An intrinsic physical content of “single photon power” − (hν∙Δν)
Considered in this paper is the possibility to use information properties of photon noise inherent to thermal radiation. Using the calculations of threshold limitations for detecting the fluctuations of thermal radiation as a signal and not disturbances only, we have adduced some arguments concernin...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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| Cite this: | An intrinsic physical content of “single photon power” − (hν∙Δν) / E.A. Salkov, G.S. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 12-20. — Бібліогр.: 27 назв. — англ. |
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| citation_txt | An intrinsic physical content of “single photon power” − (hν∙Δν) / E.A. Salkov, G.S. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 12-20. — Бібліогр.: 27 назв. — англ. |
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| description | Considered in this paper is the possibility to use information properties of photon noise inherent to thermal radiation. Using the calculations of threshold limitations for detecting the fluctuations of thermal radiation as a signal and not disturbances only, we have adduced some arguments concerning the phenomenological content of the “power of a single photon” qΔν ≡ hν∙Δν not defined earlier, where the frequency band Δν is determined by an observation spectral slit. It has been shown that a non-traditional view of “photonic noise” determined by this factor appears in relation with the fluctuations of the photon flux at the observation spectral slit. With definite measurement parameters, this kind of noise is capable to append some essential points to classical models of photon noises and even block access to measurements of fluctuations in the power of thermal radiation. Also adduced are supplementary considerations concerning the existing models of arising shot noise at the photo detector output as a result of physical-and-statistical specificity of the electron excitation process and its kinetics. Offered is a model for a background radiator, which allows us performing the numerical calculations with the use of the specific parameters of the background structure.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
PACS 44.40.+a, 74.40.-n
An intrinsic physical content of “single photon power” − (hν⋅Δν)
E.A. Salkov, G.S. Svechnikov
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine
41, prospect Nauky, 03028 Kyiv, Ukraine
Abstract. Considered in this paper is the possibility to use information properties of
photon noise inherent to thermal radiation. Using the calculations of threshold limitations
for detecting the fluctuations of thermal radiation as a signal and not disturbances only,
we have adduced some arguments concerning the phenomenological content of the
“power of a single photon” qΔν ≡ hν⋅Δν not defined earlier, where the frequency band Δν
is determined by an observation spectral slit. It has been shown that a non-traditional
view of “photonic noise” determined by this factor appears in relation with the
fluctuations of the photon flux at the observation spectral slit. With definite measurement
parameters, this kind of noise is capable to append some essential points to classical
models of photon noises and even block access to measurements of fluctuations in the
power of thermal radiation. Also adduced are supplementary considerations concerning
the existing models of arising shot noise at the photodetector output as a result of
physical-and-statistical specificity of the electron excitation process and its kinetics.
Offered is a model for a background radiator, which allows us performing the numerical
calculations with the use of the specific parameters of the background structure.
Keywords: thermal radiation, photon noise, fluctuations, power of a single photon.
Manuscript received 02.03.10; accepted for publication 02.12.10; published online 28.02.11.
1. Introduction
This work is aimed at using the fluctuations of
thermal radiation (TR) as a carrier of definite physical
information on the radiator. Considering the example of
a small size (SR) single radiator within the model of
blackbody (BB), we made an attempt to calculate
threshold limitations for detecting the intrinsic photon
noise of TR as a signal. This approach is justified, first,
by perspectives in realization of additional physical
information that is contained in the variance of TR noise
inherent to SR and has been illustrated in [1-4] as based
on the well-known relation between the dispersion 〈ΔF2〉
of a random value F and its mean value 〈F〉:
.2 FFqF Δ= (1.1)
As always, Exp. (1.1) is used in the form of the so-called
relative fluctuation FFqF
2Δ=∗ , which, to some
extent, screens the quantitative sign of its physical
content qF .
The simplest example concerns fluctuations of the
charge Q in a capacitor:
192 106.1 −⋅==→⋅=Δ eqQeQ Q where
QQQ
e
Q
Qe
qQ
1019 104106.1 −−
∗ ⋅
=
⋅
→== .
