- Email: [email protected]
Inversion of rugate spectra
579
Thin Solid Films, 181 (1989) 579-588
INVERSION OF R U G A T E SPECTRA ROGER J. BECKER AND MARK K. KULLEN
Research Institute, University of Dayton, Dayton, OH 45469-0001 (U.S.A.) (Received March 16, 1989)
Reflection spectra from inhomogeneous films can be interpreted in terms of a characteristic admittance to obtain the dielectric profile of a complex film. All of the standard formulas used for film analysis remain valid if the film index is replaced by the characteristic admittance. The reflection spectrum consists of a rapidly oscillating function contained within two envelopes. The envelopes yield the characteristic admittance, which can be transformed to obtain the shape of the index profile. The periodicity of the fast oscillations give the number of layers in the film and phase information. In most cases a simple Fourier transform of the characteristic admittance gives a sufficiently accurate inversion. Phase information is needed at no more than one point in the spectrum.
I. INTRODUCTION
Modern film deposition technology has the potential for the fabrication of films with continuously varying refractive indices, such as rugate filters. Many problems with manufacturing control remain in the development of this potential. Therefore, a non-destructive means of rapidly and accurately determining the index profile would greatly facilitate film analysis as well as real-time monitoring of the deposition process. As the reflection and transmission spectra of optical films are rich in information content, and as spectroscopy is a rugged diagnostic tool, the inversion of spectra from inhomogeneous films is of considerable interest. A direct analytic connection can be made between the reflection or transmission spectrum of a multilayered film and modulation profile of the dielectric constant e in the film. This relationship can be expressed in terms of an effective admittance r/. Although the concept of an effective or characteristic admittance has been in existence for a long time ~'2, it has not yet been applied to the design of multilayered films with nonhomogeneous profiles (e.g. Rugate filters), let alone the inversion of the reflection spectra from these films. This may be because of the lack of simple formulas for calculating r/for inhomogeneous films, which will be given below. The advantages of the impedance concept are that it makes contact with a vast body of literature, both within and from outside the optical community; it 0040-6090/89/$3.50
© ElsevierSequoia/Printedin The Netherlands
580
R . J . BECKER, M. K. K U L L E N
provides a direct means of obtaining swift inversions of spectral performance; it gives formulas for reflection and transmission spectra yielding extremely accurate predictions of performance in rapid (less than 1 s) time scales; it provides a natural framework for describing index matching; and it can be used in formulas for reflectance that have proved useful in the analysis of homogeneous films. The most useful formulas for reflectance and transmittance concern the propagation of light across three regions. In keeping with the notation of Born and Wolf a we denote the initial region by the subscript 0 and the final region by the subscript L. As most cases of interest apply to periodic modulation in the intermediate region, the subscript c will be used for this region, although a characteristic admittance can also be used for either the first or third region if they are non-uniform. The reflection amplitude p for light passing from a uniform region with admittance qo across a film of physical length L into a region with input admittance qL is (r/0 - - qL)r/c coskL + i(qoqL - y/c 2) sinkL P = (qo + qL)qc coskL + i(r/or/L+ r/c2) sinkL
(1)
where k is the propagation constant in the film. I f k is complex then the cosine and sine functions will be replaced by their hyperbolic equivalents. In particular, k will be pure imaginary within a stop band and r/c will be pure imaginary as well. Equation (1) is equivalent to the statement that the input admittance qi at the ambient/film interface is given by ~/i
qL coskL + i~l~sinkL q~ coskL + iqL sinkL q~
(2)
Equation (1) can be reformulated through the Fresnel-type definitions qo -- q~
P12 -- - -
q0 -~- Y~c
r/c -- qL /923 ----- Y~c"~ qL
(3)
to yield the familiar Airey formula [4,5] P12 + P 2 3 e21~r
fl -
(4)
l +plEP23e21a
where a -- kL. 2. THEORY The validity of eqn. (1) and the equivalent eqns. (2)-(4) require the proper definition of qc, or of qo and qL, in the event that they pertain to inhomogeneous regions. This is achieved through the solution to the wave equation for the region in question. This is a second-order equation of the form 1'4 d2~b ~_ko2{e(z)_s2}~9 _ d In(e) d~b dz dz
