Carrier fringes and a non-conventional rotational shear

in a triangular cyclic-path interferometer Rosaura Kantun-Montiel* and Cruz Meneses-Fabian

Facultad de Ciencias Fisico-Matematicas, Benemerita Universidad Autonoma de Puebla, Av. San Claudio y 18 Sur, C. U. San

Manuel, Apdo 165, 72000 Puebla, Mexico

*Corresponding author: rkantun@gmail.com

Received Month X, XXXX; revised Month X, XXXX; accepted Month X,

XXXX; posted Month X, XXXX (Doc. ID XXXXX); published Month X, XXXX

Abstract

This work presents a method for generating carrier fringes and a non-conventional rotational shear in a triangular cyclic-

path interferometer, suppressing at the same time the presence of typical lateral and radial shearing. To carry out this

method a 4f optical system is implemented into this cyclic interferometer. The most important contribution of this paper is

the demonstration of linear dependence of the movable mirror displacement with the carrier frequency introduced and the

obtaining of a non-conventional rotational shearing interferometer. Additionally, we think that one of its possible potential

applications is the observation of the angular derivative of parallel projections of a phase object placed at the output plane,

generating a great advantage in edge-enhancement optical tomography. In this paper, we develop a theoretical model and

show experimental results.

1. Introduction

A cyclic path optical configuration (CPOC) has been used extensively to measure many physical quantities in many fields

in science and technology [1-21]. Typical schemas of a CPOC have been carried out principally in lateral shearing

interferometry (LSI) [3-8] (Fig. 1(a)) and radial shearing interferometry (RSI) [9-15] (Fig. 1(b)). These kinds of setups

contain few elements and are relatively insensitive to external mechanical vibrations, which are due to the beams

travelling along the same optical circuit. The retrieved phase in these kinds of interferograms is done by applying phase-

shifting interferometry (PSI) [4,6,12] or carrier fringe interferometry (CFI) [13]. Since this phase is proportional to the

directional or radial derivative of the object phase [16], this object phase is obtained by a convenient integration

algorithm [16]. In order to carry out CFI in RSI for phase extraction, some methods have been proposed such as tilting

the beam splitter by a very small angle [17-21], but this tilt decentres the beams making it necessary to apply a special

algorithm to correct this defect [17-21]. It is important to note that this effect of decentring the beams is interpreted as a

lateral displacement, and thus it is accepted that the lateral and radial shear are implemented simultaneously in a

CPOC by tilting the beam splitter [17-21]. On the other hand, we want to note that all known shearing interferometers,

superposing an object beam with itself that is displaced respecting to the original object beam by a very small distance on

the detector plane, while their propagation directions (axial directions) are not changed. For instance, supposing that the

beam travels on z-axis, in LSI the beam is translated on x or y direction [16], in RSI the beam is scaled on r direction

with 2 2r x y [17-21]. In rotational shearing interferometry (ROSI) the beam is rotated on direction with

tan /y x [16,22,23] whose rotational axis is parallel to the z-axis, and in the reversal shearing interferometry (RESI)

the beam is inverted with respect to x or y direction [16,24]. In any case this displacement is not done on an axial

direction or rotational with respect to the x or y-axis. Thus it is important to explore the possibility of implementing and

applying these kinds of non-conventional shearing interferometers.

In this paper, we describe the implementation of a triangular cyclic-path interferometer (TCPI) equipped with two lenses

of the same focal length placed to form a 4f optical system in which we first demonstrate analytical and experimentally

that the typical radial and the lateral shearing are removed even though one of its mirrors is displaced, implying that a

phase object at the input plane is not observable at the image plane. Secondly, under the Fresnel diffraction theory, we

demonstrate the existence of a linear phase instead of the lateral shear when this movable mirror is displaced, and in

consequence if a phase object is placed at the image plane instead of input plane, the angular derivative of its phase

variations over the ( , )x z plane, instead of ( , )x y plane as in ROSI, are observed. This setup is a type of ROSI, but with

the object beam rotated with its rotational axis parallel to y-axis, instead of z-axis. Thus the typical ROSI could be

renamed as z-rotational shearing interferometry (z-ROSI), and the present proposal can be named as y-rotational

shearing interferometry (y-ROSI). Additionally, from the viewpoint of phase optical tomography, a collimated beam

leaving a phase object is interpreted as a parallel projection [25], and therefore, the phase observed in the present

proposal is proportional to the angular derivative of its projections, which we think could be useful in edge-enhancement

computed tomography under the approximation of parallel-projections, as it was previously demonstrate and reported by

us in Ref. [25].

