Inhomogeneous phase-visibility modulating interferometry by space

0 downloads 0 Views 3MB Size Report
L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with ... T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on ...
Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation Uriel Rivera-Ortega, Cruz Meneses-Fabian,* and Gustavo Rodriguez-Zurita Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Apdo. Postal 165, Puebla, Pue. 72000, México * [email protected]

Abstract: A new method in interferometry based on on-off non-quadrature amplitude modulation for object phase retrieval is presented. Although the technique introduces inhomogeneous visibility and phase variations in the interferogram, it is shown that the phase retrieval of a given object is still possible. This method is implemented by using three beams and two MachZehnder interferometers in series. One of the arms of the system is used as a probe beam and the other two are used as reference beams, yielding from their sum the conventional reference beam of a two-beam interferometer. We demonstrate that if there is a phase difference within the range of ( 0, π ) between these two beams, the effect of modulation in both amplitude and phase is generated for the case of on-off non-quadrature amplitude modulation. An analytical discussion is provided to sustain this method. Numerical and experimental results are also shown. ©2013 Optical Society of America OCIS codes: (120.3130) Interferometry; (120.2440) Filters; (120.5050) Phase measurement; (120.5060) Phase modulation.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed., (Elsevier Science, 1990), pp. 274–276. D. Malacara, Optical Shop Testing (Wiley, New York, 2007), pp. 547–550. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991). L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). X. Xu, L. Cai, H. Yuan, Q. Zhang, G. Lu, and C. Wang, “Phase shift selection for two-step generalized phaseshifting interferometry,” Appl. Opt. 50(34), H171–H176 (2011). A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005). C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994). G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994). J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). Z. Zhigang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun. 285, 521–526 (2011). Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt. 109, 103106 (2011). Q. Yang, R. Zhou, and B. Zhao, “Principle of the moving-mirror-pair interferometer and the tilt tolerance of the double moving mirror,” Appl. Opt. 47(13), 2486–2493 (2008). J. C. Wyant and R. N. Shagam, “Use of electronic phase measurement techniques in optical testing,” Proc. ICO11, Madrid, 659–662 (1978). D. Malacara, I. Rizo, and A. Morales, “Interferometry and the Doppler effect,” Appl. Opt. 8(8), 1746–1747 (1969).

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17421

15. T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt. 47(21), 3784–3788 (2008). 16. T. Susuki and R. Hioki, “Translation of Light Frequency by a Moving Grating,” J. Opt. Soc. Am. 57(12), 1551 (1967). 17. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. 36(13), 2417–2419 (2011). 18. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng. 50(7), 905–909 (2012). 19. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by amplitude modulation,” in: Interferometry research and applications in science and technology, Ivan Padron (Ed.), ISBN 978–953–51– 0403–2, InTech, (2012). 20. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982).

1. Introduction The theory of phase-shifting interferometry (PSI) is usually based on the interference of two beams [1,2], where one beam is considered as the reference wave and the other as the probe wave. Its mathematical representation can be written as I n ( x , y ) = a ( x , y ) + b ( x , y ) cos φ ( x , y ) + α n 

(1)

where a is the background illumination, b the fringe modulation, φ the object phase, and α n = 2π n N the phase shifting step, which is kept spatially constant at least during the capture time of the interferogram I n , with n = 0,  , N − 1 and N meaning the number of phase steps. For N ≥ 3 a resoluble set of equations is formed because a , b and φ are considered temporally constant [1,2], which implies that the visibility is kept constant when the phase-step is generated. A more general theory can be proposed for the case of α n unknown and arbitrary. Typically, this happens when the phase shifter has a non-linear response and hysteresis or is miscalibrated, but also such a case can appear due to influence of turbulence or environmental temperature gradients, mechanical vibrations and other possible perturbations. In this case, the method is called generalized phase-shifting interferometry (GPSI). Here N equations are obtained but N + 3 unknowns are present, and the solution cannot be obtained under the usual PSI theory. However, numerous methods for giving solutions to this problem have been successfully introduced [3–8]. Experimentally, in PSI and GPSI a phase-step can be introduced from different principles and ways. For instance, by changing the optical path using a displacing a mirror on a piezoelectric transducer [9], by inducing changes in the refractive index [10,11], by means of tilting a glass plate [12], using the frequency shifts between the two beams induced by the Zeeman effect [13] or the wavelength variations with the Doppler effect [14]. Further techniques include the modulating of the polarization [15] or by lateral displacements of a grating [16], among others. Recently, we have theoretically introduced a novel method based on a quadrature (QAM) [17] and non-quadrature amplitude modulation (NQAM) [18]. In this method, the phase modulation (PM) case was attended because PM does not modify the visibility as it is requested in PSI. The others two possible cases: amplitude modulation (AM) and phase and amplitude modulation (PAM) were avoided because they both modify the visibility, and thus, the PSI technique would not be applicable. Besides, in those papers QAM and NQAM were proposed by using two reference beams of an interference of three beams, where a phase difference between them was required to be spatial and temporally constant: π 2 [17] and within the range ( 0, π ) [18,19]. In this paper we demonstrate analytically and experimentally the utility of space NQAM for the on-off modulation case in the modality of inhomogeneous PAM for object phase retrieval in interferometry, which implies that the PSI method cannot be applicable here, because the inhomogeneous PAM modifies spatially the visibility and phase shift instead of