(5)
for p-polarized (TM) waves. In eqn. (5), ko is the propagation constant in the
581
INVERSION OF RUGATE SPECTRA
incident medium and s is the cosine of the angle of incidence. In the case of spolarized (TE) waves, the first-order term in eqn. (5) is omitted. Except for special singular values of k o, q~ has two orthogonal solutions. Denoting these solutions by 4)1 and thr2, r/c2 = r/lq2
(6)
where rli(Z' ) =
~)i'(Z')/ko~)i(Z')
i = 1,2
(7)
where z' is the location of the ambient/film interface. As the electric field strength E is given by q~ and the magnetic field strength H is given by q~', eqn. (7) is in keeping with the definition H(ko) = tl(ko)E(ko)
(8)
As q~ and q2 are derived from ~b~ and q~z it might appear that no gain has been achieved in the use of eqns. (1)-(4) and qc over the formal solution to the wave equation and the calculation of reflection spectra in terms of the elements of the transfer matrix 3 5 which are expressed in terms of ~bl, q~2, and their derivatives. However, r/c is much simpler in form than the wavefunction. The rapid oscillations in the reflectance that occur in multilayer films are described by the cosine and sine functions in eqn. (1); r/c itself is slowly varying and can be approximated by relatively simple expressions to high accuracy. For example, excluding harmonic resonances, a film with a dielectric constant e given by = Co+el cos kcz will have a value for tl~ 2 ~ So + e l k
(9)
qc2 given by
cos k c L / ( k c 2 - ko 2)
(10)
i.e. tl~ is essentially
given by the mean value of the refractive index in the film ri below a resonance at k¢, by ri minus the modulation depth An above the resonance, and by a Lorentzian-like resonance term at the resonance. The simplicity of expressions such as eqn. (10) is because of the cancellation of lower-order and rapidly oscillating terms in ql and q2 when the product given by eqn. (6) is taken. Part of the utility of eqns. (6) and (7) in eqns. (1)-(4) is that it facilitates the interpretation of spectral behavior. For example, in the absence of a passivation layer, when k L = nrt, we find Pb = (tlo -- rlL)/(~lo + rlL)
(1 1)
independent of t/c. The condition k L = nrt results in either maxima or minima in the sideband spectra. In the case of a multilayered film r/o and qL will be slowly varying compared with r/c, and one of the two envelopes enclosing the side lobes of the reflection spectra, which we will denote by R b, will be given by eqn. (11). This envelope will yield the substrate index as a function of wavelength, or the dispersion curve for the substrate refractive index nL if the ambient admittance is known. Thus it should be possible to identify not only the index of the substrate, but its chemical makeup as well. Alternatively, i f n L is known, the nature of any passivation layer can
582
R . J . BECKER, M. K. KULLEN
be discerned from R b and eqn. (11). If both n o and n L are constant, then Rb will be constant, resulting in a fiat side-lobe envelope (see Fig. 1). I f k L = (n +½)it, then Pe =
(r/Or/L --
(12)
r/c2)/(r/Or/L + r/c 2)
Equation (12) gives the amplitude of the second side-lobe envelope, which we denote by Rc. Ifr/o and r/r are known, then this envelope will yield the real part ofr/c 2 (see Fig. I). These relations are quite general, and are independent of the details of modulation profile in the film, so long as it is described by the same wave equation throughout its length. The propagation constant k will be a pure imaginary number near the middle of the stop bands. If losses in a medium are small, then k will have only a small imaginary part in the reflection spectrum pass bands. Of course, when k is complex or pure imaginary cos k L and sin k L should be replaced by cosh k L and sinh kL. Equation (1) holds exactly provided eqns. (6) and (7) are used to obtain q~c. Equation (4) follows from eqn. (1) if the definitions given by eqn. (3) are used. Equation (1) may be rewritten as
P
=
r/L)e -
ikL
(r/o-- r/c)tr/c-- r/L) e -
ikL
(/70 - - qc)(r/L Jr r/c)e ikL Jr (/70 Jr ~]c)(qc - Jr "e ikt" + . . . .
(r/o + r/c)(r/L
r/c)
(r/o - ql,)/(r/o + r/e) + (r/c - r/1,)/(r/, + r/L)e- 2ikLr/o P - 1 + {(r/o - qc)(r/c - r/L)/(r/o + r/c)(q=+ r/L)}e- 2,kr
(13) (14)
which gives eqn. (4) using the definitions ofeqn. (3).
1.00
0.86
nI =
1.0
nL = 5.5 0.71
=
'~0 = '~l
0.57
18o I0.0
= 0.5
R
O.43
0,14 0.00 I
0.60
0.70
0.80
0.90
I
I llilltLlllllllllllll 1.00 1.10 1.20
1.30
1.40
kod
Fig. 1. Reflection spectrum for single-line Rugate filter. Heavy line is prediction from eqn. (11) based on exact qc2 curve. One of the side-lobe envelopes is tangent to a horizontal line giving the substrate index.
