A CHARACTERISTIC OF A SAMPLE

This invention relates to apparatus for and a method of determining a characteristic of a sample, such as a thin film or a substrate.

Embodiments of the present disclosure enable a determination of the complex refractive index of a sample. Although in this disclosure, the convention n-ik is used, it will be appreciated that the convention n+ik may also be used.

Embodiments of the present disclosure will now be described, by way of example, with reference to the accompanying drawings, in which:

Figure 1 shows a schematic block diagram of apparatus for determining the complex refractive index of a sample, the apparatus comprising an interferometer system and a data processing and control apparatus;

Figure 2 shows a functional block diagram of computing apparatus that may be configured to provide the data processing and control apparatus shown in Figure 1 ;

Figure 3 shows a functional block diagram of the apparatus shown in Figure 1 illustrating in greater detail an example of a data processor embodying the present disclosure; and

Figure 4 shows a flow chart for explaining an example of a method of determining the complex refractive index of a sample.

In embodiments of the disclosure, the measurement operation is an interferometric measurement operation, for example an interferometric measurement operation in which one of a reference mirror and a sample is moved relative to the other in a scan direction z during a said measurement operation.

The measurement operation may be a coherence correlation, coherence (coherent) scanning or white light interferometry measurement operation. Coherence scanning interferometry (sometimes called "White-light scanning interferometry") is discussed in a paper entitled "Profilometry with a Coherence Scanning Microscope" by Byron S. Lee and Timothy C Strand published in Applied Optics Volume 29, No. 26 10 September 1990 at pages 3784 to 3788, the whole contents of which are hereby incorporated by reference. Such coherence correlation, coherence scanning or broadband scanning interferometry may use an interferometer such as a Michelson, Mirau or Linnik interferometer with a broadband spatially incoherent light source such as a quartz halogen lamp. In coherence correlation, coherence scanning or broadband scanning interferometry, as one of the test sample and a reference mirror of the interferometer is moved relative to the other along a scan path to change the relative path length, an image sensor (which may be for example a two-dimensional image sensor such as a CCD or CMOS camera), is used to sense the resulting interference pattern such that each sensing element or pixel of the image sensor senses the portion of the interference pattern for a corresponding surface region or surface pixel of the sample surface. As the test sample and the reference mirror are moved relative to one another, the amount or intensity of light received by a sensing element will vary in accordance with the change in the interference fringes resulting in a coherence envelope with a coherence peak or extremum (maximum or minimum amplitude) occurring at the position along the scan path z of zero path difference. Where different regions of a surface have different relative surface heights, then those different regions will have coherence peaks at different positions along the scan path. Accordingly, the relative positions of the coherence peaks can be used to provide surface profile data, that is data representing the relative height of the different regions of the surface of the test sample.

In coherent scanning interferometry (CSI) generally, the apparent topographical height at a particular site on the surface is, as set out above, defined by the location of the maximum of the coherence envelope. Interferograms ("frames" of image data) are obtained at scan intervals along the scan path and the maximum of the coherence envelope for a surface region or surface pixel of the sample surface is determined from these interferograms, for example using the techniques discussed in the above-mentioned paper by Lee and Strand or for example using the techniques discussed in the applicant's US7385707 and US7948634, the whole contents of which are hereby incorporated by reference. The coherence peak may not coincide with the scan position (z distance) at which an interferogram is obtained. The apparent topographical height may therefore be considered as consisting of two terms, these being firstly the product of the z-scan frame interval (that is the interval along the scan path at which interferograms are obtained) and the integer frame-number defined as that lying closest to the coherence envelope maximum and secondly a sub-z-scan interval component which represents the offset of the coherence peak in the scan direction from the integer frame- number defined as that lying closest to the coherence envelope maximum. Labelling such integer frame numbers associated with the measurement of a test surface and a reference surface S and R, labelling such sub-z-scan interval components associated with the measurement of a test surface and a reference surface )S and )R, and labelling the z-scan frame interval >, then a parameter AZ _HC F is defined herein as

Az _HCF = Z _s - S - {Z _R - R )

where Z _S and Z _R are respectively the actual surface heights of the test sample and reference sample. For some materials, such as dielectrics with very low absorption over the measurement bandwidth, the apparent topographical heights are the actual topographical heights. For such a reference sample measurement then

Z _R = {R + AR

and likewise for such a test sample measurement

Z _S = (S + AS .

