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This paper is a theoretical analysis of mirror tilt in a Michelson interferometer and its effect on the radiant flux over the active area of a rectangular photodetector or image sensor pixel. It is relevant to sensor applications using homodyne interferometry where these opto-electronic devices are employed for partial fringe counting. Formulas are derived for radiant flux across the detector for variable location within the fringe pattern and with varying wave front angle. The results indicate that the flux is a damped sine function of the wave front angle, with a decay constant of the ratio of wavelength to detector width. The modulation amplitude of the dynamic fringe pattern reduces to zero at wave front angles that are an integer multiple of this ratio and the results show that the polarity of the radiant flux changes exclusively at these multiples. Varying tilt angle causes radiant flux oscillations under an envelope curve, the frequency of which is dependent on the location of the detector with the fringe pattern. It is also shown that a fringe count of zero can be obtained for specific photodetector locations and wave front angles where the combined effect of fringe contraction and fringe tilt can have equal and opposite effects. Fringe tilt as a result of a wave front angle of 0.05° can introduce a phase measurement difference of 16° between a photodetector/pixel located 20 mm and one located 100 mm from the optical origin.

The Michelson interferometer [

Discrete photodetectors and image sensors are commonly used to detect the sinusoidal pulsing fringe pattern. The radiant flux of the fringe pattern incident on the device active area is mathematically derived by integrating the irradiance over the circular aperture of the light source [

When using a well collimated beam and plane flat mirrors that are not perfectly aligned,

As wave front angle increases, the modulation amplitude of the dynamic fringing reduces and at a specific tilt angle the modulation amplitude of the radiant flux becomes zero [

To overcome or minimise mirror tilt prevalent with flat plane mirrors, corner cube retro-reflectors [

The behaviour of the radiant flux with varying wave front angle is also affected with varying photodetector distance from the central axis of the interferometer and varying its distance from the optical model origin. Analysis of the radiant flux with varying wave front in conjunction with photodetector area, distance from beam centre, distance from origin and wavelength appears not to be covered in the literature although [

Despite modulation amplitude

This paper addresses these issues, specifically:

Behaviour of the radiant flux for variable wave front angle as a function of photodetector width and position within the fringe pattern;

Behaviour of the radiant flux on two identical photodetectors adjacent each other;

Magnitude of the radiant flux at wave front angle(s) of equal radiant flux;

Polarity reversal of the radiant flux beyond specific wave front angles;

Behaviour of the radiant flux for variable wave front angle with variable distance of the photodetector from the tilting mirror;

Speed of transition of the fringe lines across the photodetector for variable wave front angle;

Speed of fringe line tilt across the photodetector for variable wave front angle;

Damping function constant of the radiant flux for variable wave front angle.

The relevance of this theoretical analysis is to make evident how these other factors may have an undesirable effect on sensor applications using homodyne interferometry where photodetectors or image sensors are employed to sense small fractions of a fringe to achieve extremely high resolutions of measurement. It goes beyond the adverse effect of modulation amplitude reduction due to increasing wave front angle [

The analysis is carried out based on a conventional Michelson Interferometer that is configured as shown in

Light source is a collimated monochromatic beam;

Wave fronts over the area of the photodetector are approximated to be plane waves;

Flat plane mirrors are used to reflect the transmitted and reflected beams back to the beamsplitter;

Beamsplitter is lossless and is non-polarising and creates a transmitted and reflected beam of equal amplitude.

The two wave fronts are orientated as depicted in

Origin of the Cartesian coordinate system is the point at which the centre of the incident beam is reflected by mirror M2;

Mirror M2 tilts only about the

Mirror M2 translates only along the

Plane of the photodetector remains orthogonal to the

Shape of the active area of the photodetector is rectangular with variable side length

Fringe pattern irradiates the entire active area of the photodetector;

Output of the photodetector is assumed to be a 1:1 linear function of the incident radiant flux;

Distance to the photodetector from mirror M2 is variable.

