Abstract

Interferometric instruments with dispersion introduced in the reference arm have previously been created, as the controlled dispersion can be used to generate a signal that contains a clearly identifiable point, the location of which relates to the position of the scattering surface in the measurement arm. In the following, we illustrate that the linear approximations that have been used previously can lead to significant errors, and that second-order terms need to be included in order to correct this. These corrections are vital if these instruments are to be used for metrological applications.

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References

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2016 (1)

2014 (1)

2013 (1)

H. Martin and X. Jiang, “Dispersed reference interferometry,” CIRP Ann. 62, 551–554 (2013).
[Crossref]

2009 (1)

2005 (1)

2004 (1)

1997 (1)

1996 (1)

Dändliker, R.

De Groot, P. J.

P. J. De Groot, “Interferometry employing refractive index dispersion broadening of interference signals,” U.S. patent9,377,292 (28 June 2016).

Deng, Y.

Galle, M. A.

Ghosh, G.

Gray, S.

Häusler, G.

Jiang, X.

Kong, W.

Martin, H.

Mohammed, W. S.

Pavlicek, P.

Pavlícek, P.

Qian, L.

Saini, S. S.

Schnell, U.

Soubusta, J.

Tao, J.

Wang, X.

Williamson, J.

Yang, W.

Zhang, Z.

Zhou, C.

Zhu, E. Y.

Appl. Opt. (3)

CIRP Ann. (1)

H. Martin and X. Jiang, “Dispersed reference interferometry,” CIRP Ann. 62, 551–554 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Other (1)

P. J. De Groot, “Interferometry employing refractive index dispersion broadening of interference signals,” U.S. patent9,377,292 (28 June 2016).

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the instrument with dispersion being introduced via a plate of glass of thickness t . The difference in the length of the two arms is z . M is a reference mirror, and O is the object being measured.
Fig. 2.
Fig. 2. In part (a) the solid line shows the value of z needed for the minima in ϕ to occur at each value of k according to the exact equations, while the dotted dashed line shows the value of z given when the linear approximation is used. k 0 = 7.9784 × 10 6    m 1 for the linear approximation. Part (b) shows the difference between the lines at each value of k .
Fig. 3.
Fig. 3. Diagram of the instrument. G 1 / G 2 are gratings, and M 1 / M 2 are mirrors. The inset shows the form of an ideal signal on the spectrometer.
Fig. 4.
Fig. 4. Solid line shows the value of d needed for the minima in ϕ to occur at the corresponding value of k , while the dashed line shows the value given when r { k } is replaced by its linear approximation.
Fig. 5.
Fig. 5. Example of ϕ calculated using the exact value of r , shown in part (a) along with the signal this would lead to, shown in part (b). Part (d) shows the value of ϕ obtained when r is replaced by its linear approximation, with the signal that would be expected if ϕ took the form shown in part (c).
Fig. 6.
Fig. 6. Plot showing how the true value of d differs from a linear change. The linear change is given by taking a tangent to the d curve at the point k = 7.5819 × 10 6    m 1 .
Fig. 7.
Fig. 7. (a) Typical signal generated by the instrument after being rescaled to remove the envelope on the signal; (b) shows the autoconvolution of the signal in part (a) minus its mean, which is used to determine the minima in ϕ .
Fig. 8.
Fig. 8. Difference between the measured locations of k at which the ϕ minima occurs and a best-fit straight line for the experimental results (solid line) and a solution to the exact equations (dashed line).

Equations (18)

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α = d n d k | k 0 .
I d { k } = I s { k } 2 [ 1 + cos ( ϕ { k } ) ] ,
ϕ { k } = 2 k [ z t ( n { k } 1 ) ] ,
ϕ { k } = 2 [ z t ( n | k 0 α k 0 1 ) ] k 2 α t k 2 ;
z approx = t ( 1 + α k 0 n | k 0 2 k α ) ,
z approx = 2 α t ,
z = t ( 1 n { k } k n { k } ) ,
z = 2 t n { k } + k t n { k } .
n 2 ( λ ) = A + B ( 1 C / λ 2 ) + D ( 1 E / λ 2 ) .
r { k } = L 1 ( 2 π / k D ) 2 ;
ϕ = 2 k ( r L d ) ,
I { k } = I 0 { k } 2 ( 1 + V cos ( ϕ { k } ) ) ,
ϕ approx = 2 k ( r | k 0 α k 0 L d ) + 2 α k 2 = a 0 k + a 1 k 2 .
d approx = r | k 0 k 0 α L + 2 k α ,
d approx = 2 α ,
ϕ = 2 ( r + k r L d ) .
d = r + k r L .
d = 2 r + k r .

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