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17422

only the phase shift as it is appropriated for PSI. In the experiment, a three-beam MachZehnder interferometer is employed, where one of the arms is selected to be used as a probe and the others are used as reference arms. The phase difference of these references is considered to be spatially variable within the range ( 0, π ) instead of constant as considered in [17–19]. This is achieved by adjusting their optical path difference. On the other hand, the onoff modulation in the beams is done by blocking and unblocking them, thus, the amplitudes take only zero or maximum value. 2. Principle The set-up depicted in Fig. 1 shows a three-beam Mach-Zehnder interferometer. The beam leaving from the laser is filtered, expanded, and collimated in order to obtain ideally a nontilted plane wave, which is divided into two beams by a beamsplitter BS 1 . One of these two

beams is referred to as reference beam 1 ( E 1 ) . The other beam is subdivided by another beamsplitter, BS 2 , in another two beams. One of these is named “reference beam 3” ( E 3 ) ,

while the remaining beam is named “the probe beam” ( E 2 ) . The reference beams E 1 and E 3 are added by BS 5 , and the resulting beam is called “the reference beam” ( E r ) . Finally

E r = E 1 + E 3 is added with E 2 by means of BS 6 and so, the sum E = E 1 + E 2 + E 3 (or

equivalently E = E r + E 2 ) is obtained. Note that BS 3 and BS 4 could be mirrors, but they are chosen as beamsplitters to compensate the lost of amplitude in the beams at the interferometer’s output. With this idea, each beam crosses four beamsplitters, and, in this way, the three beams have similar amplitudes. Let AFk = 0,1 , with k = 1, 2,3 , represents the action of a binary amplitude filter. Thus, A k takes only zero and maximum amplitude A kM values. Then, each field in the sum leaving from BS 6 can be expressed as, E k ( x , y , z ,t ) = Ak ( x , y )e

i  k z z −ωt +φk ( x , y ) 

= AFk A kM ( x , y ) e

i  k z z −ωt +φk ( x , y ) 

(2)

Fig. 1. Three-beam Mach-Zehnder interferometer. MO Microscope objective, PH Pinhole, L Convergence lens, BS Beam splitter, AF on-off amplitude filter, A Bean amplitude, and CCD camera.

where each field is considered to be coherent, linearly polarized, and traveling mainly on the z-direction as a plane wave, k z is the wave number and ω is the temporal frequency. A k represents the spatial variations of the amplitudes because of non-uniformity of illumination, nonlinearly of the detector, or some defect in the optical components. φk indicates the phase variations because of optical-path, the optical system’s aberrations, changes in the

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17423

temperature, atmospheric turbulence, mechanical vibrations, or objects under test, among others, and i = −1 is the imaginary unit. Let us first consider the sum E r = E 1 + E 3 leaving from the BS 5 or from BS 6 but with AF2 = 0 , whose description by using Eq. (2) can be simply written as [18,19]

{

}

E r ( x , y ) = A r ( x , y ) exp i  k z z − ωt + φr ( x , y )  ,

(3)

A r2 ( x , y ) = A12 ( x , y ) + A 32 ( x , y ) + 2A1 ( x , y ) A 3 ( x , y ) cos Δφ31 ( x , y ) ,

(4a)

where,

with Δφ31 ( x , y ) = φ3 ( x , y ) − φ1 ( x , y ) , and tan φr ( x , y ) =

A1 ( x , y ) sin φ1 ( x , y ) + A 3 ( x , y ) sin φ3 ( x , y )

A1 ( x , y ) cos φ1 ( x , y ) + A 3 ( x , y ) cos φ3 ( x , y )

,

(4b)

or, by considering the phase differences Δφ31 ( x , y ) , and Δφr 1 ( x , y ) = φr ( x , y ) − φ1 ( x , y ) , tan Δφr 1 ( x , y ) =

A 3 ( x , y ) sin Δφ31 ( x , y )