The mathematical analysis is divided into the following subsections, the outcome of which is studied further in the Results section:

Derivation of the equation for radiant flux from irradiance of the fringe pattern;

Identification of specific wave front angles _{n}

Determination of the magnitude of the radiant flux at specific wave front angles _{n}

Determination of the linear equation defining the profile of the fringe pattern in the

Determination of the speed of the fringe lines with variable wave front angle

Determination of the damping function of the radiant flux with variable wave front angle

Note: The angle

The electric field of a plane wave is given by _{0}

_{1}_{2}

Also depicted in

The sum of the electric fields of wave fronts 1 and 2 is therefore:

The irradiance ^{−2}, _{RI}_{0} is the vacuum permittivity, and
_{sum}

If _{1} and _{2} and then multiplied by side length _{2} − _{1} =

The radiant flux Φ_{e} given by _{e} = 0/0 which is indeterminate, therefore applying L'Hôpital's rule to the integral solution of

Therefore, as

The radiant flux in _{f}λ_{f}

To demonstrate the behaviour of the radiant flux over differing integral boundaries, ^{−9} m therefore _{2} = 0.0005 m, _{1} = −0.0005 m, blue curve integral boundaries _{2} = 0.001 m, _{1} = 0 m.

It can be seen from the

To analyse the effect of mirror tilt angle on two separate rectangular areas of equal size and determine the node points observed in _{1}, _{2} & _{3}, _{4} such that _{2} – _{1} = _{4} – _{3} =

To solve for

To obtain the node points that satisfy _{2} – _{1} = _{4} – _{3} = _{1}, _{2} & _{3}, _{4} be the two intervals depicted along the plane of the photodetector in _{1} = −_{2} = 0; _{3} = −_{4} =

Substituting these values in

The equality of

Solving:

is satisfied when (_{p}π_{p}

If small angles are considered, implying _{np}_{np}_{np}_{np}

is satisfied when

There are two solutions that satisfy

(−_{0} = 0 where _{0} = 0. As only small angles are of concern, _{0} = 0 is co-incident with _{p}

The cosines are identical for _{s}π_{s}

_{s}

_{s}

The occurrence of secondary nodes is unique and specific to the defined integral boundaries _{1}, _{2} & _{3}, _{4}, the values of _{ns}

To determine the magnitude of the radiant flux of the fringe pattern at the _{1} = −_{2} = 0; _{3} = −_{4} =

For small angles sin

At _{0},

Repeating the above for

_{1} = −_{2} = 0; _{3} = −_{4} =

Note

To work out the value of radiant flux for all other values of _{np}_{1,2,3,…}, substitute _{np}_{p}λ/s

Repeating the above for Φ_{e}_{3}, _{4}) again yields the same result, therefore the radiant flux at _{ns}_{1,2,3,..} is:

_{ns}_{1,2,3,..} is half maximum (_{ns}_{0} given in _{e}_{(}_{θ}_{1,2,3,..)} is independent of _{np}_{1,2,3,…}

When there is an exact multiple of fringe lines within the active area [

To determine the effect of distance

Where _{f}^{th}

_{f}λ

By solving _{f}

The _{f}

_{f}_{f}

As discussed with

Where _{f}

If Δ

Taking the derivative of the

The speed of sideways deflection of the fringe lines at point (

The overall fringe movement speed is therefore:

From _{rev}

That is, when the perpendicular of the wave front from the origin points towards +_{rev}_{rev}_{rev}

From

Which has solution:

This occurs at the angle _{n}_{= 0}:

If _{n}_{= 0} = _{n}_{= 0} > _{n}_{= 0} <

The relationship between _{n}_{= 0} and _{rev}_{n}_{= 0} ≡ 2_{rev}

From

Resulting in:
_{n}_{= 0} ≡ 2_{rev}

From

For a centred photodetector of width _{2} = +_{1} = −_{e}_{(}_{x}_{1},_{x}_{2)} in