INVERSION OF RUGATE SPECTRA
583
3. INVERSION Based on the present level of understanding of the nature of r/C, it is possible to outline an approach to direct inversions of reflection spectra to obtain the dielectric profiles of multilayer films. To the extent that the film consists of several diverse regions, this procedure will become complicated. However, in most cases, the film profile will be reasonably simple (e.g. a two-line sinusoidal filter with a passivation layer and an accidental spacer layer). Indeed, in many cases, such as real-time in-situ monitoring, r/o will be identically equal to unity; r/L will correspond to an infinite homogeneous medium whose index of refraction is known as a function o f k o, and the intended profile, and hence the approximate form for r/c, will be known. Of course, to find r/C(ko) exactly from the reflection spectrum for a general case, it will be necessary to obtain the reflection amplitude p(kao), and not just the reflection intensity R. As p is complex, phase information will be needed. Fortunately, this can be obtained to reasonable accuracy from the reflection spectrum alone. It must be stressed that the various elements in the reflection spectrum are not independent quantities, but are related on the basis of fundamental principles such as causality 6"7. In particular, writing p = re i*
(15)
we have In p = In r + i lnq~
(16)
therefore, the quantities In r and In ~bare the real and imaginary parts of the response function In p, and must be Hilbert transform pairs. This means that q~(ko) can be obtained from a Kramers-Kronig type interpretation over the spectrum R(ko)6' 7. In general, it will be helpful to have an exact value for ~bat as many spectral points as possible. Fortunately, this can easily be done for a multilayer film with low absorption. For example, if no passivation layer is present and there are an integral number of cycles in the filter, the phase of all of the maxima and minima corresponding to R b is n and the phase of the maxima in R e is also n. For the case given, the phase of the minima in R e depends on whether rh(ko) is greater or less than r/o(k0). Just to the right of the stop bands r/c~0 and the phase of the peaks is zero. In general, however, the side-lobe minima in R e will in general have a phase of n. It should be possible to obtain the phase's sideband maxima and minima for more complicated cases through a study of the properties of eqn. (1). To the extent that the film profile can be described by a simple equation, th may be modeled through eqn. (1). A large number of sum rules are known for response functions 6, so it should be possible to obtain self-consistent estimates of r/c(ko) for regularly periodic films with a high degree of accuracy. As a general rule, given a response function r/(k), its spatial counterpart n(z) can be obtained by a Fourier transform:
n(z) =
f dl,elk~q(k)
(17)
with an inverse transform giving q(k) from n(z). This suggests that the inversion of
584
R . J . BECKER, M. K. KULLEN
qc(k) can readily be made. Unfortunately, the definition for r/c given by eqns. (6) and
(7) shows that q~ is actually a function of both k and the surface position z' or, equivalently, the phase ofn(z) at the surface. Therefore, it may be that in general an inversion of r/~(k) via a Fourier transform may not be unique. However, considering that qc consists essentially of a background value given by a and a resonance factor, it is probable that the functional dependence of r/¢ does not depend significantly on z', and that the primary effect of small changes in the length of a filter is to create an oscillation in the background level due to the modulation depth, An, and to produce a shift in the phase of the resonance part of r/~. Indeed, a Fourier transformation of the index profile corresponding to the dielectrix constant given by eqn. (9) leads to eqn. (10).
4.
SPECTRAL BEHAVIOR
Before outlining how a reflection spectrum may be analyzed to obtain film profiles, it will be helpful to review some important general characteristics of regularly periodic multilayered films. This review will be followed by a discussion of the effect of spacer layers in a film. The peak reflectance of a filter with matched external indices (n~ = ~ = nL) is given by RN
=
pN 2
=
tanh 2 xN
(18)
where i¢ = b 6 + c 6 2
(19)
6 = ~I/Eo
(20)
and
The parameters b and c are functions of the modulation profile only. The ratio c/b is of order 6 and c can usually be neglected if6 is small. For sinusoidal modulation, c is identically zero. The behavior of b and c as a function of various a/d profiles for square-well modulation is shown in Fig. 2. Figure 3 shows how x varies with 6 for several cases. Equation (18) is exact and applies to all profiles. If the filter has an incomplete surface layer, N will assume a non-integer value. If the external indices are not matched, then the effective reflectance amplitude p can be calculated in an analogous manner to that used for films of multiple homogeneous layers. The center-band frequency of the mth harmonic is a filter with an integral number of layers given by k c = m~/~d
(21)
The peak reflectance of thejth side lobe occurs approximately at J ~ kj = ~V~ The/th minimum reflectance in the sideband structure occurs at approximately
(22)
INVERSION OF R U G A T E SPECTRA
585
0.90
0.70
0.50
0.30
0.10
-
-0.10
-
-0.30
-
I
0.50
I
0.10
0.00
0.20
I
I
0.30
0.40
1 0.50
I
I 0.60
I 0.80
0.70
I 0.90
old
Fig. 2 Variation in the coupling coefficients b and c with profile shape for square-well filters.
17.5
15.0
0 ~0 E0 + a/d X old 0 o/d old
~ 4.0 • 9.0 =0.1 = 0.2 =0.~. = 0.5
I .-
~ aid • 0.7 z aid - O.B ¥ a/el = 0.9
/
jj
i'°
IZ.5 o x I0.0
7.5
5.0
z.61
0.000
0.0;~5
0.050
0.075
0,100
0.123
0,150
0.175
0.2C
El / E 0
Fig. 3. Dependence of coupling parameter ~ on modulation depth and profile shape.