A1 ( x , y ) + A 3 ( x , y ) cos Δφ31 ( x , y )

,

(4c)

This indicates that both A r and φr depend on the amplitudes A1 and A 3 and the phases φ1 and φ3 (or its phase difference Δφ31 ). From Eq. (4c) it is easy to demonstrate that Δφr 1 = Δφ31 2 when A1 = A 3 . In this work, we will consider φ1 and φ3 spatially variable and temporally constant, but their difference Δφ31 has to be within the range ( 0, π ) . Thus, once

established φ1 and φ3 (or Δφ31 ), then the respective values of A r and φr (or Δφr 1 ) will only depend on the amplitudes A1 and A 3 . Note that, while Δφ31 is restricted within the range

( 0, π ) , Δφr 1

can take any value in the range [ 0, 2π ] , but the amplitudes have to take

negatives values as well. This is achieved by a change of phase by π radians in the fields. A description of practical interest occurs when A1 and A 3 are expressed in terms of A r and Δφr 1 [18] A1 ( x , y ) =

Ar ( x , y )

sin Δφ31 ( x , y ) A3 ( x , y ) =

sin  Δφ31 ( x , y ) − Δφr 1 ( x , y )  ; Δφ31 ≠ 0, π Ar ( x , y )

sin Δφ31 ( x , y )

sin Δφr 1 ( x , y )

(5a)

(5b)

since for a desired phase and amplitude value, the reference amplitudes are obtained directly from Eq. (5). However, the inconsistencies that appear when Δφ31 = 0, π do not allow the amplitude modulation, which describes the reference waves to be in phase or antiphase. The amplitude modulation is possible for three special cases when Δφ31 ≠ 0, π . For instance, if A r remains constant and Δφr 1 does not, the reference amplitudes in Eq. (5) have to obey the equation of an ellipse, as expressed in Eq. (4). In such equation, the reference amplitudes are seen as points on the plane, a situation known as the case of phase-only modulation (PM). On the contrary, if Δφr 1 remains constant and A r does not, we say we have the case of

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17424

amplitude-only modulation (AM). If A r and Δφr 1 are simultaneously variables, the corresponding case is named as the case of phase and amplitude modulation (PAM). Figure 2 shows these three cases for Δφ31 = 3π 4 , where the values of A r = 0.4, 0.7,1.0,1.3,1.6 and Δφr 1 = 2π n N with n = 0, , N − 1 for N = 16 have been chosen to illustrate these modulation cases. In PM the reference amplitudes A1 and A 3 are to be found as points over the perimeter of the ellipses for each value of A r with Δφr 1 as parameter, whereas in AM the reference waves are on a straight line where each pair of points is equivalent to be in another element of the same ellipse family obeying to A1 sin Δφr 1 + A 3 sin ( Δφr 1 − Δφ31 ) = 0 with Δφr 1 constant. This relation can be deduced from Eq. (5). In both cases, these lines are traced with solid lines. Finally, in the case of PAM the amplitudes are in both different ellipses and straight lines, which are indicated with dashed lines. In an experimental situation, the last case would be the predominant one because of the defects in the optical system and especially by the amplitude filters to be used.

Fig. 2. Phase, Amplitude, and Phase and amplitude modulation cases for Δφ31 = 3π 4 , where

Ar = 0.4, 0.7,1.0,1.3,1.6 for N = 16 phase-steps with a fundamental step of π 8 .

Now, let us consider the total optical field E leaving from the BS 6 given by the sum of three fields E 1 + E 2 + E 3 , whose intensity, observed by an optical detector (omitting coordinates), is given by [18] I ( A1 , A 2 , A 3 ) = A12 + A 22 + A 32 + 2A1A 2 cos φ + 2A1A 3 cos Δφ31 + 2A 2 A3 cos (φ − Δφ31 ) , (6a)

or, interpreted as the sum of only two fields E r + E 2 , the intensity is described as,

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17425

I ( A1 , A 2 , A 3 ) = A r2 ( A1 , A 3 ) + A 22 + 2A r ( A1 , A 3 ) A 2 cos (φ − Δφr 1 ) ,

(6b)

where φ = φ2 − φ1 represents the phase of the object as it is commonly done in a two-beam interferometer, such as it was indicated in Eq. (1). Δφr 1 becomes an additional phase term given by Eq. (4c), which depends on the amplitudes A1 and A 3 and the phase difference Δφ31 . From Ec. (6) the fringe visibility can be described by V ( A1 , A 2 , A 3 ) =

2A r ( A1 , A 3 ) A 2

A r2 ( A1 , A 3 ) + A 22

=

2A 2 A12 + A 32 + 2A1A 3 cos Δφ31 A12 + A 22 + A32 + 2A1A 3 cos Δφ31

,

(7)

