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Article 1
Spectral singularities and threshold gain ofa
slab laser under illumination ofafocused
Gaussian beam
MEHRDAD BAVAGHAR,RASOUL AALIPOUR*,ANDKAZEM
JAMSHIDI-GHALEH
Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran
*
aalipour@azaruniv.ac.ir
Abstract: W e study spectral singularities of an infinite planar slab of homogenous optically
active material in focus of a thin lens under illumination of a Gaussian beam. W e describe the
field distribution of the Gaussian beam under this configuration as a plane w a v e propagated
near the optical axis which it’s phase and amplitude v a r y with distance from center of the beam.
Based on this approximation, w e carry out the transfer matrix f o r the slab. W e explore the
consequences of this configuration on determining the threshold gain of the active medium and
tuning the resonance frequencies related to spectral singularities. W e show that the spectral
singularities and the threshold gain besides that v a r y with distance from center of the Gaussian
beam, also they change with relative aperture of the focusing lens. As a result, using a thin lens
with higher relative aperture, the spectral singularities corresponding resonances shift to the
higher frequencies (lower wavelengths). Numerical results confirm the theoretical findings.
1. Introduction
Spectral singularities are certain points of the continuous spectrum of Schrodinger operator
f o r a complex potential with real scattering energies at which the transmission and reflection
amplitudes of the potential diverge. Physically they correspond to scattering states that behave
like zero-width resonances [1]. They draw the attention of mathematicians because they are
responsible f o r a number of mathematical peculiarities that can never arise f o r Hermitian operators
and fascinated physicists because of their optical applications. Spectral singularity w a s discovered
b y Naimark and then studied b y some mathematicians and physicists [2
–
5]. Applications of the
spectral singularities in optical systems were introduced b y Mostafazadeh [6]. He show that a
slab involving optical gain material begins emitting purely outgoing coherent w a v e s at resonance
frequencies related to the spectral singularities, i.e., it acts as a slab laser. This observation
has provided sufficient motivation f o r the continuous study of spectral singularities and their
applications [7–16, 18, 19, 21–25].
Among the notable applications of this approach, those focusing on the structure of incident
light and variety of the optical setups, are of interest to us [17, 20, 26,27]. In recent relevant
studies, the electromagnetic w a v e incident on the optically active medium is a plane w a v e , f o r
which the transfer matrix f o r getting the transmission and reflection spectra of the medium is
easily calculated. On one hand, the field distribution of most lasers is Gaussian, therefore, it is
worthwhile to investigate the spectral singularities of the active medium under illumination of a
Gaussian beam. Also, w e know that the amplitude and phase of a Gaussian beam is a quadratic
function of the radius of beam, so makes it difficult to use the transfer matrix method. However,
b y using proper optical setups, one can use the transfer matrix method f o r the Gaussian beam.
In this paper, w e consider the spectral singularities of a slab laser in focus of a thin lens under
illumination of a Gaussian beam, f o r which the transfer matrix method is applicable. The paper
is organized as follows: In Sec. 2, w e describe the field distribution of a Gaussian beam in
Article 2
focus of a thin lens. In Sec. 3, the transmission and reflection coefficients of a Gaussian beam
from the laser slab are calculated using the transfer matrix method and the associated spectral
singularities are obtained. W e express the equation related to threshold gain of the slab laser
under illumination of a focused Gaussian beam in Sec. 4. In Sec. 5, w e account the dispersion
relations f o r the equations governing the spectral singularities and the threshold gain. Finally, w e
report the numerical results in Sec. 6.
