Spatial phase-shifting characteristic of double grating interferometer

Yang Song; YunYun Chen; Anzhi He; Zhimin Zhao

doi:10.1364/OE.17.020415

1. Introduction

Phase shifting interferometry has been considered as an effective tool to exact phase information from interferograms. It includes temporal phase shifting methods and spatial phase shifting methods. The temporal phase shifting records these phase-shifted interferograms sequentially [1,2]. However, the measured object should remain unchanged during the recording process, which can lead to disturbances by thermal and mechanical fluctuations. In addition, the temporal phase shifting is difficult to apply in measuring objects which changes rapidly.

These problems can be solved by application of the spatial phase shifting which can generate and record a series of phase-shifted interferograms simultaneously. There are various methods have been developed to generate spatial phase-shifted interferograms [3–7]. Among all these methods, the gratings played an important role [8–12]. Kwon [12] firstly presented the multichannel spatial phase shifted interferometer which consists of a point-diffraction interferometer fabricated on a transmission grating. The pinhole on the grating creates the reference beam and three diffraction orders generate by the grating generate the desire phase shift. In this optical configuration, the diffraction phenomena of gratings are used. Another way for spatial phase shifting with gratings is based on the beam splitting property of gratings. M. Kujawinska [11] used a diffraction grating to triple the probe light and polarization optics was used to implement the appropriate phase shifts. Therefore, three or four fringe patterns can be generated simultaneously [9–11].

The double grating configurations used as the lateral shearing interferometers have been thoroughly studied and widely used [13–15]. The phase shifting characteristic of the double grating interferometer is an important research field. I. Amidror [16] has studied the moiré phenomena with spectral approaches and found that the phase shift occurs by lateral moving one or more gratings. M. Kujawinska [17] made a detail consideration to double grating systems and showed that the phase shifting could be introduced by both lateral and longitudinal moving of gratings. In fact, the double grating systems have been widely used in temporal phase shifting as the double grating lateral shearing interferometer [18–20]. In these optical configurations, the second grating is moved laterally for changing phase shift, while the desired shear is achieved by adjusting the distance between the two gratings.

The purpose of this paper is to present the spatial phase shifting characteristic of double grating interferometer. No polarizer and wave plate are necessary, and it is no necessary to moving any gratings especially. It is known that there are phase shift between different diffraction orders of one grating. In a recent paper [21], the angular spectrums during field propagation through double gratings are carefully analyzed from the scalar diffraction theory. In this paper, intensity distributions of the plus-first, minus-first and zero order interferograms caused by double gratings are derived and phase shift is found between these interferograms. In section 2, intensity distributions of phase shift interferograms without phase objects are presented. In section 3, the distorted fringe patterns caused by phase objects are obtained, based on which the influence of phase objects to the wanted phase information and phase shift between interferograms are discussed in detail. In section 4, the triple grating interferometer is presented, while section 5 gives some concluding remarks.

2. Principle

2.1 Theory

The optical configuration shown in Fig. 1 is very simple. G1 and G2 are two identical gratings that are illuminated with collimated monochromatic coherent light. They are set a distance $Δ_{1}$ apart and oriented at angles $+ α / 2$ and $- α / 2$ , respectively, relative to the y axis. The distance between grating G2 and the observation Plane is $Δ_{2}$ . In order to avoid the overlapping of different diffraction orders, $Δ_{2}$ should satisfy

Δ_{2} \geq \frac{F d}{λ} .

where λ is the wavelength, d is the grating period, and F is the aperture of collimated beam. In practical, the distance

Δ_{2}

is usually set to

Δ_{2} = 1.1 F d / λ

. The mutual lateral shift perpendicular to grating lines is approximated as b and shown as Fig. 2 .

Fig. 1 Optical configuration of the double grating interferometer.

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Fig. 2 the shift b between Grating G1 and G2.

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The incident plane wave before G1 will be distorted when a refracting object O is placed before the grating G1. Because the phase distribution of the distorted wave front is actually the phase projection, the field before G1 is:

u_{1}^{-} (x, y) \propto \exp [i k φ (x, y)] .

where

k = 2 π / λ

and λ is the wavelength. The grating G1 forms an angle of

+ α / 2

with respect to the y axis and the transmittance can be described by its Fourier expansion as:

g_{1} (x, y) = \sum_{(m)} a_{m} \exp [i \frac{2 π m}{d} (x \cos \frac{α}{2} - y \sin \frac{α}{2})] .