Comprehension of Eq. (1.1) can be founded using
the respective analysis of recognized scientific sources
[5-16], etc. So, adduced in the papers [1, 2] are the
examples of validity for Eq. (1.1) both for the ideal gas
thermodynamic and electric current fluctuations in the
cases of resistor thermal noise, current shot noise, and
the generation – recombination one. It was shown that in
the case of adequate statistics the value qF (i.e., the
“intrinsic micro-quantity of chaos” [1] that will be
renamed as “eigen-parameter”≡“e-p” in what follows) is
a physical concept which is fully defined by its content
and dimensionality and includes concrete characteristics
of a fluctuating physical system.
Based on the analysis of the references [5-8, 13-21]
and results of researches [1-4] the following improved
list of “e-p”≡ qF for the BB TR is proposed:
( )1+= nqn is the “eigen-number”
of photons in (BB cavity) TR field, (1.2)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
12
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
( ) qf nq ν1 Δ⋅+= is the “eigen-flow”
BB TR photon within a band of Δνq, (1.3)
( )1ν += nhqE is the “eigen-energy”
of TR inside (BB cavity), (1.4)
( )
qV
hnq ν1+=ε is the “eigen-density energy”
of TR inside (BB cavity), (1.5)
( ) qhnq ν1P Δ⋅ν⋅+= is the “eigen-power”
of BB TR within a band of Δνq, (1.6)
( ) q
q
p V
hnq ν1 Δ⋅
ν
⋅+= is the “eigen-power density”
of BB TR within a band of Δνq, (1.7)
νΔ⋅ν=νΔ hq is “eigen-power-Δν”
of BB TR power flow which is restricted
in spectral slit with a frequency band Δν. (1.8)
Physical parameter Vq in (1.5) and (1.7) is a
specific volume which in the case of SR coincides with
its real volume. As to the frequency band Δνq in (1.3),
(1.6) and (1.7), this has to be defined in accordance with
a specific physical situation (problem).
The above expressions for qF (1.3)-(1.7) are in
accordance with the quantum-mechanical “TR
fluctuation factor” ( )1+= n [8, 14-19].
To establish accordance of Eqs. (1.5)-(1.7) to the
fluctuations of TR energy density in the BB cavity,
one can use the classic formula for the variance
( )λε
( )[ ]2λEΔ [5-7] of the total TR energy
( ) ( )λλ ѓГ⋅= qVE in the BB cavity of a small constant
volume1 qV . In accord with the definition [22], the
variance of the product of the constant qV and
fluctuating ( )λѓГ values is
( )[ ] ( )[ ]222 λεΔ⋅=λΔ qVE . Using qε (1.5) to
determine ( )[ ]2λεΔ in accordance with (1.1), one can
obtain the following expression:
( )[ ] ( ) =×⋅=Δ ελελ qVE q
22
( ) ( ) =+⋅
⋅λ
×λε⋅ n
V
hcV
q
q 1
2
(1.9)
= ( ) ( )nhcE +⋅
λ
×λ 1 ,
which is the Einstein formula [5-7]. Being based on (2.8)
and using Eqs (1.5), (1.7), considered in the works [3, 4]
was the calculation model of the principle for distant
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
1Volume that did not lose its space-time relation with the number of
modes of an actual frequency in its cavity yet [13].
identification of both single SR and the stochastic set
(“cloud”) of SR [4]. It is noteworthy that while
determining the thresholds for detecting the intrinsic
fluctuations of TR SR BB power the “problem of the
single photon power” is solved.
Thus, in the literature devoted to TR and its
fluctuations (e.g. [8] and so on) one can find the
understandable by its dimensionality self-sufficient
expression [ ]-1sergѓЛѓЛ ⋅Δ⋅h , but it, as an acting physical
value, does not find its specific phenomenological
interpretation. As it will be shown below, this enigmatic
single-photon power νν Δ⋅h can play a rather essential
role and obtain definite phenomenological interpretation.
Expediency of using the property (1.1) in the case of TR
SR seems to be obvious as, in spite of wide applications
of laser technologies, TR is inherent to any objects both
natural and technological cloud-like SR systems.
2. Fluctuations of TR in terms
of the “eigen-parameter”
2.1. Initial conceptions.
It is obvious that only using measurements of the TR
power emitted by SR and fluctuations of this power one
can obtain some information included in TR of SR. As
to TR, the property (1.1) can be applied beyond
controversy, and first of all, due to the fact that photons
do not interact with each other (see e.g. [7, 15-17]).
