Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures
The method to obtain analytical expressions for envelope functions in spectra of normal incidence light reflection and transmission by single-layer structures is proposed
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
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nasplib_isofts_kiev_ua-123456789-1177772025-06-03T16:25:24Z Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures Kosobutskyy, P.S. The method to obtain analytical expressions for envelope functions in spectra of normal incidence light reflection and transmission by single-layer structures is proposed 2007 Article Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures / P.S. Kosobutskyy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 1. — С. 67-71. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS 78.30-j https://nasplib.isofts.kiev.ua/handle/123456789/117777 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The method to obtain analytical expressions for envelope functions in spectra
of normal incidence light reflection and transmission by single-layer structures is
proposed |
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Kosobutskyy, P.S. |
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Kosobutskyy, P.S. Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures Semiconductor Physics Quantum Electronics & Optoelectronics |
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Kosobutskyy, P.S. |
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Kosobutskyy, P.S. |
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Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures |
| title_short |
Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures |
| title_full |
Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures |
| title_fullStr |
Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures |
| title_full_unstemmed |
Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures |
| title_sort |
simulation of applied principles of envelope functions for fabry-perot spectroscopy of plane wave for single-layer structures |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2007 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117777 |
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Simulation of applied principles of envelope functions for Fabry-Perot spectroscopy of plane wave for single-layer structures / P.S. Kosobutskyy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 1. — С. 67-71. — Бібліогр.: 28 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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| first_indexed |
2025-11-24T09:58:47Z |
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2025-11-24T09:58:47Z |
| _version_ |
1849665338400047104 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 1. P. 67-71.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
67
PACS 78.30-j
Simulation of applied principles of envelope functions
for Fabry-Perot spectroscopy of plane wave
for single-layer structures
P.S. Kosobutskyy
National University “Lviv Polytechnic”, 13, Bandery str., 79646 Lviv, Ukraine
E-mail: petkosob@yahoo.com
Abstract. The method to obtain analytical expressions for envelope functions in spectra
of normal incidence light reflection and transmission by single-layer structures is
proposed.
Keywords: Fabry-Perot interferometry, single-layer structure, reflection, transmission,
envelope function method.
Manuscript received 11.10.06; accepted for publication 26.03.07; published online 01.06.07.
1. Introduction
The principle of multibeam interferometry of light for
plane parallel plate developed by Fabry-Perot [1] have
been widely used for solving a large number of scientific
and applied problems and have been discussed in many
textbooks [2-5]. Neveretheless, a number of new
regularities in the scope of the problem were recently
found. It was shown that Fabry-Perot interferometry is
still actual for non-destructive control of optical para-
meters of micro- and nano-size layers [6-8].
Recently, the so-called method of envelope
functions as tangential to contours of amplitude spectra
of light reflection and transmission by a single-layer
Fabry-Perot structures [10-19] was developed. So, for
example, on the basis of the method the approach for
determining the variation parameters in thickness and
influence of spectral distribution of light and light
absorption [9-15] was grounded. The envelope function
method was used for grounding the new approach for
determination of the instrumental characteristics of
interference bands [16, 17] and for reconstruction of the
phase of a reflected wave by single-layer films [16, 17,
19]. In other our papers [20-26], we developed the
envelope function method for arbitrary ratio between the
indices of refraction 3,2,1n of three media.
In the paper, authors generalized the basic
principles of application of the envelope function
method to the data analysis in amplitude-phase
spectroscopy of light interference for single-layer
structures of Fabry-Perot type (SLSFP) in the case of
normal incidence of a beam onto the interfaces for
transparent and absorptive structures. A method for
obtaining the analytical expression for energetic
coefficients ( ) minmax,,TR and phase minmax,φ is
proposed.
2. General relations
The SLSFP of a thickness d and of the complex
refraction index 222
~ χ−= inn is bounded by the semi-
infinite top transparent media with refraction index 1n
(interface 12) and bottom (transparent or absorbing)
media with refraction index 3
~n (interface 23). In the
absorbing layer, the phase change of wave is equal to
2
~4~ n
d
λ
π
=δ . It is well known also that taking into
account the multibeam interference the complex
amplitude values of light reflection r~ and transmission
t~ by SLSFP are determined from
( )
( )δ−+
δ−+
= ~exp~~1
~exp~~
~
2312
2312
irr
irr
r and
( )
( )δ−+
δ−
= ~exp~~1
2/~exp~~
~
2312
2312
irr
itt
t ,
(1)
where 23,12
~r and 23,12
~t are the well-known amplitude
Fresnel's coefficients [2] for the single interfaces with
subscripts 12 and 23. The resonant dispersion of the
media is modelled by one-oscillator function of
permittivity [27]
ε~ =
ωτ+ω−ω
ωπ
+ε
i22
0
2
0
0
4 D
. (2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 1. P. 67-71.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
68
Here 0ε is the background permittivity, Dπ4 is the
oscillator force of transition into electron state with
resonant frequency 0ω , τ is the damping factor.