Fig. 3. Non-quadrature amplitude modulation and its effects in changes of phase and visibility in an interferogram for the cases: PM, AM, and PAM.

whose variation depends principally on A1 , A 2 and A 3 , since Δφ31 is not modulated in the system. For the case of PM, the visibility V becomes unaltered when some spatial phase variations Δφr 1 are introduced in the interferogram. This becomes a phase-step when it is spatially constant, which is useful for PSI method. In contrast, in the case of AM, the visibility changes and the phase results remain unaltered, which is the most inadequate case for any PSI technique. In the case of PAM, both effects are present. A change in the visibility and the phase are observed simultaneously, which is also inadequate for a PSI approach. Figure 3 shows these three cases where, by simplicity, the same parameters of A r and Δφr 1

used in Fig. 2 and also A 2 = 1 , and φ = ( x 2 − y 2 ) 0.5 were taken into account for computing

these images, which have 300 × 300 pixels in resolution and are codified in 256 colormap hot levels. Each row shows seven interferograms changed in phase with the same visibility. Each column shows five interferograms with the same phase but with different visibility. These are the PM and AM cases respectively. The case of PAM is obtained when the interferograms are arbitrarily taken in different rows and columns. For instance, from the principal diagonal, five interferograms changed in phase but with different visibility are shown. In the numerical simulations shown in Fig. 2 and Fig. 3 for a value of Δφ31 = 3π 4 , constant spatially values for Δφr 1 and A k were considered for the three cases of modulation in NQAM: PM, AM, and PAM, and its changing effect in phase and visibility were also #183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17426

shown. As a first case, in an experimental situation, Δφ31 could be unknown and spatially variable, but with values within the range

( 0, π ) .

Thus, an important task would be to

measure it experimentally. As a second case, Δφr 1 and A k also must be considered spatially variable. In a third case, A k will be only digitally modulated but especially in binary mode, zero or maximum or on-off keying (OOK), which in general is known as quantized or digitalized NQAM. The modulation effects shown in Fig. 3 are useful to explain what will happen in a particular manner at each point ( x , y ) or in each pixel on the CCD camera used. For an experimental demonstration, the analytical deductions must be done from the measured intensities, as it would happen if an optical detector observes the fields leaving from BS 6 . Then, for the case of on-off amplitude modulation and by taking into account Eq. (2), the followings intensities can be stored, I ( A1 , 0, 0 ) = A12M = I 1 ,

(8a)

I ( 0, A 2 , 0 ) = A 22M = I 2 ,

(8b)

I ( 0, 0, A 3 ) = A 32M = I 3 ,

(8c)

I ( A1 , A 2 , 0 ) = A12M + A 22M + 2A1M A 2 M cos φ = I 12 ,

(8d)

I ( A1 , 0, A 3 ) = A12M + A 32M + 2A1M A 3 M cos Δφ31 = I r ,

(8e)

I ( 0, A 2 , A 3 ) = A 22M + A 32M + 2A 2 M A 3M cos (φ − Δφ31 ) = I 23

(8f)

I ( A1 , A 2 , A 3 ) = A r2 ( A1M , A 3M ) + A 22M + 2A r ( A1M , A 3 M ) A 2 M cos (φ − Δφr 1 ) = I (8g)

where it has been assumed only two values for AFk , 0 or 1. The trivial case I ( 0, 0, 0 ) = 0 has been omitted. From Eqs. (8a), (8c), and (8e) the phase difference Δφ31 is easily calculated as cos Δφ31 =

I r − I1 − I 3 2 I 1I 3

; Δφ31 ∈ ( 0, π ) ,

(9a)

with I 1 , I 3 ≠ 0 and by substituting this result and Eqs. (8a) and (8c) into Eq. (4c), it is possible to rewrite Δφr 1 as tan Δφr 1 =

4I 1 I 3 − ( I r − I 1 − I 3 ) I r + I1 − I 3

2

,

(9b)

here, if I 1 = 0 , from Eq. (4a), I r = I 3 . Therefore, Eq. (9b) is undefined, but from Eq. (4c) the solution gives Δφr 1 = Δφ31 , and for I 3 = 0 , from Eq. (4a) I r = I 1 . Therefore, from Eq. (9b) Δφr 1 = 0 , as it is expected. Alternatively, a simpler expression for Δφr 1 can be deduced from Eqs. (4c) and by using Eqs. (8a) and (8c), cos Δφr 1 =

#183957 - $15.00 USD (C) 2013 OSA

I r + I1 − I 3 2 I r I1

; Δφr 1 ∈ [ 0, Δφ31 ] ,

(9c)

