Polarisation-entangled photon pairs from a two crystal, type-I spontaneous parametric down conversion source are used to make accurate measurements of the ellipsometric angles of a silicon dioxide film on silicon and of internal and external reflection from BK7 glass. Since our source produces an entangled state with some mixture, a novel technique based on quantum tomography was developed to estimate the components of the density matrix for the state before and after reflection from the samples. The ellipsometric angles are readily calculated from these components and experimental measurements made on the samples were found to be in good agreement with their expected values.
© 2006 Optical Society of America
Ellipsometry [1, 2] is a well-established technique used to determine the optical properties of substrates and thin films. The sample is characterised by its ellipsometric angles ψ and Δ, which are defined as :
where rp,rs are the Fresnel reflection coefficients for p- and s-polarised light respectively, at a given angle of incidence. Making ellipsometric measurements over a range of wavelengths  or incident angles  allows more complex samples to be investigated. In particular, for a substrate coated with thin films, the optical constants of the substrate and the films, and the thickness of the films, can be determined accurately and non-destructively. This has many practical applications, particularly in the semiconductor industry, for which accurate characterisation of the sample is essential.
In this paper we describe an ellipsometer in which polarisation-entangled photon pairs are used to determine the ellipsometric angles of a sample. Particles are said to be entangled if the quantum state describing them cannot be factored into single particle states. Polarisation-entangled photon pairs can be produced and measured relatively easily ,  and have been used to test local-realism , for work towards quantum computers , and to demonstrate quantum cryptography . Our polarisation-entangled photon pairs are produced by type-I spontaneous parametric down-conversion in two, non-linear crystals pumped by a laser . Since this source produces an entangled state with some mixture, we have developed a technique that is based on the partial reconstruction of the components of the density matrix for the state with and without reflection of one photon from a sample. Ellipsometric measurements made using this technique are presented and compared with the expected values calculated from the known properties of the samples.
2. The entangled photon ellipsometer
An entangled photon ellipsometer exploits the quantum interference of polarisation-entangled photon pairs to determine the ellipsometric angles [10, 11]. Measurement of tan ψ using polarisation-entangled photon pairs has been demonstrated by Sergienko et al  while Toussaint et al  have demonstrated ellipsometry using correlated, but not polarisation-entangled, photon pairs. The latter authors have also shown that if high efficiency detectors are used, the errors in the measurement due to the fluctuations of the light source are reduced over those in a classical ellipsometer. It is expected that this would apply equally to ellipsometry with polarisation-entangled photon pairs.
Figure 1 shows a schematic of the entangled photon ellipsometer . To make ellipsometric measurements, one photon of each pair is reflected from the sample and the polarisations of both photons analysed with linear polarisers. The photons are detected with single photon detectors and coincidences recorded. The ellipsometric parameters can then be calculated from the coincidence count rates for different settings of the polarisers. Suppose that the photon pairs are initially in the polarisation-entangled state
which is approximately the state produced by our source, where H (V) denotes horizontal (vertical) polarisation. Abouraddy et al [10, 11] start with the state |HV⟩ + |VH⟩, with similar results. After reflection of one photon of the pair from a sample, the state will be
If the photons are analysed with linear polarisers set to angles θ 1 and θ 2 with respect to the horizontal and detected with single photon detectors, then the coincidence count rate Nc, that is, the rate at which two photons are detected simultaneously, will be
where C is a constant determined by the source brightness, detector efficiencies and any losses of photons. If the initial state is known, that is ε and ϕ are known, tan ψ and Δ can be calculated from the coincidence count rates for 3 different polariser settings, e.g., θ = 45°, θ 2 = 0°,45c and 90°. If the initial state is not known, a separate calibration measurement must be made with the sample removed.
