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Theoretical modeling of a detection system based on optical coherence

contrastRicardo C. Coutinho

a, David R. Selviah

b, Hugh D. Griffiths

b

a

Brazilian Navy Weapon Systems Directorate, Rua Primeiro de Maro, 118, 20

o

andar, Rio deJaneiro, Brazil, 2314@dsam.mar.mil.brbDepartment of Electronic and Electrical Engineering, University College London, London WC1E

7JE, United Kingdom, d.selviah@ee.ucl.ac.uk

ABSTRACT

Optical detection systems usually rely on the intensity contrast (visible) or temperature difference (infrared) betweentarget and background. Adding new dimensionality to the detection process is essential to enhance the sensitivity. Thispaper presents a novel theory for modeling the performance of an optical detection technique called Interferogram PhaseStep Shift (IPSS), which relies on the coherence contrast between target and background to perform discrimination. Thetechnique uses an interferometer to measure the self-coherence function of the input radiation, forming an interferogram,

and an interference filter to produce an event marker (phase step) in it. The model predicts the displacement of the phasestep in the interferogram, when a coherent target enters the system field of view, which is the kernel of the IPSStechnique. The paper assesses the effects of the target to optical filter bandwidth ratio in the system responsivity, foroptimization purposes, and models the experiments presented in a previous publication, predicting the experimentalresults theoretically to perform a comparison. It also includes the analytical derivation of the self-coherence functions oftarget and background as measured by the systems interferometer, and the computer modeling of the same self-coherence functions for an interference filter, with any arbitrary spectral response, considering the effects of the

polarization of the light sources and optical components in the experiments. Finally, the theoretical curves fordisplacement vs. target-to-background power ratio, among others, are compared with the experimental results. Goodagreement is demonstrated, and the causes of differences are discussed.

Keywords: Coherence, target detection, target discrimination, clutter reduction, signal enhancement

1. INTRODUCTIONOne of the most powerful resources to mitigate the effects of clutter in optical detection systems is the use of otherdimensions apart from intensity in the detection process

1. The phase, polarization and coherence

2properties of light have

been increasingly used in order to improve target discrimination in environments exposed to high clutter or

countermeasures. In previous publications,2,3,4,5

we have presented a technique named Interferogram Phase Step Shift(IPSS), which exploits phase changes in the coherence interferogram of a scene to detect coherent or partially coherenttargets in incoherent backgrounds with very high sensitivities. Coherence-based approaches are specially useful to detectnarrowband emission or absorption features, which occur in various wavebands due to several phenomena6. In this paper

we describe the theoretical and computational modeling of the experiments previously published, which were used tovalidate the models developed. It is expected that, given this validation, the design of a perfected detection system can go

on to allow its use in a wide range of applications, such as biomedical engineering, remote sensing and long rangeaircraft and missile detection.

2. COHERENCE-BASED DETECTION SYSTEM BRIEF DESCRIPTIONThe technique whose modeling is the object of this paper was described in our previous publications2,3,4,5. In this sectionit is briefly described to allow a reasonable understanding of the upcoming sections. The technique relies on three maincomponents: an interferometer which produces an interferogram from the radiation coming from the scene, whose width

is proportional to its degree of coherence; a narrowband interference filter to remove clutter outside the band of interestand to produce a marker in the interferogram; and a computer algorithm, to recover the position of the marker from themeasured interferogram. The marker just mentioned consists of an abrupt transition (phase step) caused in the phase ofthe coherence interferogram by the steep roll-off characteristic of the narrowband filters frequency response. Whenpointing the sensor to an incoherent scene, such as a portion of the sky, the marker remains at a fixed position, given by

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the optical bandwidth of the interference filter. As a coherent target enters the field of view of the sensor, the net degreeof coherence of the scene is increased, and the position of the marker in the path difference axis of the interferogramchanges accordingly. This displacement of the marker is then used by the computer algorithm as a signal to declare the

detection of a target. Referring to our previous publications, [2] describes experiments with the detection of a He-Nelaser, of light from a tungsten halogen lamp bandlimited by a second interference filter, and of light from an LED; [5]shows experiments assessing the optimization of the target-to-filter bandwidth ratio; and [7] is a doctoral thesis

comprising the whole theoretical and experimental study performed so far.