586
R . J . BECKER, M. K. K U L L E N
j + 1/2 rt N ri
kf -
(23)
Equations (22) and (23) must be corrected for the dispersion due to the modulation profile in the vicinity of k c. For small 6 this dispersion is small. Filters with sinusoidal profiles are completely dispersionless for small 6. At normal incidence points well removed from the harmonic peaks the sideband maxima and minima are given by R
nlnL _~q2 e
- -
m
n in L+ r/z
(24)
nl - n L 2
(25)
and R b --
nl +nL
The normalized line width of the filter is given approximately by Ak k~
-
~6
(26)
for large N. If N is small, eqn. (26) must be corrected. Thus far the best correction formula obtained is Ak/k¢ = ~6 + fl/N + o¢/N 2
(27)
The above comments suggest a systematic procedure for interpreting the spectra of periodic multilayered films. The number of cycles in the film can always be found by counting the number of sideband maxima between harmonics or, equivalently, the mean spacing between sideband maxima in relation to the frequency at which the main peak occurs. Approximate values for nL and v/can be obtained from the asymptotic values of the sideband envelopes at low frequencies. These estimates may be slightly offdue to the presence o f a passivation layer and/or an incomplete top layer, which will introduce a slowly varying modulation of the reflectance. However, exact values for nL and v/can be obtained measuring the shift in the frequencies of the reflection maxima, especially for the stop band and the change in the sideband reflectance as a function of the angle of incidence for either sor p-polarized light. The details in these changes for each sideband peak will give v/ and n L as a function of wavelength. Any discrepancy between the index values and peak frequencies due to measuring shifts with the angle of incidence and taking values at a single polarization and angle of incidence may be attributed to an incomplete top layer and/or passivation layer. The passivation layer of width w will have only a small effect when kw ,~ n/2. The periodicity and intensity variation in the slowly varying part of Rb will then give the mean index of the passivation layer/'/p and w. In short, Rb can be examined to give a self-consistent and reasonable estimate for N, r/, w, np, and nL. As N is known, the optical density at the stop-band peak and the width of the stop band can be used to find the parameters k and c~ from eqns. (18) and (26). Assuming that 6,~ 1, the parameter c in eqn. (19) may be omitted, so that the ratio
587
INVERSION OF R U G A T E SPECTRA
b/ct can be used to estimate the approximate mean shape of the modulation profile using Table I. This estimate can be checked by measuring the intensities of any stopband harmonics. Once the approximate shape of the modulation profile has been assumed, eqns. (18) and (19) can be used to obtain 6, given the measured value of x. Alternately, 6 can be estimated from the measured line width and an assumed value for ~, or from the difference in the asymptotic limits to Re in the upper and lower sideband spectra. Of course, given k¢, N, and r~, the physical width d of each cycle and the filter thickness L can easily be calculated. TABLE I DEPENDENCE OF COUPLING PARAMETERb AND LINE-WIDTH PARAMETER~t ON PROFILE SHAPE. Rp = Ianh2KN
a/d
b
ct
b/ct
0. l 0.2 M 0.3 0.5 0.7 0.8 0.9 S
0.323 0.608 rt/4 0.828 1.000 0.805 0.566 0.296 0.248
0.306 0.722 0.500 1.010 1.220 0.959 0.661 0.263 0.258
1.050 0.842 1.570 0.820 0.820 0.840 0.856 I. 130 1.040
~c = b (AZ/~o) = ki___1%.
.;(2 for b(a/d)
= 0.019.
5. CONCLUSIONS
The characteristic admittance expressed in terms of the solution for the wave equation gives exact expressions for the reflectance of inhomogeneous films which can be used both for the analysis of film properties and rapid calculations of spectra. Although the characteristic admittance can be derived from the solution to the wave equation, it is generally simpler in form than the wavefunction and can often be approximated with little error. It appears that the characteristic admittance offers a useful approach for inverting the reflection spectra of multilayered films. The appropriate for~ for r/c2(k) can be obtained from a Fourier transform of the modulation profile, e(z). The full reflectance and transmission spectra of a multilayered film are rich in information content which can contribute greatly to the non-destructive characterization of a multilayered film. The analysis strongly suggests that a complete inversion of the reflection and transmission spectra of a multilayered inhomogeneous film can be attained using only s-polarized light. This would minimize the need for ellipsometric measurements. However, the availability of p-polarized spectra would at least provide redundancy and enhance accuracy of the spectral interpretation. Unfortunately, the analysis of the propagation of p-polarized light is considerably less tractable than that of s-polarized light. This problem will be the subject of future research.
588
R.J. BECKER, M. K. KULLEN
ACKNOWLEDGMENTS T h e software d e v e l o p m e n t a n d c o m p u t a t i o n s for this w o r k were d o n e by M i c h a e l Creed, D a v i d G r o t e , Jeffrey Diller, Steven Muilens a n d W a y n e J o h n s o n . T h e a u t h o r is grateful to Dr. M e h m e t R o n a for his m a n y helpful c o m m e n t s . This w o r k was f u n d e d by the A i r F o r c e W r i g h t A e r o n a u t i c a l L a b o r a t o r i e s , M a t e r i a l s L a b o r a t o r y , and by A r t h u r D. Little C o r p o r a t i o n , on C o n t r a c t No. F33615-86-C5132, REFERENCES 1 2 3 4
H.A. Macleod, Thin-Film OpticalFilters, MacMillan, New York, 1986. L.I. Epstein, J. Opt. Soc. Am., 42 (1952) 806. M. Born and E. Wolf, Principles of Optics, Pergamon, New York, 1956. R.J. Becker, Analysis of Rugate Filter Behavior, in Proc. 1986 Boulder Laser Damage Syrup., National Bureau of Standards, Boulder, CO, 1986. 5 R.N. Hill and B. O. Rosland, Phys. Reo. B, 11 (1975) 2913. 6 D.Y. Smith, Dispersion Theory, Sum Rules and Their Application to the Analysis of Optical Data, in Handbook of Optical Constants of Solids, Academic Press, New York, 1985. 7 B. Harbecke, Appl. Phys. A,, 40(1986) 151.