2. Illumination an infinite planner slab b y a focused Gaussian beam
In Fig. 1, an infinite planar slab of thickness
𝐿
is installed in back focal plane
(𝑥,𝑦)
of a
convergent thin lens. Suppose that the slab contains a homogeneous material with the complex
refractive index,
𝑛
. A Gaussian light beam propagating along the
𝑧
-axis is focused on entrance
face of the slab through the lens. The transverse component of the electric field describing such
an electromagnetic w a v e can be written as follows [28]
−→ E(r,𝑡)=𝜓(r)𝑒
−𝑖𝜔𝑡ˆ𝑒
𝑥
,(1)
where r:
=(𝑥,𝑦,𝑧)
,
ˆ𝑒
𝑥
is the unit vector along the positive
𝑥
-axis,
𝜔
is the frequency of the light,
and
𝜓(
r
)
is a continuously differentiable scalar w a v e function. Inserting this component of the
field in the familiar Helmholtz equation leads to the following time-independent Schrodinger
equation:
−∇2
𝜓(r) + 𝑣(𝑧)𝜓(r)=𝑘
2
𝜓(r),(2)
with the complex potential
𝑣(𝑧)=𝑘
2
(
1
−𝔷
2
)
, where
𝑘
:
=𝜔
𝑐
stands f o r the w a v e number,
𝑐
is the
speed of light in vacuum, and
𝔷:=
𝑛|𝑧| ≤ 𝐿
2
1|𝑧|>𝐿
2
.(3)
The main characteristic of a Gaussian beam is that it concentrates near the propagation direction
(𝑧-axis), allowing it to be described b y a scalar w a v e of the f o r m [29]
𝜓(r)=´ 𝜓(r)𝑒
𝑖𝔷𝑘𝑧 (4)
i.e., the w a v e does not propagate in the
𝑥
or
𝑦
direction and
´ 𝜓(
r
)
is slowly varying with
𝑧
.
Substituting Eq.(4) into Eq. (2) and ignoring
𝜕2´ 𝜓
𝜕𝑧
2
leads to the following paraxial w a v e equation:
𝜕
2´ 𝜓
𝜕𝑥
2+
𝜕
2´ 𝜓
𝜕𝑦
2+2 𝑖𝔷𝑘𝜕´ 𝜓
𝜕𝑧
=0.(5)
It’s solution gives the field distribution of the Gaussian beam as follows [29]:
𝜓(r)=𝐸
0
𝑤
0
𝑤(𝑧)e x p [− 𝑟
2
𝑤
2
(𝑧)]𝑒
𝑖𝔷𝑘𝑧e x p [𝑖𝜙(𝑧)]e x p [−𝑖𝔷𝑘𝑟
2
2𝑅(𝑧)],(6)
where
𝑟
2=𝑥
2+𝑦
2
,
𝑅(𝑧)=𝑧[
1
+(𝑧
0
/𝑧)
2
]
,
𝑤(𝑧)=𝑤
0
p
1+(𝑧/𝑧
0
)
2
, and
𝜙(𝑧)=𝑡𝑎𝑛
−1(𝑧/𝑧
0
)
in
which
𝑤
0
and
𝑧
0
are the minimum beam waist and Rayleigh range, respectively. The Rayleigh
range can be written in terms of the minimum beam waist as 𝑧
0=𝜋𝑤
2
0
𝜆[29].
When the Gaussian beam is followed through a lens of diameter
𝐷
and focal length
𝑓
, the
minimum beam waist lies to the right of the lens, a distance
𝑓
from the lens. The minimum beam
waist and the Rayleigh range are given b y [29]
𝑤
0≈2 𝜆(𝑓/𝐷)(7)
Article 3
and
𝑧
0≈4𝜋𝜆(𝑓/𝐷)
2
,(8)
where the ratio 𝑓/𝐷is defined as the relative aperture of lens [30].
According to Fig. 1, w e choose the origin of the coordinates system at back focal plane of the
f
L
Z
X
2w
0
DY
n =1
0n =1
2
n
Fig. 1. Sketch of an infinite planar slab of thickness
𝐿
and complex refractive index
𝑛
in focus of a convergent thin lens of diameter
𝐷
and focal length
𝑓
. The slab is
illuminated b y a Gaussian beam through the lens.