In summary, this paper presents a technique for observing the phase variations of a phase object because of the

rotational shear with respect to y-axis in the presence of a carrier frequency generated with the same displacement of the

movable mirror.

Fig. 1 (Color online) Traditional setups of triangular cyclic path interferometers, (a) lateral shearing, (b) radial shearing with 1 2f f

, and (c-d) non-lateral and non-radial shearing, same focal length of lenses with 1 2f f , (c) 0N , the beams are not tilted, and (d)

0N , the beams are tilted.

2. Theoretical analysis

The present proposal is based on a TCPI as depicted in Fig. (1c). It consists of a beam splitter (BS) and two plane mirrors

1M and 2M . Additionally in this proposal two lenses

1L and 2L of the same f focal length are placed to form a 4f optical

system in such form that the input plane and the image plane coincide with the input and output of TCPI, respectively.

To explain our proposal, we first describe a typical TCPI (without lenses) as depicted in Fig (1a), which shows how the

wavefront under test passes through the BS producing in a refracted beam and a reflected beam that travel along the

same optical path, but in opposite directions named ( , )cwE x y (red lines) and ( , )acwE x y (blue lines), respectively. Both are

reflected in 1M and 2M , and finally are reintegrated by the same BS to generate their interference.

As known, when an LSI is illuminated with a non-tilt plane wave, monochromatic with wavelength , linearly polarized,

and travelling on z-direction [16], a mechanical displacement of 1M by an amount N on the normal direction N to the

mirror surface, such as shown in Fig. (2), produces in the reflected beam a lateral displacement u on the x-direction,

which aims on the direction z' forming a angle with respect to N . We can prove

2 sinu N , (1)

Fig. 2 (color on line) Mechanical displacement of 1M .

and simultaneously the optical path with respect to 0N is incremented per some value denoted by z . Then when a

displacement of 1M is yielded, a lateral shear u is induced in each arm, becoming ( , )cwE x u y and ( , )acwE x u y

respectively. Then the resultant optical path difference at the output plane can be written as [16]

( , ) ( , ) ( , ) ( , ) ( , )cw acwW x y W x y W x y W x u y W x u y . (2a)

Eq. (2a) represents the lateral shearing with collimated light, which can be approximated to the first term of

the Taylor series if u is small enough. Thus a differential change in the wavefront becomes proportional to

its first partial derivative on the x-direction [16]. Then rewriting Eq. (2a) we have

( , )

( , ) 2W x y

W x y ux

, (2b)

where the input optical field is described by

1( , ) ( , ) ( , )

2cw acwE x y E x y E x y , (3a)

with

( , )( , ) ( , ) i x yE x y A x y e , (3b)

and ( , ) ( , )x y kW x y with 2k denoting the wavenumber, and ( , )W x y is the optical path. Then the

interference pattern observed by an optical detector at the image plane can be described by using Eqs. (2b), (3),

that is

( , )( , ) ( , ) ( , )cos 2

W x yI x y a x y b x y k u

x

, (3c)

where 2 24 ( , ) ( , ) ( , )a x y A x u y A x u y , and 2 ( , ) ( , ) ( , )b x y A x u y A x u y .

On the other hand, in the closed path of the CRSI, as depicted in Fig. (1b), a Keplerian telescope is placed

comprised of 1L and

2L . As known, if the focal lengths of 1L and

2L are 1f and

2f , respectively, then,

1 2/ 1s f f is the radial shearing ratio of this CRSI [11-13].

( , ) ( , ) ( , ) ( / , ) ( , )cw acwW W W W s W s . (4)

Nevertheless, in this work we consider the case where two lenses of the same focal length f are collocated

into the TCPI shown in Figs. (1c) and (1d). The ratio of the radial shear is 1s , and hence the RSI is removed.