The periodic grating is considered infinite in extent. If we define

U_{1}^{-} (u, v)

is the angular spectrum of

u_{1}^{-} (x, y)

and

(u, v)

which are the Fourier spatial frequency components, the field

u_{1}^{+} (x, y)

behind the grating G1 should be

\begin{array}{l} u_{1}^{+} (x, y) = u_{1}^{-} (x, y) \sum_{(m)} a_{m} \exp [i \frac{2 π m}{d} (x \cos \frac{α}{2} - y \sin \frac{α}{2})] \\ = F^{- 1} {U_{1}^{-} (u, v) * \sum_{(m)} a_{m} δ (u - \frac{m}{d} \cos \frac{α}{2}, v + \frac{m}{d} \sin \frac{α}{2})} \\ = F^{- 1} {\sum_{(m)} a_{m} U_{1}^{-} (u - \frac{m}{d} \cos \frac{α}{2}, v + \frac{m}{d} \sin \frac{α}{2})} . \end{array}

So the angular spectrum of

u_{1}^{+} (x, y)

is

U_{1}^{+} (u, v) = \sum_{(m)} a_{m} U_{1}^{-} (u - \frac{m}{d} \cos \frac{α}{2}, v + \frac{m}{d} \sin \frac{α}{2}) .

The concept of the angular spectrum propagation is applied to calculate the field

u_{2}^{-} (x, y)

before the grating G2. The diffraction phenomenon is a multiplicative quadratic phase factor

\exp {i k z {[1 - λ^{2} (u^{2} + v^{2})]}^{\frac{1}{2}}}

in the Fourier domain, which increases with the propagation distance z. The angular spectrum

U_{2}^{-} (u, v)

for the plane just preceding G2

(z = Δ_{1})

is

\begin{array}{l} U_{2}^{-} (u, v) = U_{1}^{+} (u, v) \exp [i k Δ_{1} \sqrt{1 - λ^{2} (u^{2} + v^{2})}] \\ = \exp [i k Δ_{1} \sqrt{1 - λ^{2} (u^{2} + v^{2})}] \sum_{(m)} a_{m} U_{1}^{-} (u - \frac{m}{d} \cos \frac{α}{2}, v + \frac{m}{d} \sin \frac{α}{2}) . \end{array}

The grating G2 forms an angle of

- α / 2

with respect to the y axis and the transmittance can be described by its Fourier expansion as:

g_{2} (x, y) = \sum_{(n)} a_{n} \exp [i \frac{2 π n}{d} (x \cos \frac{α}{2} + y \sin \frac{α}{2} - b)] .

Let

U_{2}^{+} (u, v)

is the angular spectrum of the field

u_{2}^{+} (x, y)

behind the grating G2.

\begin{array}{l} U_{2}^{+} (u, v) = \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) U_{1}^{-} (u - \frac{m + n}{d} \cos \frac{α}{2}, v + \frac{m - n}{d} \sin \frac{α}{2}) \\ \times \exp {i k Δ_{1} \sqrt{1 - λ^{2} [{(u - \frac{n}{d} \cos \frac{α}{2})}^{2} + {(v - \frac{n}{d} \sin \frac{α}{2})}^{2}]}} . \end{array}

With the Fresnel approximation, the above equation can be rewritten as

\begin{array}{l} U_{2}^{+} (u, v) = \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) U_{1}^{-} (u - \frac{m + n}{d} \cos \frac{α}{2}, v + \frac{m - n}{d} \sin \frac{α}{2}) \\ \times \exp {i k Δ_{1} [1 - \frac{λ^{2}}{2} (u^{2} + v^{2})]} \exp (- \frac{i Δ_{1} λ n^{2} π}{d^{2}}) \\ \times \exp [i \frac{2 π Δ_{1} λ n}{d} (u \cos \frac{α}{2} + v \sin \frac{α}{2})] . \end{array}

With the angular spectrum propagation equation and the Fresnel approximation, the angular spectrum

U_{3}^{-} (u, v)

for the observation plane is

\begin{array}{l} U_{3}^{-} (u, v) = \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) U_{1}^{-} (u - \frac{m + n}{d} \cos \frac{α}{2}, v + \frac{m - n}{d} \sin \frac{α}{2}) \\ \times \exp {i k (Δ_{1} + Δ_{2}) [1 - \frac{λ^{2}}{2} (u^{2} + v^{2})]} \exp (- \frac{i Δ_{1} λ n^{2} π}{d^{2}}) \\ \times \exp [i \frac{2 π Δ_{1} λ n}{d} (u \cos \frac{α}{2} + v \sin \frac{α}{2})] . \end{array}