In what follows, we shall consider the problem of
physical informativity inherent to the TR noise variance
in the most accessible form. Here, we originate from the
opportunity to use the following approach: TR power
variance is written in accordance with relations (1.1) and
(1.5)-(1.7) when determining the thresholds for
registration of the TR power variance for SR within the
model of BB with the dimension corrections [23, 24].
Further, we shall use the following designations:
qF ≡ “e-p” ; c – light speed, ν − frequency (and the
respective wavelength − λ = c/ν) of observed TR, which
corresponds to the middle of Δν (or Δλ) which is a
spectral band for observations;
1
1exp
−
⎥
⎦
⎤
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛
⋅
=
kT
hcn
λ
is the Planck function,
( ) 3
2 ν8νν
c
VZ Δν⋅π
⋅=Δ⋅ is the number of TR spatial
modes within the spectral frequency band Δν inside the
BB cavity with the volume V = R3;
( ) i
i
i n
R
hc
⋅⎟
⎠
⎞
⎜
⎝
⎛ Δ⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⋅⋅=
λ
λλ
λλ
λε 2
2
3 4
1ѓО8 is the TR energy
density inside the SR cavity of a cubic shape with the
dimensional correction
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
− 2
2
4
1
iR
λ [23, 24] that limits
the number of TR modes in the volume ; the 3
ii RV =
13
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
© 2011, V. Lashkaryov Institute of Semiconductor o ces of e
14
)
Physics, National Academy f Scien Ukrain
ratio in the expression for is kept
constant during spectral calculations or measurements
(hereinafter, the subscript “i” means belonging to SR) .
( λλΔ / ( )λε i
2.2. The model of background TR radiator
To calculate fluctuations of an arbitrary large
background radiator, it is necessary to know the size
“eigen-volume” for this background (Vq – in formulas
(1.5) and (1.7)), which is not described in literature
accessible to the authors (for instance, [6-8, 14-21] and
so on).
To solve the problem, one needs the model of a
background radiator, which is capable to provide more
or less adequate results for numeral calculations of the
fluctuation variance for TR background 2
jBPΔ
(hereinafter, the subscripts “j” and “B” mean belonging
to background elements).
Let us assume the following simplified model: in a
warm background (B) an ideal photodetector (PD)
registers TR from a single SR of a cubic shape with the
constant cavity volume Vi = Ri
3 cm3; let the size of
radiating area be equal Si ≈ Ri
2.
The model of such a background2 (Fig. 1) that
provides the possibility for a well grounded calculations
of the TR background power seems as follows. Within
the boundaries of the observed background area ,
the area , serves as the emitting one (see
the subscript “B”) that consist of identical and
small in their
obsS
jBjBB NPS ⋅Δ= 2
jBN
sizes stationary radiators (with their non-fluctuating
volume ). When using the dimensional corrections 3
jBR
jBN
2 Any other model suitable for numeral calculations of the background
TR dispersion (except the offered one) is not known for the authors;
therefore they did not make respective references.
[23, 24] to small radiators, we consider their size in
calculations as close to ( ) ( )22 2 ijB RR ≈ .
In calculations below, we shall use the following
numeric parameters for the model of the background
(Fig. 1):
( )
( ) 22
22
cm000064.0002.0004.0002.0
cmSpacingSpacing
=−++=
=++= jBjB RS
jBN obsS
.
Density inside is
( )
2-
2 cm15625
cm008.0
11
===Γ
jB
jB S obsjBjB SN, Γ ⋅=
iD BDL
BS
Dr
22 Dr⋅π
.
To simplify the formulas, we use the “lensless3 ”
choice of the system “radiators (SR + B) - photodetector
(PD)”. The values l and correspond to distances
from SR and its background to PD, respectively. The
input PD hole can be defined via the radius ; the value
will serve as an area of the PD photosensitive
element. Joint action of the TR system (SR + B) on PD
will depend on spatial apertures in accordance with the
above approximations that do not contradict to the
conventional ones [25]: the aperture, within the
boundaries of which PD registers TR of a single SR is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅≅ 2
2
2 iD
D
iiD l
rSA
jBN
, and the background aperture given
the number of elements is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⋅
⋅⋅= 22
2
BDL
Dr
jBNjBSBDA
. In the offered model, the
spectrum of TR power that is emitted by SR into semi-
Fig. 1. The model of a background (a), and a geometry of the system (b).
