therefore, it has been used as a very efficient spectralfiltering device in many signal-detection systems [10–13], recently in the free-space quantum key distribution (QKD) system [14, 15], which could extract very weak signals from strong background noise and operate well in daytime and at night due to adopting the Faraday dispersive atomic filters.
A Faraday dispersive atomic filter was first described and demonstrated by Ohman (1956) [16]. The theoretical framework of a FADOF considered fine structures of atomic system was established by Yeh (1982) [1]. Yin and Shay (1991) [2] (Harrell et al., 2009 [17]) developed the theoretical framework of the FADOF based on the hy-
perfine structures of atomic system and groundstate transitions in the weak (any) magnetic field by using a quantum-mechanical treatment of the resonant Faraday effect. The transmission characteristics of several alkali atomic FADOFs have been observed and discussed by Menders et al. (1991, Cs D2 line) [3], Peng et al. (1993, Rb D2 line) [18], Hu et al. (1998, Na D lines) [5], Zhang et al. (2001, K D lines) [6], and Zielin′ ska et al. (2012, Rb D1 line) [19], etc. Though those filters achieved good performance, e.g., above 70% transmission (Zhang et al., 2001 [6]), and approximately 1 GHz ultra-narrow bandwidth (Zielin′ ska et al., 2012 [19]), they couldn’t have signal gain or amplification ability for the weak light. As for the weak light amplification, since the early 1990s, the mechanisms, which is based on the Raman gain attributed to stimulated Raman scattering and/or the gain from the dressed state inversion, have been extensively studied (Imamoglu et al., 1991 [20], Zibrov et al., 1995, [21], Zhu et al., 1996 [22], Zhu, 1997 [23], Wanare, 2002 [24], Kang et al., 2003 [25], Kilin et al., 2008 [26], etc.).
In 2008, Shan et al. [27] demonstrated experimentally an ultranarrow-bandwidth Ramanamplified atomic filter (RAAF) that was
“Оптический журнал”, 81, 4, 2014
11
a composite of a Raman light amplifier (RLA) and a dispersive atomic optical filter (i.e. FADOF), which achieved a weak light amplification of about 55 times, a bandwidth of ~ 60 MHz, and frequency tunability of ~ 1 GHz in an incoherent pump field. Compared with a conventional FADOF, the RAAF could be more suitable for operating in the free-space QKD or laser-communication systems; however, the detailed theoretical model and the analyses of transmission characteristics for this category of filter in the coherent (incoherent) pump field have not been reported.
In this paper, taking into account Raman gain mechanism in the Doppler-broadened atomic system, the theoretical model for the transmission characteristics of a Raman-amplified Faraday dispersion atomic filter is presented by basing on Raman gain involving the coupling field and pump field and Faraday rotation in the magnetic field. The gain properties, transmission and passband width of the filter subjected to the coherent and incoherent pump fields are analyzed and discussed respectively. We obtain not only the filter frequency tunability of over 3.5 GHz, which is important for free-space optical communication and lidar systems subjected to large Doppler shift. In addition, the other characteristics such as the transmission, gain, and bandwidth of the filter under the action of the coherent and incoherent pump fields are discussed.
1. Theory
The schematic diagram of the RAAF apparatus is shown in Fig. 1a. The coupling, probe (i.e. signal) and pump lasers are emitted from three different lasers, respectively. The coupling laser and probe laser have linear orthogonal polarizations and propagate in the same direction, opposite to that of the pump laser. After the three laser beams enter into the cell 1, the signal gain occurs under the combined action of coupling laser and pump laser; the probe field is amplified. The amplified probe light passes through the P4 and P5 to the Faraday cell 2, while the coupling light is distinguish by the P4. Due to the Faraday rotation effect in an axial magnetic field, the amplified signal light which frequency lies only in the transmission passband of the dispersive atomic filter, can experience a 90° rotation of the polarization and emerge apparently low absorption and high transmission.
