The major difficulty in precise spectrophotometricmeasurement at oblique light incidence with themost widely used accessories is the need for exactknowledge of the absolute reflection of the referencemirror. The accuracy of the sample reflection is de-termined mainly by the accuracy of the absolute re-flection of the reference mirror. In addition, it isdesirable that the mirror reflection be known for anypolarization, wavelength, and angle of incidence.The preliminary tabulation of all these values is anextremely labor-consuming task, even if some stan-dard is available.
A more efficient approach is to calculate the abso-lute reflection of the reference mirror for any condi-tion. For this purpose the optical constants andthicknesses of all layers of the mirror have to beknown.
In this study we describe a procedure for preparingand characterizing a mirror of protected aluminum�Al� and show how to obtain a reference mirror withan absolute specular reflection known to have an ac-curacy of better than �0.01 for polarized and nonpo-
The authors are with the Central Laboratories of Photopro-cesses, Bulgarian Academy of Sciences, Acad. G. Bonchev Street,Building 109, Sofia 1113, Bulgaria. S. Kitova’s e-mail address [email protected].
larized light at angles of incidence from 0° to 70° inthe 400–800-nm spectral range.
2. Preparation of the Protected Aluminum Mirror
The Al film, 50–60 nm thick, was deposited in avacuum installation Z 700 P2 �Leybold-Heraeus� bydc magnetron sputtering of Al in an argon �Ar� atmo-sphere at a pressure of 2 � 10�1 Pa. Optical boron-silicate glass �BK-7� plates that have been cleanedand highly polished were used as substrates. Theprotective AlN film, �30 nm thick, was deposited inthe same vacuum cycle on the Al mirror by rf mag-netron sputtering in a gas medium of Ar and N2 witha pressure of 2 � 10�1 Pa and a ratio of partialpressures Ar:N2 � 5:9. Auger electron spectroscopyanalysis of the AlN film showed that it is stoichio-metric.
It is known that an oxide film several nanometersthick grows rapidly on the surface of freshly depos-ited Al films even at a vacuum higher than 10�6 Pa�see, for example, Ref. 1�. Therefore the structure ofour mirror consists of a reflective Al film coveredspontaneously by an Al2O3 on which an AlN film issputtered.
3. Characterization of the Protected Aluminum Mirror
To calculate with sufficient accuracy the absolute re-flection of the mirror as a function of � and , we needto know the dispersion relations of the optical con-stants of the Al and the protective films as well astheir thicknesses.
The refractive index n��� and the thickness d of theAlN film were determined by the �TRbRm�, �TRb�,�TRm�, and T�k � 0� methods.2,3 The transmission T,reflections Rf and Rb of the film deposited on a non-absorbing substrate BK-7, and reflection Rm of thefilm on an absorbing Si-wafer substrate were mea-sured in the 400–800-nm spectral range at normallight incidence with a Cary 5E spectrophotometer.Subscripts f and b denote reflection from the frontand back sides of the film. It should be pointed outthat the AlN films were obtained in the same vacuumcycle and conditions of mirror preparation. Figure 1displays the dispersion curve of n derived by themethod T�k � 0� at d � 30�1.2 nm. The solid curveillustrates the refractive index described bySellmeier’s dispersion formula4 Eq. �1��, with the co-efficients A � 3.0417 and B � 139.836:
n � �1 �A
1 � �B���2�1�2
. (1)
For the refractive index of the Al2O3 we used datafrom the literature,5 which were fitted by Sellmeier’sdispersion formula Eq. �1�� with coefficients A �1.7088 and B � 97.626.
As a result of deriving n��� and d of AlN and n��� ofAl2O3, the problem of characterizing our three-layered mirror is reduced to determining n��� andk��� of Al and the thickness of the Al2O3. The un-known parameters can be obtained by the minimiza-tion of the objective function F including measuredRmeas��� and calculated Rcalc��� reflections:
F � �q�1
M ��j�1
N
�Rq, jcalc � Rq, j
meas�2� , (2)
N � 81 is the number of points in the 400–800-nmspectral range, and M is the number of suitably cho-sen spectral relationships of the reflection.