Performing the inverse Fourier transform to

U_{3}^{-} (u, v)

, the field

u_{3}^{-} (x, y)

just before the OP plane can be deduced:

\begin{array}{l} u_{3}^{-} (x, y) = \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) \exp [- i π λ Δ_{2} (\frac{m^{2} + n^{2} + 2 m n \cos α}{d^{2}})] \\ \times \exp (- \frac{i π λ Δ_{1} m^{2}}{d^{2}}) \exp {i 2 π (\frac{m + n}{d} \cos \frac{α}{2} x - \frac{m - n}{d} \sin \frac{α}{2} y)} \\ \times {u_{1}^{-} (x - \frac{λ Δ m}{d} \cos \frac{α}{2} - \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}, y + \frac{λ Δ m}{d} \sin \frac{α}{2} - \frac{λ Δ_{2} n}{d} \sin \frac{α}{2}) \\ * \frac{\exp (i k Δ)}{i λ Δ} \exp [\frac{i π}{λ Δ} (x^{2} + y^{2})]} . \end{array}

where ∗ denotes a convolution operation. The convolution part in the above equation is just the form of Fresnel diffraction with a distance Δ

(Δ = Δ_{1} + Δ_{2})

. The theory of asymptotic expansions of double integrals [22] is used to expand the integration part in the above equation. So we have

\begin{array}{l} u_{3}^{-} (x, y) = \exp (i k Δ) \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) \exp [- i π λ Δ_{2} (\frac{m^{2} + n^{2} + 2 m n \cos α}{d^{2}})] \\ \times \exp (- \frac{i π λ Δ_{1} m^{2}}{d^{2}}) \exp {i 2 π (\frac{m + n}{d} \cos \frac{α}{2} x - \frac{m - n}{d} \sin \frac{α}{2} y)} \\ \times u_{1}^{-} (x - \frac{λ Δ m}{d} \cos \frac{α}{2} - \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}, y + \frac{λ Δ m}{d} \sin \frac{α}{2} - \frac{λ Δ_{2} n}{d} \sin \frac{α}{2}) . \end{array}

The above equation reveals that the fringes patterns are actually multiple-shearing interferences. The shear introduced by the double grating system to the distorted wavefront is

[(λ Δ m / d) \cos (α / 2) + (λ Δ_{2} n / d) \cos (α / 2), - (λ Δ m / d) \sin (α / 2) + (λ Δ_{2} n / d) \sin (α / 2)]

2.2 Spatial phase shift without phase objects

If no phase object exists before grating G1, the phase projection $φ (x, y)$ is a constant C. Hence, the field $u_{3}^{-} (x, y)$ before the OP plane can be rewritten as

\begin{array}{l} u_{3}^{-} (x, y) = \exp (i k Δ) \exp (i k C) \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) \\ \times \exp [- i π λ Δ_{2} (\frac{m^{2} + n^{2} + 2 m n \cos α}{d^{2}})] \\ \times \exp (- \frac{i π λ Δ_{1} m^{2}}{d^{2}}) \exp {i 2 π (\frac{m + n}{d} \cos \frac{α}{2} x - \frac{m - n}{d} \sin \frac{α}{2} y)} . \end{array}

The field

u_{3}^{-} (x, y)

consists of many grating diffraction orders. The diffraction orders which satisfy

n + m = p

will generate a multiple-shearing interferogram. The diffractive energy mainly focuses on these interferograms satisfying

m + n = (-2, -1, 0, +1, +2)

and other order interferograms can be ignored. Significant contributions to the zeroth order interferogram satisfying

m + n = 0

comes from three diffraction orders including

(m =0, n =0)

,

(m =1, n = - 1)

and

(m = - 1, n = 1)