The three-level Λ-type system for Rubidium
atoms with the laser-coupled D2 transitions is depicted in Fig. 2. Rubidium has only one stable isotope, 85Rb, with the isotope 87Rb, which com-
poses almost 28% of naturally occurring Rubidium (72.17% 85Rb and 27.83% 87Rb). The
levels |1ñ, |2ñ and |3ñ represent the ground states |5S1/2, F = 2ñ, |5S1/2, F = 3ñ and the exited state 5P3/2 of 85Rb atom, and like wise, the ground
Pump Laser
(а) M
Coupling Laser Signal Laser
P1 M
HW P2
P5 B P6 D
A P3M Rb cell1 M P4 Rb cell2
Signal Laser
x
y
(b)
z2 = L2 z z1 = 0, z1 = L1 , z2 = 0
Fig. 1. Schematic diagram of the RAAF apparatus (a). P1−4 stand for polarizing beam splitters (PBS), M for a mirror, P5−6 for Glan− Thompson prism polarizers, HW for a halfwave plate, A for an attenuator, and D for a photodiode detector. Schematic of the rotation of the signal-light plane of polarization (b).
F = 1,2,3,4
5P3/2
∆
3 ∆1
F= 3 5S1/2
Pump Laser
Signal Laser
3.036 GHz
2 1
Coupling Laser
F= 2 5S1/2
Fig. 2. 85Rb far-detuned three-level Λ system. Δ1(Δ) is the coupling (probe) detuning. For 87Rb atoms, the ground hyperfine states are 5S1/2, F = 2 and 5S1/2, F = 1, and the excited hyperfine states are 5P3/2, F = 3, 2, 1, 0.
12 “Оптический журнал”, 81, 4, 2014
states |5S1/2, F = 1ñ, |5S1/2, F = 2ñ and the exited state 5P3/2 for 87Rb atom, respectively. A coupling laser is used to connect the ground state 5S1/2 (F = 2 for 85Rb or F = 1 for 87Rb) ↔ the excited
state 5P3/2 transition, and a probe laser couples the 5P3/2 ↔ 5S1/2 (F = 3 for 85Rb or F = 2 for 87Rb) transition. A pump laser with a rate Г is applied to the Rb 5S1/2 (F = 3 for 85Rb or F = 2 for 87Rb) ↔ 5P3/2 transition and excites the atoms to the Rb 5P3/2 excited state.
According to above three-level system, and the
electric-dipole and the rotating-wave approxima-
tions, the Hamiltonian in the interaction picture
can be expressed as
H1
=
-
2
êêëéêêΩ00
0
-2(∆1 - ∆) g
Ω g -2∆1
úúûúúù
.
(1)
Here Δ1 is the detuning of the coupling beam from the atomic transition ω31 and Δ is the detuning of the probe field from the atomic transition
ω32; Ω = μ31E1/ћ and g = μ32E2/ћ are the Rabi frequencies of the coupling laser (with amplitude
E1) and probe laser (with amplitude E2); and μ31 and μ32 are the relevant dipole moments. For convenience’s sake, the Rabi frequencies are taken
as real parameters, and the quantum dynamics of
the three-level atomic system can be described by
the Liouville equation
¶ρ ¶t
=
1 i
[H,ρ]-
1 2
{γ
,ρ}.
(2)
The full statements of Equation (2) are expressed as follows:
ρ11 = γ 31ρ33 + iΩ(ρ31 -ρ13) 2,
ρ22 = -Γρ22 +(Γ + γ 32 )ρ33 + ig(ρ32 -ρ23) 2,
ρ33 = Γρ22 -(Γ + γ 32 + γ 31)ρ33 + +iΩ(ρ13 -ρ31) 2 + ig(ρ23 -ρ32) 2,
ρ12 = -[Γ 2- i(∆1 - ∆)]ρ12 - igρ13 2 + iΩρ32 2, ρ13 = -[Γ 2 + γ 32 2 + γ 31 2- i∆1 ]ρ13 -
- igρ12 2 + iΩ(ρ33 -ρ11) 2,
ρ23 = -[Γ + γ 32 2 + γ 31 2- i∆]ρ23 -
- iΩρ21 2 + ig(ρ33 -ρ22) 2,
ρ11 + ρ22 + ρ33 = 1.