3. Ellipsometry with mixed states
For our ellipsometer, polarisation-entangled photon pairs are produced by a two crystal, type-I spontaneous parametric down-conversion source , pumped by a diode laser. This source produces entangled pairs in states of the form given by Eqn. (3). The relative amount of H and V polarised components in the pump beam is controlled by a waveplate and determines the value of ε. The phase ϕ is determined by the relative phase of the H and V components of the pump beam as well as details of the phase matching and the thickness of the crystals. But ϕ also depends on the angle at which the photons are emitted and, for non-degenerate pairs, their wavelength. Due to the angle and wavelength dependence of ϕ, the observed state will not be a pure state but will rather be a mixture of entangled states of the form |HH⟩ + εeiϕ|VV⟩ with different ϕ. Apertures and bandpass filters can be used to separate out pairs of photons with similar ϕ, thereby approximating a pure state. However, we found that ellipsometric measurements made with our source using Eqn (5), which assumes a pure state, gave poor results, with measured values of tan ψ and Δ differing from their expected values by 3 to 15 standard deviations. We believe this is caused by the mixed nature of the state produced by our source due to the variation in ϕ. We estimated experimentally that ϕ varies over ~ 0.8 radians for the states detected (see Section 4 for details). The coincidence count rate Nc from which the ellipsometric parameters are calculated is nonlinear with respect to ϕ and thus the value of Nc obtained is not necessarily the same as that for a pure state with ϕ equal to the average ϕ for our state. For example, it can be shown that the count rates with no sample for states of this form with ϕ uniformly distributed over 0.8 rad and with average 0 and π/2 radians will be the same as those for pure states with ϕ= 0.228 rad and ϕ = π/2 rad, respectively.
In this section we present a new technique to make ellipsometric measurements using mixed states rather than an ideal pure state. The mixed state is taken to be an incoherent mixture of pure, two photon polarisation states of the form
and is described by the density matrix
where ∑iWi = 1. Our source actually produces a continuous spectrum of pure states as ϕ changes continuously with the opening angle and wavelength. However, the sum form of the density matrix, rather than the integral form, will be used as it makes the following discussion clearer. This has no effect on the result as the integral form can be taken as the limit of an infinite sum. For the purposes of this paper, we are interested in the amounts of |HH⟩ and |VV⟩ and the phase between them, so the following will be useful later:
Reflection of one photon of a pure, two photon polarisation state from a surface, with the plane of incidence horizontal such that rH = rp, rV = rs, produces
where R is a normalisation constant. The components ρ 11, ρ 44 then become
where ρr is the density matrix for the state after reflection of one photon from the sample. Since tan ψ= |rp|/|rs|,
The component ρ 41 becomes
So if ρ 11,ρ 44 (or ρ 11/ρ 44) and ρ 41 (or at least its phase) can be measured, both with and without reflection from the sample, the ellipsometric parameters tan ψ and Δ can be calculated.
The density matrix could be completely reconstructed by quantum state tomography with 16 coincidence rate measurements  but it is simpler to just reconstruct the components needed. For a two photon coincidence measurement with the analyser for photon 1 having the operator P 1 and for photon 2 having the operator P 2, the operator for the two photon measurement is P 1 ⊗ P 2 . Measuring ρ 11 and ρ 44 is simple. Let Nc(P 1 ⊗ P 2) be the coincidence count rate for analyser settings P 1 and P 2 with the H analyser set to transmit horizontally polarised light and V analyser set to transmit vertically polarised light:
where C is a constant determined by the source brightness, detector efficiencies and any losses of photons.
To determine arg(ρ 41) requires the real and imaginary parts of ρ 41. These can be found from the following combinations of coincidence count rate measurements:
where D,A,R and Z are analysers set to transmit diagonally, anti-diagonally, right hand and left hand circularly polarised light, respectively. The ellipsometric parameters can thus be calculated from 10 coincidence count rate measurements each for (i) the state with reflection of one photon from the sample, and (ii) without reflection from the sample (calibration). The advantage of this technique over that presented in Section 2 is that it will work for any mixed state where the density matrix has significant ρ 11, ρ 44 and ρ 41 components, that is, where a significant fraction of the pairs are in states of the form |HH⟩ + εeiϕ |VV⟩ with limited variation in ε and ϕ. These can be easily prepared , unlike an ideal pure state. The measurement will be unaffected if the state has some |HV⟩ or |VH〉 components which can occur if the crystals of the source are misaligned .