Embodiments of the present disclosure provide apparatus for and a method of determining the complex refractive index, n-ik, of a test sample. Embodiments of the method comprise a processor determining first ratio data comprising a ratio between first intensity data representative of a first series of intensity values resulting from a measurement operation on the test sample and second intensity data representative of a second series of intensity values resulting from a measurement operation on a reference sample of known spectral n and k. The first ratio data are used to calculate values for spectral n and k, thereby providing candidate value pairs for spectral n and k.

The data processor determines second ratio data comprising a ratio between third intensity data representative of a third series of intensity values resulting from a measurement operation on a coated test sample (produced by coating at least a part of the test sample with a material of known n and k) and the second intensity data or comprising a ratio between the third intensity data and fourth intensity data representative of a fourth series of intensity values resulting from a measurement operation on the reference sample or another reference sample of known spectral n and k. The data processor then carries out a fitting procedure using the second ratio data and the candidate value pairs for n and k by fitting the second ratio data to synthetic ratio data which is a function of n and k, thereby determining spectral n and k for the test sample.

The reference sample may be a separate substrate of known spectral n and k. As another possibility, the reference sample could consist of a region on the test sample that has been coated with an opaque layer of material of known spectral n and k such that a measurement on the reference sample is unaffected by the underlying test sample.

The coated test sample may be provided by coating at least part of the test sample surface after the measurement operation to obtain the first series of intensity values, or may be provided by a pre-coated region of the test sample surface. The sample may be coated by, for example, a Langmuir-Blodgett process, a photosensitive resist spin-on process or an aerosol- based process or any other suitable process that provides a film of known spectral n and k. Preferably this film is a removable film. The film may be a photo-resist. The first ratio data may be used calculate values for spectral n and k, by relating the first ratio data to a synthetic function comprising the (unknown) electric field reflectance of the test sample multiplied by a phase term, where the electric field reflectance term may be expressed in terms of spectral n and k. In an example, the first ratio data comprises a ratio between transforms (for example frequency transforms such as Fourier transforms) of the intensity data multiplied by the electric field reflectance of the reference sample. Other functions relating the intensity data ratio to electric field reflectance multiplied by a phase term may be used.

In embodiments of the disclosure, for a low numerical aperture objective, the first ratio data may be represented as a 'determined' HCF function:

where

3 _5s+ is the Fourier Transform with SB ₊ indicating the positive sideband,

/ _j and I ₂ are the respective series of intensity values for measurement operations on surfaces of first, test, sample and second, reference, sample of unknown and known spectral n and k, respectively, and r ₂ (v) is the electric field reflectance of the second, reference, sample of known n and k.

This 'determined' HCF function may be re-expressed as a 'synthetic' HCF function:

HCF ^D {v) = HCF ^s {v) where

H ⁵ ( ) = r ₁ ( ^i4¾ - {n v) - ik v)) _{i4m ZHcF}

\ + {n _x {v) - ik _x {v))

where

r _! (v) is the electric field reflectance of the first, test, sample of unknown spectral n and k, a parameter AZHCF is defined herein as

where Z _S and Z _R are respectively the actual surface heights of the sample and reference, S and R are the integer frame (interferogram) numbers closest to the interference envelope maxima for the test sample and reference sample, respectively, and ξ is the distance interval (scan interval) between acquisition of frames of intensity values in a measurement operation, and ΓΗ (v) and (v) are the real and imaginary parts, as a function of frequency v, of the complex refractive index of the first, test, sample.

In embodiments of the disclosure, using the first ratio data to calculate candidate value pairs for n and k comprises letting an assumed value for Az _HCF in the synthetic HCF function range from to - 2ξ to 2ξ.

In embodiments of the disclosure, for a low numerical aperture objective, the first ratio data is represented by a 'determined' HCF function:

where

3 _5s+ is the Fourier Transform with SB ₊ indicating the positive sideband,

I _SUB and I _REF are the first and second series of intensity values,

r _ref (v) is the electric field reflectance for the reference sample.

