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Extrema Envelope Function Multibeam Interference Fabry-Perot. Part I. Properties and Applied Aspects for Plane-Parallel Single-Layer
Systems
P. S. Kosoboutskyy, M. S. Karkulovska
Abstract— In the first part of the work the regularities of the envelope functions of the amplitude-phase spectra of the Fabry-Perot multiple-beam interference for electromagnetic are generalized. Basic physical principles extending the possibilities of the envelope function method for the determination of structure parameters for the single layer are formulated.
Index Terms— antireflection, Fabry-Perot interference, envelope function.
I. Introduction
IT is known that by itself the problem of Fabry-Perot interference was formulated and began to be theoretically studied long ago [1]-[2]. The recurrence formulas for the analysis of reflecance and transmittance curves in the interference extrema were found on the basis of taking into account multiple reflections of beams in films with the subsequent coherent combination by Vlasov [3], and later Lisitsia [4] obtained concrete expressions for reflectance of a plane wave by a system by plane parallel interfaces.
An important aspect in the interference approach to the research of reflection and transmission properties of film surfaces is the analysis in the area of the Fabry-Perot interference extrema formation, on the basis of which the method of the envelopes of their intensity has been developed. In the case of a single-film, for the particular models of reflection an obvious type of the analytical expressions of the envelope function at normal reflection was found in the works [5]-[9] and only recently [10], [11] this method has been generalized for the arbitrary geometry of experiment, s- and p- polarized wave, and arbitrary level of absorption, and for extrema ellipsometry spectra [12] -[14].
Manuscript received January 17, 2012.
Petro Sydorovych Kosoboutskyy is with the Lviv Polytechnic National University, S. Bandera 12 Str., 79646, Lviv, Ukraine (e-mail: petkosob@yahoo.com)
Mar’yana Savivna Karkulovska is with the Lviv Polytechnic National University, S. Bandera 12 Str., 79646, Lviv, Ukraine (corresponding author e-mail: MKarkulovska@ukr.net)
Taking into account a wide range of practical applications of systems with multiple-beam coherent signal combining in optical filters [15], sensors [16], [17], the properties of the envelopes of the FPIE are generalized in the first part of this paper on the basis of the results of the original research, hence the physical principles of the diagnosis of single-layer coating parameters have been formulated.
II. Model, main relations and basic conclusions
Let the plane light wave of arbitrary s- and p-polarization propagate in the semi-infinite medium (index 0) with refractive index no and fall at an arbitrary incidence angle a on the surface of a layer (index 1). The layer have a geometrical thickness d, complex a refractive index ~1 = П1 — ix, in which a wave undergoes a complex phase
shift S =
4nd, ~
-n1
cos в , where в is a complex angle of
refraction on the ambient-layer interface.
It is known that for single interfaces the amplitudes of complex coefficients of reflection ~ = CTexp(—ф) and
transmission t = rexp(—ip) at an arbitrary angle of light
incidence for both s- and p-polarization are determined by the known formulas
~ = ~01 + exp(—iS) ~
ЮЙ2^ exp
S
-i —
2
1 + r01~2^exp(—iS) ’ 1 + ~)1~2^exp(—iS)
where Q = exp(lmS), S = ReS , ~)112 and ~Ц2 are
(1)
wellknown Fresnel amplitude coefficient for each single interface, ф and cp are the respective phase shifts. The
resonant complex coefficients of reflection and transmission for both polarizations are modeled by the Lorentz single-oscillator function
£ (a) = £0 +
4n%a>0
2 2
a>0 — a — ta>y
(2)
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where Sq is the background dielectric constant at low frequencies; 4nD is the oscillator force on the resonant frequency ®q ; у is the damping parameter.