(3)
Here γij (i, j = 1–3) are the radiative decay rates from state |iñ to state |jñ. Since the atomic
filter is used to detect a weak optical signal, we
neglect the second order in g and solve Equation
(3) in the steady state to get
ρ32 =
ig{(ρ22 -ρ33)(Γ + γ 31 + γ 32 -2i∆1)
(Γ + γ 31 + γ 32 -2i∆1)
´
{ } }´
éëΓ -2i(∆1 - ∆)ùû + Ω2(ρ33 -ρ11) (2Γ + γ 31 + γ 32 + 2i∆)éëΓ -2i(∆1 - ∆)ùû + Ω2
,
(4)
ρ11
=
Ω2
Ω2Γ(Γ (Γ + γ 31
+ +
γ 31 γ 32
+ γ 32) )(3Γ +
+ γ 32
)
+
+ Γγ 31 +Γγ 31
ëêé4∆12 ëéê4∆12
+ (Γ + (Γ
+ +
γ 31 γ 31
+ +
γ 32)2 ûùú γ 32)2 úùû
,
ρ22
=
Ω2
(Γ
Ω2 (Γ + γ 31 + γ 32)´ + γ 31 + γ 32)(3Γ + γ 32
)
+
+
Γγ
31
´ éëê4∆12
(Γ +
+ (Γ
γ 32 +γ
)
31
+
γ
32
)2
ùúû
,
ρ33
=
Ω2
(Γ
+
γ
31
Ω2Γ ´ + γ 32)(3Γ
+
γ
32 )
+
+Γγ
31
´(Γ + γ 31 + γ 32) êéë4∆12 + (Γ + γ 31 +
γ
32 )2
ûúù
.
(5) (6) (7)
Considering the atoms interacting with the
optical field, the complex susceptibility χ± at the probe field frequency can be obtained from the
polarization
P±
(z,t)
=
1 2
ε0χ±
E±
(z,t)eˆ±
exp
éëi(kz
-
ωt)ûù
+
c.c.
=
(8)
( )= N0 µ±ρ3±2eˆ± + c.c. .
Consequently, the atomic susceptibility χj± of the jth σ± probe transition in the cell 1 can be
written as
χj±
=
µ2j± N0 ε0 g±
ρ±32,
(9)
where +(–) denotes the left(right)-hand circular component, ћ is the Plank’s constant divided by 2p, ε0 is the permittivity of free space, μj± are the relevant dipole moments, and N0 (with unit atom/cm3) is the total atomic number density, given by [28]
lg(N0) = -4529.6 T -3.991lgT + + 0.00059T + 34.8325.
(10)
“Оптический журнал”, 81, 4, 2014
13
We consider the coupling laser and probe laser
passing through a Doppler-broadened atomic cell
1 in the same direction. An atom moving towards
the probe beam (with frequency ωp) with velocity ν is affected by the probe frequency detuning
upshifted to (Δ + ωpν/c) and the frequency detuning of the coupling beam (with frequency ωc) upshifted to Δ1 + ωcν/c. Considering that atoms are in classical thermal equilibrium with the one-
dimensional Maxwellian velocity distribution of N(ν)dν = N0exp(–ν2/u2)/(u√`p)dν (u is the most probable velocity and is defined as u = (2kBT/M)1/2), the final value of χ± is obtained by integrating over the atomic velocity ν and summing over all
the components of Equation (9)
å òχ± =
2N0 πuε0
j
µ2j± g±
+¥ -¥
ρ3±2e-ν2
u2 dν.
(11)
1.1. RAAF gain
We assume that an x-direction linearly polarized probe beam enters the atomic gas cell 1 of length L1 (Fig. 1b) at z1 = 0 and travels along the positive z-direction. The electric radiation field of the probe beam can be expressed in terms of its right- and left-circularly polarized component as
E (z)
=
E0 2
{ëéexp(ik+z)
+
exp(ik-z)ûù
xˆ
+
+i ëéexp(ik+z) - exp(ik-z)ùû yˆ} exp(-iωt),
(11)
where k± are the circular wave-number components, given by
k±
=
ω c
n±
+
i 2
α±.