lens, where installed the slab. Suppose that the thickness of the slab is v e r y small in respect to
the Rayleigh range,
𝐿𝑧
0
. Therefore, w e will be allowed to use the approximation
𝑧𝑧
0
f o r
the Gaussian beam parameters as follows:
𝑅(𝑧) ≈ 𝑧
2
0
𝑧𝑤(𝑧) ≈ 𝑤
0𝜙(𝑧) ≈ 0.(9)
Substituting these relations in Eq. (6) leads to the following equation f o r the field distribution of
the Gaussian beam in 𝑧𝑧
0approximation:
𝜓(r)=𝐴𝑖
(𝑟)𝑒
𝑖𝔷𝑘 𝜉 (𝑟)𝑧
,(10)
where
𝐴𝑖
(𝑟)=𝐸
𝑖
𝑒
−𝑟
2
𝑤
2
0,(11)
and
𝜉(𝑟)=[1+𝑟
2
2𝑧
2
0].(12)
The scalar w a v e function in Eq. (10) describes a plane w a v e which it’s phase and amplitude are
𝑟
-dependent [31]. Using these results, the general solution of the Schrodinger equation, Eq. (2),
can be expressed as follows:
𝜓(𝑟,𝑧)=
𝐴0
(𝑟)𝑒
𝑖𝑘 𝜉 (𝑟)𝑧
+𝐵
0
(𝑟)𝑒
−𝑖𝑘 𝜉 (𝑟)𝑧f o r 𝑧 < −𝐿
2
𝐴1
(𝑟)𝑒
𝑖𝑛𝑘𝜉(𝑟)𝑧
+𝐵
1
(𝑟)𝑒
−𝑖𝑛𝑘𝜉(𝑟)𝑧f o r |
𝑧
|≤𝐿
2
𝐴2
(𝑟)𝑒
𝑖𝑘 𝜉 (𝑟)𝑧
+𝐵
2
(𝑟)𝑒
−𝑖𝑘 𝜉 (𝑟)𝑧f o r 𝑧 > 𝐿
2,
(13)
where
𝐴𝑖
and
𝐵
𝑖
, with
𝑖=
0
,
1
,
2are
𝑟
-dependent forward and backward w a v e amplitudes
according to the Eq. (11).
3. Transfer matrix and spectral singularities
The transfer matrix f o r the system w e consider is the 2×2matrix Msatisfying
© «
𝐴2
𝐵
2
ª ® ¬=M© «
𝐴0
𝐵
0
ª ® ¬.(14)
Article 4
W e explore the transfer matrix b y using the boundary conditions at
𝑧=−𝐿
2
and
𝑧=𝐿
2
based on
continuity of the tangential components of electric and magnetic fields. The transfer matrix after
some straightforward calculations is obtained as follows:
M=1
4𝑛© «
𝑒
−𝑖𝑘𝑛𝜉(𝑟)𝐿
𝐹(𝑛,−𝑘𝐿
2)2 𝑖(𝑛
2−1)sin[𝑘𝑛𝜉(𝑟)𝐿]
−2 𝑖(𝑛
2−1)sin[𝑘𝑛𝜉(𝑟)𝐿]𝑒
𝑖𝑘𝑛𝜉(𝑟)𝐿
𝐹(𝑛, 𝑘𝐿
2)ª ®
¬,(15)
where f o r all 𝑝,𝑞∈𝐶
𝐹(𝑝,𝑞)=𝑒
−2𝑖𝑝𝑞
(1+𝑝)
2
−𝑒
2𝑖𝑝𝑞
(1−𝑝)
2
.(16)
From Eq. (15) it is clear that the transfer matrix elements depend on the transverse coordinate
𝑟
.
F o r a case that w e consider the center of beam, i.e.,
𝑟=
0, Eq. (15) turns to the common transfer
matrix f o r the slab laser under illumination of a plane w a v e .