3. THEORETICAL MODELIn this section we derive an analytical expression for the self-coherence function of the combined target-background

radiation that reaches a detector placed after the interferometer mentioned in the previous section. We assume that thebackground spectrum is flat across the interference filter bandwidth, the spectral response of the interference filter is a

top hat function with unity amplitude, bandwidth and centered at 0, and the target is an emission with a Gaussian

spectral profile, also centered at 0 with a total powerPT and full width half maximum (FWHM), , much smaller than

. As the target and background powers add before being filtered by the detection system, we may write the net powerspectrum after narrowband filtering, shown in figure 1, as:

+=+=

20

2 )(

0 ..

/)/()()()(

e

PPrectTBS T

B (1),

where the first term in the brackets is the background power, with power spectral density of PB/k, so that thebackground total power through the filter (area under the curve), B(k), equals PB; the second term in the brackets is the

target Gaussian power spectrum, T(k), with a maximum amplitude ofPT. /k, so that the total target power equals PT,

and is a factor inversely proportional to . The rect function term indicates a rectangular function with unity

amplitude, width k, centered at k0, and=1/ is the wavenumber.

0

Wavenumber

TotalSpectral Power

Density

PB/

PB/+PT/

Figure 1 - Target and background power spectra

It is known8 that the Fourier transform (FT) of the self-coherence function () is the power spectrum. Hence, applyingan inverse FT to the net power spectrum in eq.1 yields:

)()()( TB += (2),where is the path difference andB() is the self-coherence function of the filtered background andT() of the target.

The calculation of the inverse FT follows. The relationship between and can be derived applying the definition of

FWHM to the Gaussian spectrum. It can be shown that 2ln4= . Due to the system linearity, which applies inthis case due to the low powers involved in the detection process, one may calculate the Fourier transforms of

background and target separately:

00

0

0

22.2.

2/

2/

1 ).(.)sin(

..)()}({j

B

jBjBB e.sincPe

Pde

PB =

=

== +

,

a carrier modulated by a sinc function envelope. For the target:

dee

PT

jT

T...

.)()}({

2/

2/

2.)(10

0

20

2

+

==

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A change of variables is required. Let 1=-0. Then:

+

+

+ == 22

1

)2(22

21

)(2.....

.)( 1

21

2001

21

2

dee

Pdee

P jjTjTT

To evaluate the integrand it is necessary to manipulate further, adapting a derivation from Kreyszig9. For the sake of

simplicity, the wavenumber is termed rather than 1:

Let

jv = and consider

2

222222 2.)

.(

== jjv . Then defining

+==

d

jdjA )).

.)

.((exp()).2.(exp(

2

22222

, where for simplification the

integral is represented as indefinite and the notation exp is used for exponential.Taking out the exponential that does not

depend on one obtains:

= dkj

A ).)(exp()exp( 22

22

. But if

jv = , then

dvd = , thus the integral

= dkj

I ).)(exp(2

becomes:

JdeI

j

j

.11 2

2

2

==

, where the integration limits were also expressed in terms of v. This is an area under a

Gaussian curve. Recalling the definition of the error function =u

u dueuerf0

.2

)(2

, the integralJcan be evaluated

as:

)]2

()2

([2

)(.2

2

2

jerf

jerfuerfJ

j

j

+=

=

+

(3)

In the expression above, an approximation can be made by neglecting the imaginary part of the error function argument.

This can be done if2

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Analyzing the effect of the error function in eq. 4, for

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As the purpose of this analysis is to assess the effects of the target to background bandwidth ratio on the phase step shift,as the power ratio is increased, three cases of bandwidth ratio are studied, corresponding to three regions of the

intersection shown in fig. 2 in the target Gaussian curve: the regions close to zero and to infinity, where the slope of theGaussian is very low, which were analyzed through the behavior of the error function, and the region where the Gaussian

slope is considerable. In this latter case, the target and filter bandwidths are comparable, that is, , and the

intersection between the target and background curves occurs in the region where the target curve has the maximumslope. By setting the second derivative of the Gaussian spectrum to zero, to find its maximum slope, one obtains

)/(2ln2max = , the subscript meaning maximum slope. The maximum slope of the target curve increases thespeed of the movement of the intersection (or the phase step shift), thus increasing the responsivity. There will be an

optimum responsivity point at max= refor

2ln21 = , giving a target to filter optimum bandwidth ratio of:

375.02ln2

==

(8)

Returning to the narrow target case, and referring to fig. 2, an observability cut-off occurs when, with the increase in

power ratio, the two curves no longer have an intersection. The feature is located in the path difference axis at the first

negative minimum of the backgroundsinc function (in fig. 2) at =1.43/ (found by setting the derivative of the sincfunction to zero), from where no intersection between the target and background curves occurs. Thus, the phase step

position varies between =1/ and=1.43/. To analyze the dynamic range one has also to consider the width of thetarget self-coherence Gaussian. In the narrow target situation, the target curve would be nearly a horizontal line, and the

upper limit of the dynamic range would be simply the value of the sinc function at the first negative minimum

(=1.43/), or 2/(3)=0.212 (see fig. 2), corresponding to a power ratio of 6.73 dB. The range of path differencechange equals 0.43/, thus inversely proportional to the filter bandwidth.

Still in the narrow target case, it is possible to find how the phase step shift varies with the power ratio PR. Analyzing indetail the region where the intersection of the target and filter self-coherence functions occurs, one notices that, for a

given phase step path difference s , and as the horizontal line from the target self-coherence function moves down with

the increase in PR, the infinitesimal variation ofs with power ratio, ds,is governed by the slope of the sinc function at

that point, as depicted in fig. 3. The power ratio can then be expressed as a function of

s using the shape of the sincfunction. In the region of the sinc curve in fig. 2 between =1/ and=1.43/, one has )..(sin = scPR .

Unfortunately, the sinc function is not reversible, and as our interest is how s varies as a function of PR, the sinc

function in the interval PR=[0, 2/3] ands =[1/, 1.43/] is plotted in fig. 4 with the axis reversed. The responsivity

PRs / , defined here as the amount of phase step shift in m, obtainable from an increase in power ratio in dB, isthus given by an inverse sinc function, with no analytic expression, plotted in figure 4. In plot a), it can be seen from

the line drawn on the plot that the change in s is linear for low power ratios, which is useful in an application such as gasdetection, but this does not apply for values higher than approximately 0.1 (-10dB), where a correction based on a look-

up table would be required.

dPR

dss

Sinc slope at =s

Fig. 3 - Graphical interpretation of rate of change of the phase step shift

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0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14

16

Power Ratio

PhaseStepShift(micr

ons)

40 35 30 25 20 15 10 50

2

4

6

8

10

12

14

16

Power Ratio (dB)

PhaseStepShift(micr

ons)

Fig. 4 - Variation of the phase step shift with PR of a narrow target, for a linear power ratio in a) and a

logarithmic ratio in b). The wavelength is 633 nm and the filter bandwidth is 11 nm.

a) b)

It is important to assess the question of which type of targets can be considered narrow targets. By taking the

derivative ofTN()= )2ln(..22

2

2ln4

erfePR (the target normalized self-coherence function, part of eq. 6), one

obtains:

222

2ln4

22

..2ln2

).2ln(.)(

=

eerfPR

d

dTN

In order to have a narrow target condition, the region of the target Gaussian that intersects the filtersinc function must benearly horizontal, or have a very small slope or derivative. If one calculates the maximum value of this derivative with

respect to the product ., it can be said that this condition is true whenever the derivative for every path difference is

much smaller than this maximum value. By setting in the product above to be the maximum value of the interval ofinterest, or 1.43/ (worst case), one obtains the condition for a narrow target:

016.0

(9)

When considering targets with widths comparable to that of the filter, one has also to include the influence of the targetself-coherence Gaussian slope in the region where it intersects the filtersinc function. As shown earlier in this section,for bandwidth ratios higher than 0.416 the loss of target power in the filter starts to happen, hence one may investigate

the region in between 0.016 and 0.416 looking for a change in the responsivity PRs / . As previously calculated,

the range of possible path differences (s=[1/, 1.43/]) is solely given by the filter bandwidth; however, the range

of possible power ratios (PR) does vary with the target to filter bandwidth ratio, as different regions of the Gaussian

will intersect the filtersinc function. When this intersection occurs at a point where the Gaussian has a non-horizontalslope, the power ratio is not limited to 2/(3), the value of the sinc function at its first negative minimum. This isillustrated in fig. 2, where the power ratio (labeled in the figure) is given by the value of the Gaussian for zero path

difference. The responsivity was defined as PRs / , and if we approximate it as s/PR, we have that the

responsivity is inversely proportional to the range of power ratios PR. Hence, for a given filter with bandwidth , the

best possible target is the one that fulfils the narrow target condition, to keep the range of power ratios limited to 2/(3),otherwise this range would be longer, and the responsivity lower.