In Fig. (1c) the input plane is before BS at a f distance from 1L and the image plane is after BS at a f

distance from 2L . The refracted beam ( , )cwE x y (red lines) goes through the 4f system by

1 1 2 2BS L M M L BS . In this setup the second focus plane of 1L matches with the first focus plane of

2L . The optical field on this plane can be described by the Fourier transform of the input optical beam

. For this reason this plane is called the Fourier plane cwF and is located between the mirrors 1M and 2M .

Similarly ( , )acwE x y goes through the same 4f system but in opposite direction 2 2 1 1BS L M M L BS .

The Fourier plane acwF matches with

cwF , and thus the general Fourier plane is cw acwF F F . If at the Fourier

plane no filter is placed, the refracted optical field ( , )cwE x y and the transmitted optical field ( , )acwE x y are

superposed at the image plane by the same BS, and therefore the total output of the optical field is given by

the sum of the optical fields ( , )cwE x y and ( , )acwE x y . Because of the equality given in Eq. (3a) these beams are

not displaced, so their optical path difference is null, and therefore the obtained interference pattern does not

show interference fringes.

On the other hand, when 1M is displaced by an amount N along N direction, a lateral displacement u in

each arm but in opposite directions is induced, as LSI case, see Figs.(1d) and (2). For the ( , )cwE x y beam,

because of this lateral displacement occurs between 1L and cwF , the 4f setup is changed in two ways, a) the

optical path is incremented by an amount z , and b) throughout the rest of the path it is laterally displaced by

an amount u . As cwF is translated by an amount z from F , then the second lens 2L is placed a distance

f z of cwF . Additionally, the spectrum of the entrance optical field that travels in clockwise direction

( , )cwE is displaced laterally an amount u f in -direction becoming ( , )E , where and

are spatial frequencies. Following the Goodman analysis of diffraction [26] it is possible to prove that the

output optical refracted beam is

2 2

2' ' '

2( ', ') ( ', ')

k z ki x y i ux

f f

cwE x y E x y e e

. (5)

An alternative method to prove Eq. (5) is by using the Fourier transform propriety 2( ) ( )i xg x e g .

Similarly in the case of the ( , )acwE x y beam, the lateral displacement occurs between acwF and 1L per an amount

u , and acwF is translated by an amount z from F . The Fourier transform of ( , )acwE x y can be interpreted as

( , ) ( , )acwE E , with 1L placed a distance f z from

acwF . Analogous to Eq. (5), by using the analysis

of [26] again, the output optical reflected field is given by

2 2

2' ' '

2( ', ') ( ', ')

k z ki x y i ux

f f

acwE x y E x y e e

. (6)

Eqs. (5) and (6) represent the clockwise and anticlockwise optical field respectively at the output plane, but as

we can note between ( ', ')cwE x y and ( ', ')acwE x y there is no displacement in any direction, only a linear and a

quadratic phase is added. Then by Eqs. (3a) and (3b), and analogously to Eq. (2a) the optical path difference at

the output plane is

( ', ') ( ', ') ( ', ') 0cw acwW x y W x y W x y , (7)

but in this case it is always null mainly because the beams at the image plane are not displaced even though

one of interferometer‟s mirrors is displaced. This effect occurs because of the lenses, since their presence cause

that the displacement of the movable mirror produces an axial and lateral displacement of the probe field

spectrum, being an important fact that has as a consequence the elimination of the lateral shearing and the

obtaining of a linear phase. The intensity observed by an optical detector at the image plane can be expressed

after some manipulations of Eq. (3b), (5) and (6) by means of

0( ', ') '( , ) '( , )cos 2 'I x y a x y b x y x , (8)

where 2'( , ) '( , ) ( , ) 2a x y b x y A x y meaning that the interferogram has an ideal visibility equal to the unity,

and on the other hand, the carrier frequency is given by

0

2 4 sinu N

f f

. (9)

As can be noted in Eq. (8), only the linear phase term is present depending directly on the displacement of the

mirror, since the object phase is not possible to be observed because the elimination of the lateral shearing.