Figure 2 shows the apparatus for the entangled photon ellipsometer and its arrangement on an optical table. The polarisation-entangled photon pairs are produced by a two crystal, type-I spontaneous parametric down-conversion source . The crystals are β barium borate (BBO), 0.1 mm thick, cut with the optic axis at 29° to the crystal face. These are pumped by a Toptica iBeam-405 laser, nominally 50 mW at 405.3 nm which was supplied as a self-contained unit housing the violet diode, collimating optics and control electronics. The laser is temperature controlled and power stabilised, with internal optics that include an anamorphic prism pair to circularise the output beam. The maximum pump power was measured at 40.3 mW. Two kinematically-mounted mirrors steer the beam and a half-wave plate controls the polarisation state of the pump beam and thus the relative amounts of |HH⟩ and |VV⟩ pairs in the state produced [5, 15]. To improve the collection efficiency , a lens focuses the pump beam 350 mm behind the crystals, near the position of the first aperture in the down converted beams. At the crystals, the beam has a diameter of approximately 2 mm. Apertures and lenses select and approximately collimate the photons that form the entangled pairs. Lenses in the sample (f = 300 mm) and reference (f = 400 mm) arm are placed at their respective focal lengths from the crystals. The apertures, which were made of black card, are approximately 2 mm wide and 6 mm high and are placed 570 mm apart in the sample arm and 700 mm apart in the reference arm. The detectors have 25 mm diameter, f = 35 mm achromatic lenses in front of them and are approximately 1.2 m from the crystals. Polarisation analysers placed in each arm can be set to transmit any polarisation state (elliptical as well as linear) and consist of quarter- and half-wave plates mounted in Newport SR50CC rotation stages, followed by fixed polarising beam splitter cubes. Custom bandpass filters centred on 810 nm and with 10 nm FWHM are used to reject unwanted light. The photons are detected with single photon counting modules and coincidences detected by a time to amplitude converter and recorded by a PC. The coincidence window is approximately 10 ns.
In the absence of a sample, our apparatus provided a singles count rate of ~ 3500 counts per second (cps) in the reference arm and ~ 3300 cps in the sample arm. The coincidence count rates with both polarisation analysers set either horizontally or vertically were ~ 500 cps. The rate of accidental coincidences involving two photons from different pairs or dark counts was negligible at < 0.2 cps. To obtain an estimate of the purity of the state, a birefringent quartz plate was placed in the pump beam and it and the pump half-wave plate were adjusted to make the state as close to ε= 1 and ϕ= 0 as possible . The measured coincidence fringe visibility in the diagonal basis was 97.5 %. From Eqn (5), the count rate for a pure state in this case is
For a pure state with ε = 1 the coincidence fringe visibility in the diagonal basis is cos ϕ . The equation for Nc for a pure state is linear with respect to cos ϕ and therefore a mixed state for which the average value of cos ϕ is the same as that for the pure state will give the same value for Nc and thus the same fringe visibility. The average value of cos ϕ for ϕ uniformly distributed between -0.4 rad and 0.4 rad (0.974) is close to the visibility obtained. This suggests that the values of ϕ for the pure states contributing to the mixed state may vary over ~ 0.8 rad. This is obviously an approximate estimate and neglects the possibility that ϕ is non-uniformly distributed, or that the average value has not been adjusted exactly to zero as well as any other effects that might reduce the visibility. A Bell’s inequality measurement was also made   , with 100 second counting times for each of the 16 coincidence rate measurements, resulting in a value for the correlation parameter S of S = 2.760 ± 0.005. This is close to the maximum allowed value of S= 2 √2 = 2.828 and in clear violation of the Bell inequality, S ≥ 2.
5. Experimental results
Ellipsometric measurements were made of a silicon dioxide layer on a silicon substrate, and of both total internal and external reflection from glass samples. The results of these measurements are compared to the expected values calculated from the Fresnel equations  and the known properties of the materials.