According to (1) the energy coefficient of reflection R and transmission T , and the tangent of phases tan^ of wave are defined as follows
ctqi + 0 + 20"qi@ cos 2 A
R =------------------------------
1 + o"qi0 + 2ctqi0 cos 2A+
T:
>01
«2 cos Y
T01T12^
and
tan^
n0 cos a 1 + o"o102 + 2oq10 cos 2A+
Im ~
Re ~
= о01~ ~ 02)sinф01 + 0~~о01 )sm~2 —s) о01~ + 02 )cos^)1 + 0~ + <°01 )cos ~2 — s)
where T01,12 = ~01,12 ’ ~0*1,12 , 0 = °12^ ,
± S ± Ф01 — Ф12
A =--------2-----~. Then expression for energetic
coefficients reflection and transmission are defined as [7]: Rm + b2 cos2 A— Rm — a2 sin2 A—
R = ■
1 + b2 cos2 A+ 1 — a2 sin2 A+
(5)
T =
T
-L m
TM
1 — a2 sin2 A+ 1 + b2 cos2 A+
where the values of these power coefficients in the extrema RmM and Tm M (m is the index of the minimum
extrema, and M is the index of the maximum extrema),
2 4ctq1©
a =
b2 =-
4oO10
(1 + ^010)’ (1 — ^010)
incidence a , angle of refraction в at the boundary 12 and angle of refraction y at the boundary 23 are related by known Snell's law «q sin a = «1sin в = «2 sin y . For the phase spectra the analytical form of envelope expression could be determined only for the reflected wave as:
^0l(l — o-12® ).sin Ф01 ■+ g|2(1 — ори P
,(6)
1. Functions (5) and (6) are the general analytical expressions of FPIE envelopes of multibeam interference of electromagnetic plane and gaussian beam light of s- and p-polarized waves; they are valid for both transparent and absorbing structures at the normal and oblique incidence of light. The phase spectra are described correctly by the envelope method only in the reflection geometry.
2. The points of contact of envelope functions with the Fabry-Perot contours on the side of the maxima R = Rm
(3) and minima R = Rm are determined by
conditions a2 = 0, sin2 A± * 0, (a)
a2 * 0, sin2 A± = 0, (b)
(4) b2 = 0, cos2 A± * 0, (c)
b2 * 0, cos2 A± = 0, (d)
(7)
and they do not necessarily have to be the points of the extrema. Conditions (a) and (c) correspond to the manifestations of Brewster effect (pseudo-effect) on the opposite single interfaces, and (b) and (d) correspond to phase compensations in the points of contact with the envelope functions of the maxima A1 = mn (7b) and
f 1 ^
(7d) in multiple-beam
minima A
m +—
V 2.
n
Here the angle of
interference [23].
3. 2n periodicity of the Fabry-Perot spectra makes it possible to define an area under the contour of an arbitrary maximum, excluding the area under the envelope of the minima, as
1 2n
Sm = П (R — Rm )dS. (8)
0
Within the limits of one extrema the integral (8) has the 2n 2n
form Sm =
1 — Rm
2n
! dS~!
dS
0
0 1 + b 2 cos2
S
S,
фМ ,m = 2n±^^--------'M '"Л
°01\1 + 0 /cos^)1
which under the condition Ф0112 = mn is simplified to the f Ф * 2_, (1 — oo21)0
form фМ,m * 2n ±-----------2: .
001(1 + 02)
The essence of the most important regularities of envelopes (5) and (6) consists in the following.
M
= (1 — Rm )1 —
•y/1 + b 2
and equals
(9)
Assuming that within the limits of the 2 n -band of the interference the parameters Ф0112, b, Rm change
insignificantly and replacing the value of the power reflection coefficients Rm in the nearby minima with an average value we will find that the expression for an area under the contour of reflection is simplified to the form
2
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91
SM * RM
1 -
Vl + b 2
In the area of a single oscillator
Д+ have
resonance transition (2), the functions synchronous oscillations, therefore the area limited by the maximum of reflection will equal
2°01°i2 ( — ст01 Xf — © )
Sm = 2n -----------V \ \2 Q (10)
(i+CToi®K1 —©2
-T
M = 2©
1 -^01
1 -ff()1© 2
ДТ =
cos f «2
T01T12^
4ст01©
cosa "0 (m^oi©2 (i-021®2)
determine the ranges of the energy coefficients that correspond to the changes in geometrical layer thickness, under condition of invariable optical parameters of the media. As it is shown in Fig.l, there occur at the frequency a i the points of dielectric contrast disappearance between the film and substrate (isotropic point [21]) where Fabry -Perot oscillations will collapse out and ДЯ = 0 , ДТ = 0, © = 0, and envelopes Rm = Rm , Tm = Tm will contact each other. At the isotropic point the Fabry-Perot contours will invert.