In an embodiment, for a low numerical aperture objective, the first ratio data HCF ^D (v) may be used to calculate a range of a candidate value pairs for n and k by letting an assumed value \ox z _HCF (designated Az _HCF ) range from - °A to °A (where λ ₀ is the mean spectral wavelength) in the function pair

where "Re" and "lm" indicate the real and imaginary parts of the function.

In embodiments of the disclosure, for a low numerical aperture objective, the second ratio data is represented by a 'determined' HCF function:

HCF _film (v) =

where

3 _5s+ is the Fourier Transform with SB ₊ indicating the positive sideband, / , and /„, are the second and third series of intensity values,

r _ref (v) is the electric field reflectance for the reference sample.

The fitting of the second ratio data may comprise minimising an error function, for example: ε ² (AZ _h ^s _cf ) ),Az

where HCF _lm (v) represents the second ratio data and HCF _f ^s _ilm (v) represents a synthetic HCF function, where the synthetic HCF function is a function of the film thickness, d, on the test sample, Az _HCFflim and the candidate complex refractive index pair, n{y, Az _H ^s _CF ) and k{y, Az _H ^s _CF ). At this function minimum, Az _HCF - Az _H ^s _CF , the n and k spectral values are defined by n(v,Az _HCF ) and k(v,Az _HCF ) , respectively, and in addition, both d and are now known.

Referring now the drawings, Figure 1 shows a simplified schematic block diagram of an example of apparatus 1 apparatus for determining the complex refractive index.

The apparatus 1 shown in Figure 1 has a coherence correlation interferometer (CCI) system 2 and data processing and control apparatus 3. The CCI interferometer system 2 is based on a conventional interferometer and, typically, has a Mirau, Michelson or Linnik configuration, for example. Instead of having a monochromatic spatially coherent light source, the CCI interferometer system 2 has a broadband light source 4 which may be, for example, a white light source such as a quartz halogen lamp coupled to a regulated DC power supply having a light intensity output user control 400 provided, for example, in the form of a user-rotatable knob. The light source 4 provides broadband light L which is directed by a beam splitter 12 towards an objective lens assembly 13 which includes, in addition to an objective lens 14, a beam splitter 5 and a reference mirror 6. The beam splitter 5 splits the light beam provided by the beam splitter 12 into a first, reference beam that is directed along the reference path RP towards the reference mirror 6 and a second, sample beam that is directed along the sample path SP from the interferometer I towards the surface 7 of a sample 8 mounted on a sample support stage 9. Light returning from the surface 7 and from the reference mirror 6 to the beam splitter 5 interfere and the interference is directed by the beam splitter 12 to a focussing element 3 which focusses an image of the region of interference onto a detector 10.

In this embodiment, the detector 10 has a 2D (two-dimensional) array SA of image sensing elements SE, one row of which is shown very diagrammatically in Figure 1 . The array SA images an area of the sample surface 7 falling within the field of view of the detector 10. Each individual sensing element SE of the 2D sensing array of the detector 10 detects the portion of the interference pattern falling within the acceptance cone of that element and resulting from a corresponding surface region or surface pixel of the area of the sample surface 7 so that, effectively, the imaged area of the surface can be considered as a 2D array of surface regions or surface pixels. The detector 10 may be, for example, a CCD (Charge Coupled Device) digital camera or a CMOS detector having a 2D (two-dimensional) xy array of CMOS sensing elements. Generally each of the sensing elements may be square to provide the same resolution in both directions (x and y) of the array.

A motion controller 1 1 is provided to effect relative movement between the test sample 8 and the reference mirror 6 so as to change the difference between the lengths of the paths travelled by light reflected from the reference mirror 6 and light reflected from the sample surface 7. As shown in Figure 1 , the motion controller 1 1 is arranged to move the objective lens assembly along the reference path RP. This is equivalent to moving the sample surface 7 along a scan path in the z direction shown in Figure 1 .