3 As the considered version of the problem has an illustrative character
of calculations, small numeral errors will not impact on results and
conclusions of principle.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
© 2011, V. Lash miconductor Physics, National Academy of Sciences of Ukraine
15
( )
karyov Institute of Se
space can be defined as ( ) 2
2 4 iii RcP ⋅⋅λε=π (see [8],
p. 5, (1.9)), and its fraction that excites PD can be
written as:
( ) ( ) 2
2
2
24 iD
D
iiiD l
r
RcP ⋅⋅⋅λε=λ . (2.1)
The variances of the TR power both from SR and
background single element are defined in
accordance with [22] as variances of the product of
fluctuating ( and
jBN
( )λεi ( )λε Φj ) and constant ( and
) values. The spectrum of the signal, i.e., the
variance of the SR TR power (2.1) with account of (1.1)
and (1.7) becomes (when ):
iDA
BDA
3
iq RV =
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅⋅=Δ
2
2
2
222
24
ѓГ
iD
D
iiiD
l
r
RcP
( ) .
24
1
24
ѓГ 2
2
2
32
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅⋅+⋅
⋅
⋅⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅⋅
iD
D
ii
iiD
D
ii l
r
Rcn
R
hc
l
r
Rc
λ
(2.2)
By its analogy with (2.1) and (2.2), let us define the
power ( )
π2jBP and the variance of TR emitted by small
background elements . In the case of statistical
independency inherent to TR processes from background
elements, a constant value of their number , sizes
and location within the boundaries of , the total
power of the background TR at the PD input can be
written as
jBN
jBN
jBR obsS
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅⋅⋅=
π
2
2
2
2
24
ε
BD
D
jBjBjBBD L
r
RcNP . (2.3)
The dispersion of the sum (2.3) defined as the sum
of the dispersions for separate background elements
2
jBPΔ using the Burgess4 theorem for = const in
accord with (1.1) and (1.7) acquires the following form
jBN
( ) .
24
1
24
ѓГ
2
2
2
3
2
2
22
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅⋅+⋅
⋅
×
×⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅⋅⋅=Δ
iD
D
jBjB
jB
iD
D
jBjBjBBD
L
r
Rcn
R
hc
L
r
RNcP
λ
(2.4)
3. Photocurrents at the output of an ideal PD
Descending to the photoelectric currents corresponding
to the SR TR powers (2.1), (2.2) and SB TR (2.3), (2.4),
let us remind you about the important property of the
principle adduced in [26]:
4Burgess R.E. Faraday, Soc.Discussions, 1959, v.28, p.151; reference
cited in [20].
“… the spectrum of photocurrent power
reconstitutes the optical spectrum independently of the
optical field statistics”. It is this condition that allows us
prolonging our consideration of the problem formulated
in the Introduction. In the model adopted above for the
background and the scheme of joint exciting the
photodetector by the system (SR + B), acting at the PD
output are the currents (3.1)-(3.6) defined in accordance
to the conventional rules [8].
Originating from relations (2.1)-(2.4), at the output
of ideal PD we have (e is the electron charge,
η - quantum efficiency of PD):
− the stationary current caused by SR TR power
(2.1)
iDiD P
h
eI ⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
=
ν
η ; (3.1)
− the respective iDI (3.1) shot noise
iDisn IfeI ⋅Δ⋅⋅=Δ 22 ; (3.2)
− the signal corresponding to the SR TR variance
(2.2)
⎟
⎠
⎞
⎜
⎝
⎛
νΔ
Δ
⋅⋅Δ⋅⎟
⎠
⎞
⎜
⎝
⎛
ν
⋅η
=Δ
fP
h
eI iDiD 22
2
2 ; (3.3)
− the current corresponding to the background TR
power (2.3)
BDBD P
h
eI ⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
=
ѓЛ
ѓЕ ; (3.4)
− the corresponding BDI (3.4) shot noise
BDBsn IfeI ⋅Δ⋅⋅=Δ 22 ; (3.5)
− the current corresponding to the background TR
dispersion (2.4)
⎟
⎠
⎞
⎜
⎝
⎛
νΔ
Δ
⋅⋅Δ⋅⎟
⎠
⎞
⎜
⎝
⎛
ν
⋅η
=Δ
fP
h
eI BDBD 22
2
2 . (3.6)
The coefficient ⎟
⎠
⎞
⎜
⎝
⎛
νΔ
Δ
⋅
f2 [8] in (3.3) and (3.6)
defines, to some extent, the fraction of chaotic
fluctuations inherent to TR within the band Δν at the PD
input, which is really registered as a chaotic in time
current at the PD output within the frames of its electron
band of frequencies Δf. When calculating and measuring
the current and power spectra corresponding to TR, the
values ⎟
⎠
⎞
⎜
⎝
⎛
νΔ
Δ
⋅
f2 in formulas (3.3), (3.6) can be kept
constants, by changing the electron band of frequencies
(e.g., at 210−=⎟
⎠
⎞
⎜
⎝
⎛
νΔ
Δf and 210−=⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ one has
4
6
10
103
−⋅λ
⋅
=Δf ).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
4. Thresholds for detection of single SR TR
fluctuations by an ideal PD
4.1. Detection thresholds 2
iDPΔ
on the background of intrinsic noises
(single-noise-limited detection – SNLD mode [8])
Before starting our consideration of the main task (as to
the threshold values (3.3)) formulated above, let us
remind you about the problems that contain some
conceptions poorly ascertained from the physical
viewpoint.
