1564 OPTICS LETTERS / Vol. 34, No. 10 / May 15, 2009

Spectral interferometry and reflectometry used forcharacterization of a multilayer mirror

Petr Hlubina,* Ji1í Luná~ek, and Dalibor CiprianDepartment of Physics, Technical University Ostrava, 17. Listopadu 15, 708 33 Ostrava-Poruba, Czech Republic

*Corresponding author: petr.hlubina@vsb.cz

Received March 4, 2009; accepted April 14, 2009;posted April 21, 2009 (Doc. ID 108254); published May 12, 2009

A white-light spectral interferometric technique is used to retrieve a relative spectral phase and group delayof a multilayer mirror from the spectral interferograms recorded in a dispersive Michelson interferometer.The phase retrieval is based on the use of a windowed Fourier transform in the wavelength domain, andcharacterization of the multilayer mirror is completed by a three-step measurement of the reflectance spec-trum of the mirror in the same interferometer. © 2009 Optical Society of America

OCIS codes: 120.3180, 120.2650, 120.5700, 230.4040.

Characterization of a multilayer mirror involves de-termination of the complex reflection coefficient thatincludes the mirror reflectance and phase change onreflection from the mirror. Moreover, the group delay(GD) introduced when a pulse is reflected by the mir-ror is also of great importance. There exist many dif-ferent methods reported in the literature to measurethese quantities. Interferometric methods, includingthe time-domain and spectral-domain implementa-tions, belong to the most powerful techniques of mea-suring the phase changes and the GDs characterizingoptical components [1–6]. In this Letter, a white-lightspectral interferometric method [7–9] for character-ization of multilayer mirrors is presented. The tech-nique, which is rather simple, utilizes a Michelson in-terferometer with one of its mirrors replaced by amultilayer mirror. From the spectral interferogramsrecorded by a fiber-optic spectrometer a relative spec-tral phase and a GD (the first derivative of the phase)characterizing the mirror are retrieved. The phaseretrieval is based on the use of a windowed Fouriertransform [8]. Moreover, in the Michelson interferom-eter the reflectance spectrum of the mirror is alsomeasured by a three-step procedure [9].

Let us consider the mutual interference of twobeams from a broadband source at the output of aslightly dispersive Michelson interferometer with acube beam splitter of the effective thickness teff [7–9]when mirror 1 of the interferometer is replaced by amultilayer mirror as shown in Fig. 1. We assume thatthe mirror is characterized by a complex reflection co-efficient

r��� = �R��� exp�i�r����, �1�

where R��� is the wavelength-dependent reflectanceand �r��� is the wavelength-dependent phase changeon reflection. Moreover, we assume that the geo-metrical path lengths of the light rays in dispersiveglass of the beam splitter are not the same for bothinterferometer arms so that the beam splitter can berepresented by an ideal beam splitter and a plate ofthe same dispersion and of the thickness teff (see Fig.1). The complex reflection coefficient r2��� of mirror 2

is represented by a relation similar to that of Eq. (1)

0146-9592/09/101564-3/$15.00 ©

with R2��� and �2��� as the reflectance and the phasechange on reflection, respectively. The wavelength-dependent optical path difference (OPD) ���� be-tween two beams in the Michelson interferometer isgiven by

���� = 2L + 2n���teff − ����r��� − �2�����/�2��, �2�

where 2L is the difference of path lengths betweenthe interfering beams in the air whose dispersion isneglected and n��� is the wavelength-dependent re-fractive index of the beam-splitter material.

The procedure for the interferometric measure-ment [7,9] of a relative spectral phase is based on re-cording of the spectral interferograms in the disper-sive Michelson interferometer and retrieving theabsolute OPD ����. The absolute OPD ���� is used toconstruct for a chosen mirror position L=L0 the rela-tive spectral phase ���� given by the relation

���� = �2�/���2L0 + 2n���teff − ����� + �2���. �3�

To compensate for the phase change �2���, a nextmeasurement step with the reference sample used in-stead of the multilayer mirror needs to be applied [9].

The procedure for the reflectance measurement iswith the arm of mirror 2 blocked and it consists ofthree steps: first, with the source blocked, the back-ground spectrum Ibkg��� is measured; second, with areference sample used instead of mirror 1, the refer-ence reflection spectrum Iref��� is measured; andthird, with the multilayer mirror used instead of the

Fig. 1. (Color online) Experimental setup with a Michel-

son interferometer to measure a multilayer mirror.

2009 Optical Society of America

May 15, 2009 / Vol. 34, No. 10 / OPTICS LETTERS 1565

reference sample, the reflection spectrum Imeas��� ofthe mirror is measured. The absolute reflectance R���of the multilayer mirror is given by

R��� =Imeas��� − Ibkg���

Iref��� − Ibkg���Rref���, �4�

where Rref��� is the theoretical reflectance of the ref-erence sample, which can be computed by use of ex-perimental data available from optical handbooks[10].

The experimental setup used for measurement of amultilayer mirror is shown schematically in Fig. 1and it consists of a white-light source: a halogen lampHL-2000 (Ocean Optics) with launching optics, an op-tical fiber and a collimating lens, a bulk-optic Mich-elson interferometer with a cube beam splitter madeof BK7 optical glass, a sample of multilayer dielectricmirror, metallic mirror 2 connected to a microposi-tioner, a microscope objective, micropositioners, aread optical fiber, a fiber-optic spectrometer S2000(Ocean Optics), an A/D converter and a personal com-puter. The spectrometer has a spectral operationrange from 350 to 1000 nm.

