random errors in interferometry with the least-squares method

Random errors in interferometry withthe least-squares method

Qi WangDepartment of Physics, Huazhong University of Science & Technology,

Wuhan, 430074, China

Contact address: 15 Cascades Road, Pakuranga, Auckland, New Zealand

Received 1 October 2010; revised 24 November 2010; accepted 23 November 2010;posted 3 December 2010 (Doc. ID 136011); published 12 January 2011

This investigation analyzes random errors in interferometric surface profilers using the least-squaresmethod when random noises are present. Two types of random noise are considered here: intensity noiseand position noise. Two formulas have been derived for estimating the standard deviations of the surfaceheight measurements: one is for estimating the standard deviation when only intensity noise is present,and the other is for estimating the standard deviation when only position noise is present. Measurementson simulated noisy interferometric data have been performed, and standard deviations of the simulatedmeasurements have been compared with those theoretically derived. The relationships have also beendiscussed between random error and the wavelength of the light source and between random error andthe amplitude of the interference fringe. © 2011 Optical Society of AmericaOCIS codes: 120.3940, 120.3180, 120.2830.

1. Introduction

Optical interferometry has long been a techniqueused for surface profiling [1]. Surface height can bedetermined by measuring the phase of the interfero-metric data that are recorded when the referencemirror of the interferometer is shifted to a numberof positions. However, in any real optical system, ran-dom noises are present in interferometric data andthis will degrade otherwise accurate measurements.Many researchers have analyzed the effect of randomnoises in optical interferometric systems [2–5].

There are usually two types of random noise in anoptical interferometric system. We call them inten-sity noise and position noise. An example of intensitynoise is that caused by the random fluctuation ofthe light source, and an example of position noiseis that caused by the vibration of a mirror in theinterferometer.

Rogala and Barrett have analyzed the effect of in-tensity noise on the phase measurements previously,and their analysis leads to the result that many pop-

ular algorithms for phase measurements actuallyminimize the effect of intensity noise [2]. The effectof vibration noise, which is called position noise here,has also been analyzed by de Groot [3]. His researchwas performed in the frequency domain, and the fre-quency components of vibration noise that causephase measurement errors were given. Fleischeret al. investigated the best possible resolution achiev-able by any phase measurement algorithm [4]. Mil-man and Basinger also investigated the bias andvariance of the phase measurement algorithms [5].

In this investigation, random errors in interfero-metric surface profilers will be analyzed in the pre-sence of intensity noise and position noise. Formulasfor estimating standard deviations of surface heightmeasurements will be derived, and the theoreticalresults will be compared with those of simulatedmeasurements.

A mathematical model of noisy interferometricdata will be introduced first, and then assumptionsare made about the characteristics of the noise pre-sent in the optical system; furthermore, two formulashave been derived for estimating the standard devia-tion of the surface height measurements, and the

0003-6935/11/030246-07$15.00/0© 2011 Optical Society of America

246 APPLIED OPTICS / Vol. 50, No. 3 / 20 January 2011

theoretical results are discussed. Next, surfaceheight measurements are performed on simulatednoisy interferometric data by using the algorithmproposed by Morgan [6], and the simulation resultsare compared with the theoretical results. Finally,conclusions of this investigation are given.

2. Noisy Interferometric Data

In order to determine a surface profile, a series of in-terferograms needs to be recorded by an image sen-sor when the position of the reference mirror in theinterferometer is shifting. Every photodetector in theimage sensor will record a series of intensities of in-terferometric data. The intensity received by one op-tical detector in the image sensor may be modeledmathematically as

IðxÞ ¼ bþ a cos�4πλ ðx − x0Þ

�; ð1Þ

where I is the intensity received by the detector, a isthe amplitude of the interference fringe, b is the biasintensity, x is the optical length difference of the armsin the interferometer and it can be varied by shiftingthe position of the reference mirror, x0 is the surfaceheight at the point of the surface, and λ is the wave-length of the light source. The visibility of the inter-ference fringe given by Eq. (1) is a=b.