The left and right reflection and transmission amplitudes are given b y [1]
𝑅𝑙=−𝑀21
𝑀22
𝑅𝑟=𝑀12
𝑀22
𝑇 𝑙=det 𝑀
𝑀22
𝑇 𝑟=1
𝑀22
,(17)
where
𝑀𝑖𝑗𝑖,𝑗=
1
,
2are the elements of transfer matrix
𝑀
and
det
stands f o r determinant of the
matrix which one can show that
det 𝑀=
1. Inserting transfer matrix elements from Eq. (15) in
Eq. (17) yields the following relations:
𝑅𝑙=𝑅𝑟=𝑒
−𝑖𝑘 𝜉 (𝑟)𝐿√r(1−𝑒
2𝑖𝑘 𝜉 (𝑟)𝑛𝐿
)
1−r𝑒
2𝑖𝑘 𝜉 (𝑟)𝑛𝐿 (18)
and
𝑇 𝑙=𝑇 𝑟=𝑒
−𝑖𝑘 𝜉 (𝑟)𝐿(1−r)𝑒
𝑖𝑘 𝜉 (𝑟)𝑛𝐿
1−r𝑒
2𝑖𝑘 𝜉 (𝑟)𝑛𝐿,(19)
where
r
:
=(𝑛−1
𝑛+1)
2
. Expressing the refractive index of the slab as
𝑛=𝜂+𝑖𝜅
, substituting it into
Eqs. (18) and (19), and multiplying them b y their complex conjugates give the reflection and
transmission coefficients as follows
R=|r|[(1−𝐺)
2
+4𝐺sin2𝛿/2]
(1−𝐺|r|)
2
+4𝐺|r|sin2(𝛿+𝜙)/2(20)
T=|1−r|
2
𝐺
(1−𝐺|r|)
2
+4𝐺|r|sin2(𝛿+𝜙)/2
,(21)
where
|r|:=(𝜂−1)
2
+𝜅
2
(𝜂+1)
2
+𝜅
2, 𝐺 :=𝑒
−2𝑘 𝜉 (𝑟)𝜅𝐿
, 𝛿 :=2𝑘𝜉(𝑟)𝜂𝐿, 𝜙:=2 tan−1(2𝜅
𝜂
2+𝜅
2−1
).(22)
The spectral singularities correspond to the real and positive values of the wavenumber
𝑘
f o r
which
𝑀22 =
0[1]. This implies that the reflection and transmission amplitudes diverge. So it
occurs when
𝑒
−2𝑖𝑘 𝜉 (𝑟)𝑛𝐿=r.(23)
Taking the natural logarithm of both sides of this equation yields
𝑘𝜉(𝑟)𝐿=−1
2 𝑖𝑛
ln r,(24)
Article 5
and in follow b y inserting 𝑛=𝜂+𝑖𝜅in Eq. (24), w e g e t
𝑘𝜉(𝑟)𝐿=−1
2 𝑖(𝜂+𝑖𝜅)ln (𝜂−1+𝑖𝜅
𝜂+1+𝑖𝜅
)
2(25)
Because
𝑘𝜉(𝑟)𝐿
is real, therefore it equates with the real part of the right-hand side of this
equation and so the imaginary part of the right-hand side is set to zero. This gives
𝑘𝜉(𝑟)𝐿=1
2𝜅ln |r|(26)
and
𝜂ln |r| + 𝜅(𝜙−2𝑚𝜋)=0,(27)
where 𝑚is an arbitrary integer.
4. Threshold gain
The gain coefficient of an active medium is defined b y [32]
𝑔:=−4𝜋𝜅
𝜆,(28)
where
𝜅
is imaginary part of the refractive index
𝑛
and
𝜆
:
=
2
𝜋/𝑘
is the wavelength. The value
of gain coefficient f o r which the spectral singularity takes place is called threshold gain, which is
the lasing threshold condition. F o r getting the threshold gain of the active medium, w e derive
𝜅
from Eq. (26)and then substitute it in Eq. (28) which yields
𝑔=𝑔
𝑡ℎ
:=1
2[1+𝑟
2
2𝑧
2
0]𝐿
ln[(𝜂+1)
2
+𝜅
2
(𝜂−1)
2
+𝜅
2],(29)
where w e have used Eq. (12) f o r 𝜉(𝑟).