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17427

Thus, the phase changes can be calculated from the measured intensities. From Eq. (8) the following normalized interferograms can be easily deduced I 12 = cos φ =

I 12 − I 1 − I 2 2 I 1I 2

(10a)

,

I = cos (φ − Δφr 1 ) =

I −Ir −I2

I 23 = cos (φ − Δφ31 ) =

I 23 − I 2 − I 3

2 IrI2

2 I 2I 3

(10b)

,

(10c)

,

Fig. 4. Numerical simulation of optical fields at the output of three-beam Mach-Zehnder interferometer: (a)-(c) Amplitudes and phases, (d1) Phase difference φ = φ2 − φ1 considered as the phase object, and (d2) Phase-difference Δφ31 = φ3 − φ1 within the range

( 0, π ) .

Fig. 5. Numerical simulation of phase variations because of binary NQAM: (a) Δφr 1 = 0 when

AF1 = 1 and AF3 = 0 , (b) Δφr1 ≈ Δφ31 2 when AF1 = AF3 = 1 , and (c)

Δφr1 = Δφ31 when AF1 = 0 and AF3 = 1 .

These patterns have the additional phases, 0, Δφr 1 , and Δφ , which can be known by means of Eq. (9). Note that, when A 3 = 0 from Eq. (9b) or (9c), Δφr 1 = 0 , which is the case of Eq. (10a). Similarly, Δφr 1 = Δφ31 takes place when A1 = 0 , which is the case of Eq. (10c). Thus, these two values of Δφr 1 are special cases of amplitude modulation that are located on the positive axes in Fig. 2, being the general case (Eq. (10b)) when A1 and A 3 are different

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17428

from zero. With these patterns, the desired phase φ can be obtained. For example, from Eq. (9), Eq. (10a) and (10c), we have tan φ =

I 23 − I 12 cos Δφ31 2I 1 ( I 23 − I 2 − I 3 ) − ( I 12 − I 1 − I 2 )( I r − I 1 − I 3 ) = , (11a) 2 I 12 sin Δφ31 ( I − I − I ) 4I I − ( I − I − I ) 12

1

2

1 3

r

1

3

or from Eq. (9), Eq. (10a) and (10b), it is possible to demonstrate that tan φ =

I − I 12 cos Δφr 1 2I 1 ( I − I r − I 2 ) − ( I 12 − I 1 − I 2 )( I r + I 1 − I 3 ) = , (11b) 2 I 12 sin Δφr 1 ( I − I − I ) 4I I − ( I − I − I ) 12

1

2

1 3

r

1

3

Also from Eq. (9), Eq. (10b) and (10c), it is possible to express tan φ =

I cos Δφ31 − I 23 cos Δφr 1 I sin Δφ31 − I 23 sin Δφr1

=

(I − I

r

− I 2 )( I r − I1 − I 3 ) − ( I 23 − I 2 − I 3 )( I r + I1 − I 3 )

(I − I

r

− I 23 − I 3 ) 4 I1 I 3 − ( I r − I1 − I 3 )

2

(11c)

These expressions tell us that the phase object can be extracted from the normalized interferograms and their additional phases, which are occasioned by the binary nonquadrature amplitude modulation of the reference fields 1 and 3. 3. Numerical simulation The theory exposed above is numerically verified by assuming for simplicity  ( x − x k )2 + ( y − y k A kM ( x , y ) = exp  −  σ k2 

)

2

 (x − ck  ; φk ( x , y ) =  

) − ( y −dk ) 2

hk2

2

, (12)

x 1 = 1, x 2 = 3, x 3 = 2 ; y 1 = 0.5, y 2 = 2, y 3 = 1.8 ; σ 1 = 14, σ 2 = 16, σ 3 = 18 ; with c1 = 2, c 2 = 0, c 3 = 1.5 ; d 1 = 3, d 2 = 0, d 3 = 3.5 ; and h1 = 50, h2 = 1.2, h3 = 13 , the fields leaving from BS 6 are particularly well described with a Gaussian amplitudes and hyperbolical phases, which take into account the spatial variations in the fields such as it would occur in an experimental situation. These amplitudes and phases are shown in Figs. 4(a)-(c), and the phase differences φ and Δφ31 computed from these data are depicted in Fig.

