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IOP Conference Series: Materials Science and Engineering
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Correlating stimulated emission phase to the gain
spectra in semiconductor lasers
To cite this article: Abanoub Mikhail and Safwat William Zaki Mahmoud 2022 IOP Conf. Ser.: Mater.
Sci. Eng. 1269 012003
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36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
1
Correlating stimulated emission phase to the gain spectra in
semiconductor lasers
Abanoub Mikhail1, Safwat William Zaki Mahmoud2
1,2Faculty of Science, Minia University, Minia, Egypt
Email: abanoub.mikhail@mu.ed.eg , safwatzaki@mu.edu.eg
Abstract. In laser physics, the incident electric field and the stimulated field are assumed to
have the same frequency, direction of propagation, polarization, and phase —same state.
However, no formal proof of the phase identity (zero phase shift) was reported. The existing
theories on the phase shift are in conflict with each other and with the phase identity. The
phase shift in three semiconductor active media (GaAs, In0.15Ga0.85As, and
In0.48Ga0.58As0.9P0.1) has been studied using a semiclassical approach. Contrary to the
conventional belief, it has been found that the phase shift is not zero. The phase shift is not
even a single value but rather a spectrum corresponding to the gain spectrum. At a carrier
concentration of , the minima of the phase shift spectra are , , and
radians for GaAs, In0.15Ga0.85As, and In0.48Ga0.58As0.9P0.1, respectively. Stimulated
emission is shown to occur whenever the phase shift is positive and lies in the interval between
0 and π/2. Because of radiation reaction of the radiating source and the finite lifetime of excited
states, stimulated emission cannot attain the same phase as the incident field —no zero-phase
shift. These results reveal that the conventional picture of phase identity is incorrect, and phase
matching between the incident field and stimulated is not a necessary condition for stimulated
radiation. Despite the fundamentality of the concept of stimulated emission phase, such
outcomes are not reported nor discussed in the literature.
1. Introduction
The process of stimulated emission was predicted by Einstein based on mechanical and
thermodynamical arguments in 1917 [1]. He stated that the energy exchange between the incident
electromagnetic field and Plank oscillators could be positive or negative depending on the phases of
the incident radiation field and the oscillators. The problem of the phase shift (phase difference)
between the incident radiation field and the stimulated field was not dealt with in Einstein’s work.
When an electromagnetic wave passes through a medium, the wave creates oscillating dipoles.
These dipoles initially absorb energy. If the electromagnetic wave continues to propagate in the
medium, the oscillating dipoles start to radiate energy in a way that is temporary and spatially coherent
with the driving field —stimulated emission [2]. Determining the phase shift between the incident
electric field and the stimulated field is complicated for multiple reasons. Firstly, the radiation pattern
of a system in free space is substantially altered when the same system is embedded in media. For
example, in free space, the field lines of energy flow run radially outwards except for the dipole axis.
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
2
In a material medium, the energy flow lines are perpendicular to the dipole axis due to damping [3].
Secondly, the experimentally measurable physical quantity is not the phase shift but rather the
intensity of radiation. Therefore, during conventional experiments, the phase shift information is lost.
Nonetheless, Josef Kröll and coworkers managed to measure the phase and amplitude of the emitted
field and correlated it with the input signal [4]. Their phase measurement experiment showed that the
behavior of the phase at resonance gain is inverted with respect to resonance absorption.
The widespread view is that stimulated emission occurs in phase with the incident field. This point
of view is taught in our universities without any mathematical or experimental proof. Our objective is
to address this issue based on practical lasing media.
Pollnau argued that, at resonance, polarization leads the incident field by
and the electric field
scales with the distance between the two charges of the dipole. Therefore, stimulated emission must
lead the incident field by
. According to Pollnau, phase shift of
is a must for energy
conservation to hold true [5].
