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Spin Dependent Steady State and Transient Gain Characteristics of 

Stimulated Brillouin Scattering in Magnetized Quantum Plasma 

 

Gopal Chand Dangi1, Ravi Vanshpal2 Ratna Agrawal3 

1Department of Physics, Shri Vaishnav Vidyapeeth Vishwavidyalaya Indore (M.P.) India 

2Department of Physics, Shri Vaishnav Vidyapeeth Vishwavidyalaya Indore (M.P.) India 

3School of Studies in Physics, Vikram University, Ujjain 456010, India 

 

   Article History:  

 Received: 14-11-2024 

Revised: 15-12-2024 

Accepted: 21-01-2025 

ABSTRACT 

The steady state stability and the transient gain features of stimulated Brillouin gain in the 

semiconductor plasmas are examined whenever an expanded quantum magnetohydrodynamic 

formula is considered. The model includes primeval quantum corrections including the Bohm 

potential, and also spin-generated effects of magnetization to better explain the electronic fluid 

behavior under excitation of the electromagnetic field. This is because the third order nonlinear 

susceptibility is responsible in the amplification mechanism known as the Brillouin amplification 

mechanism and therefore this is as a result of the nonlinear current density and electrostrictive 

coupling of the plasma medium. Our analysis demonstrates that both spin polarization and quantum 

corrections significantly alter the SBS gain dynamics. Notably, the spin effect enhances the 

Brillouin gain profile and leads to a substantial reduction in the threshold pump intensity, thereby 

improving the efficiency of SBS generation. These results underscore the critical role of spin 

dynamics in tailoring nonlinear optical responses in semiconductor plasmas and offer valuable 

insights for the development of spin-dependent photonic systems, plasma-based amplifiers, and 

quantum sensing technologies. 

 

1. INTRODUCTION 

The interaction of intense electromagnetic fields with dense media has led to the emergence of 

nonlinear optical phenomena such as Stimulated Brillouin Scattering (SBS), a process wherein a 

monochromatic optical wave couples with an induced acoustic wave through the mechanism of 

optical electrostriction. This interaction facilitates the generation and amplification of coherent optical 

radiation with fine spectral tunability and has been extensively studied in plasmas and condensed 

matter systems [1-3]. SBS has become a vital technique in photonics, offering applications ranging 

from laser frequency stabilization and signal processing to high-resolution spectroscopy and optical 

phase conjugation [4-5]. 

In recent years, the integration of quantum mechanical effects into classical plasma models has opened 

new avenues for understanding plasma dynamics at nanoscales. Notably, the role of the electron's 

intrinsic spin [6], once considered negligible in many plasma systems, has gained attention for its 

significant impact on wave dispersion, stability, and transport properties. Electron spin effects have 

not only enriched the theoretical framework of plasma physics but also inspired emerging applications 

in spintronics, quantum computation, and spin-based diagnostic techniques [7,8]. The spin of charged 

particles introduces additional magnetic moment interactions, leading to spin-current coupling and 

spin-induced forces, which are particularly relevant in magnetized environments such as astrophysical 

plasmas and semiconductor-based quantum plasmas. 

To rigorously account for these quantum spin effects, the Quantum Magnetohydrodynamic model has 

been developed [9]. This model extends the classical Magnetohydrodynamic (MHD) framework by 

incorporating quantum statistical pressure or Fermi pressure, Bohm quantum potential (accounting 

for quantum diffraction), and spin-induced magnetization effects via the Pauli equation formalism. 



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The QMHD model thus provides a robust platform to study collective plasma behaviors influenced 

by both quantum mechanics and magnetism [10]. 

In semiconductor plasmas, particularly in piezoelectric crystals, quantum mechanical effects become 

increasingly significant due to the reduced carrier density, small effective masses, and the ability to 

confine carriers in nanostructured geometries [11]. These factors enhance the de Broglie wavelength 

of carriers, making quantum tunneling, nonlocality, and spin interactions essential to consider [12-

13]. Under high magnetic fields or near cyclotron resonance, these systems exhibit enhanced 

nonlinearities, enabling experimental access to regimes where classical models fall short. 

Despite the extensive literature on SBS in classical and quantum plasmas, the combined influence of 

quantum spin dynamics on steady-state and transient Brillouin gain characteristics has remained 

relatively unexplored. In response to this shortcoming, the proposed work examines, to the best of our 

knowledge, the spin induced quantum corrections on SBS in magnetized semiconductor plasmas 

using the QMHD model. We analytically derive the modified Brillouin gain profiles by incorporating 

spin magnetization forces and quantum potentials. Our results reveal that spin effects not only enhance 

Brillouin gain constants but also lower the threshold pump intensity for SBS excitation. 

This manuscript is structured as follows: Section 2 outlines the theoretical foundation based on the 

QMHD model and derives the governing equations relevant to stimulated Brillouin scattering (SBS) 

in spin-influenced semiconductor plasmas. The subsections 2.1 and 2.2 detail the steady-state and 

transient gain characteristics, respectively, while Section 2.3 elaborates on the nonlinear structure of 

the model and its broader implications. Section 3 presents numerical simulations carried out for an n-

InSb semiconductor crystal subjected to pulsed CO₂ laser excitation, highlighting the influence of spin 

polarization and quantum effects. Finally, Section 4 summarizes the principal findings and discusses 

their relevance to future developments in quantum plasma technologies and spin-sensitive optical 

applications. 