Let us define the threshold values
thiDP (2.9) and
thiDP2Δ (2.2) via the signal-to-noise ratio for respective
“measured” currents revealed with the background of
intrinsic noises. Extrinsic noises are absent here.
1) As always, the threshold
thiDP is defined by
the following TR power
η
Δ⋅ν⋅
=
fhP
thiD
2 [8] by
detection of the signal
thiDI (3.1) with the background
2
isnIΔ (3.2). In this case, there are no explanations why
the noises 2
iDPΔ related to the intrinsic fluctuations of
TR in a BB cavity5 of the radiator are not taken into
account. At the same time, the very combination
can be hardly explained in a
phenomenological aspect. But taking
fh Δ⋅ν⋅2
2
iDIΔ into
account together with 2
isnIΔ results in a more
informative expression (the essence of SNLD condition
[8] remains valid), namely:
( )
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅+⋅
⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ
⋅⋅
λ
⋅+⋅
Δ⋅
⋅= 2
2
2
1
4
η1
η
ν2
iD
D
i
i
thiD l
rn
R
fhP ,
(4.1)
which contains definite information upon this radiator
and its aperture (Fig. 2).
2) After respective simple algebraic operations with
account of expressions (2.1), (2.2) and (3.1)-(3.3), the
threshold values for the signal of our interest 2
iDIΔ
(3.3) against the background of the intrinsic shot noise
2
isnIΔ (3.2) can be reduced to the following expression
( )
⎭
⎬
⎫
⎩
⎨
⎧ νΔ⋅
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅⋅⋅+⋅
⋅λ η
ν
4
1 2
2
2
3
h
l
rRcn
R
hc
i
D
ii
i
. (4.2)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
5 Definition and measurement of the noise-coefficient for
semiconductor photodiodes takes into account the photon-noise stated
in the paper [27] but doesn’t consider the specific parameters of the
radiator (except for (〈n〉+1) [8, 14-19]) as well as the aperture features.
Fig. 2. Spectral dependences of
η
ν fhP
thiD
Δ⋅⋅
=
2
(curve
n(λ)) and
thiDP (4.1) at two distances to the radiator (size
Ri = 20 μm): liD = 10 cm (curve R(λ)), and liD = 5 cm
(curve r(λ)).
It is obvious that (4.2) describes equality of two
“intrinsic powers”: on the left – “e-p” (1.7), (“intrinsic
micro-quantity of chaos” in terms [1, 2]) that
corresponds to the frequency band defined via the
aperture factor and the SR intrinsic volume
(1.5), (1.7), i.e.
qνΔ
qii VRV ≡= 3
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
⋅
=Δ≡Δ
2
2
22 i
D
i
iq
l
r
R
cνν ; on the
right – “single photon power ” defined through the
optical observation band divided by the value of PD
quantum efficiency:
⎭
⎬
⎫
⎩
⎨
⎧ νΔ⋅
=νΔ η
νhq . (4.3)
This “single photon power” occurs in [8] (e.g. page
87) also as the factor defining power within a single
mode of TR, namely: “…the power per mode is
νΔνε= hfP kk …”; here, is emissivity and fε k – Bose-
Einstein occupancy probability. It seems very doubtful
that to determine the single mode TR power one use the
spectral frequency band νΔ . As it follows from the
commonly known literature, the TR power within the
band νΔ quantifies the number of modes ( ) νΔ⋅zZ (see
the above item 1.2) in the range of which it is observed
(measured) or calculated.