Thin Solid Films, 181 (1989) 579-588
INVERSION OF R U G A T E SPECTRA ROGER J. BECKER AND MARK K. KULLEN
Research Institute, University of Dayton, Dayton, OH 45469-0001 (U.S.A.) (Received March 16, 1989)
Reflection spectra from inhomogeneous films can be interpreted in terms of a characteristic admittance to obtain the dielectric profile of a complex film. All of the standard formulas used for film analysis remain valid if the film index is replaced by the characteristic admittance. The reflection spectrum consists of a rapidly oscillating function contained within two envelopes. The envelopes yield the characteristic admittance, which can be transformed to obtain the shape of the index profile. The periodicity of the fast oscillations give the number of layers in the film and phase information. In most cases a simple Fourier transform of the characteristic admittance gives a sufficiently accurate inversion. Phase information is needed at no more than one point in the spectrum.
I. INTRODUCTION
Modern film deposition technology has the potential for the fabrication of films with continuously varying refractive indices, such as rugate filters. Many problems with manufacturing control remain in the development of this potential. Therefore, a non-destructive means of rapidly and accurately determining the index profile would greatly facilitate film analysis as well as real-time monitoring of the deposition process. As the reflection and transmission spectra of optical films are rich in information content, and as spectroscopy is a rugged diagnostic tool, the inversion of spectra from inhomogeneous films is of considerable interest. A direct analytic connection can be made between the reflection or transmission spectrum of a multilayered film and modulation profile of the dielectric constant e in the film. This relationship can be expressed in terms of an effective admittance r/. Although the concept of an effective or characteristic admittance has been in existence for a long time ~'2, it has not yet been applied to the design of multilayered films with nonhomogeneous profiles (e.g. Rugate filters), let alone the inversion of the reflection spectra from these films. This may be because of the lack of simple formulas for calculating r/for inhomogeneous films, which will be given below. The advantages of the impedance concept are that it makes contact with a vast body of literature, both within and from outside the optical community; it 0040-6090/89/$3.50
© ElsevierSequoia/Printedin The Netherlands
580
R . J . BECKER, M. K. K U L L E N
provides a direct means of obtaining swift inversions of spectral performance; it gives formulas for reflection and transmission spectra yielding extremely accurate predictions of performance in rapid (less than 1 s) time scales; it provides a natural framework for describing index matching; and it can be used in formulas for reflectance that have proved useful in the analysis of homogeneous films. The most useful formulas for reflectance and transmittance concern the propagation of light across three regions. In keeping with the notation of Born and Wolf a we denote the initial region by the subscript 0 and the final region by the subscript L. As most cases of interest apply to periodic modulation in the intermediate region, the subscript c will be used for this region, although a characteristic admittance can also be used for either the first or third region if they are non-uniform. The reflection amplitude p for light passing from a uniform region with admittance qo across a film of physical length L into a region with input admittance qL is (r/0 - - qL)r/c coskL + i(qoqL - y/c 2) sinkL P = (qo + qL)qc coskL + i(r/or/L+ r/c2) sinkL
(1)
where k is the propagation constant in the film. I f k is complex then the cosine and sine functions will be replaced by their hyperbolic equivalents. In particular, k will be pure imaginary within a stop band and r/c will be pure imaginary as well. Equation (1) is equivalent to the statement that the input admittance qi at the ambient/film interface is given by ~/i
qL coskL + i~l~sinkL q~ coskL + iqL sinkL q~
(2)
Equation (1) can be reformulated through the Fresnel-type definitions qo -- q~
P12 -- - -
q0 -~- Y~c
r/c -- qL /923 ----- Y~c"~ qL
(3)
to yield the familiar Airey formula [4,5] P12 + P 2 3 e21~r
fl -
(4)
l +plEP23e21a
where a -- kL. 2. THEORY The validity of eqn. (1) and the equivalent eqns. (2)-(4) require the proper definition of qc, or of qo and qL, in the event that they pertain to inhomogeneous regions. This is achieved through the solution to the wave equation for the region in question. This is a second-order equation of the form 1'4 d2~b ~_ko2{e(z)_s2}~9 _ d In(e) d~b dz dz