4(d), where the range for Δφ31 is within ( 0, π ) as it is required in our proposal. These images were evaluated on x ∈ ( −5,5 ) and y ∈ ( −5,5 ) in a rectangular grid with 300 × 300 in

resolution and codified in 256 colormap hot levels. In this simulation, only two values of amplitude will be considered, A k = 0 and A k = A kM , which are gotten with AFk = 0 and AFk = 1 , respectively. Of particular interest, the modulation of the reference amplitudes takes place in three possible phase variations in the interferogram: 0, Δφr 1 and Δφ31 , computed by Eq. (4c) and with the use of the data given in Fig. 4. These phases are shown in Figs. 5(a)-5(c) for the three cases of on-off NQAM: ( A1 , A3 ) = ( A1M , 0 ) , ( A1 , A3 ) = ( A1M , A3M ) , and ( A1 , A3 ) = ( 0, A3M ) , respectively. It is important to note that these phase variations have been computed from the fields given in Eq. (12), which are not compatible with an experimental situation, since these fields cannot be observed, only their intensities, but they were computed in this manner to be compared with a method based on the intensities, as follows. In order to proceed as happens in the experiment; the fringe patterns observed at the interferometer’s output as indicated in Eq. (8) are computed by using the fields in Eq. (12);

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17429

what is obtained is shown in Fig. (6). Note the visibility and phase variations between the interferograms in Figs. 6(d), 6(f) and 6(g). From these intensities, by using Eq. (9) the phase variations 0, Δφr 1 , and Δφ31 are obtained and compared by subtracting with those obtained from the amplitudes in Fig. 5 as shown in Fig. 7(1), as well as, by using Eq. (10) the normalized interferograms are obtained and depicted in Fig. 7(2). Finally, Figs. 8(a) and 8(b) show the wrapped and unwrapped phases, which are obtained by using Eq. (11)c) and by using a typical algorithm for phase integration [20]; the calculation of the wrapped phase with Eqs. (11a) and (11b) (not shown) gives the same results than as computed with Eq. (11c), shown in Fig. (8a). Figure 8(c) shows a comparison by subtraction between the theoretical and unwrapped phases, which gives a plausibility of the proposal method in this paper.

Fig. 6. Intensities obtained at interferometer’s output for the all cases of on-off amplitude modulation, except the trivial case, AF1 = AF2 = AF3 = 0 .

Fig. 7. Normalized interferograms and theirs phase spatial variations indicated above of each interferogram as result of on-off NQAM.

Fig. 8. (a) wrapped phase, (b) unwrapped phase, and (c) comparison by subtraction of proposed phase and unwrapped phase.

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17430

4. Experimental verification The schema depicted in Fig. 1 was implemented in the laboratory in order to prove the theory here exposed. It was illuminated with a green laser of 532 nm in wavelength; linearly polarized and stabilized in amplitude, model Verdi-2W. As well as, the beamsplitters BS 1 − BS 6 Newport model 10B20NP.24 of 1 inch in diameter and 50/50 at π 4 of incidence, three black sheets operating as on/off amplitude filters, a 10X microscope objective, and a lens L with focal distance f = 100mm were also used in this experiment. The fields leaving the BS 6 are observed with a CCD Pixelink camera model PL-B741U. The obtained intensities are digitalized and stored in the memory of a computer. The phase distribution is constructed by placing a slightly bended acetate sheet as the object, while the phase difference between the two reference fields is obtained by adjusting the optical path difference with the displacement of BS 2 or BS 4 . The desired range for Δφ31 is achieved when the observed intensity I r does not show dark and brilliant fringes simultaneously with

A2 = 0 , as can be observed in Fig. 9(e).

Fig. 9. Intensities obtained experimentally at the output of the three-beam Mach-Zehnder interferometer for all cases of binary amplitude modulation, except the trivial case,

AF1 = AF2 = AF3 = 0 .

Fig. 10. Experimental results obtained from the intensities in Fig. (9): (a)-(c) Normalized interferograms, (d)-(f) Phase difference variations introduced between each couple of interferograms.

Fig. 11. Phase retrieval from the experimental normalized interferograms and their phase difference variations: (a) wrapped phase and (b) unwrapped phase.

The experimental procedure is done as it was explained in the above section, where for the capture of intensities, the condition of AFk = 0 is carried out by simply blocking the passage