The equality νΔ= qqp (4.2) puts more definite
phenomenological content into the value (4.3), since
the equality (4.2) also defines the value as “e-p” in
the list (1.2)-(1.8), taking into account the physical sense
of (1.7). In what follows (see the item 4.2 below),
our reasoning in regard to the physical meaning of the
“power =
νΔq
νΔq
pq
νΔνh ” confirms the validity of consideration
of the detection thresholds 2
iDIΔ (3.3) with the total
noise background: (3.2), (3.5), and (3.6).
16
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
4.2. Detection thresholds 2
iDPΔ in presence of external thermal noise
When calculating the thresholds of the signal 2
iDIΔ , the total current that takes into account external noises will be
defined using the following logic.
The fluctuating total virtual current 2
ΣΔI measured by a PD in view of statistical independency for the currents
(3.2), (3.5), (3.6) and containing only averaged values will be determined via the sum of variances in the accepted
approximation
( )
νΔ
Δ
⋅⋅⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
⋅η
+⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
⋅η
⋅Δ+
νΔ
Δ
⋅Δ⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
⋅η
+⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
⋅η
⋅Δ=Δ Σ
fPq
hc
eP
hc
efefP
hc
eP
hc
efeI BDjpBDiDiD 2
//
22
//
2
2
2
2
2 .
(4.4)
Having used (4.4) and (3.1)-(3.6) to write the signal-to-noise ratio via PD output currents expressed through the
corresponding TR power at the PD input one has
( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Δ
Δ
⋅⋅⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
+⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
⋅Δ+⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
⋅Δ
Δ
Δ
⋅Δ⋅⎟
⎠
⎞
⎜
⎝
⎛ ⋅
=
Δ−Δ
Δ
Σ
νλ
η
λ
η
λ
η
νλ
η
f
Pq
hc
eP
hc
efeP
hc
efe
fP
hc
e
II
I
BDjpBDiD
iD
iD
iD
2
//
2
/
2
2
/
2
2
2
22
2
. (4.4*)
Let us cancel (like in Eq. (4.1)) all the reducible terms in the numerator and denominator. After this
conventional procedure [8] for conversion from PD currents to TR powers, the threshold condition for
thiDP2Δ
(4.4*) can be reduced to the following form
( )
1
2
=
⋅⋅+Δ⋅⋅+Δ⋅⋅
Δ
⋅
BDjpBDiD
thiD
PqhPhP
P
ηνννν
η . (4.5)
The formula (4.5) considered definition (1.1) and physical meaning of the equality (4.2) allows to consider the
expression ( ) iDPh ⋅νΔ⋅ν in the denominator (4.5) as the variance of (2.1) related to the “photon noise at the
spectral slit”. Let us designate it as
iDP
( ) iDPhP ⋅νΔ⋅ν=Δ νΔ
2 . (4.6)
The same reasoning concerns the expression ( ) BDPh ⋅νΔ⋅ν , too.
The factor is physically defined as “e-p” (1.8) for TR power flowing through an observation
spectral slit (“e-p” for slit-fluctuation). This “Δ-noise” (4.6), created artificially to some extent, is defined via the
above list of “e-p” (1.2)-(1.8).