Zhao and coworkers studied radiation emitted from a sheet of charge [6]. They found that the
emitted field is delayed by
from the incident counterpart. Being inconsistent with practical optical
gain process, the authors proposed that if the radiated electric field from a single dipole is
advanced in phase, then the superposition of the radiated field from the infinite sheet of dipoles would
be in phase with the incident field. In the discussion section of this paper, we will show the reasons
why Pollnau and Zhao et al. analyses are not sound.
Charles Henry found a relation between the phase change due to carrier injection and the gain peak
[7]. However, the phase shift between the incident and the stimulated field due to the detuning
frequency, at constant carrier density, was not dealt with.
In this work, the phase shift in three typical gain media GaAs, In0.15Ga0.85As, and
In0.48Ga0.58As0.9P0.1 is investigated. These gain materials were chosen for their wide use in diode laser
fabrication and in photonic devices, but other candidates could have been used instead. In addition to
showing the fallacies in the work of Pollnau and Zhao et al., we illustrate why the conventional belief
of phase identity cannot be true.
2. Method
Within the semi-classical model of electromagnetic waves interacting with semiconductors, the field
is treated classically while the semiconductor is treated using the techniques of quantum statistical
mechanics such as the density matrix. An incident electric field oscillates at a
frequency and propagates in the z-direction with a wave number k. The field displaces the center of
the positive nucleus and negative electron cloud from their equilibrium position forming oscillating
dipoles. The medium polarization is related to the incident field by the constitutive relation [8]
(1)
where and are the susceptibility and vacuum permittivity in order. The radiated electric field ()
from an active medium of thickness is related to the medium polarization [9]
(2)
where and are the vacuum permeability and host medium background refractive index,
respectively. Within the free carrier theory and under the parabolic band approximation, the real and
imaginary parts of the susceptibility are obtained as [10, 11]
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
3
(3)
(4)
where is the band gap energy, is the transition energy, is relaxation time, is the
joint density of states, is the dipole moment, and are the quasifermi distribution
of electrons in the conduction and valence bands respectively. Equation (2) shows that the radiated
field is always perpendicular to the polarization —the radiated field leads the polarization in phase by
[9].
The gain coefficient is the fractional increase in the intensity of radiation per unit distance travelled
in the medium. There is gain in the lasing medium only if . This condition limits the
imaginary part of the susceptibility to the upper half of the complex plane. Thus, the phase shift
between the stimulated field and the incident field becomes
(5)
where the gain coefficient is given by [10]
(6)
The equation for the phase shift (Eq. (5)) must be solved numerically for the gain medium under
study. We have chosen GaAs, In0.15Ga0.85As, and In0.48Ga0.58As0.9P0.1 because these are common
direct band-gap semiconductor materials. They emit laser light around 850, 1300 and 1550 nm,
respectively.
The quasi-Fermi levels of the valence and conduction bands are numerically found using the charge
neutrality condition imposed on the gain medium. Charge neutrality requires the electrons excited into
the conduction band to leave an equal number of empty states behind in the valance band [12].
Afterwards, equations (3-6) are solved numerically to obtain the phase shift of the stimulated radiation
as a function of frequency. Wolfram Mathematica (https://www.wolfram.com/mathematica/) has been
used to carry out the calculations using the parameters in Table (1).
Parameter name
Symbol
GaAs
In0.15Ga0.85Asa
In0.58Ga0.42As0.9P0.1b
Units
Band gap at 0K
1.519
Band gap offset
0.34
Electron effective mass
0.08
--
Heavy hole effective mass
0.5
--
Refractive index
3.6
--
Relaxation time
s
aextracted from [13] bextracted from [14]
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
4
3. Results
Figures (1-3) show the gain spectrum and the phase shift spectrum for the three materials under study
at the threshold gain concentration. The main observations are:
Firstly, the phase shift for all the investigated materials is a spectrum corresponding to the gain
spectrum. Secondly, the phase shift spectrum is always positive and less than
. Thirdly, stimulated
radiation is produced by both resonant electromagnetic fields and quasi-resonance fields. However, the
phase shift at a quasi-resonance incident field is larger than phase shift in case of resonant
electromagnetic field. Fourthly, in the absorption region, the phase shift becomes negative. Taken
together, the four observations show that the phase shift is a spectrum whose minimum is not at zero.