To determine the surface height, x0, the optical in-tensity given by Eq. (1) can be recorded as the refer-ence mirror in the interferometer shift to a number ofpositions through a half-wavelength range, andthese sampled intensities recorded by the detectorcan be expressed mathematically by

IðjdÞ ¼ bþ a cos�4πλ ðjd − x0Þ

�; ð2Þ

where j ¼ 1; 2…N and d is the step size of the refer-ence mirror. The total shift length of the referencemirror is approximately equal to a half-wavelength,i.e., Nd ≈ λ=2. In any real optical instrument, noisesare always present. If intensity noise and positionnoise are present, the noisy sampled intensitiescan be written as

IðjdÞ ¼ bþ a cos�4πλ ðjdþ nPðjÞ − x0Þ

�þ nIðjÞ; ð3Þ

where nPðjÞ and nIðjÞ represent position noise and in-tensity noise at the jth sampling step, and their root-mean-square (rms) values are σP and σI, respectively.

We may define the signal-to-noise ratio (SNR) asthe peak-to-peak value of the interference fringe overthe rms value of the intensity noise, i.e., SNR ¼2a=σI. The concept of the SNR may be useful whencomparing the magnitude of the intensity noise withthe size of the interference fringes.

3. Assumptions about the Characteristics of NoisePresent in Interferometric Data

In order to proceed with the theoretical investigationin this paper, assumptions have to be made about thecharacteristics of the noises present in the interfero-metric data. These assumptions are:

1. The mean values of position noise and inten-sity noise are zero.

2. Position noise and intensity noise are indepen-dent from each other.

3. Noises at different sampling steps are inde-pendent from each other.

4. The rms value of the position noise is equal toσP and is much less than a half-wavelength (λ=2).

5. The rms value of the intensity noise is σI.

The above assumptions can be expressed mathe-matically as

1. EðnPðiÞÞ ¼ EðnIðiÞÞ ¼ 0; ð4aÞ2. E½nIðiÞnPðjÞ� ¼ 0; ð4bÞ3. E½nPðiÞnPðjÞ� ¼ E½nIðiÞnIðjÞ� ¼ 0; i ≠ j; ð4cÞ4.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE½n2

PðiÞ�q

¼ σP ≪ λ=2; ð4dÞ

5.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE½n2

I ðiÞ�q

¼ σI; ð4eÞ

where EðzÞ denotes the mean value of the variable z,i ¼ 1; 2…N, and j ¼ 1; 2…N. Note that the noises arenot assumed to be Gaussian distributed. This meansthat the theoretical results derived here can be ap-plied to interferometric systems where noises aresymmetrically distributed.

4. Random Errors of Height Measurements

To determine surface height, one can use a cosinefunction in the least-squares method. The cosinefunction can be written as

I0ðxÞ ¼ A cos�4πK

ðx − X0Þ�þ B: ð5Þ

There are four parameters in the function given byEq. (5): A, B, K , and X0. These parameters corre-spond to the amplitude of the interference fringe,bias intensity, wavelength of the light source, andsurface height, respectively. If the function givenby Eq. (5) is used in the least-squares method, aleast-squares penalty function can be formed:

Q ¼XNj¼1

ðIðjdÞ − I0ðjdÞÞ2; ð6Þ

where IðjdÞ represents the sampled intensities. Thesurface height can be measured by minimizing thepenalty function Q. By substituting Eq. (3) and (5)into Eq. (6), the penalty function can be written as

20 January 2011 / Vol. 50, No. 3 / APPLIED OPTICS 247

Q ¼XNj¼1

�bþ a cos

�4πλ ðjd − x0 þ nPðjÞÞ

�− B

− A cos�4πK

ðjd − X0Þ�þ nIðjÞ

�2: ð7Þ

The value of the penalty function Q varies with thevalues of the parameters A, B, K , and X0, and thesurface height can be determined when Q is mini-mized. To simplify the analysis here, the values ofthe parameters A, B, and K can be assigned to a,b, and λ, respectively. This should be a good approx-imation because we actually have A ≈ a, B ≈ b, andK ≈ λ, when Q is minimized. Besides, when Q is tobe minimized, we have X0 ≈ x0, and we also assumenP ≪ λ=2. Then a linear approximation can be ap-plied to Eq. (7), which can be written approximatelyas