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We may find an approximate shape of the variation of responsivity with bandwidth ratio, by noticing that the maximum

achievable power ratio PR is the amplitude of the Gaussian for zero path difference, when this Gaussian is touching thefiltersinc function at one single point. This is because, in this case, the value of the sinc function at its first negative

minimum (2/(3) is added to a part of the magnitude of the target Gaussian.

By using the equation of the Gaussian from eq. 5 and imposing the existence of the point at which the path difference is

1.43/ and the self-coherence equals 2/(3), one has:

3

2.

222

)/.()43.1.(2ln4 =

ePR . Solving forPR and substituting below gives:

222

222

)()43.1.(2ln4

).()43.1.(2ln4

..645.0

.3

2

43.0

==

e

ePRPR

ss(10)

We can now apply an optimization technique to eq. 10, which shows the responsivity (from now on termedR) as a

function of two variables and. The extremes inR=f(,) can be found setting the total derivative of the function tozero11:

0)()( =

+

= d

fd

fdR

As and are independent, this is the same as setting each partial derivative to zero:

0)(28.72..02.2 2)(28.7

=

=

xxef

, a condition which only happens if

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the derivation of eq. 8, if we have used

2ln243.1 = instead of 2ln21 = , we would obtain in

eq. 8 the same result given by eq. 11.

For design purposes, the optimum ratio to be chosen depends on the application: if one wants to increase the responsivityfor low signals or weak targets, then use the value in eq. 8; if, otherwise, the aim is to achieve optimum average

responsivity throughout the whole range, the value in eq. 11 is to be used.Fig. 7 shows the three-dimensional surface formed by the responsivity R = f(,). The plots in figs. 5 and 6 are theintersection of this surface with the planes with a filter bandwidth of 11 nm (fig. 5) and target bandwidth of 7 nm (fig. 6).

The pattern with a point of maximum, obtained with a constant target bandwidth, can also be seen in a lateral view in fig.7. The amplitude of the maximum decreases with the increase in the targets bandwidth.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

Target to Filter Bandwidth Ratio

Responsivity(microns)

Figure 6 - Responsivity variation with bandwidth ratio for a target with FWHM=7 nm.

1 2 3 4 5 6 7 8 9 10

0

50

100

0

20

40

60

80

100

120

Filter Bandwidth (nm)Target Bandwidth (nm)

Responsivity(microns)

Figure 7 - 3D plot of responsivity varying with target bandwidth ranging from 1 to 10nm, and with filter

bandwidth, ranging from 1 to 100nm. The wavelength is 633nm.

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4. COMPUTER MODELING

The simplified model presented in the previous section allowed the analytical derivation of the shift vs. PR curve, as wellas of the effects of the bandwidth ratio on the responsivity, offering a good insight into the Interferogram Phase Step

Shift (IPSS) technique. However, the model embeds a set of assumptions, which are only valid under specific conditions,constraining the application of the theory to this limited set of conditions, and preventing a broader agreement between

theory and experiment. As an example, the approximation given in eq. 10 is very good in the case of narrow targets, butdegrades with the increase in the target to background bandwidth ratio. A more complete model was required, which didnot present the limitations of the simplified model. These limitations were concerned with the spectral width and shapeof the target, the filter spectral shape, the spectral non-flatness of the white light source used in the experiments 2,5, an

eventual target-to-filter central wavelength offset, and the effects of polarization. As examples, the effect of a target-to-filter spectral off-set was modeled as a beating which produced an additional modulation in the carrier shown in eq. 4,

and the effect of polarization was to affect the fringe amplitudes coming from target and background differently,according to their degrees of polarization and to the angle between their orientations and that of the interferometer.