The detector 10 is arranged to capture or sense the light intensity (i.e. the interference pattern) at a distance interval ξ as the path length difference between the sample 8 and the reference mirror 6 is changed. In this example, the detector captures or senses the light intensity at a distance interval ξ corresponding to axial (z) changes in the relative position of the sample 8 of Az _step = λ ₀ /8 (equivalent to a path length difference of λ ₀ /4), where λ ₀ is the nominal mean spectral wavelength of the broadband source. For the case where the numerical aperture cannot be approximated by zero, then the distance interval ξ is given by z _step - λ ₀ /(8 cos Θ) where # is the effective mean angle of incidence over the numerical aperture. The step may be, for example, 75nm.

2D image or frame data representing the intensity pattern for the field of view of the detector 10 is acquired by the detector 10 at each distance interval ξ. The intensity of the illumination sensed by one sensing element of the 2D sensing array (that is the portion of the interference pattern provided by light reflected from the corresponding surface region or surface pixel of the sample surface 7 imaged on that sensing element) varies as the path length difference changes with movement of the reference mirror 6, resulting in a series of fringes which have a coherence peak at the position along the scan path corresponding to zero path length difference. The envelope of the intensity distribution is the Fourier transform of the spectral distribution of the broadband source, modified by the numerical aperture and the spectral transmission of the instrument together with the spectral responsivity of the detector. The stage 9 may be arranged to enable x and y movement to allow different areas of a sample surface to be measured. Although not shown, as in the case of the Talysurf CCI 3000, the stage 9 may also be tiltable about the z axis to enable the user to ensure that the sample surface is parallel to the reference mirror 6 in which case this may be achieved by adjusting the tip-tilt stage 9 to produce near-minimum fringe densities.

As shown in Figure 1 , the data processing and control apparatus 3 has control apparatus 30 for controlling operation of the interferometer system 2, an intensity data receiver 33 for receiving intensity data signals from the detector 10, a data processor 32 for processing the intensity data under the control of a controller 21 of the control apparatus 30 and a user interface 31 for enabling a user or operator to control operation of apparatus, for enabling the user or operator to be provided with a data output representing the results of processing by the data processor 32 of the data acquired during a measurement operation, and also for enabling messages such as error messages to be communicated to the user.

The controller 21 of the control apparatus 30 controls overall operation of the apparatus and communicates with the user interface 31 and data processor 32.

At least the controller 21 and data processor 32 of the data processing and control apparatus may be implemented by programming computing apparatus, for example a personal computer.

Figure 2 shows a simplified block diagram of such computing apparatus. As shown, the computing apparatus has a processor 25 associated with memory 26 (ROM and/or RAM), a mass storage device 27 such as a hard disk drive, a removable medium drive (RMD) 28 for receiving a removable medium (RM) 29 such as a floppy disk, CDROM, DVD, memory stick or the like, input and output(l/0) controllers 37 for interfacing with the components of the CCI interferometer system to be controlled by the control apparatus (for example, z, x and y movers and the detector 10) to enable the processor 25 to control operation of these components. The user interface 31 consists, in this example, of a keyboard 31 a, a pointing device 31 b, a display such as a CRT or LCD display 36a and a printer 36b. The computing apparatus may also include a communications interface (COMMS INT) 199 such as a modem or network card that enables the computing apparatus to communicate with other computing apparatus over a network such as a local area network (LAN), wide area network (WAN), an intranet or the Internet.

In this example, the intensity data receiver 33 is provided as a dedicated frame capture circuit board 230 installed within the computing apparatus. The processor 25 may be programmed to provide the data processor 32 and controller 21 by any one or more of the following ways: 1) by pre-installing program instructions and any associated data in a non-volatile portion of the memory 26 or on the mass storage device 27; 2) by downloading program instructions and any associated data from a removable medium 29 received within the removable medium drive 28; 3) by downloading program instructions and any associated data as a signal SG supplied from another computing apparatus via the communications interface 199; and 4) by user input via the user interface 31 .

Further details of an example of an interferometer system 2 that may be used can be found in, for example, the applicant's WO03/078925 and US2007/0188768, the whole contents of which are hereby incorporated by reference. Examples of commercially available apparatus that may be used are the CCI range of instruments available from Taylor Hobson Limited, Leicester, England, UK. Figure 3 shows a functional block diagram of the apparatus shown in Figure 1 illustrating an example of the data processor 32.