νΔν=νΔ hq
νΔ
To put more definite phenomenological meaning into the values and νΔq 2
νΔΔP , let us consider spectral and
quantitative (calculated) features of the signal threshold values
thiDP2Δ as compared with the spectra of
νΔ
Δ 2
iDP and the signal 2
iDPΔ . In the above accepted model for the background (§1, item 1.3) given the
definitions (3.1) to (3.6) as well as expressions (4.5), (4.6), the spectral dependence of the threshold power 2
iDPΔ for
the signal defined through the respective photocurrent at the PD output
thiDI 2Δ in terms of wavelengths λ and for
η = 1 can be expressed as follows
( ) BD
jB
D
jBjB
jB
iDthiD P
L
rRcn
R
chcPchcP ⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅⋅⋅+⋅+⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ
⋅
λ
⋅
λ
+⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ
⋅
λ
⋅
λ
=Δ 2
2
2
3
2
24
11 . (4.7)
Being targeted to compare the spectra of the noise signal (2.10) and “Δ-noise” (4.6) with those of the signal
threshold (4.7) for various temperatures, we have depicted them in Figs 3 and 4. The signal (2.2) defined via the
current (3.3) takes the detailed look:
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
17
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
( )
2
2
2
2
3
2
3
2
24
11
2
18
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅⋅⋅+⋅⋅⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⋅⎟
⎠
⎞
⎜
⎝
⎛ Δ⋅⋅=Δ
iD
D
ii
i
i
i
iD
l
r
Rcn
R
n
R
hcP λ
λ
λ
λ
π
λ
. (4.8)
“Δ-noise” (4.6), also in the detailed notation, is as follows:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅⋅⋅⋅⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ
⋅⎥
⎦
⎤
⎢
⎣
⎡
⋅
λ
−⋅
λ
π
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
λ
λΔ
⋅
λ
⋅
λ
=Δ νΔ 2
2
2
2
2
3
2
244
18
jB
D
ii
i L
rRcn
R
chcP (4.9)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
The parameters of the SR and the respective
background are chosen first of all considering the
necessity to simplify our phenomenological analysis.
The numeral values of threshold powers
thiDP2Δ for
the chosen parameters are determined for the wavelength
(shown with arrows ↓ in Fig. 3) of the intersection
points between the signal spectrum (4.8) and those of its
threshold values (4.7) for various temperatures.
Fig. 3. Illustration of spectral distributions of TR powers (4.8),
(4.9) and signal thresholds (4.7) )(2 λ=Δ BP
thiD for three
temperatures (TjB) of background radiators. The designations
and numeral parameters: ( )
iBTthiDPB 2Δ=λ ;
( ) SIGNAL
K600
2 ≡Δ=λ
thiDIS ;
( ) "noise"
K600
2 −Δ≡Δ=λΔ Δ thiI ;
( ) ( ) ( )222222 8.22002.02 ⋅=⋅= iDDiiD lrRA ; TSR = 600 K;
; -310/ =λλΔ
( ) ( ) ( )[ ]22242 600.223010563.1004.0 ⋅⋅⋅⋅=iBA ;
; T-310/ =λλΔ jB = 300, 150, 90, 60 K.
Further comparison of these spectra shows that
measuring the signal thresholds
thiDP2Δ is not always
possible.
If the observation parameters are chosen
differently, “Δ-noise” (4.9), (designated as )(λΔ in
Fig. 3), when )Q()( λ>λΔ , the signal spectra are
“blocked” by photon flux fluctuations at the spectral slit,
that is by the “Δ-noise” (4.9).
As it follows from Figs 2 and 3, there are specific
circumstances (e.g., mutual combinations of numeral
characteristics inherent to a radiating system SR+BG as
well as parameters of measuring process) when
fluctuation limitations (like the usual shot noise) for
measurements of TR signal threshold characteristics
arise in the course of TR flux passing within an optical
band Δν inside a detecting apparatus before reaching the
sensitive element of PD. In the literature (see, for
instance, [6-8, 14-21]) TR fluctuations of this kind were
not considered. However, as it follows from Figs 3
Fig. 4. Illustration of “blocked” spectral distributions of
TR powers (4.8), (4.9) and signal thresholds (4.7)
)(2 λ=Δ TP
thiD for three temperatures (TjB) of
background radiators. The designations and numeral
parameters are same as for Fig. 3.
18
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 12-20.
and 4, one should not neglect the intrinsic TR noises
(4.5) and (4.8) within the spectral band. Thus, for
example in [8] the shot noise related to a photocurrent is
calculated in the traditional way when originating only
from statistical (or in [17] from quantum-and-statistical)
notions as to appearance of electrons in the PD active
area when these are excited by photons. In relation with
the considered problem, there arises the following
question: is it possible that the measured photocurrent
shot-noise is always determined by random electrons
appearing statistically inside the photosensitive area of
PD?