(5)
for p-polarized (TM) waves. In eqn. (5), ko is the propagation constant in the
581
INVERSION OF RUGATE SPECTRA
incident medium and s is the cosine of the angle of incidence. In the case of spolarized (TE) waves, the first-order term in eqn. (5) is omitted. Except for special singular values of k o, q~ has two orthogonal solutions. Denoting these solutions by 4)1 and thr2, r/c2 = r/lq2
(6)
where rli(Z' ) =
~)i'(Z')/ko~)i(Z')
i = 1,2
(7)
where z' is the location of the ambient/film interface. As the electric field strength E is given by q~ and the magnetic field strength H is given by q~', eqn. (7) is in keeping with the definition H(ko) = tl(ko)E(ko)
(8)
As q~ and q2 are derived from ~b~ and q~z it might appear that no gain has been achieved in the use of eqns. (1)-(4) and qc over the formal solution to the wave equation and the calculation of reflection spectra in terms of the elements of the transfer matrix 3 5 which are expressed in terms of ~bl, q~2, and their derivatives. However, r/c is much simpler in form than the wavefunction. The rapid oscillations in the reflectance that occur in multilayer films are described by the cosine and sine functions in eqn. (1); r/c itself is slowly varying and can be approximated by relatively simple expressions to high accuracy. For example, excluding harmonic resonances, a film with a dielectric constant e given by = Co+el cos kcz will have a value for tl~ 2 ~ So + e l k
(9)
qc2 given by
cos k c L / ( k c 2 - ko 2)
(10)
i.e. tl~ is essentially
given by the mean value of the refractive index in the film ri below a resonance at k¢, by ri minus the modulation depth An above the resonance, and by a Lorentzian-like resonance term at the resonance. The simplicity of expressions such as eqn. (10) is because of the cancellation of lower-order and rapidly oscillating terms in ql and q2 when the product given by eqn. (6) is taken. Part of the utility of eqns. (6) and (7) in eqns. (1)-(4) is that it facilitates the interpretation of spectral behavior. For example, in the absence of a passivation layer, when k L = nrt, we find Pb = (tlo -- rlL)/(~lo + rlL)
(1 1)
independent of t/c. The condition k L = nrt results in either maxima or minima in the sideband spectra. In the case of a multilayered film r/o and qL will be slowly varying compared with r/c, and one of the two envelopes enclosing the side lobes of the reflection spectra, which we will denote by R b, will be given by eqn. (11). This envelope will yield the substrate index as a function of wavelength, or the dispersion curve for the substrate refractive index nL if the ambient admittance is known. Thus it should be possible to identify not only the index of the substrate, but its chemical makeup as well. Alternatively, i f n L is known, the nature of any passivation layer can
582
R . J . BECKER, M. K. KULLEN
be discerned from R b and eqn. (11). If both n o and n L are constant, then Rb will be constant, resulting in a fiat side-lobe envelope (see Fig. 1). I f k L = (n +½)it, then Pe =
(r/Or/L --
(12)
r/c2)/(r/Or/L + r/c 2)
Equation (12) gives the amplitude of the second side-lobe envelope, which we denote by Rc. Ifr/o and r/r are known, then this envelope will yield the real part ofr/c 2 (see Fig. I). These relations are quite general, and are independent of the details of modulation profile in the film, so long as it is described by the same wave equation throughout its length. The propagation constant k will be a pure imaginary number near the middle of the stop bands. If losses in a medium are small, then k will have only a small imaginary part in the reflection spectrum pass bands. Of course, when k is complex or pure imaginary cos k L and sin k L should be replaced by cosh k L and sinh kL. Equation (1) holds exactly provided eqns. (6) and (7) are used to obtain q~c. Equation (4) follows from eqn. (1) if the definitions given by eqn. (3) are used. Equation (1) may be rewritten as
P
=
r/L)e -
ikL
(r/o-- r/c)tr/c-- r/L) e -
ikL
(/70 - - qc)(r/L Jr r/c)e ikL Jr (/70 Jr ~]c)(qc - Jr "e ikt" + . . . .
(r/o + r/c)(r/L
r/c)
(r/o - ql,)/(r/o + r/e) + (r/c - r/1,)/(r/, + r/L)e- 2ikLr/o P - 1 + {(r/o - qc)(r/c - r/L)/(r/o + r/c)(q=+ r/L)}e- 2,kr
(13) (14)
which gives eqn. (4) using the definitions ofeqn. (3).
1.00
0.86
nI =
1.0
nL = 5.5 0.71
=
'~0 = '~l
0.57
18o I0.0
= 0.5
R
O.43
0,14 0.00 I
0.60
0.70
0.80
0.90
I
I llilltLlllllllllllll 1.00 1.10 1.20
1.30
1.40
kod
Fig. 1. Reflection spectrum for single-line Rugate filter. Heavy line is prediction from eqn. (11) based on exact qc2 curve. One of the side-lobe envelopes is tangent to a horizontal line giving the substrate index.