The data processor 32 has an intensity data store 40 configured to store intensity data and a ratio determiner 60 configured to determine first intensity data ratio which comprises a ratio between the first intensity data representative of a first series of intensity values resulting from a measurement operation on a test sample of unknown n and k and second intensity data representative of a second series of intensity values resulting from a measurement operation on a reference sample of known n and k. The ratio determiner 60 is also configured to determine second ratio data which comprises a ratio between third intensity data representative of a third series of intensity values resulting from a measurement operation on a coated test sample and the second intensity data or which comprises a ratio between the third intensity data and fourth intensity data representative of a fourth series of intensity values resulting from a measurement operation on the reference sample or another reference sample of known spectral n and k.

The coating on the test sample is a coating of a material of independently well-characterised spectral n and k. The coating may be provided on the test sample by, for example, a Langmuir-Blodgett process, a photosensitive resist spin-on process or an aerosol-based process or any other suitable process that provides a film of known n and k.

In this example, the intensity data store 40 has a reference intensity data store 41 , a substrate intensity data store 42 and a coated substrate intensity data store 43 configured to store the second (plus fourth if provided) , first and third intensity data, respectively. However of course a single data store could be provided.

The data processor 32 also has an n and k data set creator 70 to calculate, using the first ratio data and a synthetic function, a range of candidate value pairs, n(y,Az^ _CF ) and k{y, Az _HCF ), for spectral n and k for the test sample.

In this example, the first and second ratios are ratios of frequency transforms of the intensity data and so the data processor 32 has a frequency transformer 50 to transform intensity data to the frequency domain, in this example by use of a Fourier transform process. In this example, the frequency transformer 50 has respective stores 51 , 52 and 53 for the reference frequency-transform data, the test sample frequency-transform data and the coated test sample frequency-transform intensity data, respectively. However of course a single data store could be provided.

The data processor 32 has a fitter 90 configured to carry out a fitting procedure using the second ratio data and the candidate value pairs, n{y, Az _H ^s _CF ) and k{y, Az _H ^s _CF ), by fitting the second ratio data to a synthetic ratio data which is a function of n and k, thereby to determine n and k for the test sample.

The fitter 90 is governed by the minimisation of an error function which, although other error functions may be used is in this example of the form ε ¹ { z _HCF ) =∑ (v) - HCF _fi ^s _lm (d, n(v, Az _H ^s _CF ), k(v, Az _H ^s _CF ), Az _HCFfiim ,vf

where HCF° _lm {y) represents the second ratio data and HC _/m (v) is the synthetic ratio data that is a function of the film thickness, d, on the test sample, Az _HCFfii , and the candidate complex refractive index pair, n{y, Az _H ^s _CF ) an k{y, Az _H ^s _CF ). At the function minimum, Az _HCF - Az _H ^s _CF , both d and Az _HCFfii are determined and the n and k spectral values are defined by n(v,Az _HCF ) and k(v,Az _HCF ) respectively.

The data provider 1 10 may output the determined n and k values and/or the film thickness. The data provider 1 10 may be coupled to provide an output to the user interface and/or may be coupled to the controller or for example via a network to other computing apparatus. An example of a method of determining the complex refractive index, n-ik, of a test sample will now described with reference to Figure 4 which shows a very schematic flow chart. This method may or may not use the data processor shown in Figure 3. In order to enable determination of n and k of a test sample, a user causes the apparatus to carry out: 1) a scan of the test sample 200 surface to obtain a frame of intensity data (an interferogram) at each scan distance interval ξ and, for respective surface pixels, store at S1 in Figure 4 a first series of intensity values, each intensity representing the intensity value provided by a sensing element of the detector at a respective scan distance interval ξ along the scan path; and 2) a scan of a surface area of a reference sample 201 to obtain a frame of intensity data (an interferogram) at each scan distance interval ξ and, for respective surface pixels, store at S2 in Figure 4 a second series of intensity values, each intensity representing the intensity value provided by a sensing element of the detector at the respective scan distance interval ξ along the scan path.