Furthermore, the null effect in the interferogram of the quadratic phase in the beams described in Eq. (5-6) is

due to its value and sign are the same for each beam in the interferometer.

3. Experimental results

In order to prove the theory presented here, the schema depicted in Fig. (1c-d) was implemented. The RGB-655/500mW

laser with 532nm was used as an illumination source; two lenses of focal length 400f mm , and a CCD camera

Pixelink B741F were employed to capture and digitalize the obtained interferograms and an angle / 8 was used.

Fig. 3 (color on line) a) Some interferograms obtained with different N values and a pixel-millimeter relation of

1 148mm px , b) graphics of 0 ( )N theoretical model vs experimental measurement, the error bar indicate the one

standard deviation s value, and c) the plot of the variability coefficient of each measurement set.

The 1M mirror was mounted on a stage controlled by a vernier micrometer (Newport model SM 25) having a nominal

resolution of 10 m . In the first stage, without the phase object placed in the TCPI, the movable mirror was displaced in

steps given by 50N j m , for 0,...,20j . In each step an interferogram was stored (some of them are shown in Fig.

3a) and its carrier frequency 0 was measured by applying the Fourier-transform method for fringe analysis. The

procedure for measuring 0 experimentally consists in applying the Fourier‐transform method for phase extraction

introduced by Takeda et al [27], but without centring the filtered lobule of the interferogram spectrum. Fig. (3b) shows in

green dashes the theoretical model given in Eq. (9) and the black points depict the experimental measurements of 0 . As

noted, the experimental measurements of 0 coincide strongly with the theoretical predictions obtained with values of

standard deviations not greater than 0.4 / mm , which were calculated with the know formula 1/2

2

0 0

1

/n

i p

i

s N

,

where pN is the data amount, for this case each pixel of a frame of 928X856 pixels was considered as a measurement, in

others words 794368pN , 0i is the carrier frequency of each frame pixel, and

0 is the mean of all data set. It is

important to note that these experimental measurements are obtained in pixels, and the theoretical equation is given in

meters. In our experiment the relation between pixels and millimeter was of 1 148mm px . Fig. (3c) shows the variability

coefficient, that indicates the percentage of error in relation with the measured value 0 for each N , it was calculated

with the well know formula 0100 /cV s , as can be seen the maximum percentage is when 100N m eventhough

the standard deviation is not large, but this small error is equivalent to 12% of the total of 0 . This accuracy was

obtained by turning manually a micrometer screw to displace the movable mirror, thus the standard deviation and the

variability coefficient may be improved substantially if a sophisticated system such as an actuator is used to move this

mirror.

Fig. 4 a) A tilt microscope slide placed at the input plane, and, b) a tilt microscope slide placed at the output plane.

Eqs. (5) and (6) indicate that the input optic field does not suffer a lateral shear at the output plane when 1M

is displaced, and instead a linear phase is created for each arm but with opposite inclinations, generating an

interferogram with a carrier frequency. We think that if an object under test is placed at the output plane,

each beam is going to travel through the object in different directions, obtaining the difference of its parallel

projections. To prove this theory experimentally a tilted microscope slide made of glass and 31 25 75x x mm in size

was employed as an object under test. The second stage the object was placed at the input plane covering a

part of the probe aperture. The horizontal dark streak in the interferograms shown in Fig. (4a) indicates the

interface between this object and the air. These interferograms were recorded with different values of N , and

as it can be noted the interference fringes were not displaced or deformed because the phase object, as in Eq.

(8) is predicted. In contrast, the Fig (4b) shows several interferograms taken when the object was placed at the

output plane, note how the object‟s presence causes fringe displacements. These displacements can be

understood as a phase shift, and as it can be noted the value of these phase-steps depends directly on N .

Besides these phase shifts could be interpreted as the angular derivative of the parallel projection of a

microscope slide, such as was demonstrated in Ref. [25].