. Then the zeroth order interferograms can be considered as the result of triple-shearing interference. Substituting these three diffraction orders into Eq. (13), the intensity distribution can be obtained:

\begin{array}{l} I_{0} (x, y) = a_{0}^{4} + 4 a_{1}^{4} \cos^{2} (\frac{4 π}{d} y \sin \frac{α}{2} - \frac{2 π b}{d}) \\ + 4 a_{0}^{2} a_{1}^{2} \cos (\frac{4 π}{d} y \sin \frac{α}{2} - \frac{2 π b}{d}) \cos [\frac{π λ}{d^{2}} (4 Δ_{2} \sin^{2} \frac{α}{2} + Δ_{1})] . \end{array}

where

a_{m} = (1 / 2) sinc (m / 2)

is the Fourier coefficients and

a_{m} = a_{- m}

. Because the angle α is a very small value, the term

4 Δ_{2} \sin^{2} (α / 2)

can be omitted. Significant contributions to the plus-first order interferogram satisfying

m + n = + 1

come from the two diffraction orders including

(m =+1, n =0)

and

(m = 0, n = + 1)

. Substituting these two diffraction orders into Eq. (13), the intensity distribution can be given:

I_{+ 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + \frac{π λ Δ_{1}}{d^{2}} - \frac{2 π b}{d}]} .

Similarly, the minus-first order interferogram satisfying

m + n = - 1

is the result of double shearing interference

(m=-1, n =0)

and

(m = 0, n = - 1)

. So:

I_{- 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) - \frac{π λ Δ_{1}}{d^{2}} - \frac{2 π b}{d}]} .

Equation (15) and Eq. (16) indicate strict cosinusoidal fringe patterns. The term $(4 π y / d) \sin (α / 2)$ denotes the fringe structure, and the term $2 π b / d$ denotes the phase shift generate by the relative laterally shift between grating G1 and grating G2. If the position of grating G2 is fixed, the lateral shift b generates the same phase shift to + 1 and −1 order interferograms.

The term $π λ Δ_{1} / d^{2}$ , which only depends on the distance $Δ_{1}$ between two gratings and the period d of gratings, denotes the spatial phase shift. If $Δ_{1}$ and d are defined, the plus-first order and minus-order order interferogram have the phase shift $π λ Δ_{1} / d^{2}$ and $- π λ Δ_{1} / d^{2}$ respecting to the zeroth order interferogram, respectively.

With the Talbot distance ( $Δ_{1} = K d^{2} / λ$ ), the intensity distributions of zero order, plus-first order and minus-first order interferogram are

I_{0 t} (x, y) = {\begin{array}{c} {[a_{0}^{2} + 2 a_{1}^{2} \cos (\frac{4 π}{d} y \sin \frac{α}{2} - \frac{2 π b}{d})]}^{2} Δ_{1} = 2 l \frac{d^{2}}{λ} \\ {[a_{0}^{2} - 2 a_{1}^{2} \cos (\frac{4 π}{d} y \sin \frac{α}{2} - \frac{2 π b}{d})]}^{2} Δ_{1} = (2 l + 1) \frac{d^{2}}{λ} . \end{array}

I_{+ 1 t} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + K π - \frac{2 π b}{d}]} .

I_{- 1 t} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) - K π - \frac{2 π b}{d}]} .

The fringe pattern of zeroth order interferogram has the best contrast at the Talbot distance. But because the spatial phase shift term $π λ Δ_{1} / d^{2} = K π$ , no phase shift exists between these three interferograms. Figure 3 shows the plus-first, zero and minus-first order interferograms when the second grating G2 locates at the Talbot distance. The zero order interferogram has the best contrast and no phase shift exists between these interferograms.

Fig. 3 Interferograms with the Talbot distance.

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If the distance between two gratings is a sub Talbot distance ( $Δ_{1} = (K + 1 / 2) d^{2} / λ$ ), the intensity distributions are

I_{0 s t} (x, y) = a_{0}^{4} + 4 a_{1}^{4} \cos^{2} (\frac{4 π}{d} y \sin \frac{α}{2} - \frac{2 π b}{d}) .

I_{+ 1 s t} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + \frac{π}{2} - \frac{2 π b}{d}]} .

I_{- 1 s t} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) - \frac{π}{2} - \frac{2 π b}{d}]} .

The fringe pattern of zero order interferogram has the worst contrast at the sub Talbot distance. But the spatial phase shift takes the maximum value $π / 2$ . Figure 4 demonstrates that the second grating G2 located at the sub Talbot distance procedures three interferograms with a phase shift of $π / 2$ . In fact, the spatial phase shift $π λ Δ_{1} / d^{2}$ can be found between these three order interferograms with any distance $Δ_{1}$ except the Talbot distance. Because this phase shift is generated by the diffraction characteristic of gratings, the absolute phase shifts of the plus-first and minus-first order interferograms respecting to that of the zero order interferogram are always identical.