The test sample or an area thereof is coated with a film 202 (for example by use of a Langmuir-Blodgett process, a photosensitive resist spin-on process or an aerosol-based process or any other suitable process) that provides a film of known n and k and the user causes the apparatus to carry out a scan of the coated surface area 202 of the test sample 200 surface to obtain a frame of intensity data (an interferogram) at each scan distance interval ξ and, for respective surface pixels, store at S6 in Figure 4 a third series of intensity values, each intensity representing the intensity value provided by a sensing element of the detector at a respective one of the scan distance interval ξ along the scan path. Of course, if a part of the test sample is pre-coated, then the measurement operations on the reference sample, test sample and coated test sample may be carried out in any order. The light source should be constant for all three measurement operations.

The subsequent processes S3 to S13 are described below for a surface pixel but will generally be carried out for a number of surface pixels.

At S3, S4 and S7 in Figure 4, the data processor transforms the respective first, second and third series of intensity values (intensity data) into the frequency domain, for example using a Fourier transform process and stores the resulting first, second and third frequency-transform data. The respective frequency transforms may be carried out as and when the respective series of intensity vales are obtained or when they are required for the next process.

At S5, the data processor determines first ratio data between the first (test sample) frequency- transform data and the second (reference sample) frequency-transform data. At S8 the data processor determines second ratio data between the third (coated test sample) frequency- transform data and the second (reference sample) frequency-transform data or a ratio between the third (coated test sample) frequency-transform data and fourth intensity data frequency transform data where the fourth intensity data is representative of a frequency transform of a fourth series of intensity values resulting from a measurement operation on the reference sample or another reference sample of known spectral n and k. The respective ratio data may be determined as and when the corresponding frequency-transform data is available or when they are required for the next process. At S9, the data processor creates a family of potential spectral n and k candidate pairs (n and k data set) by varying the apparent surface height (AZ _H CF) in a synthetic HCF function. The process at S9 may be carried out once the preceding process S5 has been carried out or when required by the subsequent process. At S11 , the data processor carries out a fitting procedure using the second ratio data and the candidate value pair-functions for spectral n and k by fitting the second ratio data to synthetic ratio data which is a function of n and k, provides, at S12, values for n and k together with the film thickness and at S13 outputs the film thickness, with or without the values determined for n and k, to, for example, a user or another processor or to other functionality of the data processor.

An example of a method of determining spectral n and k will now be described where the numerical aperture (NA) is sufficiently low that it may be reasonably approximated by NA ~ 0. In this example, the ratio data is represented by the determined HCF functions mentioned above and for the first intensity data ratio is:

where I _sub and I _ref are the interference series corresponding to the unknown and known spectral n and k respectively (that is, in this example, the first intensity data representative of the first series of intensity values acquired in a measurement operation on a test sample of unknown spectral n _s (v) and k _s (v) and the second intensity data representative of a second series of intensity values acquired in a measurement operation on a reference sample of known spectral n(v) and k(v)).

The determined function HCF ^D (v) can also be represented as a synthetic function HCF ^S (v), that is:

HCF ^s (v) = HCF ^D (v)

The synthetic function HCF ^S (v) may be expressed using Fresnel coefficients as: (n,( ^y )-icf,( ^y ))

= r _s y iAnv&z _HCF l- _, iAxv z _HCT

HC »

l + {n _s {v)-ik _s {v)) ⁽

In the above equations, whilst the determined function HCF ^D (y) is known (because it corresponds to the (in this case frequency transform) first ratio data multiplied by the known electric field reflectance r _ref of the reference sample), none of the constituent parts

(ra(v), k(y) & Az _HCF ) of the synthetic function HCF ^S (v) is known. Considering Fresnel electric field reflectance in terms of spectral n & k gives: rr _sub ({vv)) = R X*e(^r (\yv)))) ₊₊ i ⁱ limmrr^ (\yy))= ^l _{l +} ^~ ^ _{n(y) ^~ _ ⁱ _i ^k _k ^ _(y)) = _{{l +} H _n{y -f^ ₊ )) _{ ^ _y)