(12)
By virtue of Equation (11), at z1 = 0 and L1, the x components of this probe beam may be writ-
ten as
E1x (0) = E0xˆ exp(-iωt),
(13)
=
E0 2
E1x (L1) = ëéexp(ik+L1) + exp(ik-L1)ùû xˆ exp(-iωt),
(14)
respectively. The transmission coefficient of the
RLA, defined by TrRLA = I1х(L1)/I1х(0), is then
given by
TrRLA
=
E1x (L1) 2 E1x (0) 2
=
=
exp(-
α+
+ 2
α-
L1)
=
exp(GL1),
(15)
where G is the mean gain coefficient of the RAAF. It can be written as
G
= - α+
+ α2
=
-
ωp 2c
Im(χ+
+
χ- ),
where c is the speed of light in free space.
(16)
1.2. RAAF transmission
When the amplified probe beam goes through
the cell 1, polarizing beam splitter P4 and Glan–
Thompson prism polarizer P5 to the Faraday cell 2,
the
x
component
of the polarized E2x (0) = E1x (L1
wave ).
at
z2
=
0 is (17)
Owing to Faraday magneto-optical effect, the
plane of polarization will rotate when the probe
beam propagates through the cell 2 of length L2. By virtue of Equations (11), (14) and (17), at
z2 = L2, the transmitted wave which is parallel to the y-direction can be written as
E2y
(L2 )
=
i
E0 4
éëexp(ik+L1)
+
exp(ik-L1)ûù
´
(18)
´ ëéexp(ik+¢ L2) - exp(ik-¢ L2)ûù yˆ exp(-iωt).
The transmission Tr of the RAAF (z1 = 0 →
→ z2 = L2), defined by Tr = I2y(L2)/I1x(0), is then
given by
Tr =
E2y (L2) 2 E1x (0) 2
=
(19)
=
1 2
exp(GL1
-
αL2
)[cosh(∆αL2
)
-
cos(2ρL2
)].
Here α , Δα and ρ are the mean absorption coefficient, circular dichroism, and rotatory power, respectively. They can be written as
α=
α+¢
+ α-¢ 2
=
ωp 2c
Im(χ+¢
+
χ-¢ ),
∆α =
α+¢
- α-¢ 2
=
ωp 2c
Im(χ+¢
- χ-¢ ),
ρ=
ωp 2c
(n+¢
-n-¢ )
=
ωp 4c
Re(χ+¢
-
χ-¢ ).
Here χ¢± are the dielectric susceptibilities seen by the left and right circularly polarized com-
ponents of the probe light in the magnetic field.
It can be given by our previous work [28]
χ±¢
=
i
3 8π
ln2e2 πmε0 ν21
´
å´
n,j
NeF f ∆νDn
CF
C3j
C6j
W
(δνn
± δνnsj
+ ia).
(20)
14 “Оптический журнал”, 81, 4, 2014
where
CF = (2J1 +1)(2F1 +1)(2F2 +1),
C3j = çèçæç-Fm2F
1 ±1
F1 mF
1ø÷ö÷÷÷2
,
C6j = ìïíïïîïJF21
1 I
F1 J2
üþïïïýï2
,
òW(δνn
±
δνnsj
+
ia)
=
i π
+¥ -¥
δνn
exp(-t2 ) ± δνnsj + ia
-
tdt,
δνn = 2
ln2 ν- νoF ∆νDn
,
δνnsj = 2
ln2 ∆νnsj , a = ∆νDn
ln2 ∆νL . ∆νDn
Here e and m are the charge and mass of an
electron, respectively, Δνnsj is the Zeeman hyperfine splitting depending on the external magnetic
field strength, NeF is the population density of the ground-state hyperfine level F of Rb atoms,
f is the absorption oscillator strength of Rb D2 line, C3j is 3j coefficient, C6j is the 6j coefficient, ΔνDn is the Doppler width, and ΔνL is the Lorentz width.
laser which is ~ 1GHz detuned from the |5S1/2, F = 3ñ → |5P3/2, F = 3ñ pump transition of 85Rb atom. The beam diameter dcoupling of the coupling light is 1.5×10–3 m. The gain cell length L1 is 0.1 m and its temperature T1 = 376 K. The Faraday cell length L2 is 0.04 m and its temperature T2 = 372 K. The axial magnetic strength B in the Faraday cell is 0.0205 T. According
to Equation (19), the peak transmission of the
Raman-amplified atomic filter as a function of
probe detuning from a resonance transition for 5S1/2(F = 3) → 5P3/2 of 85Rb in an incoherently pumped field (pump detuning Δp ≈ –1 GHz ) is shown in Fig. 3. It can be found that the RAAF
displays a single dominant peak with transmission of 18.4 (corresponding to G = 29) at 4.0 GHz relative to the absorption peak 1a with FWHM
(full width at half maximum) of 66 MHz. The an-
alytical results correspond greatly to the experi-
mental results carried out by Shan [27].