2. THEORETICAL FORMULATIONS 

Here in this section, we have talked about the field theoretical expression of the third order nonlinear 

optical susceptibility 𝜒𝐵
(3)

of the Stokes part of the scattered electromagnetic wave of the doped 

semiconductor QMHD model. We have chosen a model which is the magneto hydrodynamical model 

of a homogenous one component (electron) plasma in thermodynamic equilibrium and that satisfies 

the condition 𝑘𝑎𝑙 << 1 (𝑘𝑎 is acoustic wave vector and l is the average distance which electrons 

move between collisions). The implication that follows this assumption is that the sound wavelength 

is very long relative to the mean free path of the electrons in the structure the carrier motion dictated 

with the external fields will smooth out. It also empowers us to disregard high frequency electric field 

non uniformity in dipole approximation [14]. To obtain third order Brillouin susceptibility induced 

by the induced polarization and electrostriction, the incident pump radiation 𝐸0(𝑥, 𝑡) =
�̂�𝐸0exp⁡[𝑖(𝑘0𝑥 − 𝜔0𝑡)] is expected to be travelling along x direction and is polarized along z 

direction. The longitudinal polarized acoustic wave �⃗� (𝑥, 𝑡) = �⃗� 0exp⁡[𝑖(𝑘𝑎𝑥 − 𝜔𝑎𝑡)] is also assumed 

to be travelling along 𝑥 direction. The Brillouin back scattered Stoke wave 𝐸1(𝑥, 𝑡) =
�̂�𝐸1exp⁡[𝑖(−𝑘1𝑥 − 𝜔1𝑡)] is traveling in -x direction and is polarized along z direction. So that SBS 

can be investigated within a medium, the phase matching criteria that must be achieved in the present 

case are: ℏ𝜔0 = ℏ𝜔1 + ℏ𝜔𝑎 and ℏ𝑘0 = ℏ𝑘1 + ℏ𝑘𝑎. These conditions provide 𝜔1 = 𝜔0 − 𝜔𝑎 and 

𝑘𝑎 = 2𝑘𝑠 (since |𝑘0| ≈ |𝑘1| ). Since the crystal is supposed to be centro-symmetric, the impact of any 

pseudo-potential can be ignored in order to simplify the analysis. 

In these conditions when the density is considerably high and the plasma cooled into the quite low 
level of temperature, then the ultracold plasma will serve as the degenerate fermion gas and the 

quantum effects will be rather significant in dynamics of the charged particles [15]. For the 

appreciation of the above fact for this problem let us calculate the TF in terms of carrier density (𝑛0 =



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3 × 1024𝑚−3) through standard formula. It comes out to for InSb. We have considered the lattice 

temperature as 77 K, hence here T<<TF  (6.27 × 103𝐾) which defines a fully degenerate quantum 

regime.  

To make the discussions definite we suppose that the pressure law in one dimensional Fermi gas is 

satisfied by the plasma particles [16] 

𝑷 =
𝒎𝑽𝑭𝜶

𝟐𝒏𝟏𝜶
𝟓/𝟑

𝟓𝒏𝟎𝜶
𝟐/𝟑                                                  (1) 

Where P stands for Fermi pressure with 𝑉𝐹𝛼𝑗 = √𝜉3𝐷ℏ(𝑛0𝛼⁡1/3)/𝑚 is the Fermi velocity. 𝜉3𝐷 is 

degree of spin polarization given by 𝜉3𝐷 = [(1 − 𝜂)5/3 + (1 + 𝜂)5/3]/2 with spin polarization ( 𝜂 ) 

defined by 𝜂 = |𝑛↑ − 𝑛↓|/𝑛↑ + 𝑛↓, ↑ and ↓ denotes the electrons in the spin up and spin down states 

correspondingly [17]. The quantum force 𝐹𝑄 on an electron in Eq. (1) includes two terms. The first 

term is the Bohm potential of quantum diffusion of electrons, and the other one is the energy of spin 

magnetization due to spin interaction with magnetic field. Here 𝑆𝛼 is the spin of species 𝛼, 𝛼 =↑
𝑎𝑛𝑑 ↓ denotes the electrons in the spin up and spin down states correspondingly. 𝑛1𝛼 is the perturb 

number density of species 𝛼. It is an established fact that the quantum effects have no impacts on the 

transverse electromagnetic wave which is linearly polarized. 

According to Guha et al. [18] and Manfredi [19], the other fundamental equations that are used are 

𝜕2𝑢

𝜕𝑡2 −
𝐶

𝜌

𝜕2𝑢

𝜕𝑥2 + 2𝛤𝑎
𝜕𝑢

𝜕𝑡
=

𝛾

2𝜌

𝜕

𝜕𝑥
(𝐸0𝐸1𝑥

∗ )                        (2) 

𝜕𝜐0

𝜕𝑡
+ 𝜈𝜐0 =

𝑒

𝑚
𝐸0                 (3) 

𝜕𝜐1

𝜕𝑡
+ 𝜈𝜐1 + 𝜐1 (𝜐0

𝜕

𝜕𝑥
) = −

𝑒

𝑚
(𝐸1 + 𝜐1 × 𝐵1) −

1

𝑚𝑛0

𝜕𝑃

𝜕𝑥
+ 𝐹𝑄  (4) 

Where, ⁡⁡⁡𝐹𝑄 =
2𝜇𝐵

ℏ𝑚
𝛻(𝐵1. 𝑆𝛼) +

ℏ𝟐

4𝑚2𝑛

𝜕3𝑛1

𝜕𝑥3  

𝜐0
𝜕𝑛1

𝜕𝑥
+ 𝑛0

𝜕𝜐1

𝜕𝑥
+

𝜕𝑛1

𝜕𝑡
= ⁡0                                                 (5) 