ф (a)-2n фm(a )-2л_. фM (a )-2n . R (a)
RM (a) ..
Rm (a) '
Z R (a )
a a 0
Fig. 1
In the interval the frequency a < a, provided that for optical parameters of layer "i))xi, and Ree)) Ims , then 2 2
from equations «0 2 = "l(®i0112) for following
analytical expression for the frequency are obtained:
ai,01,12 = a0„
1 + -
4nD
' 0-"0,2
(12)
"+" — e0 ) "0,2; "-" — e0 < "0,2
At the frequency a г- for each interface the energetic
and shows dependent of the about frequency and absorption level of the wave in the layer [20].
4. The envelope contour for a single layer does not oscillate, so far as Fabry-Perot extrema limit the width of the amplitude-phase spectra oscillations as the width of the layer varies. The differences of envelopes ДЯ = Rm — Rm
and AT=Tm~
coefficient Ra ,, = and Ra = ctqi from
"0 > "ъ
©0i 12 ai2
"0 - "1
"0 + "i "1 — "0
"l + "0 "1 - "2
4
Rn
"1 + "2 "2 - "1
"0 < "1.
"i) ^, "2 < "1.
(11) [22]:
"2 + "1
5. The quadratic form of the envelope functions (5) makes it possible to redefine the visibility of the extrema as
wrt =
4(R,t )m --Щй).
t/(R, t)m +#л
(13)
In contrast to the approach adopted according to Michelson [23], the approach (13) substantially simplifies the analytical expressions for visibility:
WR = (l—СТ01 К, WT = 2ст01© . (14)
СТ0Д1 — ©2 )
Taking into consideration the condition of experimental observation of the extrema ©2 <<1, the logarithmic dependences on the frequencies scale are equal to
ln Wr = ln
^ (l -СТ02|)
ст01
4nd
+ — Xi
1
(15)
ln Wt = ln(2CT0iCTi2) + Im 5
i.e. they become practically linear, and in the region of the constancy of the absorption level in the layer Xi = co"St
the slopes ln WRt coordinate with the slope Im5 to the constant. The lineare type of the dependence of (15) makes it possible to determine the absorbing coefficient X1 as
Xi =
4nd
-ln
Wr
ст01
°12
Here the connection between the
visibilities Wr t and according to Michelson Vrt is established by means of the transformation
2x+y _4x +Jy +-Jx-<Jy x—y 4X —Jy 4x +Jy
whence V
r,t
= 2[Wt?,T + WR,T ^ .
1
1
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6. Within the limits of the 2 п -band the following
equality holds:
Rm - R _ t - Tm
R - Rm TM - T
fa^
V b J
2 Д+’-
tan2-----. Its
2
left parts change in the scope [0,+<ю] and on the Fabry -Perot contour on both sides in relation to the points of the maximum it is always possible to single out the frequencies ®Z and ®z + Д®£, which correspond to the phase thicknesses З^ and З^ + ДЗ^ , for which the following
relations hold:
RM - R _ T - Tm
R - Rm
Tm - T
_ 1.