Q ≈XNj¼1

�−4πaλ sin

�4πλ ðjd − x0Þ

�ðΔxþ nPðjÞÞ

þ nIðjÞ�

2; ð8Þ

where Δx ¼ X0 − x0 represents the difference be-tween the value of the parameter X0 and the actualsurface height. The penalty function given by Eq. (8)

varies with Δx. The measurement error of a givendataset, which will be denoted by δ, is the value ofΔx when the penalty function given by Eq. (8) isminimized. So to find the measurement error δ, wecan let the derivative of the penalty function Q van-ish, i.e., ∂Q=∂ðΔxÞ ¼ 0. Then we have

XNj¼1

4πaλ sin


�

×�4πaλ sin


�ðδþ nPðjÞÞ − nIðjÞ

�

¼ 0: ð9Þ

By rearranging Eq. (9), the measurement error δcan be given by following equation:

δ ¼ λ4πa

PNj¼1 nIðjÞ sin


�P

Nj¼1 sin

2


�

−

PNj¼1 nPðjÞ sin2


�P

Nj¼1 sin

2


� : ð10Þ

Equation (10) shows how the measurement error δis related to the intensity and position noise at eachsampling step. The measurement error δ varies asthe noise values at each sampling step change.The standard deviation of the surface height mea-surement, which will be denoted by Δ, can be foundby

Δ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðδ2Þ − ½EðδÞ�2

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiEðδ2Þ

q; ð11Þ

where EðzÞ is the expectation value of variable z. InEq. (11), EðδÞ ¼ 0 because the noises present are as-sumed to be symmetrically distributed.

In the situation that the assumptions given in Sec-tion 3 are true, by substituting Eq. (10) into Eq. (11),we can express the standard deviation of the heightmeasurement as

Δ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� λ4πa

�2 σ2IP

Nj¼1 sin

2


�þσ2P

PNj¼1 sin

4


��P

Nj¼1 sin

2


��2

vuuuuut : ð12Þ

When d ≪ λ=2, the sums in Eq. (12) can be esti-mated by integration; then

Δ ≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� λ4πa2

�dσ2I�R π

−π sin2ðyÞðdyÞ

�þ 4πdσ2Pλ

R π−π sin

4ðyÞðdyÞ�R π−π sin

2ðyÞðdyÞ�2

vuuuut :

ð13Þ

Because the N sampled intensities are recordedwhile the reference mirror is shifted through ahalf-wavelength distance, the integrations in Eq. (13)cover one period of the cycle, and the result of Eq. (13)can be written as


Δ ≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2

2π2N

�σI2a

�2þ 32N

σ2P

s; ð14Þ

where σI and σP are the rms values of the intensitynoise and position noise, respectively, λ is the wave-length of the light source, a is the amplitude of theinterference fringe, N ¼ λ=2d is the number ofsampled intensities in an interference fringe, and2a=σI is the SNR of the interferometric data.

Equation (14) gives the standard deviation of thesurface height measurements when both intensitynoise and position noise are present. The first termunder the square root operator is caused by intensitynoise, and the second term is caused by positionnoise.

When only intensity noise is present, the standarddeviation of the height measurements, ΔI, can be gi-ven by letting the second term under the square rootoperator be zero:

ΔI ≈

�λπ

��σI2a

� ffiffiffiffiffiffiffi12N

r: ð15Þ

From Eq. (15), when only intensity noise is pre-sent, the standard deviation of the height measure-ments is directly proportional to the wavelength,inversely proportional to the square root of the num-ber of sampled intensities used in the measurement,and inversely proportional to the SNR of the inter-ferometric data 2a=σI.