𝑃𝑒𝑠 = −𝛾𝐸0
𝜕𝑢∗

𝜕𝑥
                                          (6) 

𝜕𝐸1

𝜕𝑥
=

𝑛1𝑒 +
𝛾
𝐸0

𝜕𝑢∗

𝜕𝑥
                                                      (7) 

The quantum force 𝐹𝑄 of an electron in Eq. (4) includes two terms: The former one is the Bohm 

potential as the energy that causes the quantum diffusion of the electrons and the energy of the spin 

magnetization as the outcome of the energy of spin magnetization due to spin interaction with 

magnetic field. Here 𝑆𝛼 is the spin of species 𝛼, 𝛼 =↑ 𝑎𝑛𝑑 ↓ denotes the spin up and the spin down 

electrons respectively with 𝜇 = −
𝑔𝜇𝐵

2
, 𝜇𝐵 =

𝑒ℏ

2𝑚
, 𝑔 = 2.0023192 is the electron 𝑔 factor,  𝑛1𝛼 is the 

perturb number density of species 𝛼. The quantum effects are unsurprisingly known to possess no 

effects on the transverse electromagnetic linearly polarised wave. The term proportional to ℏ2 in Eq. 

(4), accounts the quantum diffraction represented by the Planck’s constant where ℏ is the Planck’s 

constant divided by the 2𝜋. The role of this term can also be perceived in the other way round; that 

they can also be described as the quantum pressure term or as the quantum Bohm potential [20]. In 

the other applications to semiconductor physics, the Bohm potential leads to tunnelling and 

contributions to differential resistances [21]. The model has incorporated quantum statistics, and it is 

incorporated by the equation of state (Eq. (1)) that takes into consideration the Fermionic nature of 

the electrons.  Eq. (2) represents the dynamic of the lattice in the crystal, where ρ is the mass density 

of the crystal, 𝑢 is the displacement of the lattice, 𝛤𝑎 is the phenomenological damping factor of 



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acoustic mode, 𝐶 the elastic constant, γ the electrostriction coefficient of the crystal respectively. The 

conservative term at the right-hand side of Eq. (2) is proportional to the electrostrictive force, which 

is generated by the pump electric field through electrostriction. Eqs. (3) and (4) are the zeroth and 

first order oscillatory velocities of the fluid of an electron of constant mass m and charge e. 𝜈 is the 

phenomenological frequency of an electronic collision. In Eq. (3) we have neglected the pump 

magnetic field by assuming 𝜔𝑝 ≈ 𝜔𝑐 ≈ 𝜔0. The quantum correction on the Eq. (4) manifests itself in 

terms of Fermi temperature and third term on the right hand-side. The conservation of charge is 

embodied by the continuity Eq. (5). Eq. (6) show that as a direct consequence of varying electronic 

strain there is a corresponding change in the dielectric constant of a material because the modulated 

or varying electrostrictive strain then leads to an electrostrictive induced polarization𝑃𝑒𝑠. When the 

frequency of the field is very high and therefore very high in comparison with the frequencies with 

which we have become familiar as related to the movement of electrons in the medium, we may say 

that the polarizability of the medium is negligible interaction of the electrons with each other and the 

nuclei of the atoms of the medium. The space charge field 𝐸1is determined by Poisson Eq. (7) where 

휀 is dielectric constant of semiconductor. 

The electro strictive force causes the carrier density perturbation within the Brillouin active medium. 

This density perturbation is accessible in a doped semiconductor in the normal way [22] as 

𝜕2𝑛1𝛼

𝜕𝑡2
+ �̅�2

𝑝𝑛1𝛼 + 𝑛0 (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2�̅�

𝜔0
𝑆𝛼0 +

𝛽 𝜕2𝑢

𝜕𝑥2) + 𝜈
𝜕𝑛1𝛼

𝜕𝑡
=⁡⁡ �̅�

𝜕𝑛1𝛼

𝜕𝑥
      (8) 

Where �̅� =
𝑒

𝑚
𝐸0(1 +

𝜔𝑐⁡(𝜈−𝑖𝜔0)

(𝜔𝑐
2−𝜔0

2−2𝑖𝜔0𝜈)
),  𝜛𝑝

2 = 𝜔𝑝
2 + 𝑘2𝑉𝐹

′2, 𝑉𝐹
′ = 𝑉𝐹√1 + 𝛾𝑒 , 𝛾𝑒 =

ℏ2𝑘2

8𝑚𝐾𝐵𝑇𝐹
 

In deriving of Eq. (8) we have neglected the Doppler shift under the assumption that  𝜔0 >> ⁡𝜈 >

𝑘𝜐0; 𝜔𝑝 = (
𝑛0𝑒

2

𝑚
)
1 2⁄

is the plasma frequency. It is obvious that the second term on left hand side of 

Eq. (8) has the composite effect of quantum correction and Fermi dispersion. 