4. Discussion

The potential utility of the theory described above is not in placing the phase object at the system‟s entrance,

but in placing it at the system output, since the beams in this configuration pass through the phase object in

different directions (phase object denoted by sO in Fig. (1d)). Thus the phase variations observed in the

interferogram are due to the variations of the refractive index over the ( , )x z plane, instead of the ( , )x y plane

as in ROSI, which can be interpreted as a rotated beam with rotational axis parallel to the y-axis, instead of

the z-axis as in the typical ROSI. Attending the rotation axis of the beam, we named the present non-

conventional rotational shearing interferometer as y-rotational shearing interferometer (y-ROSI), and the

typical ROSI as z-ROSI. Note that an important characteristic in y-ROSI is that the propagation direction of

the beam is changed linearly by N , which also produces the carrier frequency, modulated in a wide range as

shown in Fig. (3b). In summary, this displacement produces two important effects simultaneously: in the first

place, it introduces a linear phase with high reliability, which is appropriate for fringe analysis by applying

the Fourier-transform method [27]. In the second place, it obtains the angular derivative of the phase object

projections under the parallel projections approximation, which is useful for reconstructing the edge-

enhancement of the phase object slice as it was demonstrated in Ref. [25] where speckle interferometry was

used in an out-of-plane configuration. To be more precise, by analysing the setup with the object placed at the

system output, the interferogram in y-ROSI is now described by

0( ', ') ', ' ', ' cos , ' 2I p y a p y b p y p y p , (10)

where 2', ' ', ' ', ' / 2a p y b p y A p y , and cos sinp x z is known as the projection coordinate; is known as

the projection angle;

k n denotes the phase difference because of parallel projection difference

n n n

, with indicating an increment of the projection angle induced also by the mirror displacement. Optically n

means

the optical path difference between two beams, and

n is known as the parallel projection, which is described

mathematically by the Radon transform [25], given by

, ' , , cos sinn p y dxdzn x y z p x z

, (11)

where n is the distribution function of the index of refraction into the phase object. Note that if 0 , then p x , and

Eq. (10) becomes Eq. (8), therefore Eq. (8) is a particular case, and Eq. (10) is a more general case that takes into account

the angular position of the phase object. Thus with a collection of many views within the range 0, , some

proportional to /n , that is the angular derivative of n over the ,x z plane, can be reconstructed by applying a back-

projection algorithm, and an image with its enhanced edges can be obtained. However this task is beyond the scope of this

paper.

Another great advantage of the present method is that the phase difference could be retrieved by only one interferogram

by using the Fourier method for fringe analysis instead of the several interferograms by using the phase-shifting

interferometry (PSI) method [25]. Besides this carrier frequency and the phase difference are yielded by the same

displacement N , thus this method is very easy and practical in an experimental situation.

5. Conclusions and remarks

This paper demonstrated that by placing two lenses of the same focal length forming a 4f optical system into a

triangular cyclic-path interferometer a linear phase is introduced, and the radial and lateral shearing are

missed even though one of its mirrors is displaced as it is commonly done in these kinds of interferometers.

Instead of the lateral shearing a linear phase is obtained, and the phase variations of a phase object over the

,x z plane, instead of the ( , )x y plane, are observed. Thus we have introduced a non-conventional rotational

shear named y-rotational shearing interferometry, which as it was discussed theoretically, can be useful and

practical experimentally in edge enhancement computed tomography. Additionally, we think that the present

schema can also be useful in holographic microscopy since the Fourier planes cwF and acwF are separated in

both axial and lateral directions by z and u , respectively, and they are generated simultaneously by the

same displacement N . Thus these separations can be modulated in a wide range. Besides, because N also

generates a carrier frequency, in this same schema Fourier holograms out of axis of microscopic targets can be

obtained if these are placed in one of the Fourier planes while the other one is used as a reference. However,

these tasks are also beyond the scope of this paper.

Acknowledgments

R. Kantun-Montiel is grateful to the Consejo Nacional de Ciencia y Tecnología (México) for the scholarship

given under grants 160211. This work was partially supported by the ConsejoNacional de Ciencia y Tecnología

(México) under grant 166742 and by the Vicerrectoría de Investigación y Estudios de Posgrado of Benemérita

Universidad Autónoma de Puebla under grant MEFC-EXC14-I. Authors thank N. Keranen for her advice on

wording.

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