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17431

of the light with any opaque body such as a black sheet, while AFk = 1 is gotten with the free pass of the fields, so this binary amplitude filter is easy to implement in the laboratory making the use of amplitude filters such as a neutral density filter unnecessary. In these conditions the procedure begins by capturing the intensities as indicated in Eq. (8); they are shown in Fig. 9. With these experimental data, Δφ31 , Δφr 1 , I , I 12 , I 23 are computed as it is expressed in Eq. (9) and Eq. (10), the results are depicted in Fig. 10. By using Eq. (11), the wrapped phase is computed and shown in Fig. 11(a), while Fig. 11(b) shows the unwrapped phase. On the other hand, in order to verify qualitatively the phase variations obtained experimentally by using Eq. (9), we use a technique of generalized phase-shifting interferometry introduced by Farrel [7], which is based on an adjustment of ellipses, where the phase shift between two interferograms is considered to be spatially constant. Because in our case the phase variations are spatially variable, we apply Farrel’s method to measure in an approximated form the phase variations introduced experimentally by the binary NQAM. To implement this work we first choose a dominium with minimum fluctuations in Δφ31 and Δφr 1 , such as shown in Figs. 10(d) and 10(e) with blue lines. Figure 12(a) shows the estimation of the phase variations between the interferograms I 12 and I shown at its superior part, by using the Farrel’s method, and at its inferior part shown by a blue line, the ellipse formed with the experimental data extracted from the interferograms, the ellipse drawn with a red line is obtained with the adjusting method. The estimated value Δφ 'r 1 = 1.03 rad is comparable with the phase variations calculated with Eq. (10) and shown in Fig. 10(d), which are shown together in gray and blue lines, respectively. Similar comments for Fig. 12(b) and Fig. 12(c) are valid for estimations of Δφ '31 = 2.061 rad and Δφ '31 − Δφ 'r 1 = 1.053 rad, between I 12 and I 23 , and I and I 23 respectively. Note that in these three cases the estimated value Δφ 'r 1 = 1.03 rad is comparable with the value measured by using Eq. (9), this result demonstrates the viability of the experimental implementation.

Fig. 12. Qualitative estimation of phase variations between each couple of interferograms by using the adjustments of ellipses: (a)

5. Discussion and remarks We have presented a new method for phase retrieval in interferometry based on space on-off NQAM in the modality of PAM. This method introduces inhomogeneous variations in the visibility and phase. The corresponding theoretical model, numerical simulations, and experiments were successfully tested and presented in this paper. It is noteworthy that the mathematical model deduced here is valid for both analog and digital NQAM, as well as for the spatial aspect of a given optical wave, which is different from an electrical signal in communications that is considered one-dimensional with the time. For this reason, effects of coherence, diffraction, and polarization have to be taken into account in future studies, mainly because these parameters could alter the phase difference between the reference fields, but also their amplitudes could be varied by these effects, and therefore errors in the PAM could be generated by a contribution of both. First, the theory was tested with a numerical simulation, where spatial variations for the amplitudes and phase were considered. Also, a

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17432

method for obtaining a given object phase starting from the measured interferograms was shown. In the experimental tests, firstly Δφ31 was considered spatially variable within the range ( 0, π ) instead of constant as considered in [17–19]. Such conditions were obtained by

adjusting BS 2 or BS 4 . The assumption of this fact was due to the experimental conditions present in the laboratory, such as the mechanical calibration and the system’s phase local variations due to non-homogeneity of the illumination or aberrations of optical components, among other factors. Despite of this, we formulated a simple method for phase retrieval that minimizes errors produced by many of such deleterious factors. One of the consequences of considering the Δφ31 variable is the fact that the visibility and phase-shifts were also spatially variable instead of being constant as in the conventional theory of PSI and GPSI. As a second kind of experimental test, the digital NQAM was implemented in its more basic aspects such as the on-off modulation, both in two reference beams and a three-beam Mach-Zehnder interferometer. This opens the possibility of implementing NQAM in its analog version and in its digital mode for m -bits with m > 1 . Thus, it will have 2m levels of amplitude shifts for each beam, so N = 22 m − 1 both visibility and phase variations can be possible to attain. In this work, only the case m = 1 was implemented. For this reason, only three interferograms changing in visibility and phase were obtained. Also, it is important to say that on-off NQAM was carried out by using black sheets, with which the on-off amplitude filters were implemented. This shutting procedure is very practical and efficient, because the simple sheets are inexpensive and free of defects. This is because they do not introduce phase variations as a neutral density filter does due to defects of fabrication or inhomogeneity in the material, for instance. Also in this experimental verification, the maximum beams amplitude were considered to be spatially inhomogeneous in order to take into account defects in optical components, nonuniformity of illumination, or nonlinearty in the optical detector, among others. However these amplitudes had comparable values because they crossed four BS 50/50. For this reason, the phase variations introduced in the interferograms for the cases AFk = (1,1, 0 ) and AFk = (1,1,1) were about half compared with the interferogram obtained for the cases AFk = (1,1, 0 ) and AFk = ( 0,1,1) , which means Δφr 1 ≈ Δφ31 2 . This condition was verified qualitatively with the approximated measurements of the phase variations made by the method of adjustment of ellipses (Fig. 12). For the object phase retrieval by the on-off NQAM, only two conditions were required: the normalization of the interferograms that have phase variations and the measurement of these phase variations. It is worth mentioning that, in essence, the PSI method also needs the same two conditions for phase retrieval. In PSI such conditions are met assuming known phase-steps between interferograms with no visibility changes. Thus, a resoluble system of equations can be formed. This usually needs a phase shifter calibrated with high enough accuracy. Even though our proposed method does not follow these conditions in general, fortunately it could fulfill them and become a PSI method if the amplitude values are over an ellipse as it was established theoretically in [18,19], this being the case of PM generated by NQAM. In order to control this amplitude modulation, a careful calibration of the amplitude filters as well as its automated control has to be done, which will be a future work. In respect to the accuracy of this method, we can say that in general, space NQAM for PAM of the resulting field is susceptible to errors because of undesirable variations in the amplitudes and phases of reference fields, where these errors are noted in undesirable changes in both visibility and phase in an interferogram. These undesirable variations can be occasioned by several sources, for instance, by the spatial characteristics of the field, such as diffraction, coherence, polarization, scattering, or by experimental conditions such as mechanical vibrations, atmospheric turbulence, gradients of temperature, etc., and also by defects in the optical elements used in the experiment such as inhomogeneity in the materials,