Fig. 4a, b show an atomic absorption spectrum
in cell 1 and a typical transmission spectrum
Transmission
Results and discussion
It is assumed that the coupling light power Pcoupling (Pcoupling ≈ áE1〉2) is 236 mW, and the detuned value ∆1 of its central frequency is 4 GHz red detuned from the transition |1ñ = |5S1/2, F = 2ñ → |3ñ = |5P3/2, F = 3ñ of 85Rb D2 line. The coupling light also acts as the pumping
20
Tr1
16
12
8
4
Tr1a
Tr3 Tr4
0
−8 −4
0
4
Probe Detuning, GHz
Fig. 3. The transmission of the Raman-amplified atomic filter versus the probe detuning. Pcoupling = 236 mW, dcoupling = 1.5×10−3 m, T1 = = 376 K, T2 = 372 K, L1 = 0.1 m, L2 = 0.04 m, B = 0.0205 T and Δ1 = –4.0 GHz, Δp = –0.964 GHz.
Transmission
Absorption
(a)
1
0.9 1 0.8 3 0.7 2
0.6 −8 −6 −4 −2 0 2 4
4
6
8 10
1 0.8 0.6 0.4 0.2
0 −10
(b)
Tr1 FWHM
Tr1a 1.98 GHz
Tr3 Tr4
−8 −6 −4 −2 0 2 4 6
Frequency Detuning, GHz
Tr2
8
10
Fig. 4. The atomic absorption spectrum in cell
1 without coupling light at room temperature
(293 K) (a). This spectrum is used to calibrate
the transmission spectrum of the filters. The
absorption peaks 1, 2, 3, and 4 correspond to
the F= and
87Rb 3ñ → 87Rb
|5|5SP13/2/2,ñ,F85=R2bñ|5→S1/|52,P3F/2=ñ, |5S1/2, F = 1ñ → |5P3/2ñ
85Rb |5S1/2, 2ñ → |5P3/2ñ transitions,
respectively. The position of absorption peak 2
is set to zero detuning point. The transmission
spectrum of the dispersive atomic filter without
the coupling light (b). T1 = 376 K, T2 = 372 K, L1 = 0.1 m, L2 = 0.04 m and B = 0.0205 T.
“Оптический журнал”, 81, 4, 2014
15
of the FADOF in cell 2 without coupling light, respectively. The FADOF displays a 1.98 GHz bandwidth (FWHM) at a maximum peak (Tr1) transmittance of 94.6%. It should be noted that the Rb-FADOF possesses a secondary maximum peak Tr2 in the strong magnetic field [9], while it doesn’t emerge in reference [27] mainly because the transmission frequency of peak Tr2 is beyond the scan range of signal-light frequency (≈15 GHz).
Fig. 5a, b represent the gain coefficient, transmission and FWHM of the RAAF in the incoherent pump field [27] as a function of coupling-light power, respectively. It can be seen that with the coupling-light power (Pcoupling) increasing from 60 to 95 mW, the RAAF can achieve a larger gain, higher transmission and a narrower bandwidth. When Pcoupling varies from 95 to 500 mW, the filter can obtain an ultra-narrow bandwidth of 66 MHz and the transmission coefficient of over 2.5. It is worth mentioning that the saturation of transmission coefficient appears when Pcoupling attains to approximatelly 472 mW, as shown in Fig. 5a, which means that too larger coupling-laser power is unnecessary to get a higher transmission.
The outstanding characteristic of the RAAF is the light amplification by comparison with traditional FADOFs. According to Equations (6) and (7), we can get
ρ33 ρ22
=
Γ
Γ + γ 32