In contrast, when only position noise is present,the standard deviation of the height measurement,ΔP, can be given by letting the first term be zero:

ΔP ≈ σPffiffiffiffiffiffiffi32N

r: ð16Þ

From Eq. (16), when only position noise is present,the standard deviation of the height measurementsis proportional to the rms value of position noise andinversely proportional to the square root of the num-ber of sampled intensities. The equation has alsoshown that the random error caused by positionnoise is independent of the wavelength of the lightsource and the amplitude of the interference fringe.

5. Algorithm Used in Simulation Measurements

To verify the theoretical results given by Eqs. (15)and (16), the least-squares method for surface heightmeasurements presented by Morgan in Ref. [6] isadopted here for simulated measurements. Accord-ing to Ref. [6], the surface height can be calculated by

x0 ¼ λ4π tan

−1

264P

Nj¼1 IðjdÞ sin

�j 2πN

P

Nj¼1 IðjdÞ cos

�j 2πN

375: ð17Þ

In Eq. (17), the N noisy sampled intensities, IðjdÞ,were given by Eq. (3) while the reference mirror wasshifting over a half-wavelength.

To determine the standard deviation of the heightmeasurements, 500 measurements of a certainheight were made in the presence of random noise,and then the standard deviation of 500 measure-ments was evaluated and compared with the theore-tical results.

6. Effect of Intensity Noise on Random Error

In this section, we examine the random error of sur-face height measurements by simulation when onlyintensity noise is present in the interferometric data.In this situation, the theoretical standard deviationof the height measurements is given by Eq. (15).Therefore, the standard deviations of simulatedheight measurements in this section will be com-pared with Eq. (15).

The surface height of noisy interferometric data gi-ven by Eq. (3) will be measured by Eq. (17). Then thestandard deviation of 500 measurements will beevaluated and compared with that given by Eq. (15).

Equation (15) has predicted that the standarddeviation of the surface height measurements is pro-portional to the wavelength of the light source. A si-mulation measurement experiment was performedto verify whether the theoretically predicted resultis true or not. Both the standard deviations of the si-mulation measurements and that theoretically de-rived are shown in Fig. 1. The plus signs show thestandard deviation of the simulated height measure-ments, whereas the solid curve shows that given byEq. (15). Both the simulation result and the theore-tical result indicate the linear relationship betweenthe standard deviation and the wavelength of thelight source.

The theoretical result given by Eq. (15) has alsopredicted that the standard deviation is inverselyproportional to the square root of the number of in-tensities used in measurements. A simulation mea-surement experiment was also performed to verify

Fig. 1. Standard deviation of the height measurements varieswith the wavelength of the light source when only intensity noiseis present. The number of sampled intensities was nine, and theSNR of the interferometric data was 100.


this. The result of the simulation measurements sup-ports the theoretical result, which is shown in Fig. 2.The plus signs represent the result of the simulatedmeasurements, and the solid curve is the theoreticalresult given by Eq. (15).

Equation (15) has also predicted that the standarddeviation of the surface height measurements isinversely proportional to the SNR of the interfero-metric data. A simulation measurement experimentwas also performed to verify this. Figure 3 shows thestandard deviation of the simulation measurementsand that given by Eq. (15) when the SNR of the in-terferometric data varies. The plus signs show theresult from the simulation measurements and the so-lid curve is given by Eq. (15). From Fig. 3, the simu-lation result supports the theoretical result that the

random error is inversely proportional to the SNR ofthe interferometric data.

7. Effect of Position Noise on Random Error

In this section, we examine the random error of thesurface height measurements by simulation whenonly position noise is present in the interferometricdata. In this situation, the theoretically derived stan-dard deviation is given by Eq. (16). Therefore, thestandard deviation of simulation measurements per-formed in this section will be compared with Eq. (16).

The surface height of the noisy interferometricdata given by Eq. (3) will be measured by Eq. (17).Then the standard deviation of 500 measurementswill be evaluated and compared with that givenby Eq. (16).