These real (Re) and imaginary (Im) electric field reflectance equations may be inverted to give:

_{( )} _ 1-Re ² fc ))-Im ² ))

"(l + Re(^(v))) ² ₊ Im ² (^(v))

2Im(r _s »)

k(v)

(l + r _sub (v))) ² + Im ² (r _sub (v))

These electric field reflectance equations may be expressed in terms of HCF ^s (y) and advantage then taken of the fact that typically,

A family of potential spectral n and k solutions (candidate n and k pairs) may be created by

1

letting an assumed value for Az _HCF (designated Az _HCF ) vary in the range from - λ ° / to A °A /

in the function pair:

Intensity data representative of a third series of intensity values resulting from a measurement operation on the test sample after coating with a material of independently well-characterised spectral n and k is obtained and second ratio data corresponding to, in this example, the ratio between the coated test sample frequency transform data and the reference frequency transform data determined. The sample may be coated by, for example, a Langmuir-Blodgett process, a photosensitive resist spin-on process or an aerosol-based process or any other suitable process that provides a film of known n and k.

The second ratio data is given by: where I _film and I _ref are the interference series corresponding to the coated test sample and the reference sample of known spectral n & k, respectively. Now the second ratio data represented by the determined function HCF° _lm {y) may be fitted to the synthetic function HCi¾ _m (v) :

where r _fi | _m is the electric field reflectance of the film or coating which is of thickness d and AzticFfiim is as defined above.

The electric field reflectance for a layer on a substrate is discussed in Optical Properties of Thin Solid Films" by O.S. Heavens (ISBN 0-486-66924-6 published as the Dover edition in 1991 but first published in 1955) at pages 55 to 59, incorporated herein by reference. By analogy with equation 4(52) on page 58 of Optical Properties of Thin Solid Films" by O.S. Heavens for the case where n ₀ is the refractive index of air and so 1 , HCF _f ^s _ilm can be written as:

i4mAz, _HCFfilm

HCF film Vv) ) = r ' film ' where

N _F (v) is the complex refractive index of the film as a function of frequency v where

N _F {v) = n _F (v)-ik _F (v) , at\

N _F (v) is the complex refractive index of the substrate (test sample) film as a function of frequency v where N _s (v) = n _s (v) - ik _s (v) .

A fitting procedure is then carried out using the second ratio data (the determined function HCFp _lm (y)) and the candidate value pair functions n{y, Az _H ^s _CF ) and k{y, Az _H ^s _CF ), by fitting the determined function HCF _lm (y) to the synthetic ratio data, the synthetic function

(HCFf _Um (v)). The fitting is governed by the minimisation of an error function which, although other error functions may be used, is in this example of the form:

ε ² (Az _H ^s _CF ) = v - _film , n v, z _HCF v, z _HCF , ζ ^■ HCF film ' such that at the function minimum, Az _HrF = Azl _rF , d and Az _HrF f _r il,m are determined and the n and k spectral values are defined by n(v,Az _HCF ) and k(v,Az _HCF ) respectively.

The above approach may be extended to higher numerical aperture by integrating over the numerical aperture, for example in a manner similar to that discussed in US2007/0188768, the whole contents of which have previously been incorporated herein by reference.

An example of extension to the case of a higher numerical aperture will now be described.

Firstly, for a randomly polarised Coherent Scanning Interferometer (CSI) instrument of higher numerical aperture, such as 0.3, the HCF functions become:

and HCF ^D {v) = HCF ^s (v) where

HCF ^s (y) - r _{sub 4w∞sa Jff} where w^ is the angular weighting function of the objective lens used. As an example, for a uniformly filled entrance pupil, w{d) - sin ^cos i? where sin ^ NA and f _mb represents the mean of the electric field reflectance over the numerical aperture of the test sample and r , represents the mean of the electric field reflectance over the numerical aperture of the reference sample. For a derivation of w(d) - sin Θ cos Θ reference should be made to US7403289B2, the whole contents of which are hereby incorporated by reference.