5. Conclusions
1. Conversion of the common threshold expression
in current terms at the PD output to that (4.5) for the
respective TR powers at the PD input results in the
expression for an additional photonic noise that arises
due to limitations of spatial-and-time conditions for TR
flux passing through the observation slit . As a result,
the process of detection is limited by the “photonic
νΔ
νΔ -
noise” ( ) iDPhP ⋅νΔν=Δ νΔ
2 , (4.6), along with the
traditional “electron shot noise”.
2. As an “eigen-parameter” (in terms of [1, 2] it is
the “intrinsic micro-quantity of chaos”) in this case we
deal with well-known but a phenomenologically
unascertained up to date “single photon power”
. νΔν=νΔ hq
3. Inclusion of the “power” (4.3) in the list
(1.2)-(1.8) as an “intrinsic power of TR flux” (1.8)
seems to be adequate. From a phenomenological
viewpoint, the “power” defines the dispersion of
additional photonic fluctuations in the TR flux through
an observation slit .
νΔq
νΔq
νΔ
4. In the case of photoelectronic detection the TR
signals at the PD output, along with traditional photonic
(3.3) and shot noises [8, 10], it seems reasonable to take
into account the “photonic νΔ -noises” (4.6) considered
above (Figs 3 and 4).
5. Used in this work a rather risky approximation
(first of all, ignoring the correlation processes, moments
of higher orders, etc. [17]) does not reduce to zero the
above considered physical concepts as well as their role
and numeric values.
Thus, the following result is non-trivial: it is
possible to measure a numeral meaning of the physical
value “power of a single photon” under
specific physical conditions (with the cold enough
background: e.g. in Fig. 4 it can be realised at
T
νΔν=νΔ hq
jB < 120 K and high enough ratio
ν
Δν ∼ 0.1) only by
measuring (with an ideal PD) the TR flux fluctuations at
the spectral slit ѓЛΔ .
References
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© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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|
| id | nasplib_isofts_kiev_ua-123456789-117604 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T16:22:49Z |
| publishDate | 2011 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Salkov, E.A. Svechnikov, G.S. 2017-05-25T16:16:26Z 2017-05-25T16:16:26Z 2011 An intrinsic physical content of “single photon power” − (hν∙Δν) / E.A. Salkov, G.S. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 12-20. — Бібліогр.: 27 назв. — англ. 1560-8034 PACS 44.40.+a, 74.40.-n https://nasplib.isofts.kiev.ua/handle/123456789/117604 Considered in this paper is the possibility to use information properties of photon noise inherent to thermal radiation. Using the calculations of threshold limitations for detecting the fluctuations of thermal radiation as a signal and not disturbances only, we have adduced some arguments concerning the phenomenological content of the “power of a single photon” qΔν ≡ hν∙Δν not defined earlier, where the frequency band Δν is determined by an observation spectral slit. It has been shown that a non-traditional view of “photonic noise” determined by this factor appears in relation with the fluctuations of the photon flux at the observation spectral slit. With definite measurement parameters, this kind of noise is capable to append some essential points to classical models of photon noises and even block access to measurements of fluctuations in the power of thermal radiation. Also adduced are supplementary considerations concerning the existing models of arising shot noise at the photo detector output as a result of physical-and-statistical specificity of the electron excitation process and its kinetics. Offered is a model for a background radiator, which allows us performing the numerical calculations with the use of the specific parameters of the background structure. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics An intrinsic physical content of “single photon power” − (hν∙Δν) Article published earlier |
| spellingShingle | An intrinsic physical content of “single photon power” − (hν∙Δν) Salkov, E.A. Svechnikov, G.S. |
| title | An intrinsic physical content of “single photon power” − (hν∙Δν) |
| title_full | An intrinsic physical content of “single photon power” − (hν∙Δν) |
| title_fullStr | An intrinsic physical content of “single photon power” − (hν∙Δν) |
| title_full_unstemmed | An intrinsic physical content of “single photon power” − (hν∙Δν) |
| title_short | An intrinsic physical content of “single photon power” − (hν∙Δν) |
| title_sort | intrinsic physical content of “single photon power” − (hν∙δν) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/117604 |
| work_keys_str_mv | AT salkovea anintrinsicphysicalcontentofsinglephotonpowerhνδν AT svechnikovgs anintrinsicphysicalcontentofsinglephotonpowerhνδν AT salkovea intrinsicphysicalcontentofsinglephotonpowerhνδν AT svechnikovgs intrinsicphysicalcontentofsinglephotonpowerhνδν |