On these
frequencies the power coefficients are equal to
ZR,T _ ^ KT’ R[M + (R, T)m ]. (16)
As shown in fig. 1, the contours Zrt (16) do not
oscillate and for them the (16) hold, for which multiple-beam interference disappears. The spectral width of the interference bands Д®£ at (16) level is related to the remains above the 2п of the wave phase shift ДЗ^ by means of the expression
п Д®£
ДЗ
2 Д®
(17)
mM
where Д®
'mM
01
CT01©:
_ ¥ ± л/i+¥4", © _
°~12
CT12
'\lRm "
1
Tm -
(18)
the system where ^:
" ^Rm
condition of the phase compensation [19], [24] holds and it is localized on the frequency
® m_® dP1 W^i2 - P2
(19)
where
p 1 4пК
P1 _ 1 +---+
s0
л/2®о tan З
f 4ПК . Л
, p2 _ 1 + sin З)
J V s0 J
It is noted here that on the frequency com the tangent of the phase thickness of a layer is tan З _ tan ф . Therefore, the equality
s0 + Ss + s2 + 2s0s2ctg_
sin2 З
. 2 , 2 s1 +s2
holds and the dispersion equation takes the form П4 + s0 + 2s0n2ctg З +
+
4ПК ®q 4пК cop + 2(®q — ® ^(2 + spctg З
, 2 2\2 2 2 (®q -o ) +OT
)
2и4
s0 +
4п К ®Q (п К ®Q + 2sq
(0 -°2\
/2 2Л2 , 2 2 (®0 — ® ) + ® у
sin2 З ^
where s _ s1 + i s2 . In the limit of y{{ ®q we obtain that
: ®m - ®M is the spectral resolution between the adjacent minimum and maximum. Since the values Дюпм and Д®£ included in (17) are
experimentally determinable, it is possible to estimate the remains of the phase thickness ДЗ^ in relation to the phase period 2 п.
7. In an arbitrary geometry and polarization of a wave, the experimentally determined values of the power coefficients (R, T)m m and the structure parameters
°"01,°T2, ^1 are connected by means of a system of equations
-jRM
f \2 ®m
®q j
л 4пК 2 З '• 1 +-cos —
s0 2
(20)
Hence, if the phase thickness of the layer is З = 4п,4п...., the frequency ®m is localized in the vicinity of ®l (®m ^ ®l ), where ®l is the longitudinal frequency. For З _ п,3п,5п..., the minima is localized at the frequency ®q (®m ^ ®q ). Therefore, a change in the phase thickness of the layer to a periodic variation of frequency of the minima of contour reflection ®m within the limits of the longitudinal-transverse splitting Д®0Ь _ ®L - ®0 as shown on the Fig. 2.
ДМ+л1тш
making it possible to determine the optical characteristics of 1 + У RMRm
The system of
^R-M +•
equations is applicable to an arbitrary geometry of experiment and light polarization.
8. The method of envelope functions is effective in the area of resonance dispersion of the substrate for the determination of the phase layer thickness of the nanoscale З << 2п . In the point of contact of the envelope function of the minima with the reflection contour R _ Rm the
Fig. 2
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93
In the case of the vanishing layer thickness S ^ 0 the expression (20) coincides with the known Lyddane-Sachs-I 4 па
Teller relation <am = a> 0 1 +----[25].
V s0
III. Conclusion
1. Theoretical investigation of of the spectral characteristics of the envelopes of reflection and transmission spectra of light by single-layers strusture has been carried out. The general the analytical expressions for the envelopes of transparent and absorbing structures at the normal and oblique incidence of plane electromagnetic and acoustic wave of both polarization (s and p ) are found. It is shown, that the envelope function can be connected between extremum reflecting and transmitting energetic coefficients and parameters of layers.
2. In the single-film coatings the envelopes of Fabry-Perot spectra intercept at the expense of spectral dispersion of refraction index one of the media which form plane-parallel film.
3. The main conclusions about the envelopes of the Fabry-Perot extrema for single-layer coatings we should note that the above-mentioned conclusions are also valid for Gaussian beams, within the validation of the Fabry-Perot principle for them.
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P.S. Kosoboutskyy, Professor of CAD department, Lviv Polytechnic National University. Research interests: Solid-state
physics and semiconductors, optical interferometry, oscillating processes.
E-mail: petkosob (at) polynet.lviv.ua
M. S. Karkulovska, Assistant professor of CAD department, Lviv Polytechnic National University. Research interests: Fabry-Perot interferometer. E-mail: Mkarkulovska (at) ukr.net.
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