5. Accounting f o r dispersion
F o r a more realistic situation w e take into account the effect of dispersion, i.e., the refractive
index
𝑛
and so
𝜂
and
𝜅
are not independent from the frequency. In this regard, suppose that
the slab contains a doped host medium of refraction index
𝑛
0
that w e can model b y a two-level
atomic system with lower and upper level population densities
𝑁
𝑙
and
𝑁
𝑢
, resonance frequency
𝜔
0
, damping coefficient 𝛾, and the dispersion relation [33]
𝑛
2=𝑛
2
0+𝜔
2 𝑝
𝜔
2
0−𝜔
2−𝑖𝛾𝜔
,(30)
where
𝜀
0
is the permeability of the vacuum,
𝜔
2 𝑝
:
=(𝑁𝑙
−𝑁𝑢
)𝑒
2
𝑚𝑒𝜀0
, and
𝑒
and
𝑚
𝑒
are electron’s charge
and mass, respectively. Substituting the refractive index of the slab
𝑛=𝜂+𝑖𝜅
into Eq. (30) and
taking
𝜅=𝜅
0
at
𝜔=𝜔
0
,w e g e t the following approximate equations f o r the real and imaginary
parts of the refractive index
𝜂'𝑛
0−𝜅
0
𝐹 1(31)
and
𝜅'𝜅
0
𝐹 2
,(32)
where
𝐹 1:=(1−ˆ 𝜔
2ˆ)𝛾
(1−ˆ 𝜔
2
)
2
+ˆ 𝛾
2ˆ 𝜔
2,𝐹 2:=ˆˆ 𝜔𝛾
2
(1−ˆ 𝜔
2
)
2
+ˆ 𝛾
2ˆ 𝜔
2,(33)
Article 6
ˆ 𝜔=𝜔
𝜔0, and ˆ 𝛾=𝛾
𝜔0.
Next, b y inserting Eqs. (31) and (32) in Eq. (26) and using Eqs. (12) and (8), w e find
𝜅
0'
1
2ln r
0
2𝐹 1
𝑛2
0−1+[2𝜋(ˆ 𝜔
𝜆0) + 𝑟
2
16𝜋(𝑓/𝐷)
4(ˆ 𝜔
𝜆0)
3
]𝐿𝐹
2
.(34)
where r
0=(𝑛0
−1
𝑛0
+1)
2
. By this w a y , threshold gain is obtained from Eq. (28) as follows
𝑔
𝑡ℎ
:=−4𝜋𝜅
0
/𝜆
0
,(35)
where
𝜆
0
:
=
2
𝜋𝑐/𝜔
0
, and
𝜅
0
is evaluated from Eq. (34). In deriving Eqs. (31)-(35) the quadratic-
and higher-order terms of
𝜅
0
are neglected. Using the same approximation to calculate the
reflection and transmission coefficients b y Eqs. (20) and (21), w e can use the following relations
instead of Eq. (22)
|r| ' r
0
(1+4𝜅
0
𝐹 1
𝑛
2
0−1
), 𝐺 '𝑒
−2𝑘 𝜉 (𝑟)𝜅0
𝐿𝐹 2
, 𝛿 '2𝑘𝜉(𝑟)𝐿(𝑛
0−𝜅
0
𝐹 1
), 𝜙 '4𝜅
0
𝐹 2
𝑛
2
0−1.(36)
Then, extracting
|r|
from Eq. (27), substituting in Eq. (26), and using Eqs. (8), (12) and
(31)-(33), w e find the following mode equation
𝑟
2
𝑛
0
𝐿
16𝜋(𝑓/𝐷)
4
𝜆
3
0
ˆ 𝜔
4+ [2𝜋𝑛
0
𝐿
𝜆
0+ln r
0
ˆ2𝛾
2]ˆ 𝜔
2−𝜋𝑚ˆ 𝜔−ln r
0
ˆ2𝛾
2=0.(37)
The solutions of this equation are the resonance frequencies related to the spectral singularities
which, in addition to depending on the physical characteristics of the active medium, they are
𝑟-dependent and change with the relative aperture of lens
6. Numerical results
In order check the numerical results of the theoretical findings in the previous sections, w e
consider a slab composed of a semiconductor gain medium with the specifications:
𝑛
0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02, in the back focal plane of a thin convergent lens with
variable relative aperture under illumination of a Gaussian beam. The reflection and transmission
coefficients from the slab are calculated from Eqs. (20) and (21), using Eq. (36). Fig. 2 illustrates
the logarithmic plots of the reflection (dashed curve) and transmission (full curve) coefficients of
a light at center of the focused Gaussian (
𝑟=
0) from the specified slab versus the wavelength.