This conclusion is true independent of the injected carrier concentration and attains validity for the
three materials under study.
Figure (4) illustrates the phase shift minimum and the gain peak plotted against the carrier
concentration. While the gain peak increases with carrier concentration, the minimum of the phase
shift spectra decreases. At any fixed carrier concentration, the phase shift minimum of
In0.48Ga0.58As0.9P0.1 is smaller other two materials. For example, at a carrier concentration of
, the phase shift spectrum minimum is , , and radians for GaAs,
In0.15Ga0.85As and In0.48Ga0.58As0.9P0.1, respectively. The reason for the low minimum of phase shift
spectrum in case of In0.48Ga0.58As0.9P0.1 may be attributed to the smaller effective mass of the charge
carriers in In0.48Ga0.58As0.9P0.1 as compared with the other two materials.
Fig. 1: Phase shift (solid) and gain spectrum of GaAs (dashed) at lowest gain carrier concentration of
and temperature of K
8.57x10-7 8.60x10-7 8.62x10-7 8.65x10-7 8.68x10-7
1.5603
1.5624
1.5645
1.5666
1.5687
Dq (rad)
l (m)
phase shift
gain
GaAs
0.00
0.08
0.16
0.24
0.32
0.40
gain 104(m-1)
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
5
1.00x10-6 1.01x10-6 1.01x10-6 1.02x10-6 1.02x10-6
1.5600
1.5625
1.5650
1.5675
1.5700
Dq (rad)
l (m)
phase shift
gain
In0.15Ga0.85As
0.0
0.1
0.2
0.3
0.4
0.5
gain 104(m-1)
Fig. 2: Phase shift (solid) and gain spectrum of In0.15Ga0.85As (dashed) at lowest gain carrier concentration of
and temperature of K
1.57x10-6 1.59x10-6 1.60x10-6 1.61x10-6 1.62x10-6
1.5600
1.5625
1.5650
1.5675
1.5700
Dq (rad)
l (m)
gain
phase
In0.48Ga0.52As0.9P0.1
0.0
0.1
0.2
0.3
0.4
0.5
gain 104(m-1)
Fig. 3: Phase shift (solid) and gain spectrum of In0.48Ga0.58As0.9P0.1 (dashed) at lowest gain carrier
concentration of and temperature of K
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
6
1.14x1024 1.52x1024 1.90x1024 2.28x1024 2.66x1024
1.35
1.40
1.45
1.50
1.55
1.60
Dqmin (rad)
Carrier density (m-3)
Phase shift minimum (solid)
Gain peak (dashed)
GaAs
In0.15Ga0.85As
In0.48Ga0.52As0.9P0.1
0.0
1.3
2.6
3.9
5.2
gain peak 104(m-1)
Fig.4: Phase shift minimum (solid), and gain peak (dashed) versus carrier concentration for the three materials under
study at temperature of K
4. Discussion
The problem of stimulated emission phase is intimately linked to foundational concepts in physics.
Stimulated emission results from the interference of the incident field with the field radiated from the
atom [15]. Scattering and interaction between carriers could destroy the phase shift. Despite of
adopting the free carrier theory, which neglects scattering and coulomb interaction between carriers,
the phase shift turns out to attain nonzero values. There is a minimum in the phase shift spectrum, but
its value is greater than zero. We believe there are compelling reasons that inhibit the phase shift from
vanishing.
The first reason is the finite lifetime of excited states. Typically, GaAs-based-heterostructure
semiconductors have carrier lifetime around s [16, 17]. This means that it takes the stimulated
photon time equal to the lifetime of the state to get fully emitted, but, by then, the phase of the emitted
radiation would be different from the phase of the incident radiation. This argument can be extended
to low dimension devices such as quantum dot lasers, and small lasers [16, 18, 19].