In

Normalising the radiant flux to 2

As the normalised radiant flux ranges from 0 to +1, it is converted to a range from −1 to +1 in order to compare it to a standard sine function:
_{D}

_{D}_{D}

The radiant flux and the fringe count are influenced by a range of parameters, namely the wave front angle

The effect of

_{np}_{np}_{np}

According to

At this point it has to be mentioned that the same radiant flux curves are obtained if _{f}

Photodetector positions

As the angle

When introducing the distance

Increasing

The dotted lines in

The number of fringe lines _{f}

Assuming Δ_{f}

The

With Δ

The speed of the fringe line tilt (from the

_{rev}

If Δ

The effect of fringe tilt and _{0}_{0} = 1 is the maximum flux amplitude, _{f}_{f}_{f}

The focus of this paper has been to establish the behaviour of the radiant flux of the interferogram over a photodetector of rectangular aperture that is variable; in size; in displacement across the interferogram; and, in axial distance from the source of interference, for variable angle between the two wave fronts and variable wavelength.

The most apparent observation from the mathematical analysis in this study is that the radiant flux decays rapidly with increasing wave front angle with the recurrence of

What is so far not apparent from the literature, nor is it evident in the figures, is that between each

In the literature [

A further finding from this study, which to the best of our knowledge, is not mentioned in the literature is how the radiant flux is affected when the fringe lines tilt and contract/expand with varying wave front angle. Having included as an integral parameter the axial distance of the photodetector from the interferometer, it is found for a centred photodetector that fringe tilt initially lags fringe contraction, but then fringe tilt becomes increasingly dominant on the radiant flux with increasing distance of the photodetector from the beamsplitter.

Interferometry applications that use a plane flat mirror with translation stage, and discrete photodetectors [

If the wave front angle varies beyond

Alignment of the photodetector with the centre of the interferogram reduces the susceptibility of the fringe lines crossing over the photodetector as they contract as a linear function of distance from centre with increasing wave front angle.

Finally, the tilt angle of the fringe lines increases with increasing wave front angle and they cross over the central axis of the interference beam with increasing distance from the beamsplitter. Therefore to limit the error in measurement that this produces, the photodetector should be located as close as possible to the beamsplitter.

The radiant flux Φ across the active area of a photodetector is a damped sine function (

If the radiant flux magnitude at wavefront angle

The larger the distance

The movement of fringe lines with increasing

Consequently, significant fringe count errors occur if the photodetector is operated near or beyond

The authors thank Christophe Fumeaux for invaluable comments on the manuscript.

This appendix provides a detailed derivation of the equations in the paper “Theoretical Analysis of Interferometer Wave Front Tilt and Fringe Radiant Flux on a Rectangular Photodetector” by authors R.M. Smith and F.K. Fuss.

The equation numbers in this appendix follows the equation numbering in the parent paper, however, where additional equations are included they are designated an alphanumeric reference.

The electric field of a plane wave is given by

where _{0}

_{1}_{2}

Also depicted in

The sum of the electric fields of wave fronts 1 and 2 is therefore

The irradiance ^{−2}.

Where _{RI}_{0} is the vacuum permittivity, and
_{sum}

If _{1} and _{2} and then multiplied by side length _{2} –_{1} =

The radiant flux Φ_{e} given by

Therefore, as

It is worth noting that at

The irradiance and radiant flux in _{f}λ_{f}

To demonstrate the behaviour of the radiant flux over differing integral boundaries, ^{−9} m therefore _{2} = 0.0005 m, _{1} = −0.0005 m, blue curve integral boundaries _{2} = 0.001 m, _{1} = 0 m.