The plots show the zero-width resonance frequencies with the free spectral range is of the order
of 𝛿𝜆 ≈1𝑛𝑚.
T o find out how the spectral singularities and the corresponding resonance frequencies depend
to the transverse distance
𝑟
,w e calculate the reflection and transmission coefficients at various
transverse distances on the beam spot. Fig. 3 shows the logarithmic curves of the calculated
transmission coefficient at various transverse distances. Based on these plots, the spectral
singularities corresponding resonances are
𝑟
-dependent, and as
𝑟
increases, shift to the higher
wavelengths. In fact, in a Gaussian beam the outer rays, in comparison to the central ray, are
obliquely propagating, that is, the incident angle of the rays striking to the front face of slab
increases, radially and so the optical path lengths experienced b y the rays in passing through the
slab increase, radially. Recently, it has been illustrated that the resonance frequencies related to
spectral singularities depend strongly on the incident angle [17, 20].
N o w , w e study the dependence of the threshold gain on the wavelength at different transverse
distances from the center of the beam. F o r this purpose, w e calculate the threshold gain from Eq.
(34) using Eq. (35). Fig. 4. shows the plots of the calculated threshold gain versus the wavelength
Article 7
Wavelength(nm)
1480 1482 1484 1486 1488 1490
T&R
100
104
108
Fig. 2. Logarithmic plots of the reflection (dashed curve) and transmission (full
curve) coefficients versus the wavelength of a Gaussian beam from a slab composed
of a semiconductor gain medium with the specifications
𝑛0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02 in focus of a thin lens with the relative aperture of
𝑓/𝐷=
15 that have been calculated using Eqs. (20), (21) and (36) at center of the
beam spot.
Wavelength (nm)
1484 1484.2 1484.4 1484.6 1484.8 1485
T
100
104
108
r=0
r=0.02 mm
r=0.03 mm
r=0.04 mm
r=0.05 mm
Fig. 3. Logarithmic plots of the transmission coefficient versus the wavelength of
a Gaussian beam from a slab composed of a semiconductor gain medium with the
specifications
𝑛0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02 in focus of a thin lens
with the relative aperture of
𝑓/𝐷=
15 that have been calculated using Eqs. (21) and
(36) at various transverse distances from the center of the beam spot.
Article 8
at various transverse distances from the center of the beam. Upon these plots the threshold gain
of the slab becomes maximum at the center of the beam and decreases as
𝑟
increases.. This is
due to that the amplitude of the field distribution across the Gaussian beam decreases radially
according to Eq. (6).
In the proposed configuration, the relative aperture of the thin lens
𝑓/𝐷
is a factor that influences
Wavelength (nm)
1499.8 1499.9 1500 1500.1 1500.2
gth (cm-1 )
40.406
40.407
40.408
40.409
40.41
40.411
40.412
r=0
r=0.02 mm
r=0.03 mm
r=0.04 mm
r=0.05 mm
Fig. 4. The plots of the threshold gain versus the wavelength of a Gaussian beam f o r
a slab composed of a semiconductor gain medium with the specifications
𝑛0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02 in focus of a thin lens with the relative
aperture of
𝑓/𝐷=
15 that have been calculated using Eqs. (34) and (35) at various
transverse distances from the center of the beam spot.