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17433

fabrication defects, nonhomogeneity in the illumination, nonlinearly and nonhomogeneity of the detector, among others. A detailed study of errors accumulated by the effects cited above will be carried out in a future task. However, in a qualitative manner, we can say that this proposal such as discussed in this paper, is easy to implement and has high accuracy since the condition for Δφ31 is easy to generate experimentally and can be measured from the intensities. Additionally it is not limited to be spatially constant as in PSI method, and as a consequence it can have a spatial distribution. On the other hand, the implantation of on-off NQAM for introducing the phase variations in the interferogram was very inexpensive and easy to carry out, since it was done by unblocking and blocking the passage of the beam with some dark sheets. Additionally, the phase variations introduced and the normalization of interferograms were also obtained from the measured intensities. Thus the spatial distribution of the visibility introduced between each interferogram was removed and therefore the phase retrieval was successfully possible. Notice that, this phase retrieval was done without any sophisticated calibration of any optical or mechanical element to obtain Δφ31 or to generate the on-off amplitude modulation. Consequently, since the on-off amplitude modulation was done manually, each frame was captured at different times. Thus Δφ31 were exposed to temporal variations, which is the most significant source of error in the present method. The interferometric system is based on two Mach-Zehnder placed in series, which results in a system susceptible to mechanical errors because its beams travel through different components. However, we now believe that such error sources can be notably reduced if common-path cyclic interferometers or Sagnac interferometers are used as a base instead of Mach-Zehnder blocks. Also, the influences from the temporal variations of Δφ31 will be strongly reduced with on-off amplitude modulation carried out with an automatized system, such as optical switches or a modulation of a laser diode. Then, although the proposed set-up seems to have been sufficient to show the viability of the basic method, it also reveals some sensibilities that could be further reduced by redesigning more appropriate blocks as suggested. Such a possibility relies on the fact that the mentioned error sources are not inherent to the method, but are a consequence of its preliminary experimental implementation. On the other hand, the optical detector is a factor that also has influence in the error in phase and amplitude modulation because of its nonlinearity and its protection plate. The last introduces undesirable amplitude variations in the measured intensity that depends on the beam alignment. However this can be overcome and minimized when the interferogram is normalized as it is basically done in PSI that is carried out by subtracting two interferograms to eliminate a and by dividing the subtraction to eliminate b . In our case we captured the individual intensities I 1 , I 2 , I 3 and the composed intensities I r , I 12 , I 23 , and I ; in order to normalize the interferogram, which is carried out by subtracting and dividing as indicated in Eq. (10), then each spatial defect introduced by the detector is minimized as it is done in PSI method. Authors thank N. Keranen for her advice on wording. Acknowledgment U. Rivera-Ortega appreciates the scholarship from Consejo Nacional de Ciencia y Tecnología (México) under grant 209494. This work was partially supported by Consejo Nacional de Ciencia y Tecnología (México) under grant 166742 and by Vicerrectoría de Investigación y Estudios de Posgrado of Benemérita Universidad Autónoma de Puebla under grant MEFCEXC13-I. This work was experimentally carried out in the FCFM-BUAP’s “Laboratorio de Interferometria” facilities, which has been developing thanks to the CONACyT grants 124145 and 154984.

#183957 - $15.00 USD (C) 2013 OSA

Received 22 Jan 2013; revised 3 Mar 2013; accepted 4 Mar 2013; published 15 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017421 | OPTICS EXPRESS 17434