Table 1 shows the experimentally determined ellipsometric angles of a commercial reference wafer comprising 102.7 nm of SiO2 on a silicon substrate. Three separate, independent calibration measurements were made, labelled M1, M2 and M3. After each calibration, 10 coincidence count measurements were made with appropriate settings of the waveplates at each of three angles of incidence, 40.0°, 50.0° and 60.0°. This sample has reasonable reflectivity at 810 nm and, at 40.0° angle of incidence, provided around 65 to 75 cps with the polarisers in the H⊗H setting and approximately 90 cps for the V ⊗ V setting. As a consequence, we used counting times of 200 s for each coincidence count rate measurement and for the calibration measurements. The measurements used to calculate tan ψ were repeated to give a total counting time of 400 s. Since entangled photon coincidences follow Poisson statistics , the errors shown in the table for tan ψ and Δ are one standard deviation errors calculated from the Poisson statistics (shot noise) of the coincidence counts. These errors are expected to be optimistic since all other sources of error are neglected, of which uncertainty in the angle of incidence due to inaccurate sample alignment and imperfect beam collimation, is the likely major contributor. The rotation stages have an absolute accuracy of 0.035° and were used to determine the axes of the waveplates in a separate experiment. Standard error propagation methods were used to determine the uncertainty in tan ψ and Δ from the Poisson statistics of the coincidence count measurements.
The expected values of the ellipsometric angles were calculated at each angle of incidence from the known properties of the film and substrate. The refractive indices of SiO2 (n = 1.453) and Si (n = 3.685 - 0.006i) at 810 nm were both obtained from curve fitting to data in , while the thickness of the SIO2 film, which was determined ellipsometrically at 632.8 nm by the manufacturer, was given on the calibration certificate. We find that the majority of the values of tan ψ and Δ are indeed within one standard deviation of their expected values. One notable exception is the value of Δ at 60.0° for calibration M2 and which we believe arises from a misalignment. The value of the phase ϕ of the entangled state varies across the beam so that if the apparatus is misaligned and part of the beam is undetected, the measured average value of ϕ will change. Since the apparatus has to be realigned to change the angle of incidence or perform a calibration, this is an obvious source of error in Δ. This sensitivity to misalignment could be greatly reduced by employing compensating crystals that remove the variation in ϕ across the beam .
Table 2 shows experimental and expected values of the ellipsometric angles for reflection from the surface of a clean BK7 glass wedge at three angles of incidence. The reflectivity of glass at these angles of incidence is rather small and the count rates were correspondingly low. At 70.0° angle of incidence, the H ⊗ H setting provided around 20 cps while the V ⊗ V setting yielded about 135 cps. At 80.0° angle of incidence, these count rates increased to around 120 cps and 260 cps, respectively. In view of these low count rates, 400 s counting times were used for these setting of the polarisers. The other 8 polariser settings required to determine the phase used 200 s counting times at 70.0° and 75.0°, but 100 s counting times at 80.0°. 100 s counting times were used during the calibration, except the H ⊗ H and V ⊗ V settings for which 200 s was used. The experimental values given in the Table are one standard deviation errors based on the Poisson statistics of the count rates and are seen to be in good agreement with their expected values.
Finally, measurements were made of Δ following total internal reflection from one face of an equilateral prism of BK7 glass. We made no attempt to measure tan ψ as we expect this to be 1. Some photon loss is inevitable on entering and leaving the prism and this reduced the observed count rate slightly but had no other effect on the measurements.
Three independent calibrations were performed, labelled M1, M2 and M3 with 100 s counting times. After each calibration, the 8 measurements needed to determine Δ were made, each with 100 s counting times. The experimentally determined values for Δ are given in Table 3 together with one standard deviation errors calculated as for the previous samples. We find that the experimental values for Δ are in very good agreement with the expected value, being within one standard deviation of that value in each case.
A novel technique to make ellipsometric measurements using entangled states with some mixture, as well as ideal pure states, has been developed. Our method is based on reconstructing the components of the density matrix from the observed coincidence count rates for different polarisation analyser settings, with and without reflection from the sample. Our technique is particularly advantageous since it does not require the preparation of a pure state and works well with the more easily generated mixed states available via parametric down conversion in a nonlinear crystal. Ellipsometric measurements made using this approach for samples of glass and silicon dioxide on silicon show good agreement with the expected values of the ellipsomet-ric parameters calculated from the known properties of the materials.
We are pleased to acknowledge many helpful discussions with A. White and G. Pryde at the University of Queensland, Australia. Partial funding for this project was provided by the Research Committee of the University of Auckland.
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