We have found that for accurate determination ofn��� and k��� of Al, several reliable spectral relation-ships of the reflection at oblique incidence should beknown. However, for the measurements at any an-
gle a reference mirror with reflection, known as ac-curately as possible at angles between 10° and 70°, isrequired. We chose a Si wafer as a reference mirror,since its optical constants are known with sufficientaccuracy and reproducibility in a wide spectral rangeif its surface is cleaned by the prescribed procedure.6,7
A. Silicon Wafer as Reference Mirror
Since we calculate the reflection RSi of the Si wafer
at oblique incidence from its optical constants, it isnecessary to make some assessments of the reliabil-ity and accuracy of RSi
and of the influence of the RSi
values and the corresponding errors RSi on the
measurements of the oblique reflection RAl of the Al
mirror.Figure 2 shows the measured reflections RSi at
normal incidence of eight Si wafers after the first andthe second cleaning; i.e., the solid curve includes 16experimental curves. The dashed curve illustratesthe reflection calculated with the optical constantstaken from the literature.6 The maximum differ-ence RSi between the measured and the calculatedvalues of RSi is �0.005 at � � 600 nm. The questionis whether this difference remains the same atoblique incidence of polarized light. To check this,we used the maximum absolute errors of RSi
ex-pressed by
RSi � ��RSi
�n � n � ��RSi
�k � k. (3)
Equation �3� does not take into account the error inangle , since the value of RSi
is calculated fornominal values of with n and k, with known errors n and k.
In Fig. 3 the contours of the error ratio RSi� RSi
calculated by Eq. �3� for a Si wafer are plotted in the–� plane. It is seen that the ratio is not higher thanunity. This means that for obliquely incidents-polarized light the deviations RSi
from the realvalues will not exceed 0.005 provided that the devi-ations RSi between the measured and the calculated
Fig. 1. Dispersion curves of n derived by spectrophotometric mea-surements at normal incidence �points� and fitted with the disper-sion formula of Selmeier �curve� for AlN film with d � 30�1.2 nm.
Fig. 2. Reflections of eight Si wafers measured at normal inci-dence after the first and the second cleaning �solid curve includes16 experimental curves� and calculated �dashed curve� with theoptical constants taken from the literature.6
values of RSi at normal incidence are the result ofimprecise knowledge of the optical constants. If thedeviations RSi result from the several-nanometer-thick oxide film on the Si-wafer surface, similar as-sessments reveal that at oblique incidence possibleoxide film with thicknesses as great as 5–6 nm wouldnot change RSi
values by more than 0.005.Obviously, the problem now concerns which angles
and polarization types would ensure the most accu-rate measurement of RAl
, i.e., the lowest errors RAl
.From two consecutive measurements R1
and R2
of the standard reference mirror with known reflec-tion Rst
and of the sample with Rsp, respectively, the
absolute reflection of the sample Rsp is calculated by
Rsp �
R2
R1 Rst
, (4)
at
R1 � ZRst
, R2 � ZRsp
, (5)
where Z is a coefficient that is proportional to theproduct of the reflections of the accessory mirrors andthe transmission of the polarizer and depolarizer inthe object beam and inversely proportional to thetransmission of the attenuator in the reference beam.Since the quantities R1
and R2 are independent
measurements with errors R1 and R2
, and Rst is
known with an error Rst, the error Rsp
is calcu-lated by the following expression:
� Rsp�2 � ��Rsp
�R1 R1
�2
� ��Rsp
�R2 R2
�2
� ��Rsp
�Rst Rst
�2
. (6)
Using Eqs. �4� and �5� at R1 � R2
� R, from Eq.�6� we obtain
� Rsp�2 � �Rsp
Rst Rst
�2
� � R
Z �2�1 � �Rsp
Rst�2� .
(7)
The error R in reflection measurements at obliqueincidence is a result of the instrumentation error Rinstr
and the experimental error in the angle ;i.e.,
� R�2 � � Rinstr�2 � ��R
� �2
. (8)
For Rinstr � 0.001, � 0.25°, and ��R��� � 0.01
�the change in sample reflection is 0.01 for 1°�, theerror in R, calculated from Eq. �8�, is R � 0.0027.