The perturbed concentration electron 𝑛1 will consist of two parts which can be separated as slow and 

fast ( 𝑛1 = 𝑛1𝑠 + 𝑛1𝑓 ). It is assumed that the slow part 𝑛1𝑠 is related to the low frequency acoustic 

wave ( 𝜔𝑎 ), whereas the fast part 𝑛1𝑓 oscillates only at the electromagnetic waves ( 𝜔0 ± 𝜔𝑎 ). The 

higher order terms with frequencies 𝜔0 ± 𝑝𝜔𝑎(𝑝 = 2,3,4, … ) being off resonant are neglected. In this 

case we have taken the energy of the photons ( ℏ𝜔1 ) slightly below the band gap energy (ℏ𝜔𝑔); this 

approximation enables the optical energy to be treated in terms of a small perturbation. 7 

characteristics of the sample to be altered significantly by the free charge carriers and not to be altered 

The quantum magnetohydrodynamic is used to model the stimulated Brillouin scattering (SBS) 

process in this work, where quantum pressure, spin effects, and electrostrictive feedback are 

incorporated. The resultant system is a coupled non-linear system of governing equations. The present 

formulation is highly applicable in applied nonlinear analysis because of its highly structured nature 

and sensitivity to parameters. 

2.1. Steady-state characteristics 

On combining Eq. (8) we have the following coupled equations. Regarding rotating wave 

approximation 

𝜕2𝑛1𝑠

𝜕𝑡2 + 𝜈
𝜕𝑛1𝑠

𝜕𝑡
+ 𝜛𝑝

2𝑛1𝑠 = −𝐸
𝜕𝑛1𝑓

∗

𝜕𝑥
                               (9) 

and 

𝜕2𝑛1𝑓

𝜕𝑡2 + 𝜛𝑝
2𝑛1𝑓 + 𝜈

𝜕𝑛1𝑓

𝜕𝑡
− 𝑛0 (

2𝜇𝐵

ℏ𝑚

𝑖𝑘2�̅�

𝜔0
𝑆𝛼0 +

𝛽 𝜕2𝑢

𝜕𝑥2) = −𝐸
𝜕𝑛1𝑠

∗

𝜕𝑥
     (10) 



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subscripts 𝑠 and 𝑓 represents slow and fast components respectively. Asterisk (*) is used to indicate 

the complex conjugate of the quantities. 

Based on Eqs. (9) and (10) that the slowly and rapidly varying components of the density perturbed 

are locked together by the pumping electric field. It thus becomes noticeable that the existence of the 

pump field forms the irreducible necessity of the SBS occurrence to take place. 

Using the above equations, we obtain 

𝑛1𝑠 = 𝑛0 (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2�̅�

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −
(𝛿1

2−𝑖𝜔𝑎𝜈)(𝛿2
2+𝑖𝜔1𝜈)

𝑘2|𝐸|
2 ]

−1

𝐸0
2𝐸1(𝜔1)     (11) 

where 𝑣𝑎 = √𝐶/𝜌 is the acoustic velocity in the medium. 𝛿1
2 = 𝜛𝑝

2 − 𝜔𝑎
2 and 𝛿2

2 = 𝜛𝑝
2 − 𝜔1

2. 

As can be read in the above expression (11), 𝑛1𝑠 is based on magnitude of the pump intensity (𝐼𝑖𝑛), 

where 𝐼𝑖𝑛 =
1

2
𝜂휀0𝑐|𝐸0|

2 with 𝜂 and 𝑐 being the refractive index of background of the crystal and 

velocity of light respectively. These induced density anguish affect the propagation aspect of the 

developed waves. The induced current density has a resonance on the Stoke component which is 

always given as 

𝐽1(𝜔1) = 𝑛0𝑒𝜐1 + 𝑛1𝑠
∗ 𝑒𝜐0                                             (12) 

Which on using eq (11) may be obtained as 

𝐽1(𝜔1) =
𝑖 𝜛𝑝

2�̅�+𝑛1𝑒𝑘2𝑉𝐹
′2

𝜔1
− 𝑛0 (

2𝜇𝐵

ℏ𝑚

𝑖𝑘2�̅�

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −
(𝛿1

2+𝑖𝜔𝑎𝜈)(𝛿2
2−𝑖𝜔𝑎𝜈)

𝑘2|𝐸|
2 ]

−1

𝐸0
2𝐸1(𝜔1)                         

(13) 

The first term at the right-hand side of above expression is linear term of produced current density. 

The second term holds the non-linear density current caused by the interaction of the three waves 

which interact with each other. In deriving Eq. (13) the contribution of the velocities of the oscillatory 

electron fluid in the presence of the pump and the disturbed fields can be obtained out of following 

Eqs. (3) and (4). 

Now with the induced polarization 𝑃𝑐𝑑 as time integral of induced nonlinear current density 𝐽𝑛𝑙(𝜔1), 
using eq. (13) we may obtain the following relation 

𝑃𝑐𝑑(𝜔1) = [
𝜔𝑃

2(𝑣−𝑖𝜔0)

(𝜔𝑐
2−𝜔0

2
−2𝑖𝜔0𝑣)

] (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2�̅�

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −
(𝛿1

2+𝑖𝜔𝑎𝜈)(𝛿2
2−𝑖𝜔1𝜈)

𝑘2|𝐸|
2 ]

−1

𝐸0
2𝐸1(𝜔1)

 (14) 

As we well know, the SBS process originates in that part of 𝑃𝑐𝑑(𝜔1) which is proportional to 𝐸0
2𝐸1, 

with that third-order susceptibility being the Brillouin susceptibility 𝜒𝐵
(3)

. 