The original candidate solution equation pair given above is close but not identical to the actual candidate solution pair; this needs to be determined through iteration. Providing the original candidate solution pair with zero subscripts to indicate this,

which as a pair is the starting solution to: which may be solved, for instance, through minimising the function χ ¹ {&z _H ^s _CF )where

where the weighting A{y) = |3 _5s+ (l _ref )| is the coherent signal amplitude, and where

an

As before, the final step is to coat the test sample with a material of independently well- characterised spectral n and k and re-measure it in the same way as for the test sample of unknown spectral n and k. This then provides a determined HCF function given by

and a corresponding synthetic HCF function defined by i v cos Az _HCF

HCF v) = f _film {y)e ⁱ

A fitting procedure is then carried out using the second ratio data (the determined function HCF° _[m (v)) and the candidate value pair-functions n{y, Az _H ^s _CF ) and k{y, Az _H ^s _CF ), by fitting the determined function HCF° _lm {y) to the synthetic ratio data where, in this case, the synthetic function( HCF _f ^s _ilm (v)):

_HCF s _{y

The fitting is governed by the minimisation of an error function which, although other error functions may be used, is in this example of the form: ε ² (Az _H ^s _CF ) =∑ |tfC¾ (y) - HCF _f ^s _ilm (d, n(v,Az _H ^s _CF \ k(v, Az _H ^s _CF \ Az _HCFfilm , v] ² such that at the function minimum, Az _HCF = Az _H ^s _CF , d and Az _HCFfilm are determined and the n and k spectral values are defined by n(v,Az _HCF ) and k(v,Az _HCF ) respectively.

The data processor described above may be a processor or processors of computing apparatus programmed by program instructions or could for example be dedicated hardware or a digital signal processor or digital signal processors or any combination of these.

Figure 3 shows the data processor 32 as being made up of a number of different modules. The functionality of each of these modules may be provided by respective different processors, programs or sub-routines. As another possibility, the functionality may be distributed so that the individual modules, or some of the individual modules, may not exist as separate functional entities.

It will, of course, be appreciated that whether an equation contains a quantity or its complex conjugate will be dependent on the polarity of the exponent of the Fourier Transform and, that if the other polarity is used (the Inverse Fourier Transform in this case), then the complex quantities will be replaced by their conjugates. Where reference is made above to the positive sideband, it will be appreciated that the negative sideband could be used.

As described above, the function requires transformation of the intensity data into the frequency domain using a Fourier transform. It will be appreciated that other frequency transformations may be used. It will also be appreciated that other approaches may be used that do not require transformation into the frequency domain or which use other transformations. An embodiment of the disclosure provides a method and apparatus that determine the complex refractive index, n-ik, of a test sample by: determining first ratio data related to a ratio between first intensity data representative of a first series of intensity values resulting from a measurement operation on the test sample and second intensity data representative of a second series of intensity values resulting from a measurement operation on a reference sample of known n and k; using the first ratio data to calculate values for n and k, thereby providing candidate value pairs for n and k; determining second ratio data related to a ratio between third intensity data representative of a third series of intensity values resulting from a measurement operation on a coated test sample, that is the test sample after coating with a material of known n and k, and the second intensity data or related to a ratio between the third intensity data and fourth intensity data representative of a fourth series of intensity values resulting from a measurement operation on the reference sample or another reference sample of known spectral n and k; and carrying out a fitting procedure using the second ratio data and the candidate value pairs for n and k by fitting the second ratio data to synthetic ratio data which is a function of n and k, so as to determine n and k for the test sample.

The above approach should be applicable to any sample or substantially opaque layer, such as a metal layer that behaves effectively as bulk metal, where opaque means opaque in the spectral range of the light source used. The above approach may be straightforwardly extended to the case of determining the spectral n and k of a thin film on a substrate.

The above description assumes that the spectral range of light source is in the visible. It could, however, extend into or lie within the infra-red or ultra violet regions of the spectrum.

The above description assumes that a single reference measurement is made and its corresponding series of intensity values is used in the determination of both the first and second ratio. As another possibility, a second reference measurement could be made for the second ratio and on the same or a different reference sample.

The above description assumes that an adequate range for AZ _H CF to vary to encompass the actual spectral n and k values is - ° to ° . There may be some materials for which this range should be extended.