According to Eq. (1), the energy coefficient of
reflection R and transmission T, and the tangent of
phases φtan and Φtan of light are defined as follows
+
−
Θσ+Θσ+
Θσ+Θ+σ
=
F
F
R
cos21
cos2
12
22
12
12
22
12 ,
+Θσ+Θσ+
Ω
=
F
TT
n
n
T
cos21 12
22
12
2312
1
3 , (3)
and
( ) ( ) ( )
( ) ( ) ( ) ,~Recos1cos1
~Resin1sin1
~Re
~Im
tan
23
2
1212
2
12
23
2
1212
2
12
δ−φσ+Θ+φΘ+σ
δ−φσ−Θ+φΘ−σ
=
==φ
r
r
(4)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
δ−Θσ+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
δ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
δ−Θσ+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
δ
−=
==Φ
+
+
~Re
2
1
cos~Re
2
1
cos
~Re
2
1
sin~Re
2
1
sin
~Re
~Im
tan
12
12
F
F
t
t
(5)
where ( )δ−=Ω ~Imexp , ∗⋅= 23,1223,1223,12
~~ ttT ,
( )23,1223,1223,12 exp~ φσ= ir , ( )δ−φ±φ=±
~Re2312F .
Then expression for energetic coefficients of
reflection and transmission are defined as [21, 24-26]:
2
sin1
2
sin
2
cos1
2
cos
22
22
max
22
22
min
+
−
+
−
−
−
=
+
+
=
F
a
F
aR
F
b
F
bR
R and
2
cos1
2
sin1 22
max
22
min
++ +
=
−
=
F
b
T
F
a
T
T , (6)
where
( )212
122
1
4
Θσ+
Θσ
=a and
( )212
122
1
4
Θσ−
Θσ
=b ,
Ωσ=Θ 23 . The functions
2
12
12
minmax, 1 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
Θσ±
Θ±σ
=R and
( )212
2312
1
3
minmax,
1 Θσ
Ω
=
m
TT
n
n
T
(7)
are the envelope functions of Fabry-Perot spectra.
3. Discussion
1. SLSFP with resonant dispersion (2). In many
practical problems, medium with refraction 1n is air or
vacuum and 11 =n . For a free layer with resonant
dispersion of )(~ ωε , the calculated spectra )(ωR , )(ωT
)(ωφ , )(ωΦ and their envelope functions
( ) minmax,,, φTR are shown in Fig. 2. In the region of
resonant absorption near the resonant frequency 0ω , it is
possible to separate a frequency interval with width
pω∆ , which is bounded by an interval with significant
absorption where 0→Ω . Inside the interval
minmax RR ≈ , and the spectra are formed as if the light
wave is reflected from a semi-infinite medium with
resonant dispersion (2) [21, 24-26]. The distance
between envelope functions minmax RRR −=∆ and
minmax TTT −=∆ are equal to
R∆
( )
( )( )22
12222
12
12 11
1
4
Θ−σ−
Θσ−
Θσ
= and
( ) 2312
1
3
222
12
12
1
4
TT
n
n
T
Θσ−
ΘΩσ
=∆ . (8)
The absorption level increases with approaching to
the resonant frequency 0ω , and the value 0→∆R
because of damping factor approaching to null 0→Ω .
Near the resonant region beyond the interval pω∆ , the
functions (Fig. 1b)
( )
( ) 1212
2
1223122312
minmax, cos1
)1(sin1
2
φΘ+σ
Ωσ−σ+φΘσ−σ
±π≅φ (9)
are the envelope functions of the phase spectrum of
reflection. It is problematic to apply the envelope
function method (Fig. 1b) to describe the phase spectrum
)(ωΦ for light transmitted through the layer.