Now the induced polarization at frequency 𝜔1 may also be defined as 

𝑃𝑐𝑑(𝜔1) = 휀0(𝜒𝐵
(3)

)
𝑐𝑑

𝐸0
2𝐸1(𝜔1)                                            (15) 

Using Eqs. (14) and (15) the Brillouin susceptibility with quantum correction becomes 

(𝜒𝐵
(3)

)
𝑐𝑑

= [
𝜔𝑃

2(𝑣−𝑖𝜔0)

(𝜔𝑐
2−𝜔0

2
−2𝑖𝜔0𝑣)

] (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2𝐸

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −
(𝛿1

2+𝑖𝜔𝑎𝜈)(𝛿2
2−𝑖𝜔1𝜈)

𝑘2|𝐸|
2 ]

−1

     (16) 

Equation (15) actually represents the intensity dependent Brillouin susceptibility of the medium. One 

may infer from it that (𝜒𝐵
(3)

)
𝑐𝑑

is dependent upon material parameters, including equilibrium carrier 

density n0 through the electron plasma frequency 𝜔𝑝. 



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The Brillouin susceptibility in Equation (16) contains nonlinear dependencies on the spin polarization 

𝜂, carrier density 𝑛, and field intensity 𝐸0. These nonlinearities directly influence the threshold 

behavior and gain properties of the medium. 

He-Liu [23] reported that the nonzero induced polarization directly gives the onset of SBS. Hence 

following He-Liu and others, from the Eq. (14) we can find out what the character of the threshold 

leading to the occurrence of SBS by setting𝑃𝑐𝑑(𝜔1) = 0as. This threshold corresponds to the 

vanishing nonlinear polarization condition defined in Eq. (17). 

|𝐸𝑜𝑡ℎ| =
𝑚

𝑒𝑘
√(𝛿1

2 + 𝑖𝜔𝑎𝜈)(𝛿2
2 − 𝑖𝜔1𝜈)       (17) 

The threshold pump field as the SBS starts is highly perturbed by the quantum correction with 𝛿1
2 and 

𝛿2
2. So the relations inside the pump and centrosymmetric crystal will be determined by the 

mechanisms of stimulated Brillouin scattering at the power level of the pump which is many times 

exceeds the power level of the threshold field 𝐸oth .So the acoustical and the scattered optical beam 

are released in fixed directions, and can only be created above 𝐸oth but in the common dielectric media, 

the effect of beam trapping causes the intensities to rise above the critical threshold of SBS at 

relatively modest input powers. Consequently, the threshold experimentally measured in majority of 

semiconductor materials encompasses beam trapping [24]. 

The electrostrictive strain interacts with the pump wave into the Brillouin active media generating an 

electrostrictive polarization 𝑃𝑒𝑠(𝜔1). Therefore, in addition to the induced polarization that is not 

linear, because of the disturbed current density, the system ought to have electrostrictive polarization. 

This electrostrictive Polarization 𝑃𝑒𝑠(𝜔1) is derived as one of the applications from the following 

equations (6) and (7) as 

𝑃𝑒𝑠(𝜔1) =
−𝛾2𝑘2𝐸0

2𝐸1(𝜔1)

2𝜌(𝜔𝑎
2−𝑘2𝜐𝑎

2−2𝑖𝛤𝑎𝜔𝑎)
= 휀0(𝜒𝐵

(3)
)
𝑒𝑠

𝐸0
2𝐸1(𝜔1)    (18) 

An induced nonlinear polarization per unit volume is proportional to 𝐸0
2𝐸1(𝜔1) n a centrosymmetric 

crystal doped, in which the electrostrictive terms contributing to the nonzero coupling to the square 

power are only of second order is: 

𝑃𝑛𝑙(𝜔1) = 𝑃𝑒𝑠(𝜔1) + 𝑃𝑐𝑑(𝜔1) 

= 𝐸0
2𝐸1(𝜔1) [

−𝛾2𝑘2

2𝜌(𝜔𝑎
2−𝑘2𝜐𝑎

2−2𝑖𝛤𝑎𝜔𝑎)
+

𝜔𝑃
2(𝑣−𝑖𝜔0)

(𝜔𝑐
2−𝜔0

2
−2𝑖𝜔0𝑣)

] (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2𝐸

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −

(𝛿1
2+𝑖𝜔𝑎𝜈)(𝛿2

2−𝑖𝜔1𝜈)

𝑘2|𝐸|
2 ]

−1

 = 휀0𝜒𝐵
(3)

𝐸0
2𝐸1(𝜔1)                  (19) 

Hence the total third order Brillouin susceptibility can be obtained from Eq. (19) as 

(𝜒𝐵
(3)

) = [
−𝛾2𝑘2

2𝜌(𝜔𝑎
2−𝑘2𝜐𝑎

2−2𝑖𝛤𝑎𝜔𝑎)
+

𝜔𝑃
2(𝑣−𝑖𝜔0)

(𝜔𝑐
2−𝜔0

2
−2𝑖𝜔0𝑣)

] (
2𝜇𝐵

ℏ𝑚

𝑖𝑘2𝐸

𝜔0
𝑆𝛼0 +

𝑒𝛽

𝑚

𝜕2𝑢

𝜕𝑥2) [1 −

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡
(𝛿1

2+𝑖𝜔𝑎𝜈)(𝛿2
2−𝑖𝜔1𝜈)

𝑘2|𝐸|
2 ]

−1

                                                                     (20) 

where 𝜒𝐵
(3)

= 𝜒𝐵𝑟
(3)

+ 𝜒𝐵𝑖
(3)

, the quantities are written with subscripts 𝑟 and 𝑖, hence signifying real and 

imaginary parts respectively. 