It can be seen from the

To analyse the effect of mirror tilt angle on two separate rectangular areas of equal size and determine the node points observed in Figure A1, consider only the integral solution of _{1}, _{2} & _{3},_{4} such that _{2} – _{1} = _{4} – _{3} =

To solve for

Applying the identity below and simplifying

To obtain the node points that satisfy _{2} – _{1} = _{4} – _{3} = _{1}, _{2} & _{3}, _{4} be the two intervals depicted along the plane of the photodetector in Figure A1 with values defined as _{1} = − _{2} = 0; _{3} = − _{4}

Substituting these values in

The equality of

Solving:

is satisfied when
_{p}_{np}_{np}_{np}_{np}

is satisfied when

There are two solutions that satisfy

(− _{0} = 0 where _{0} = 0. As only small angles are of concern, _{0} = 0 is co-incident with _{p}

_{s}

_{s}_{s}

From

As the left- and right-hand terms refer to the same angle

Therefore:

The 2_{s}

Therefore:

This eliminates the unknown term Δ and provides the function for

Rearranging and grouping terms yields:

Let

By way of demonstrating the presence of _{1} = − _{2} = 0; _{3} = − _{4} =

Then, the variables in

Finally, _{s}_{ns}

_{s}_{ns1} ≥ 0 and when 0 ≤ _{s}_{ns}_{1} < 0.

Also
_{s}_{ns}_{2} < 0.

Consider the solution defined by _{s} = _{−}_{1} = 0.07409° and confirmed by substituting the above conditions into the integral part of

Mirror M2 in Figure A1 can tilt left or right of the _{ns}_{ns}

From the above derivation of the _{1} = _{2} = 0; _{3} = _{4} = _{1}, _{2} & _{3}, _{4}, the values of _{ns}

To determine the magnitude of the radiant flux of the fringe pattern at the _{1} = _{2} = 0; _{3} = _{4} =

For small angles

At _{0},

Repeating the above for

For small angles

At _{0},

_{1} = _{2} = 0; _{3} = _{4} =

Note

To work out the value of radiant flux for all other values of _{np}_{1, 2, 3,..}, substitute inline from

Repeating the above for

_{np}_{1,2,3,..} and therefore the radiant flux at these wave front angles is given by

_{np}_{1,2,3,..} is half maximum (cf. _{np}_{0} given in Equation _{e}_{1,2,3,..}) is independent of _{np}_{1,2,3,…}

When there is an exact multiple of fringe lines within the active area, the radiant flux across the active area is the mean of the maximum constructive and destructive interferences,

To determine the effect of distance

_{f}^{th}

By solving _{f}

The

For small angles

The polarity of _{f}

When _{f}_{f}

As discussed with _{f}

If

Taking the derivative of the

The speed of sideways deflection of the fringe lines at point (

The overall fringe movement speed is therefore

From

And therefore the angle of movement reversal _{rev}

That is, when the perpendicular of the wave front from the origin points towards +_{rev}_{rev}_{rev}

From

This occurs at the angle _{n}_{= 0}

If

The relationship between _{n}_{= 0} and _{rev}_{n}_{= 0} ≡ 2_{rev}

From

Resulting in

And proving the identity of _{n}_{= 0} ≡ 2_{rev}

From

For a centred photodetector of width _{e}_{(}_{x}_{1},_{x}_{2)} in

In

Normalising the flux to 2_{n}

As the normalised flux ranges from 0 to +1, it is converted to a range from −1 to +1 in order to compare it to a standard sine function

For small

This equation defines a damped sine function. The equivalent undamped sine function has the form of

Where

The decay function _{D}

After simplifying and considering that

Considering that

_{D}_{D}

The authors declare no conflict of interest.

Michelson interferometer.

Wave fronts 1 and 2 with Mirror M2 tilted at angle

Normalised radiant flux _{2} = 0.5 mm & _{1} = −0.5 mm and _{2} = 1 mm & _{1} = 0 mm.

Normalised radiant flux against wave front angle; (a) node points; (b) wave front angle at 99% radiant flux;

Contour plot of equal wave front angles of 99% radiant flux as a function of the side length

Normalised radiant flux against wave front angle at

Normalised radiant flux across an

Normalised radiant flux against wave front angle at two different

Normalised radiant flux against wave front angle (a) and

Normalised radiant flux against wave front angle

The effect of variable

Fringe counting as a function of