INVERSION OF RUGATE SPECTRA
583
3. INVERSION Based on the present level of understanding of the nature of r/C, it is possible to outline an approach to direct inversions of reflection spectra to obtain the dielectric profiles of multilayer films. To the extent that the film consists of several diverse regions, this procedure will become complicated. However, in most cases, the film profile will be reasonably simple (e.g. a two-line sinusoidal filter with a passivation layer and an accidental spacer layer). Indeed, in many cases, such as real-time in-situ monitoring, r/o will be identically equal to unity; r/L will correspond to an infinite homogeneous medium whose index of refraction is known as a function o f k o, and the intended profile, and hence the approximate form for r/c, will be known. Of course, to find r/C(ko) exactly from the reflection spectrum for a general case, it will be necessary to obtain the reflection amplitude p(kao), and not just the reflection intensity R. As p is complex, phase information will be needed. Fortunately, this can be obtained to reasonable accuracy from the reflection spectrum alone. It must be stressed that the various elements in the reflection spectrum are not independent quantities, but are related on the basis of fundamental principles such as causality 6"7. In particular, writing p = re i*
(15)
we have In p = In r + i lnq~
(16)
therefore, the quantities In r and In ~bare the real and imaginary parts of the response function In p, and must be Hilbert transform pairs. This means that q~(ko) can be obtained from a Kramers-Kronig type interpretation over the spectrum R(ko)6' 7. In general, it will be helpful to have an exact value for ~bat as many spectral points as possible. Fortunately, this can easily be done for a multilayer film with low absorption. For example, if no passivation layer is present and there are an integral number of cycles in the filter, the phase of all of the maxima and minima corresponding to R b is n and the phase of the maxima in R e is also n. For the case given, the phase of the minima in R e depends on whether rh(ko) is greater or less than r/o(k0). Just to the right of the stop bands r/c~0 and the phase of the peaks is zero. In general, however, the side-lobe minima in R e will in general have a phase of n. It should be possible to obtain the phase's sideband maxima and minima for more complicated cases through a study of the properties of eqn. (1). To the extent that the film profile can be described by a simple equation, th may be modeled through eqn. (1). A large number of sum rules are known for response functions 6, so it should be possible to obtain self-consistent estimates of r/c(ko) for regularly periodic films with a high degree of accuracy. As a general rule, given a response function r/(k), its spatial counterpart n(z) can be obtained by a Fourier transform:
n(z) =
f dl,elk~q(k)
(17)
with an inverse transform giving q(k) from n(z). This suggests that the inversion of
584
R . J . BECKER, M. K. KULLEN
qc(k) can readily be made. Unfortunately, the definition for r/c given by eqns. (6) and
(7) shows that q~ is actually a function of both k and the surface position z' or, equivalently, the phase ofn(z) at the surface. Therefore, it may be that in general an inversion of r/~(k) via a Fourier transform may not be unique. However, considering that qc consists essentially of a background value given by a and a resonance factor, it is probable that the functional dependence of r/¢ does not depend significantly on z', and that the primary effect of small changes in the length of a filter is to create an oscillation in the background level due to the modulation depth, An, and to produce a shift in the phase of the resonance part of r/~. Indeed, a Fourier transformation of the index profile corresponding to the dielectrix constant given by eqn. (9) leads to eqn. (10).
4.
SPECTRAL BEHAVIOR
Before outlining how a reflection spectrum may be analyzed to obtain film profiles, it will be helpful to review some important general characteristics of regularly periodic multilayered films. This review will be followed by a discussion of the effect of spacer layers in a film. The peak reflectance of a filter with matched external indices (n~ = ~ = nL) is given by RN
=
pN 2
=
tanh 2 xN
(18)
where i¢ = b 6 + c 6 2
(19)
6 = ~I/Eo
(20)
and
The parameters b and c are functions of the modulation profile only. The ratio c/b is of order 6 and c can usually be neglected if6 is small. For sinusoidal modulation, c is identically zero. The behavior of b and c as a function of various a/d profiles for square-well modulation is shown in Fig. 2. Figure 3 shows how x varies with 6 for several cases. Equation (18) is exact and applies to all profiles. If the filter has an incomplete surface layer, N will assume a non-integer value. If the external indices are not matched, then the effective reflectance amplitude p can be calculated in an analogous manner to that used for films of multiple homogeneous layers. The center-band frequency of the mth harmonic is a filter with an integral number of layers given by k c = m~/~d
(21)
The peak reflectance of thejth side lobe occurs approximately at J ~ kj = ~V~ The/th minimum reflectance in the sideband structure occurs at approximately
(22)
INVERSION OF R U G A T E SPECTRA
585
0.90
0.70
0.50
0.30
0.10
-
-0.10
-
-0.30
-
I
0.50
I
0.10
0.00
0.20
I
I
0.30
0.40
1 0.50
I
I 0.60
I 0.80
0.70
I 0.90
old
Fig. 2 Variation in the coupling coefficients b and c with profile shape for square-well filters.
17.5
15.0
0 ~0 E0 + a/d X old 0 o/d old
~ 4.0 • 9.0 =0.1 = 0.2 =0.~. = 0.5
I .-
~ aid • 0.7 z aid - O.B ¥ a/el = 0.9
/
jj
i'°
IZ.5 o x I0.0
7.5
5.0
z.61
0.000
0.0;~5
0.050
0.075
0,100
0.123
0,150
0.175
0.2C
El / E 0
Fig. 3. Dependence of coupling parameter ~ on modulation depth and profile shape.