the threshold gain of the active medium and the corresponding spectral singularities, according
to Eqs. (34), (35), and (37). Therefore, to consider this effect w e can use various lenses with
different relative apertures. W e calculated the transmission coefficient of a Gaussian beam from
the specified slab in focus of the lenses with different relative apertures have been calculated
using Eqs. (21) and (36) and plotted them, as shown in Fig. 5. According to these plots, the use
Wavelength (nm)
1484.2 1484.4 1484.6 1484.8 1485 1485.2
T
100
104
108
F/D=7.5
F/D=8.5
F/D=10
F/D=12
F/D=15
Fig. 5. Logarithmic plots of the transmission coefficient versus the wavelength of
a Gaussian beam from a slab composed of a semiconductor gain medium with the
specifications
𝑛0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02 in focus of the thin
lenses with different relative apertures that have been calculated using Eqs. (21) and
(36) at 𝑟=0.02𝑚𝑚from the center of the beam spot.
of the lenses with higher relative apertures leads shifting the spectral singularities to the lower
Article 9
wavelengths (higher frequencies). One can use this ability to tune the spectral singularities. Also,
Fig. 6 shows the plots of the threshold gain calculated from Eqs. (34) and (35) f o r the specified
slab in focus of the lenses with different relative apertures. These plots illustrate that using the
Wavelength (nm)
1499.6 1499.8 1500 1500.2 1500.4
gth (cm-1 )
40.402
40.404
40.406
40.408
40.41
40.412
40.414
F/D=7.5
F/D=8.5
F/D=10
F/D=12
F/D=15
Fig. 6. The plots of the threshold gain versus the wavelength of a Gaussian beam f o r
a slab composed of a semiconductor gain medium with the specifications
𝑛0=
3
.
4,
𝐿=
300
𝜇𝑚
,
𝜆
0=
1500
𝑛𝑚
, and
ˆ 𝛾=
0
.
02 in focus of the lenses with different relative
apertures that have been calculated using Eqs. (34) and (35) at
𝑟=
0
.
02
𝑚𝑚
from the
center of the beam spot.
thin lenses with higher relative apertures, the threshold gain of the slab increases.
7. Conclusion
W e studied the spectral singularities and the threshold gain of a laser slab in focus of a thin lens
under illumination of a Gaussian beam. Our purposes of this study are that, first, to extend the
studies done on the spectral singularities of the gain medium f o r the Gaussian beams, and second,
tuning the spectral singularities corresponding resonances b y use of the focusing elements of the
configuration. The novel and worth-mentioning features of this study are expressed as follows:
1-The field distribution of a Gaussian beam in focus of a thin lens is approximated b y a plane w a v e
which it’s phase and amplitude are
𝑟
-dependent, according to Eq. (10). Physically, in a Gaussian
beam the outer rays, in comparison to the central ray, are obliquely propagating, therefore the
Gaussian beam in focus can be described as a set of plane w a v e s which their’s w a v e vectors make
larger angle to the optical axis, as 𝑟increases.
2- The transfer matrix f o r a slab composed of optically active material under illumination of a
focused Gaussian beam is obtained in Eq. (15) which is 𝑟-dependent.
3- Because the amplitude of the field distribution across the Gaussian beam decreases radially,
according to Eq. (6), therefore the corresponding threshold gain of the medium also decreases
radially, as illustrated in Fig. 4.
4- In a Gaussian beam the rays that propagate through the slab experience larger optical path length
b y moving away from the center of the beam, therefore the corresponding spectral singularities
take place at the higher wavelengthes, as shown in Fig. 3.
5- Threshold gain and spectral singularity v a r y with the relative aperture,
𝑓/𝐷
, of the focusing
lens, according to Eqs. (35) and (37). Increasing the relative aperture leads to shift the
spectral singularities corresponding resonances to the lower wavelengths and to increase of the
corresponding threshold gain of the active medium, as illustrated in Figs. 5 and 6.
This study can be extended to the other features of the electromagnetic w a v e s in the gain and
loss mediums such as coherent perfect absorption and invisibility. Also, this study can be done
Article 10
experimentally b y providing highly sensitive intensity detectors.
8. Disclosures
The authors declare no conflicts of interest.
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