Fig. 4 Interferograms with the sub Talbot distance.

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3 Spatial phase shift with phase objects

3.1 Theory

If phase objects exist before grating G1, the wave front before grating G1 is distorted. However, the influence of the phase object to interferograms is a little complex. Substituting Eq. (2) into Eq. (12), we have

\begin{array}{l} u_{3}^{-} (x, y) = \exp (i k Δ) \sum_{(m)} \sum_{(n)} a_{m} a_{n} \exp (- i \frac{2 π n b}{d}) \exp [- i π λ Δ_{2} (\frac{m^{2} + n^{2} + 2 m n \cos α}{d^{2}})] \\ \times \exp (- \frac{i π λ Δ_{1} m^{2}}{d^{2}}) \exp {i 2 π (\frac{m + n}{d} \cos \frac{α}{2} x - \frac{m - n}{d} \sin \frac{α}{2} y)} \\ \times \exp [i k φ (x - \frac{λ Δ m}{d} \cos \frac{α}{2} - \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}, y + \frac{λ Δ m}{d} \sin \frac{α}{2} - \frac{λ Δ_{2} n}{d} \sin \frac{α}{2})] . \end{array}

Performing the Taylor series expansion to the phase term

φ (x, y)

\begin{array}{l} φ (x - \frac{λ Δ m}{d} \cos \frac{α}{2} - \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}, y + \frac{λ Δ m}{d} \sin \frac{α}{2} - \frac{λ Δ_{2} n}{d} \sin \frac{α}{2}) \\ = φ (x, y) - \frac{\partial φ (x, y)}{\partial x} (\frac{λ Δ m}{d} \cos \frac{α}{2} + \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}) \\ - \frac{\partial φ (x, y)}{\partial y} (- \frac{λ Δ m}{d} \sin \frac{α}{2} + \frac{λ Δ_{2} n}{d} \sin \frac{α}{2}) \\ + \frac{1}{2} \frac{\partial^{2} φ (x, y)}{\partial x^{2}} {(\frac{λ Δ m}{d} \cos \frac{α}{2} + \frac{λ Δ_{2} n}{d} \cos \frac{α}{2})}^{2} \\ + \frac{1}{2} \frac{\partial^{2} φ (x, y)}{\partial y^{2}} {(- \frac{λ Δ m}{d} \sin \frac{α}{2} + \frac{λ Δ_{2} n}{d} \sin \frac{α}{2})}^{2} \\ + \frac{\partial^{2} φ (x, y)}{\partial x \partial y} (\frac{λ Δ m}{d} \cos \frac{α}{2} + \frac{λ Δ_{2} n}{d} \cos \frac{α}{2}) \\ \times (- \frac{λ Δ m}{d} \sin \frac{α}{2} + \frac{λ Δ_{2} n}{d} \sin \frac{α}{2}) + \dots . \end{array}

Substituting

(m =+1, n =0)

and

(m = 0, n = + 1)

to Eq. (24) and Eq. (23), finally we have the plus-first order interferogram with disturbing phase object

I_{+ 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + P 1 + (\frac{π λ Δ_{1}}{d^{2}} - P 2) - \frac{2 π b}{d}]} .

Similarly we have the minus-first order interferogram with disturbing phase object

I_{- 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + P 1 - (\frac{π λ Δ_{1}}{d^{2}} - P 2) - \frac{2 π b}{d}]} .

P 1

consists of the first order partial derivative terms.

P 1 = \frac{\partial φ (x, y)}{\partial x} \frac{2 π Δ_{1}}{d} \cos \frac{α}{2} - \frac{\partial φ (x, y)}{\partial y} \frac{2 π (Δ + Δ_{2})}{d} \sin \frac{α}{2} .