Figure 4 depicts the relationships Rsp � f�Rsp
�Rst
�, calculated by Eq. �7� with the foregoing errors, Rst
� RSi � 0.005 and R � 0.0027, and the
indicated values of Z. It is seen that the error Rsp
of the measured sample �in this particularcase RAl
� will be less for higher reflection of Rst,
i.e., of RSi. Since the Si wafer has higher reflection
for s-polarized light, we chose measurements at an-gles of incidence � 30°, 50°, and 70° for charac-terizing the Al mirror. For the reflection valueRAl
of �0.8, expected for the Al mirror in the worstcase at � 30°, we obtain
Rsp30
Rst30 �
RAl30
RSi30 � 2.
In this case, as seen in Fig. 4 at Z � 0.6, the error inthe measurement of RAl
will be RAl � 0.01–0.012.
In Figure 5 the solid curves are the values of RAl,
measured with a Si wafer as a reference mirror fors-polarized light at the corresponding angles of inci-dence and at normal light incidence.
Fig. 3. Contours of the error ratio RSi� RSi with indicated val-
ues in the –� plane, calculated for a Si wafer at n � 0.03 and k� 0.02. RSi and RSi
were calculated from Eq. �3� at normaland obliquely incident s-polarized light, respectively.
B. Deriving n��� and k��� of Aluminum and theThicknesses of the Protective films
As a result of the foregoing measurements and as-sessments for the reflections, we chose Rq � R, Rs
30,Rs
50, Rs70, i.e., M � 4 in the objective function Eq. �2�.
For describing the dispersion relations n��� andk��� of the Al film, we use the following model of thedielectric constant ��, which explicitly separates theintraband effects from the interband8:
�� � ��Dr��� � ��Lor���. (9)
The intraband �free-electron� contribution ��Dr to theoptical properties can be described by a Drude-modeldielectric constant:
��Dr��� � 1 ��p
2
�2 � i�����. (10)
Here �p is plasma frequency for intraband transi-tions and � is the intraband relaxation time.
The interband part ��Lor of the dielectric constant ��is a simple semiquantum model resembling theLorentz result for insulators:
��Lor��� � � �j�1
K fj�p2
��2 � �j2� � i��j
, (11)
where �p is the plasma frequency and K is the num-ber of interband transitions with frequency �j, oscil-lator strength fj, and lifetime 1��j.
Knowing the real �r and the imaginary �i parts ofthe dielectric constant ��, we determine the opticalconstants nAl and kAl of the Al film from the expres-sions
nAl � 0.5��r � ��r2 � �i
2��1�2,
kAl � 0.5� � �r � ��r2 � �i
2��1�2. (12)
According to Rakic,8 the interband transitions in Al,interpreted with oscillators in a semiquantum model,are located at approximately 0.4 eV �3100 nm�, 1.56eV �795 nm�, 2.1 eV �590 nm�, and 4.5 eV �275 nm�.Our calculations confirmed that in the spectral range� � 400–800 nm the contribution of the last oscillatorto the dielectric constant is negligible �less than 1%�;hence only the sum of three oscillators with approx-imate frequency values of �1 � 0.4 eV, �2 � 1.56 eV,and �3 � 2.1 eV is sufficient for describing nAl and kAlin this spectral range.
We have used a nonlinear subspace trust regionmethod combining the interior-reflective Newtonmethod with a preconditioned conjugate-gradientmethod for the minimization of the goal function.9The initial guess values of the Drude- and oscillator-model parameters and the thicknesses of the protec-tive films, as well as the solutions obtained for thecorresponding parameters, are given in Table 1.Figure 6 compares the dispersion curves of nAl andkAl, obtained with the initial and the fitted parame-
Fig. 5. Reflection of protected Al mirror, calculated by the matrixmethod �points� and measured �curves� with a reference mirror ofSi wafer at angles of incidence � 30°, 50°, and 70° of s-polarizedlight as well as at normal light incidence.
Table 1. Initial and Fitted Values of the Drude- and Oscillator-ModelParameters and the Thicknesses of AlN and AlO Films
ters from Table 1 and the literature data for Al de-posited in ultrahigh vacuum.10
The points in Fig. 5 represent the calculated valuesof Al mirror reflection for s-polarized light incident atangles of 0°, 30°, 50°, and 70°. The solid curves inthe figure represent the corresponding values of themeasured reflections Rmeas��� used for minimizingthe goal function F in Eq. �2�. The figure illustratesthe fairly good agreement between the measured andthe calculated values of the mirror reflection. Thisgives us reason to assume that we could use theparameters thus determined for describing the mir-ror reflection of both types of light polarization at allangles of incidence with an accuracy not less than0.01.