At the arbitrary frequency, we have the following
expression:
2
tan 2
2
max
min
min
max ±⎟
⎠
⎞
⎜
⎝
⎛
=
−
−
=
−
− F
b
a
TT
TT
RR
RR
. (10)
The ratios
min
max
RR
RR
−
−
and
min
max
TT
TT
−
−
vary from 0 to
+∞ , and at the frequencies 2,1ω from both side of
extreme contour peak the ratios are equal to 1. Hence,
coefficients for reflected and transmitted light at the
frequencies 2,1ω equal (Fig. 2)
[ ]minmax2
1
RRR +=Σ and [ ]minmax2
1
TTT +=Σ . (11)
It is the approach (11) that enables to ground the
validity of interferogram apparatus characteristic deter-
mination [16, 17]. The extreme contour width ω∆ under
the condition (11) equals δ∆=ω∆
dn
c
2
0
2
, where δ∆ is
the width contour extreme in phase units (phase distance
between frequencies 2,1ω on both sides from its peak).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 1. P. 67-71.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
69
0.75
0.2
R ω( )
ΣR ω( )
T ω( )
ΣT ω( )
0.99970.9985 ω
0
Fig. 2. Calculated spectra of )(ωR , )(ωT in the resonant
region and contours )(ωΣR and )(ωΣT .
1
0
R (ω)
Rmax (ω)
Rmin (ω)
Ω (ω)
T (ω)
Tmax (ω)
Tmin (ω)
1.010.97 ω
ω 0
a)
2.5 π ⋅
1.5 π ⋅
φ (ω)
φ max (ω)
φ min (ω)
Φ (ω)
1.010.99 ω
ω 0
b)
Fig. 1. Calculated spectra of )(ωR , )(ωT in the resonant
region and their envelope functions ( ) minmax,,TR (a) and )(ωφ ,
)(ωΦ (b).
The phase width δ∆ equals
min
2
ω∆
ω∆
π=δ∆ , where
minω∆ is the distance between the frequencies on both
sides from the maximum.
The π2 -periodicity of Fabry-Perot spectra allows
to determine the area restricted by the contour of a
maximum minus the area restricted by envelope function
of adjacent minima through the structure parameters as
[ ] ζ−ζ= ∫
π
π
dRRS R
2/3
2/
min)(2 , (12)
where
2
~Re δ
=ζ . For transparent structures the area RS
is equal
=RS ( ) ⎥
⎦
⎤
⎢
⎣
⎡
ζ+
ζ
−π− ∫
π
π
2/3
2/
22min cos1
1
b
d
R .
This integral is the standard integral [28].
In the region of absorption the integration
procedure can be simplified by replacement minR at the
maximum frequency by the mean value of two adjacent
minima ( )1min,1min,2
1
+− + mm RR [22]. Clearly, this way can
be applied to the transmitted waves.
2. Free layer and the fixed one on substrate
surfaces with constant absorption const2 =χ . The
values of energetic coefficients ( ) minmax,,TR in the
absorptive films are a function of λ . Even for absorptive
layers with constant absorption, energetic coefficients
( ) minmax,,TR vary with variation of λ . According to the
definition given by Michelson the visibility of the
interferogram
minmax
minmax
RR
RR
V
+
−
= . Then, according to the
expression (7) we have
( )
( )
( )
( )
1
2
12
2
12
2
12
2
12
1
1
1
1
2
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Θ−σ
Θσ−
+
Θσ−
Θ−σ
=V . (15)
Using the transformation
yx
yx
yx
yx
yx
yx
+
−
+
−
+
=
−
+
2 , (16)
we obtained that [ ] 112
−− += WWV , where =W
2
2
12
12minmax
minmax
1
1
Θ−
σ−
σ
Θ
=
+
−
=
RR
RR
. Therefore, when the
absorption index 2χ does not depend on frequency, the
relative slope between linear dependences 1ln −W and
δ~Im does not depend on frequency for an arbitrary ratio
between the refraction indices 3,2,1n (Fig. 3) and the
relation
1ln −W = ( )δ+ ~Imexpconst (17)
ω0
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 1. P. 67-71.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
70
0.3
0.3−
ln V ω( ) 1−( )
ln W ω( ) 1−( )
ln Q ω( ) 1−( ) 4.4−
Im δ ω( )( )− 1.46−
32 ω
Fig. 3. Calculated dependences of 1ln −W , 1ln −V and
δ~Im for parameters: 11 =n , 25.12 =n , 5.33 =n ,
0038.02 =χ , m15 µ=d .
is valid with accuracy to constant. Slope of Vln
depends on the ratio between the refraction indices
3,2,1n .
4. Principal conclusions
Amplitude-phase Fabry-Perot spectra for single-layer
structures at the normal can be described by the
envelope functions ( ) minmax,,TR .
The width of the interference band as a phase
separation between the points, for which the reflectance
are ( ) ( )[ ]minmax ,,
2
1 TRTR + , is determined through
instrumental characteristics of the Fabry-Perot
interferogram of single-layer structures.
In the region of constant light absorption in film,
the slope of
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+
−
minmax
minmaxln
RR
RR
equals to that of δ~Im .
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