The procedure used to record spectral intensities,from which a relative spectral phase ���� of amultilayer mirror can be retrieved, comprises twosteps. In the first step, the spectral intensity I���with spectral interference fringes (channeled spec-trum) is recorded by the spectrometer. To obtainthe channeled spectrum, one manually translates themicropositioner with mirror 2 until spectral fringesappear. The position of the micropositioner corre-sponds to the adjusted OPD ����. In the second step,the background spectral intensity I0���, i.e., the in-tensity without spectral interference fringes, is re-corded by the spectrometer. To obtain the backgroundspectrum, one manually translates the microposi-tioner until spectral fringes disappear completely.From the two spectral intensities, the spectral signalS���=I��� /I0���−1 is evaluated, which is used to re-trieve the relative spectral phase ���� by using a suit-

Fig. 2. (Color online) Background spectrum and chan-neled spectrum recorded for a multilayer mirror in thesetup shown in Fig. 1.

able processing procedure [7]. To reach the finite

spectral signal, the background spectrum needs to beabove the zero. Figure 2 shows examples of the spec-tral intensities with and without the spectral fringesrecorded for the sample of a multilayer dielectric mir-ror in the wavelength range from 450 to 900 nm.There is apparent effect of the multilayer mirrorcausing the abrupt changes in spectral intensitiesnot present for standard metallic mirrors [8].

We use a simple and effective procedure for retriev-ing the relative spectral phase ���� from the spectralsignal S���. We process the spectral signal S��� by awindowed Fourier transform [8] to retrieve the over-all spectral phase ���� with the ambiguity of m2�,where m is an integer. This processing of the spectralintensities gives the overall spectral phase with ahigher accuracy than a standard Fourier-transformprocessing. Moreover, its implementation is easier incomparison with a wavelet-based processing that de-termines the overall spectral phase from the ridge ofthe wavelet transform of the recorded channeledspectrum [11]. Next, we use a procedure [7] for theabsolute retrieval of the adjusted OPD based on thelinear dependence of the OPD ���� on the refractiveindex n��� of the beam-splitter material, which isknown (BK7 glass). The slope of the linear depen-dence is given, as Eq. (2) indicates, by the effectivethickness teff of the beam splitter, which can be deter-mined by a similar procedure from the spectral signalmeasured in the same Michelson interferometer butwith two identical metallic mirrors [8]. The linear de-pendence also gives an estimate of the mirror posi-tion L=L0, which is needed in the evaluation of therelative spectral phase ���� according to Eq. (3). Thephase change �2��� due to mirror 2 is measured in thesame setup with the reference sample (a bare Si wa-fer) [9]. Another estimate of the mirror position L=L0 results from the approximation ���0��0, whichmeans that the relative spectral phase of amultilayer mirror is approximately zero at a centerwavelength �0 of the bandpass of the mirror reflec-tance spectrum. Fortunately, there is no problemwith determining the center wavelength �0, becausethe reflectance spectrum R��� can be measured by athree-step procedure presented above in the samesetup as the relative spectral phase ����. Figure 3shows the reflectance spectrum R��� recorded for thesample of a multilayer mirror. Once again, there isapparent effect of the multilayer mirror causing theabrupt changes in the reflectance spectrum notpresent for standard metallic mirrors [9]. The reflec-tance is nearly 1 for the mirror spectral bandpass(from 619 to 820 nm), and the corresponding centralwavelength is 719.5 nm. The relative spectral phase���� of the mirror satisfying the condition ���0�=0 isalso shown in Fig. 3. It is characterized by both theabrupt phase changes at the wavelengths with mini-mum reflectances and the steep phase decreasewithin the mirror spectral bandpass. It should benoted that the retrieved relative spectral phase ����is much greater than the phase change �2��� due tomirror 2 [9]. Moreover, the spectral phase retrieved

for the first estimate of the mirror position L0 is

1566 OPTICS LETTERS / Vol. 34, No. 10 / May 15, 2009

shifted by a constant in comparison with that re-trieved for the second estimate of the mirror positionL0.

Finally, Fig. 4 shows the GD of the multilayer mir-ror evaluated as the derivative of the relative spec-tral phase ���� with respect to the angular frequency�=2�c /�, where c is the speed of light in vacuum.This figure clearly shows spikes in the GD that cor-respond to minima in the reflectance spectrum of themultilayer mirror.

In conclusion, a simple white-light spectral inter-ferometric technique for measuring the relative spec-tral phase and the reflectance spectrum of amultilayer mirror at the same time has been pre-sented. The technique utilizes a cost-effective instru-mentation with a Michelson interferometer and afiber-optic spectrometer. The spectrometer recordsthe background and channeled spectra to obtain thespectral signal whose processing by a windowed Fou-rier transform gives a relative spectral phase. Thefeasibility of the technique has been demonstrated inmeasuring the GD and the reflectance of a sample ofa multilayer dielectric mirror. We confirmed that thewavelength positions of extrema of both quantitiesagree well. The use of the technique can be extended,

Fig. 3. (Color online) Retrieved relative spectral phaseand measured reflectance spectrum of a multilayer mirror.

for example, for measuring the GD of narrow-

bandpass filters employing high-resolution fiber-opticspectrometers or optical spectrum analyzers.

The research has been partially supported by theMinistry of Education, Youth and Sports of the CzechRepublic through grant MSM6198910016.

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Fig. 4. (Color online) Retrieved group delay and measuredreflectance spectrum of a multilayer mirror.

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