P 2

consists of the second order partial derivative terms.

\begin{array}{l} P 2 = \frac{\partial^{2} φ (x, y)}{\partial x^{2}} \frac{π λ (Δ^{2} - Δ_{2}^{2})}{d^{2}} \cos^{2} \frac{α}{2} + \frac{\partial^{2} φ (x, y)}{\partial y^{2}} \frac{π λ (Δ^{2} - Δ_{2}^{2})}{d^{2}} \sin^{2} \frac{α}{2} \\ - \frac{\partial^{2} φ (x, y)}{\partial x \partial y} \frac{2 π λ (Δ^{2} + Δ_{2}^{2})}{d^{2}} \sin \frac{α}{2} \cos \frac{α}{2} . \end{array}

According to Eq. (25) and Eq. (26), it is interesting that the intensity distribution of the plus-first order interferogram and the minus-first order interferogram behave the surprising symmetry. The term

(π λ Δ_{1} / d^{2}) - P 2

denotes the new spatial phase shift with disturbing phase objects. As a matter of fact,

P 2

depends on the second order partial derivative of

φ (x, y)

, which means that the spatial phase shift varies on the interferograms. However, the phase shift at a special point of the plus-order and minus-order interferograms are equal. Although there are algorithms which can extract phase information from arbitrary phase shifts [23], the term

P 2

should be decreased as small as possible. The term P1 denotes the fringe deviation with phase objects and contains the partial derivative of

φ (x, y)

in two orthogonal directions. In order to extracting the deflection projection from the phase information [24,25], we only need the partial derivative in x direction and the term in y direction should be as small as possible. For the sake of convenience, the two coefficients before the partial derivative in Eq. (27) are named with

P x 1

and

P y 1

. In Eq. (28), the three coefficients before the second order partial derivative are named with

P x 2

,

P y 2

and

P x y

. There are two different optical configurations that can be used to records different interfering diffraction order interferograms of double gratings simultaneously. The first configuration use spatial filtering to select interferograms. The second configuration which has been shown in Fig. 1 use high frequency gratings or set the distance between grating G2 and the OP plane at a proper position to avoid the overlapping of different interferograms. Two optical configurations take absolutely different effect to the Taylor series expansion of Eq. (24) and five coefficients in Eq. (27) and Eq. (28).

3.2 Double grating interferometer with spatial filters

The optical configuration shown in Fig. 5 is very simple. It consists of the double gratings which have the same optical setup with that in Fig. 1 and two 4-f systems. The distorted wave front is divided into two parts by a cube beam splitter. With the 4-f systems, the field before the OP plane is the same with the field just behind grating G2. Perform spatial filtering operations, Filter F1 selects the plus-first order interferogram and Filter F2 selects the minus-first order interferogram.With this optical configuration, the distance $Δ_{2}$ equals zero and the Eq. (27) and Eq. (28) are rewritten as

P 1 = \frac{\partial φ (x, y)}{\partial x} \frac{2 π Δ_{1}}{d} \cos \frac{α}{2} - \frac{\partial φ (x, y)}{\partial y} \frac{2 π Δ_{1}}{d} \sin \frac{α}{2} .

P 2 = \frac{\partial^{2} φ (x, y)}{\partial x^{2}} \frac{π λ Δ_{1}^{2}}{d^{2}} \cos^{2} \frac{α}{2} + \frac{\partial^{2} φ (x, y)}{\partial y^{2}} \frac{π λ Δ_{1}^{2}}{d^{2}} \sin^{2} \frac{α}{2} - \frac{\partial^{2} φ (x, y)}{\partial x \partial y} \frac{2 π λ Δ_{1}^{2}}{d^{2}} \sin \frac{α}{2} \cos \frac{α}{2} .

The coefficient of

\partial φ (x, y) / \partial x

is the same with Eq. (27). The coefficients in Eq. (29) and Eq. (30) and corresponding parameters are shown in Table 1 . Compared with

P x 1

, the other coefficients are very small and can be ignored.So the intensity distributions of plus-first order and minus-first order interferograms are

I_{+ 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + \frac{\partial φ (x, y)}{\partial x} \frac{2 π Δ_{1}}{d} \cos \frac{α}{2} + \frac{π λ Δ_{1}}{d^{2}} - \frac{2 π b}{d}]} .

I_{- 1} (x, y) = 2 a_{0}^{2} a_{1}^{2} {1 + \cos [4 π (\frac{1}{d} \sin \frac{α}{2} y) + \frac{\partial φ (x, y)}{\partial x} \frac{2 π Δ_{1}}{d} \cos \frac{α}{2} - \frac{π λ Δ_{1}}{d^{2}} - \frac{2 π b}{d}]} .

The term

(\partial φ (x, y) / \partial x) (2 π Δ_{1} / d) \cos (α / 2)

is the phase information that we want to extract from these interferograms. The interferograms are the result of laterally shearing interference in x direction. A propane flame caused by a Bunsen burner was placed before the grating G1. Figure 6 is the interferograms with sub Talbot distance. The white cross indicates the origin of each image. It is easy to find phase shifting between these two images.