The primary aim of the present article is to take into consideration the sensitivity of the threshold 

power SBS and the gain coefficient of the backward scattered mode Brillouin 𝑔𝐵 In doing so, the 

following expression [25] is used 

𝑔𝐵 = −
𝑘

2 1
[𝜒𝐵𝑖

(3)] |𝐸0|
2                                            (21) 



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The expression of steady state Brillouin gain given above can further be written in terms of the input 

pump power when considering centrosymmetric semiconductor plasma as 

𝑔𝐵 = 1.54 × 10−7𝐼𝑖𝑛                     (22) 

The relevant physical parameters are given in section 3. Based on the above equations, one can see 

that quantum effect has significant effect on the third-order nonlinearity of the medium and 

subsequently the steady state properties of the Brillouin active medium. 

Table 1 shows the list of key physical and mathematical symbols used throughout the manuscript, 

along with their corresponding descriptions. 

 

Table 1. Definitions of key symbols used in the paper. 

Symbol Description 

𝜂 Spin polarization parameter 

𝑛0, 𝑛1 Equilibrium and perturbed carrier density, respectively 

𝐯0, 𝐯1 Equilibrium and oscillatory (perturbed) electron fluid velocities 

𝐸0, 𝐸1 Pump (incident) electric field and scattered (Stokes) electric field 

𝜔0, 𝜔1 Frequencies of the pump and scattered electromagnetic waves 

𝐤 Wave vector associated with propagating fields 

𝜒(3) General third-order nonlinear susceptibility 

𝜒𝐵
(3)

 Third-order Brillouin susceptibility 

𝑔𝐵 Brillouin gain coefficient 

𝜏opt Optimum pulse duration for maximum transient gain 

𝑃𝑐𝑑 Induced nonlinear polarization at the scattered frequency 

𝜈 Electron collision frequency (damping rate) 

𝜔𝑝 Electron plasma frequency 

𝜇𝐵 Bohr magneton (magnetic moment of the electron) 

𝛿1, 𝛿2 Detuning parameters used in the gain model 

𝑢 Longitudinal displacement of the ion acoustic wave 

𝛽 Pressure-related coupling parameter in the quantum force model 

 

2.2. Transient characteristics 

This section gives the dynamics of SBS i.e. the time behavior of the Stoke wave intensity. In the 

macroscopic perspective, the transient coherent phenomena are caused by the ability of the material 

system in the capacity of the ability of retaining the certain phase of a coherent excitation over certain 

period of time. Such effects have transient features that determine their speed in which different types 

of optical functions are executed. From Eqs. (21) and (22), it can be assumed that only high-power 

laser source exists that will produce considerable gain of the SBS mode. Therefore, the laser pump 

source must be in the pulsed mode with a time range of the order of 10−12 s or in the pulse-train mode 



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with a pulse duration of the order of 10−9 s in the case of Q Switched lasers and mode locked lasers, 

respectively. These durations of time are either comparable to or smaller than the phonon life times 

(for acoustic phonon ≥ 10−9 s ); and thus, the transient effect study comes to the fore. Conversely the 

steady state formulations are quite handicapped both in their ability to correctly predict the threshold 

pump intensity (𝐼𝑡ℎ) at which SBS will start to occur with positive gains as well as in their ability to 

correctly predict the optimum pulse duration over which such instabilities may be observed. These 

will indicate that SBS has to be studied with the inclusion of transient effect. Overall, according to 

[26] the transient gain factors are a linear combination of the steady-state gain coefficients with the 

following relation 

𝑔𝑇𝐵 = [2𝑔𝐵𝑥𝛤𝐵𝜏𝑝]
1 2⁄

− 𝛤𝐵𝜏𝑝                      (23) 

where Γ𝐵 is the acoustic phonon lifetime, and 𝑥 is interaction length, 𝜏𝑝 is pulse duration. In case of 

an extremely short pulse duration (𝜏𝑝 ≤ 10−10𝑠) the interaction length can be substituted by (𝑐𝑙𝜏𝑝 2⁄ ) 

where 𝑐𝑙is the velocity of light in crystal lattice. 

 

2.2.1. Threshold pump intensity and optimum pulse duration 

By making⁡𝑔𝐵 = 0 in the Eq. (23) we are able to get the threshold pump intensity of the initiation of 

SBS as 

𝐼𝑡ℎ =
𝛤𝐵𝜏𝑝

2𝐺𝐵𝑐𝑙
                  (24) 

with 𝐺𝐵 =
𝑔𝐵

𝐼𝑖𝑛
, the gain per unit pump intensity. 

The threshold condition under which SBS will occur is at the point corresponding to Brillouin gain 

coefficient 𝑔𝐵 becomes zero. One of the key factors influencing this threshold is the spin polarization 

𝜂. As shown in Figure 1, an increase in spin polarization results in a nonlinear decrease in the required 

threshold pump intensity. This suggests that spin-polarized systems allow for SBS to initiate at 

significantly lower energy inputs, which is beneficial for low-power device applications. 

 

Figure 1: Threshold intensity vs spin polarization 

 

But, in the case of the relatively long pulse duration (𝜏𝑝 ≥ 10−9𝑠), the cell length can be considered 

equal to x and under such circumstances, one finds 



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𝑔𝑇𝐵 = (𝛤𝐵𝜏𝑝)
1

2⁄ [−(𝛤𝐵𝜏𝑝)
1 2⁄

+ [𝑔𝐵 𝑥 )1 2⁄ ]]        (25) 

With the help of the above equation, we can get the impression of optimum pulse duration 

(𝜏𝑝)𝑜𝑝𝑡
beyond which no gain can be attained by equating 𝑔𝑇𝐵 to zero as 

(𝜏𝑝)𝑜𝑝𝑡
≈ (

𝑔𝐵⁡𝑥

𝛤𝐵
)                                (26) 

It can be noted that reversible gain characteristics of the Brillouin scattered mode are suggested to be 

modified by the quantum terms. 