A computer program was written in Matlab language to evaluate the phase step shift as a function of power ratio,

considering the effects just described. The program performed tasks like: manual input of target data; input ofinterference filter response; generation of the blackbody curve of the tungsten halogen bulb used in the experiments;

calculation of the filtered background spectrum; calculation of two target - filtered background combined spectra, one for

a Gaussian target, using eq. 1, and another for a Lorentzian target; calculation of two self coherence functions (Gaussianor Lorentzian target) by inverse complex Fourier transformation of the calculated spectra; calculation of theinstantaneous frequencies of the two calculated self coherence functions; location of the path difference where themaximum instantaneous frequency occurs, one for each target and each power ratio; and construction of two phase stepshift vs. power ratio curves, one for each type of target.

The program developed also allowed the simulation of the observability cut-off observed in the experiments andpredicted by the simplified theory. Fig. 8 presents, in the upper plot, the amplitude (upper curve) and instantaneousfrequency (lower curve) of the target-background self-coherence function, for a power ratio of 10 dB. The frequencymaximum (the derivative of the phase step) is very weak, but still above zero. Although not visible in the plot, thenegative peak occurs at the second minimum of the amplitude function. The lower plot presents the same curves, now for

a power ratio of 9 dB. The frequency maximum is below zero, hence no longer detectable. The cut-off then occurs at apower ratio between 9 and 10 dB, a value smaller than the one predicted in the simplified theory (6.73 dB), due to the

finite roll-off characteristic of the filter, which makes the frequency spike weaker than in the top hat case. Nevertheless,the experimental value of 8.5 dB7 was in good agreement with the figures shown here.

0 10 20 30 40 50 60 70

-0.1

0

0.1

0.2PR=-10 dB

0 10 20 30 40 50 60 70

-0.1

0

0.1

0.2PR=-9 dB

path difference (microns)

Fig. 8 - Simulation of the observability cut-off. Upper plot: Self-coherence function amplitude (upper curve) and

instantaneous frequency (lower curve) for a power ratio of 10 dB. Lower plot: the same functions for PR=-9 dB.

5. COMPARISON BETWEEN SIMULATED AND EXPERIMENTAL RESULTS AND DISCUSSIONThe simulation model presented in the previous section was used to generate phase step shift vs. power ratio (or PR)

curves for comparison with the experimental results presented in [2], [5] and [7]. Fig. 9 depicts the curves obtained for

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the He-Ne laser. The experimental curve, shown with error bars, was taken from [2]. Two simulations were performed:one for a Gaussian, and another for a Lorentzian target. As predicted by the simplified theory, in the narrow target casethe two curves are coincident. The single curve is shown without error bars. A very good agreement between this curve

and the experiment exists up to a PR of 13 dB, with the theoretical curve passing within the error bars of theexperimental curve. From this PR upwards, the curves have the same derivative, but there is a horizontal displacement ofcirca 1 dB. No curve fitting was employed.

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5-5

0

5

10

15

20

signal to clutter ratio (dB)

phasestepshift(microns)

Figure 9 - Comparison between simulation model (without error bars) and experiment (with error bars) ofdetection of a He-Ne laser using the IPSS technique and algorithm.

The power measurements taken during the experiments7 to calculate the PR had an uncertainty affected by differences in

collimation between target and background, the resolution of the optical power meter employed to measure power , andthe power fluctuation of the light sources during the measurements. The calculated overall PR uncertainty in the He-Ne

laser case was 1.47 dB, not varying with the PR. By considering this uncertainty, not shown in fig. 9 for better clarityof the plot, the experimental behavior is fully predicted by the theory to within experimental accuracy. Finally, the

simulated curve bends slightly to the right at 8 dB, an effect due to the proximity to the observability cut-off explainedpreviously.

The response of the detection system to several partially coherent targets were studied, an example being a resonant

cavity light emitting diode (RCLED), whose comparison of results is shown in fig. 10. The simulated curves wereobtained using a target FWHM of 6.5 nm. This value was obtained from spectral measurements done by Oulton and

others12

. The device investigated in this paper was manufactured by the authors of that paper, thus having similarcharacteristics with the devices studied there. Two theoretical curves are presented, the upper considering a Gaussiantarget spectrum, and the lower considering a Lorentzian spectrum, the experimental curve lying between the two. Thespectral behavior of the device is presented in [12].