586
R . J . BECKER, M. K. K U L L E N
j + 1/2 rt N ri
kf -
(23)
Equations (22) and (23) must be corrected for the dispersion due to the modulation profile in the vicinity of k c. For small 6 this dispersion is small. Filters with sinusoidal profiles are completely dispersionless for small 6. At normal incidence points well removed from the harmonic peaks the sideband maxima and minima are given by R
nlnL _~q2 e
- -
m
n in L+ r/z
(24)
nl - n L 2
(25)
and R b --
nl +nL
The normalized line width of the filter is given approximately by Ak k~
-
~6
(26)
for large N. If N is small, eqn. (26) must be corrected. Thus far the best correction formula obtained is Ak/k¢ = ~6 + fl/N + o¢/N 2
(27)
The above comments suggest a systematic procedure for interpreting the spectra of periodic multilayered films. The number of cycles in the film can always be found by counting the number of sideband maxima between harmonics or, equivalently, the mean spacing between sideband maxima in relation to the frequency at which the main peak occurs. Approximate values for nL and v/can be obtained from the asymptotic values of the sideband envelopes at low frequencies. These estimates may be slightly offdue to the presence o f a passivation layer and/or an incomplete top layer, which will introduce a slowly varying modulation of the reflectance. However, exact values for nL and v/can be obtained measuring the shift in the frequencies of the reflection maxima, especially for the stop band and the change in the sideband reflectance as a function of the angle of incidence for either sor p-polarized light. The details in these changes for each sideband peak will give v/ and n L as a function of wavelength. Any discrepancy between the index values and peak frequencies due to measuring shifts with the angle of incidence and taking values at a single polarization and angle of incidence may be attributed to an incomplete top layer and/or passivation layer. The passivation layer of width w will have only a small effect when kw ,~ n/2. The periodicity and intensity variation in the slowly varying part of Rb will then give the mean index of the passivation layer/'/p and w. In short, Rb can be examined to give a self-consistent and reasonable estimate for N, r/, w, np, and nL. As N is known, the optical density at the stop-band peak and the width of the stop band can be used to find the parameters k and c~ from eqns. (18) and (26). Assuming that 6,~ 1, the parameter c in eqn. (19) may be omitted, so that the ratio
587
INVERSION OF R U G A T E SPECTRA
b/ct can be used to estimate the approximate mean shape of the modulation profile using Table I. This estimate can be checked by measuring the intensities of any stopband harmonics. Once the approximate shape of the modulation profile has been assumed, eqns. (18) and (19) can be used to obtain 6, given the measured value of x. Alternately, 6 can be estimated from the measured line width and an assumed value for ~, or from the difference in the asymptotic limits to Re in the upper and lower sideband spectra. Of course, given k¢, N, and r~, the physical width d of each cycle and the filter thickness L can easily be calculated. TABLE I DEPENDENCE OF COUPLING PARAMETERb AND LINE-WIDTH PARAMETER~t ON PROFILE SHAPE. Rp = Ianh2KN
a/d
b
ct
b/ct
0. l 0.2 M 0.3 0.5 0.7 0.8 0.9 S
0.323 0.608 rt/4 0.828 1.000 0.805 0.566 0.296 0.248
0.306 0.722 0.500 1.010 1.220 0.959 0.661 0.263 0.258
1.050 0.842 1.570 0.820 0.820 0.840 0.856 I. 130 1.040
~c = b (AZ/~o) = ki___1%.
.;(2 for b(a/d)
= 0.019.
5. CONCLUSIONS
The characteristic admittance expressed in terms of the solution for the wave equation gives exact expressions for the reflectance of inhomogeneous films which can be used both for the analysis of film properties and rapid calculations of spectra. Although the characteristic admittance can be derived from the solution to the wave equation, it is generally simpler in form than the wavefunction and can often be approximated with little error. It appears that the characteristic admittance offers a useful approach for inverting the reflection spectra of multilayered films. The appropriate for~ for r/c2(k) can be obtained from a Fourier transform of the modulation profile, e(z). The full reflectance and transmission spectra of a multilayered film are rich in information content which can contribute greatly to the non-destructive characterization of a multilayered film. The analysis strongly suggests that a complete inversion of the reflection and transmission spectra of a multilayered inhomogeneous film can be attained using only s-polarized light. This would minimize the need for ellipsometric measurements. However, the availability of p-polarized spectra would at least provide redundancy and enhance accuracy of the spectral interpretation. Unfortunately, the analysis of the propagation of p-polarized light is considerably less tractable than that of s-polarized light. This problem will be the subject of future research.
588
R.J. BECKER, M. K. KULLEN
ACKNOWLEDGMENTS T h e software d e v e l o p m e n t a n d c o m p u t a t i o n s for this w o r k were d o n e by M i c h a e l Creed, D a v i d G r o t e , Jeffrey Diller, Steven Muilens a n d W a y n e J o h n s o n . T h e a u t h o r is grateful to Dr. M e h m e t R o n a for his m a n y helpful c o m m e n t s . This w o r k was f u n d e d by the A i r F o r c e W r i g h t A e r o n a u t i c a l L a b o r a t o r i e s , M a t e r i a l s L a b o r a t o r y , and by A r t h u r D. Little C o r p o r a t i o n , on C o n t r a c t No. F33615-86-C5132, REFERENCES 1 2 3 4
H.A. Macleod, Thin-Film OpticalFilters, MacMillan, New York, 1986. L.I. Epstein, J. Opt. Soc. Am., 42 (1952) 806. M. Born and E. Wolf, Principles of Optics, Pergamon, New York, 1956. R.J. Becker, Analysis of Rugate Filter Behavior, in Proc. 1986 Boulder Laser Damage Syrup., National Bureau of Standards, Boulder, CO, 1986. 5 R.N. Hill and B. O. Rosland, Phys. Reo. B, 11 (1975) 2913. 6 D.Y. Smith, Dispersion Theory, Sum Rules and Their Application to the Analysis of Optical Data, in Handbook of Optical Constants of Solids, Academic Press, New York, 1985. 7 B. Harbecke, Appl. Phys. A,, 40(1986) 151.