The optimum pulse duration 𝜏 opt is not fixed and varies with the pump intensity 𝐼. This relationship 

is nonlinear due to gain saturation and phonon response times in the plasma. As shown in Figure 2, 𝜏 

opt τ opt  increases slowly with 𝐼, suggesting a logarithmic-type behavior. This helps determine 

suitable pulse durations for transient SBS gain in spin-sensitive plasmas. 

 

Figure 2: Optimum pulse duration vs pump intensity 

 

2.3 Mathematical Structure and Nonlinear Properties 

The QMHD model described above gives rise to set of coupled nonlinear partial differential equations 

characterizing the dynamics of the charge density perturbations, acoustic wave propagation and 

nonlinear induced polarization. The spin-polarized Fermi pressure adds cubic nonlinear terms, 

whereas the Bohm potential adds quantum corrections, which are functions of second derivatives of 

density. Such effects render the system analytically very rich and would be quite applicable to 

additional research in terms of bifurcation analysis, perturbation methods and theory of nonlinear 

stability. 

3. RESULTS AND DISCUSSION 

The given section is a study of the SBS gain behavior with the calculated QMHD model. The 

nonlinear dependence on wave vector, spin polarization, magnetic field, and carrier density is also 

illustrated in Figures 3 through 5 and the transient characteristics are illustrated in Figures 6 and 7. 

These findings show the efficacy of the model as a nonlinear dynamical system and indicates the 

contribution of quantum network interactions and of spins in changing the optical response. 



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To explain the validity of the model, an inward-looking numerical quest of the threshold condition 

and the gain behaviour of the SBS process with the addition of the quantum correction term and the 

use of the narrow-bandgap semiconductors has been presented in the present section. The above 

semiconductor bulk crystal is n-InSb that is a narrow direct gap semi-conductor which has gained zinc 

blende structure that is virtually a cube type of pattern. We have considered the irradiation of n-InSb 

medium by a pulsed 10.6 𝜇𝑚 CO2 laser at liquid nitrogen temperature (77 0K). Absorption coefficient 

of a sample under such a temperature is therefore very low and we can neglect the transition 

mechanism of band to band. Scattering of the electron in the acoustic phonon in InSb is the dominant 

mechanism of transferring the momentum and energy of the electron in the scattered state of the kind 

[27]. 

The representative values of the following material parameters have been taken into account to make 

up the theoretical formulation: 𝑚 = 0.015𝑚0, 𝑚0being the free electron mass,  휀1 = 15.8, 𝛾 =
5 × 10−10𝐹𝑚−1,𝜌 = 5.8 × 103𝑘𝑔𝑚−3, 𝜔1 = 2 × 1011𝑠−1, 𝜔0 = 1.78 × 1014𝑠−1, 𝜈 = 4 ×
1011𝑠−1. 

 

Figure 3: Variation of steady state gains with wave vector 𝒌 at 𝒏𝟎 = 𝟑 × 𝟏𝟎𝟐𝟒𝒎−𝟑 and 𝑬𝟎 =
𝟖 × 𝟏𝟎𝟕𝑽𝒎−𝟏. 

 

In Figure 3, steady-state gain characteristics of the SBS are displayed including quantum spin effect. 

The solid, dashed line showed the variation for fully spin polarized η=1 and partially spin-polarized 

i.e η=0.5 respectively. In this case, gain increases with increasing spin-polarization. It is also observed 

that the nature of both the curve is same and increases with increasing value of wave vector 𝑘. This 

graph demonstrates how changes in wave vector 𝑘 directly impact the gain constant in the system. 

1 2 3 4 5 6 7 8 9 10

0.1

1

10

wave vector (k) 10
7
 m

-1

G
a
in

 C
o

n
s
ta

n
t 

g
 x

 1
0

5

 

 

 = 

 = 



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Figure 4: Variation of steady state gains with number density 𝒏𝟎 at 𝒌 = 𝟑 × 𝟏𝟎𝟖𝒎−𝟏 and 

𝑬𝟎 = 𝟖 × 𝟏𝟎𝟕𝑽𝒎−𝟏. 

 

Figure 4 shows how 𝑔𝐵 varies as a function of free carrier density 𝑛0. The steady state SBS gain 

properties of the medium are found sensitive to the concentration of doping. It can be concluded that 

gain decreases parabolically as the carrier density rises 

The SBS in partially spin polarized decreases linearly whereas fully spin polarized curve increases 

parabolically with free carrier density. It is found that  = 0.5 favorable in achieving the larger gain 

constants and therefore is beneficial in for the construction of SBS cell. 

 

Figure 5: Variation of steady state gains with cyclotron frequency 𝒄 at 𝒌 = 𝟑 × 𝟏𝟎𝟖𝒎−𝟏 and 

𝒏𝟎 = 𝟑 × 𝟏𝟎𝟐𝟒𝒎−𝟑. 