Figure 7 presents the reflectivity characteristics ofthe protected Al mirror calculated by the matrixmethod.11 The contours of the reflection of s- andp-polarized light �a and b, respectively� with denotedvalues are depicted in the –� plane. It is seen thatthe reflection values vary between 0.7 and 0.9 de-pending on the light wavelength and angle of inci-dence.
5. Verification of the Protected Aluminum Mirror
Thin Ag films were used for verifying the accuracy ofRAl
of the protected Al mirror. Ag films were de-posited on cleaned glass substrates �BK-7� and Siwafers by thermal evaporation under vacuum higherthan 10�4 Pa. We should note that this is an unfa-vorable case in terms of the expected measurementerror �see Fig. 4�, because at all angles of incidencethe reflection of the Ag film is higher than that of theAl mirror; i.e., Rsp
�Rst � 1.
Since Ag films undergo changes on storage in air,all necessary photometric measurements were car-ried out immediately after taking the films out of thevacuum installation.
The measurements of T, Rf, Rb, and Rm at normallight incidence were used to determine n���, k���, andd of the Ag film. Figure 8 shows the dispersioncurves of n and k, obtained by the �TRfRm�, and �TRf�,and �TRm� methods.2,3 Rs
and Rp of the same film
at 30°, 50°, and 70° were measured with our referencemirror of protected Al. The values obtained aregiven with points in Fig. 9. In the figure the solidcurves denote the corresponding reflection values,calculated by the Fresnel equations12 with n���, k���,and d determined by the photometric measurementsat normal incidence �see Fig. 8�. It is seen that thereis good agreement between the calculated and themeasured reflections from the Ag film for both s-andp-polarized light and at all angles of incidence. Thegreatest difference, 0.011, between the calculatedand the measured values is obtained at � � 800 nmfor s-polarized light, incident at an angle of 70°.Hence we can assume that the reflection of the pro-
Fig. 6. Dispersion curves of nAl and kAl obtained with the initialand the fitted parameters from Table 1, as well as data from theliterature for Al deposited in ultrahigh vacuum.10
Fig. 7. Contours of the reflection of s- and p-polarized light �a andb, respectively� with denoted values in the –� plane for the pro-tected Al mirror.
Fig. 8. Dispersion curves of n and k of 36-nm-thick Ag film ob-tained by photometric measurements at normal light incidence.
tected Al mirror is determined with an accuracy of upto 0.01 at all angles of incidence from 0° to 70° in the400–800-nm spectral range.
6. Conclusion
In the present study we have described the procedurefor preparing and characterizing a reference mirror ofprotected Al. As a result the absolute specular re-flection with an accuracy better than 0.01 for p- ands-polarized light at angles of incidence from 0° to 70°in the 400–800-nm spectral range can be calculatedby the matrix method.
The mirror characterization is based on a reliableand precise measurement of the mirror reflection atangles of light incidence 0°, 30°, 50°, and 70° in the400–800-nm spectral range with a Si wafer as a ref-
erence mirror. The reflections of the Al mirror, cal-culated by the matrix method, are fitted to theexperimental data. The dispersion relations em-ployed reduce the variable parameters in the optimi-zation procedure to a reasonable and feasible numberand thus make the problem solvable. The simplephenomenological Drude–Lorentz model was appliedfor modeling the dielectric constant ����� and corre-sponding n��� and k��� of the Al film. The modelparameters as well as the thicknesses of the Al2O3and AlN film were derived by minimization of thegoal function. Once we have determined all neededparameters, the desired values of the reflection of theprotected mirror can be easily calculated.
The exact determination of the mirror reflectionswas confirmed by comparison of calculated and mea-
Fig. 9. Reflections of Ag film. Points are Rs and Rp
values measured at � 30°, 50°, and 70° with the reference Al mirror. Solid curvesare the values calculated by the Fresnel equations with n���, k���, and d � 36 nm, determined by the photometric measurements at normallight incidence.
sured reflections of an evaporated Ag thin film withknown n���, k���, and d, obtained by photometricmeasurements at normal light incidence.
Finally, we would like to emphasize that the pro-cedure described can be applied for the characteriza-tion of mirrors not only of Al but of other metals orwith other protective films as well as for the prepa-ration of mirrors with higher reflections or for otherspectral ranges.
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