-15 -14 -13 -12 -11 -10 -9 -8 -7 -60

2

4

6

8

10

12

signal to clutter ratio (dB)

phasestepshift(microns)

Fig. 10 - Comparison of simulation model and experiment for an RCLED using IPSS. Upper curve simulation

with Gaussian target. Lower curve simulation with Lorentzian target. Curve with error bars experiment.

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An experiment to investigate the optimum target-to-background bandwidth ratio was performed in [5]. Its results,obtained using a translation stage as a path difference driving scheme, were copied to fig. 11, which also presents a pointand error bar obtained with a fast piezoelectric transducer used instead of the translation stage, at a bandwidth ratio of

0.3, and a theoretical curve obtained using the simplified model described in section 3. The responsivity in m/dB, foreach bandwidth ratio, was obtained by dividing the maximum phase step shift obtained with the model by the difference

in dB between the minimum and maximum PRs, giving the average slope of the shift vs. PR curve. This method of

calculation explains the difference between the curves, once that the minimum theoretical PR does not account for noise,being smaller than the minimum experimental PR, yielding a larger denominator and a smallerm/dB ratio. In spite ofthis difference, there is agreement between the theoretical curve and the lower limit of the error bars, which veryapproximately corresponds to no experimental noise. More importantly, the point of maximum responsivity in theexperimental data is fairly close in bandwidth ratio to that of the theoretical curve, validating the finding that there is anoptimal target to background bandwidth ratio, at about 0.3, which yields maximum responsivity.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1

Target to Background Bandwidth Ratio

Responsivity(microns/dB

theory

piezoelectrictransducer

transl. stage

Figure 11 - Comparison between bandwidth optimization theory and experiments

Finally, in [5], the effects of an offset between target and filter central wavelengths were studied. Fig. 12 presents acomparison between the experimental data, copied from there, and theoretical results for a Gaussian target, converting

the maximum shift in m to a responsivity in m /dB, as done previously in this section. The experimental data, shown

as squares, was interpolated to produce the corresponding curve. The agreement between the experiment and the theory(without squares) is good, with a slight asymmetry of the experimental curve, possibly due to an asymmetry in the

response of the interference filter used to perform the measurements. It can be seen that, in order to retain the highresponsivity IPSS can offer, careful optical filter design has to be done, to minimize this offset. By limiting the offset to+/- 10% of the filter bandwidth, the responsivity is kept at approximately 80% of its maximum value.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

offset (fraction of filter bandwidth)

responsivity(microns/dB)

Figure 12 - Comparison between central wavelength offset theory and experiments (with squares)

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6. CONCLUSIONS

This paper presented the modeling and performance prediction of the Interferogram Phase Step Shift (IPSS) approach. Asimplified model was derived analytically and graphically, which was valid for spectrally narrow targets, and under

additional assumptions. This model provided a good insight into the mechanisms affecting the performance, and themain conclusions were: for a given filter, the responsivity increases as the target bandwidth decreases; given a target,

there is an optimum target to filter bandwidth ratio of approximately 0.3, which yields maximum responsivity. As ananalytical derivation including the effects on the performance of detailed characteristics of components was not possible,a simulation model was developed using the understanding of the behavior of the detection system, given by thesimplified model. This extended model was capable of predicting the behavior of the experiment with both coherent and

partially coherent light sources. The uncertainties in power measurements were the main causes of the differences foundbetween phase step shift theory and experiments. An important finding taken from the use of the simulation model was

the confirmation that, for narrow targets, the target spectral shape does not influence the results.On the other hand, whenconsidering spectrally larger targets, the ones modeled as a Gaussian spectrum produced higher responsivities than theLorentzian targets (as shown in fig. 10). The simulation model also accounted for features such as the polarization of theHe-Ne laser and the non-flatness of the blackbody curve of the tungsten halogen lamp, and allowed the confirmation of

effects such as the observability cut-off that occurs at high PRs. The simplified model was also used to validate theexperimental results from our previous publications, concerning the optimum target to filter bandwidth ratio, and the

effects of the target to filter central wavelength offset, both producing a very good agreement between experiment and

theory. Now, the models developed have been validated, and their use is recommended for future design of detectionsystems employing the IPSS technique and algorithm.

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