1.0 1.2 1.4 1.6 1.8 2.0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

 = 

 = 

G
a
in

 C
o

n
s
ta

n
t 

g
 x

 1
0

6

Number density (n) 10
24

 m
-3

 

 

7 8 9 10 11 12 13

4

8

12

16

20

G
a
in

 C
o

n
s
ta

n
t 

g
 x

 1
0

6


c
/

0
 (10

7
)

 

 

 = 

 = 



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Figure 5 shows how the Brillouin gain changes with cyclotron frequency 𝑐 that relies on the outer 

applied magnetic field. Spin is important in the exposure of the plasma to an external magnet field; 

this interact with the magnetization in the plasma because of the electron spin. The solid, dashed line 

showed the variation for fully polarized i.e η = 1 and partially polarized i.e η = 0.5 respectively. The 

nature of both curves is the same, but with the partially spin-polarized curve, medium reaching a 

higher susceptibility compared to the fully spin-polarized. The spin of the electrons is coupled with 

the direction of the external magnetic field in the magnetized plasmas in such a way that it increases 

the intensity of the external magnetic field. When this spin is aligned it results in enhancement of the 

overall magnetic characteristics of the plasma. Within the analysis the steady state SBS gain is 

determined to be of the order of 1.96 × 10−10 SI units while considering 𝑠↑0 = −𝑠↓0 and 𝑛 = 𝑛↑ −

𝑛↓ = 3𝑛
𝜇𝐵𝐵0

2𝑘𝐵𝑇𝐹
   with carrier density 𝑛0 = 1024𝑚−3. 

 

Figure 6: Variation of 𝒈𝑻𝑩 with 𝒈𝑩in SBS process at different pump pulse durations. 

 

Figure 6 displays the rivalry between transient Brillouin gain and steady state Brillouin gain across 

various pulse durations. It can be seen from the graph that 𝑔𝑇𝐵 increases linearly with 𝑔𝐵 at one 

specific pulse length of three different values for the pump. On the contrary, as the pump pulse is 

increased in duration, 𝑔𝑇𝐵 increases at a particular value of 𝑔𝐵. At 𝜏𝑝 ≈ 10−2/Γ𝐵 and smaller values 

of 𝑔𝐵, the transient gain 𝑔𝑇𝐵 is found less than 1. But on increasing the steady state Brillouin gain, 

transient Brillouin gain becomes larger than 1 For any longer pulse duration 𝑔𝑇𝐵 is always larger than 

1. 

0.1 1 10

1

10

100

 = 

III

II

I

g
T
(m

-1
)

gain (10
4
m

-1
)



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Figure 7: Variation of transient gain coefficients with pump pulse duration𝝉𝒑. 

Figure 7 depicts the dynamical behaviour of transient gain factor with the pump pulse duration 𝜏𝑝. 

The acoustic phonon lifetime ≈ 10−9𝑠, hence, to draw this behavior, we have considered pulse 

duration in the range 10−12 ≤ 𝜏𝑝 ≥ 10−9 s. For backward Brillouin mode the interaction length (that 

is the cell length) is 𝑐𝑙𝜏𝑝/2 or x, whichever is shorter. For fixed 𝐼𝑖𝑛, (𝑔𝑇𝐵)𝑄𝐸 increases with rise in 

pulse duration and at a particular value of 𝜏𝑝, (𝑔𝑇𝐵)𝑄𝐸 reaches an optimum. The maximum value stays 

almost similar to some range of 𝜏𝑝. These zones may be considered both as quasi steady state or quasi 

saturation zones. When 𝜏𝑝 is further raised outside of quasi saturation regime 𝑔𝑇𝐵 decreases extremely 

rapidly and eventually falls to zero. This figure also indicates that the consideration of quantum effects 

causes the maximum gain point to shift to the larger value of 𝜏𝑝 and expands the 𝜏𝑝 range over which 

transient phenomena may be measured. Therefore, when examining the transient behaviour of 

Brillouin mode, it would be desirable to include terms which correct the quantum condition 

From Eq. (26), one can get the numerical approximation of optimum pulse duration (𝜏𝑝)opt 
 for nearly 

centrosymmetric crystal ( n − InSb ) (using the values of 𝑔𝑇𝐵 and 𝑔𝐵 obtained earlier and 𝑥 =
3.5 × 10−5𝑚) above which no gain is possible, as 

(𝜏𝑝)𝑜𝑝𝑡
= 1.2 × 10−10𝑠 with spin effect 

Resting on these values an inference can be drawn that the rise in magnitude of optimum pulse 

duration can be attributed to both quantum correction term and such that the optimum pulse duration 

can also be increased by increasing the pump intensity. 

4. CONCLUSION 

The implication of spin on the steady state and transient characteristics of a stimulated Brillouin 

scattering light on the semi-conductor plasma medium has been discussed in the present paper based 

on quantum magneto-hydrodynamic model which considers Fermi pressure, Bohm potential and spin 

terms. The analysis demonstrates that electron spin significantly influences plasma's response to an 

external magnetic field, altering its gain characteristics, thereby impacting electron transport and 

enhancing magnetic properties. The role of spin polarization, which reflects the distribution of 

unpaired electrons, is found to be critical in governing the magnetic behavior of the semiconductor 

plasma under such conditions. Furthermore, the transient Brillouin gain profile reveals that SBS 

develops efficiently for pulse durations shorter than the phonon lifetime, while it is suppressed for 

longer pulses. Notably, the quantum effects reduce the SBS threshold, allowing for substantial 

Brillouin gain at lower laser powers, which has practical implications in minimizing power 

requirements and reducing the cost of SBS-based device fabrication. 

1 10 100

0

5

10

15

20

25

30

35

 = 

(10
10

 sec)

 

T
ra

n
si

en
t 

g
a
in

 (
x
 1

0
6
)



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Acknowledgment 

The authors express their sincere gratitude to Prof. S. K. Ghosh and Prof. Swati Dubey for their 

invaluable guidance, insightful suggestions, and unwavering support throughout this research. Their 

combined efforts have been instrumental in shaping this work, and we deeply appreciate their 

significant contributions. 

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