Fig. 5 Optical configuration of the double grating interferometer with spatial filters

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Table 1. Coefficients before the partial derivatives in Eq. (29) and Eq. (30)^a

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Fig. 6 Interferograms with spatial filters (a) plus-first order (b) minus-first order

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3.3 Double grating interferometer without spatial filters

The optical configuration with spatial filters is more complex than that of without spatial filters. Furthermore, it requires two cameras to record interferograms simultaneously, which creates a challenge in synchronization and adjustment. However, one camera is sufficient to the system shown in Fig. 1. Unfortunately, for the purpose of avoiding the overlapping of different interferograms, the distance $Δ_{2}$ between grating G2 and the OP plane is usually a large value which means some coefficients in Eq. (27) and Eq. (28) cannot be ignored. Table 2 shows these coefficients. The distance $Δ_{2}$ is about 4.7m. $P y 1$ is bigger than $P x 1$ and $P x 2$ is a value that cannot be ignored. $P y 2$ and $P x y$ are still be to small value, which means the second order partial derivative in x direction will be markedly amplified in interferograms. Figure.7 shows the interferograms without spatial filters. The phase object is the propane flame. Compared with interferograms shown in Fig. 6, interferograms in Fig. 7 have vertical bright lines. Because of the amplification of $P y 1$ and $P x 2$ , these bright lines indicate regions in which the partial derivative $\partial φ (x, y) / \partial y$ and $\partial^{2} φ (x, y) / \partial x^{2}$ rapid fluctuate.To minimizing the coefficient $P y 1$ and $P x 2$ , we can adjust some parameters, such as the grating period d, the angle α and the distance $Δ_{1}$ . The coefficient $P y 1$ and $P x 2$ can be rewritten as

P y 1 = - \frac{2 π Δ_{1}}{d} \sin \frac{α}{2} - \frac{2 π N}{λ} \sqrt{d^{2} - λ^{2}} .

P x 2 \approx \frac{π λ Δ_{1}^{2}}{d^{2}} \cos^{2} \frac{α}{2} + \frac{2 π F Δ_{1}}{d^{2}} \sqrt{d^{2} - λ^{2}} .

where N denotes the fringe number in the aperture. The first terms in Eq. (33) and Eq. (34) are usually the same with these small coefficients in Eq. (29) and Eq. (30).

Table 2. Coefficients before the partial derivatives in Eq. (27) and Eq. (28)^a

View Table | View all tables in this article

Fig. 7 Interferograms with the propane flame.

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Figure.8 shows the variations of the coefficient $P y 1$ and $P x 2$ with different parameters. The coefficient $P y 1$ mainly depends on the fringe number N and the grating period d. If higher frequency gratings are used, the grating period d decreases. So the required minimum distance $Δ_{2}$ decreases and $P y 1$ decreases. But as shown in Fig. 8 (b), the coefficient $P x 2$ increases with higher frequency gratings. So the selected grating period should make a balance between $P y 1$ and $P x 2$ . To minimize $P x 2$ , the only way is to decrease the distance $Δ_{1}$ between two gratings. According to Eq. (33), $P y 1$ is related to the fringe number N and the angle α. The fringe width increase when N decreasing, which means the angle α will decrease. So, $\sin (α / 2)$ decreases as well as $P y 1$ . The perfect condition is that the fringe number N equals zero. The interferograms have the infinite width fringe width. Then $P y 1$ decreases to zero.

Fig. 8 (a) Relation between $P y 1$ and number of fringes with different grating period. (b) Relation between $P x 2$ and the distance between gratings with different grating period.

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However, the infinite width fringe is actually impossible. Figure.9 is the approximate infinite width fringe interferograms. The phase object is the propane flame and grating G2 locates at the sub Talbot distance. Although no horizontal fringes exist in Fig. 9, the phase shift can also be found between the plus-first and minus-first order interferograms. Another advantage of these phase shift interferograms is that after the point by point calculation, the phase wrapping image just contains the wanted phase information and do not have the modulated fringe information.

Fig. 9 Approximate infinite width fringe interferograms with propane flame.

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4 Triple grating interferometer

The double grating interferometer discussed in above chapters can generate two phase shift interferograms simultaneously. But to achieve phase information, at least three or four phase shift interferograms are required. To solve this problem, we use three gratings. The optical configuration is similar to this shown in Fig. 1. The third grating G3 is located between the phase object and grating G1 and the direction of the grating G3 is perpendicular to the other two gratings. So the testing beam is divided into many diffraction order beams in vertical direction by grating G3. Each diffraction order beam propagates through grating G1 and G2 and generates shearing interference.