The second reason is the extended nature of atoms. Stimulated radiation cannot instantly detach
itself from the radiating source. The incident field modulates the source’s charge and current
distributions at first. In the region surrounding the atoms (near field region), the stimulated electric and
magnetic fields are attached to the current source. It is only after travelling a large distance compared
to the source size that they become completely detached from the radiating source. Moreover,
electromagnetic retardation may be another factor in delaying the emitted field. Retardation is a result
of the finite speed of light. The state of radiation at time reflects the state of the source of radiation at
a prior time , where is the distance travelled by the stimulated radiation [8].
The third and most important reason is radiation damping. The emitted radiation exerts a back
reaction on the atom itself which has a friction-like damping effect [20]. Even if one managed to
2
36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
7
remove other dephasing mechanisms such as scattering of charge carriers. The stimulated radiation of
electromagnetic waves undergoes a form of damping which is inherent to the radiation process and
cannot be eliminated [21, 22].
What was the flaw in Pollnau’s thinking? Let us first rephrase his argument. The number of photons
is proportional to the intensity of the electromagnetic wave of a particular optical mode. If
and
represent the amplitude of the incident and radiated fields at a particular frequency, then
(7)
Prior to Pollnau’s work, Melissa cray et al. showed that the first term in Eq. (7) is the energy of the
incident field, the second term is the energy associated with the radiated field due to scattering and
spontaneous emission, and the third term represents the interaction energy due to stimulated emission
[15]. Pollnau main point is that unless the interference term (the 3rd term in Eq. (7)) vanishes, the
conservation of energy principle is violated [5]. Therefore, he concluded that the phase shift must be
.
However, this conclusion is incorrect because electric dipoles radiate waves in a manner such that the
energy lost in one direction could be gained in other directions. This latter idea is supported by W. J.
Cocke who studied the interaction of an incident plane wave with a charge distribution and found that
the interaction term due to stimulated emission gains a net contribution exactly in the forward-
scattering direction [23].
There exist flaws in the analysis of Zhao and coworkers—as hinted at in the introduction section of
this paper. For a start, the authors did not provide any proof for the assumption that stimulated
emission from a single atom (dipole) is advanced in phase with respect to the incident field [6].
Their approach seems to be an ad hoc attempt to make stimulated emission from an ensemble of atoms
agrees with the conventional view of phase identity. In addition to that, no damping mechanism which
causes transients to decay was included in their model. In fact, Einstein argued that stimulated
radiation is a directed process. Thus, the recoil of the atoms must exist [1].
5. Conclusion
An investigation of the phase shift of stimulated emission of radiation and its relation to the gain in
various lasing media was carried out. The phase shift of stimulated radiation is found to be a spectrum
that follows the gain spectral profile. The gain spectrum and phase shift spectrum are related to each
other. The optical gain is greater than zero only when the phase shift is in the interval .
Therefore, stimulated emission is concluded to be possible over this range only.
The minimum of the phase shift spectrum seems to be a characteristic feature of any gain material.
Generally, the smaller the minimum of the phase shift spectrum is, the higher is the gain peak. This
observation suggests that the most fundamental quantity in classifying lasing materials is the phase
shift. In such a scheme, the best lasing material that could ever be found is the one with the lowest
possible phase shift minimum.
We have also elucidated the reasons for the unfeasibility of the conventional belief of zero
stimulated emission phase shift. The causes lie in the finite lifetime of the excited states, the extended
nature of the radiating sources, and most importantly radiation reaction. Future work in this direction
of research may include the dependence of the phase shift on the temperature, interaction between
charge carriers, and the study of the phase shift in the near field and in nanodevices.
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36th Eg-MRS International Conference, 24- 5 September 2022, Cairo, Egypt
IOP Conf. Series: Materials Science and Engineering 1269 (2022) 012003
IOP Publishing
doi:10.1088/1757-899X/1269/1/012003
8
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