Fig. 2. Standard deviation of the height measurements varieswith the number of the sampled intensities used in measurementswhen only intensity noise is present. The SNR of the interfero-metric data was 100, and the wavelength was 0:78 μm.

Fig. 3. Standard deviation of the height measurements varieswith the SNR of the interferometric data when only intensity noiseis present. The number of sampled intensities was nine, and thewavelength was 0:78 μm.

Fig. 4. Standard deviation of surface height measurements var-ies with the rms value of the position noise when only positionnoise is present. The linear relationship is apparent betweenthe standard deviation and the rms value of the position noise.The number of sampled intensities was nine.

Fig. 5. Standard deviation of surface height measurements var-ies with the number of the sampled intensities used in measure-ments when only position noise is present. The rms value of theposition noise was 0:01 μm.


Equation (16) has predicted that the standard de-viation of surface height measurements is propor-tional to the rms value of the position noise. Thisis confirmed by simulation measurements, and thestandard deviations of the simulation measurementsare shown in Fig. 4. The plus signs represent the re-sult of the simulated measurements, and the solidcurve is given by Eq. (16). Apparently, the simulationresult confirmed the linear relationship that was the-oretically predicted when the rms value of the posi-tion noise varied from 0.001 to 0:01 μm.

The theoretical result given by Eq. (16) has alsopredicted that the standard deviation is inverselyproportional to the square root of the number of in-tensities used in measurements. The result of a si-mulation measurement experiment supports the

theoretical result, which is shown in Fig. 5. The plussigns show the standard deviation of the simulatedmeasurements, and the solid curve shows the theo-retical standard deviation given by Eq. (16).

Equation (16) has also predicted that the randomerror is independent of the wavelength of the lightsource when only position noise is present. A simula-tion measurement experiment was also performed toverify this. Both the simulation result and the theo-retical result are shown in Fig. 6 when the wave-length varies from 0.50 to 1:50 μm. The plus signsrepresent the result of the simulated measurements,and the solid curve is given by Eq. (16). The simula-tion result shown in Fig. 6 confirmed that the ran-dom error is independent of the wavelength of thelight source.

Equation (16) has also predicted that the randomerror is independent of the amplitude of the interfer-ence fringe when only position noise is present. Thisis confirmed by a simulation measurement experi-ment, and the result is shown in Fig. 7. The plussigns represent the result of the simulated measure-ments, and the solid curve is given by Eq. (16).

8. Conclusions

Random errors have been analyzed for interfero-metric surface profilers using the least-squaresmethod. Two types of noise are considered: intensitynoise and position noise. Formulas have been derivedthat may be useful for designing a surface profilerwith low random error.

When only intensity noise was present in the inter-ferometric system, the result given theoretically andthose from simulation measurements indicated thatthe standard deviation of the height measure-ments was

• directly proportional to the wavelength of thelight source,

• inversely proportional to the signal to noiseratio of the interferometric data, and

• inversely proportional to the square root ofthe number of sampled intensities used in themeasurements.

When only position noise was present in the inter-ferometric system, the result given theoretically andthose from simulation measurements showed thatthe standard deviation of the surface height mea-surements was

• proportional to the rms value of the posi-tion noise,

• inversely proportional to the square root of thenumber of sampled intensities used in measure-ments, and

• independent of the wavelength of the lightsource and the amplitude of the interference fringe.

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A. D. White, and D. J. Brangaccio, “Digital wavefront

Fig. 6. Standard deviation of surface height measurements var-ies with the wavelength of the light source when only positionnoise is present. The number of sampled intensities was nine,and the rms value of the position noise was 0:01 μm.

Fig. 7. Standard deviation of surface height measurements var-ies with the amplitude of the interference fringe when only posi-tion noise is present. The number of sampled intensities was nine,and the rms value of the position noise was 0:01 μm.


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random errors in interferometry with the least-squares method

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