Figure 10 is the shearing interferograms generated by the triple grating interferometer. Nine main components are recorded. With the similar method in above chapters, phase shift is found between these interferograms. From top to bottom, phase shifts of the left side interferograms are $- P_{a} - P_{b}$ , $- P_{a}$ and $- P_{a} + P_{b}$ . From top to bottom, phase shifts of the right side interferograms are $+ P_{a} - P_{b}$ , $+ P_{a}$ and $+ P_{a} + P_{b}$ , where $P_{a}$ is $- π λ Δ_{1} / d^{2}$ and $P_{b}$ is $4 π λ Δ / d^{2} \sin (α / 2) \sin (β / 2)$ , Δ is the distance between grating G3 and the OP plane, $β / 2$ is the angle between the new grating G3 and y axis. Four interferograms that locate at top left, bottom left, top right and bottom right are the result of double shearing interference. Two interferograms that locate at middle left and middle right are the result of triple interference. So at least we can achieve four phase shift interferograms simultaneously. Then phase information can be extracted by spatial phase shifting algorithms. Detailed discussion to the triple grating interferometer will be presented in our future paper.

Fig. 10 Shearing interferograms generated by the triple grating interferometer.

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5 Conclusions

With the scalar diffraction theory, the intensity distributions of diffraction order interferograms of double grating interferometer are analyzed, based on which the phase shift is found between the plus-first, zero and minus-first order interferograms. If no phase object exists, this phase shift only depends on the grating period and the distance between two gratings. This phase shift takes the minimum value zero at the Talbot distance and takes the maximum value $π / 2$ at the Sub Talbot distance. If phase object exists, because of the characteristic of shearing interference, the phase information of the distorted wave front and the phase shift between interferograms are complex. If the spatial filter is used, the phase information is the partial derivative of distorted wave front in x direction and the phase shift is the same with this without phase objects. Without the spatial filter, the phase information extracted from interferograms contains the partial derivative of distorted wave front in both x and y directions. At this time, the phase shift varies on the interferograms. But, at any special point of interferograms, the phase shifts are equal. Although there are the above disadvantages of the double grating system without spatial filters, the best advantages of this configuration is simple and only one camera is required. We can decrease these disadvantages by selecting configuration parameters. To decrease the variation of phase shift, the distance between two gratings should be decreased. To decrease the partial derivative in y direction, the angle between two gratings should be as small as possible and this means the interferograms are the infinite fringe width interferograms. However, to extract phase information from the phase shift interferograms, three or four interferograms are required at least. Consequently, the triple grating interferometer which can generate at least four phase shift interferograms simultaneously is presented finally.

Acknowledgment

This Work is supported by the National Natural Science Foundation of China (Grant No. 10804052), China Postdoctoral Science Foundation (Grant No. 20080431096), and Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 0801010C).

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Coefficients in Eq. (29)			Coefficients in Eq. (30)
$P x 1$	$P y 1$		$P x 2$	$P y 2$	$P x y$
1476.31	−3.69		0.092	5.76×10⁻⁷	−0.00023

Coefficients in Eq. (27)			Coefficients in Eq. (28)
$P x 1$	$P y 1$		$P x 2$	$P y 2$	$P x y$
1476.31	−2956.14		73.90	0.00046	−0.18

Coefficients in Eq. (29)			Coefficients in Eq. (30)
$P x 1$	$P y 1$		$P x 2$	$P y 2$	$P x y$
1476.31	−3.69		0.092	5.76×10⁻⁷	−0.00023

Coefficients in Eq. (27)			Coefficients in Eq. (28)
$P x 1$	$P y 1$		$P x 2$	$P y 2$	$P x y$
1476.31	−2956.14		73.90	0.00046	−0.18

Spatial phase-shifting characteristic of double grating interferometer

Abstract

1. Introduction

2. Principle

2.1 Theory

2.2 Spatial phase shift without phase objects

3 Spatial phase shift with phase objects

3.1 Theory

3.2 Double grating interferometer with spatial filters

3.3 Double grating interferometer without spatial filters

4 Triple grating interferometer

5 Conclusions

Acknowledgment

References and links

Cited By

Figures (10)

Tables (2)

Equations (35)

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