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Wave optics analysis by phase-shifting real-time holographic interferometry
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Optik
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Optik 121 (2010) 80–88 www.elsevier.de/ijleo
Wave optics analysis by phase-shifting real-time holographic interferometry M.R.R. Gesualdia,, M. Muramatsub, D. Sogab, R.D. Paiva Jrb a Centro de Engenharia, Modelagem e Cieˆncias Sociais Aplicadas, Universidade Federal do ABC, Rua Santa Ade´lia 166, CEP 09.210-170, Bairro Bangu, CEP 09210-170, Santo Andre´ – SP, Brazil b Instituto de Fı´sica, Universidade de Sa˜o Paulo, Rua do Mata˜o, Travessa R 187, CEP 05508-900, Cidade Universita´ria, Sa˜o Paulo, SP, Brazil
Received 18 December 2007; accepted 26 May 2008
Abstract The purpose of this work is to study the potentialities in the phase-shifting real-time holographic interferometry using photorefractive crystals as the recording medium for wave-optics analysis in optical elements and non-linear optical materials. This technique was used for obtaining quantitative measurements from the phase distributions of the wave front of lens and lens systems along the propagation direction with in situ visualization, monitoring and analysis in real time. r 2008 Elsevier GmbH. All rights reserved. Keywords: Wave optics; Lens testing; Phase shifting; Real-time holography; Photorefractive crystals
1. Introduction The most important components of optical imaging and optical data processing systems are the lens or lens systems. A wave-optics analysis of the lens is in accordance with the geometrical optics theory, with the additional advantage that in the wave-optics approach is completely accorded for diffraction effects [1–2]. The interferometric methods are typically used in wave front analysis for lens and imaging quality evaluation. Two-beam interferometers have been used for the testing of optical components and optical systems. They present many advantages over other test methods: it is usually non-contact and does not damage fragile optical surfaces; in most cases, the interference Corresponding author. Tel.:+55 11 4437 8457.
E-mail address: [email protected] (M.R.R. Gesualdi). 0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.05.019
fringe patterns are diagnostic of the errors in the test piece and are easily interpreted; the quantitative and qualitative analysis in wave front aberrations; further, can be null tests. The Twyman–Green and Mach–Zehnder are the most popular interferometers in classical wave aberration measurements [2–6]. On the other hand, the holographic interferometry techniques are powerful optical methods for wave front analysis in the field of non-destructive testing [5–7]. In recent years, the use of the photorefractive sillenite crystals [8,9] as holographic recording medium has added a new dimension to holographic interferometry since the writing–reading hologram process is made in real time [10]. In this direction, the phase-shifting real-time holographic interferometry (PS-RTHI) technique [11–18] using sillenite crystal serves to capture holographic interferograms in real time and in the interferogram analysis by spatial phase measurement:
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the phase-shifting technique. In this technique, a variation of the wave front is observed directly in real time, with sequential reading of holographic interferograms and the phase map calculation. It can be used on quantitative measurements in static and dynamic processes in surfaces of opaque [11–14,18] and transparent objects [15–18]. So, the possibilities in optical testing can be interesting applications of this technique, where it is possible to obtain the intensity and phase distribution from the wave front in any arbitrary plane located between the optical element and the recording plane. To the best of our knowledge, possibility of this technique for reconstructing wave front optical phase has not been fully explored in the wave front sensing area for optical testing. Quantitative determination of the wave front phase allows investigation of the modifications suffered by the wave front through phase-distorting media: optical elements and systems [16–17], lens with aberrations or ground-glass screen [7–18], non-linear optical materials [19] and other cases. In this paper, we describe a phase-shifting real-time holographic interferometer using photorefractive sillenite crystal as the holographic recording medium, which is suitable for performing quantitative measurements in lens from the intensity and phase distributions of the wave front for different locations along the propagation direction in real time. Also, we introduced, for the first time to our knowledge, the potentialities of this technique to study light-induced lens effect in non-linear optical materials with in situ visualization, monitoring and analysis in real time.
2. Theoretical background 2.1. Real-time holography using photorefractive sillenite crystals In real-time holographic interferometry (RTHI) the hologram is recorded, and during the readout process the object is also illuminated. Since the hologram and the object remain in their original positions, the resulting image is the superposition of the holographically reconstructed object wave front and the wave front coming directly from the object. Each perturbation on the object produces fringes on its image due to the interference between the diffracted wave front and the modified object wave front. Let us consider the incidence of two coherent monochromatic waves onto a sillenite crystal. For holographic recording by diffusion, the hologram diffraction efficiency is given by [10–14] Z¼
pDn sin rL l cos y rL
2
m2
(1)
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where l is the recording wavelength, r is the crystal rotatory power, L is the crystal thickness, m is the modulation of the incident interference pattern and 2y is the angle between the interfering beams. The refractive index modulation of the hologram is written as Dn ¼ n3o r41 E sc =2, where n0 is the refractive index, r41 is the linear electro-optic coefficient of the sillenite crystal and Esc is the electric field generated by the redistributed space charges in the crystal. If t is the hologram response (build-up or erasure) time, the light intensity I0 at a point (x, y) resulting from the overlap of the wave diffracted by the hologram with the transmitted wave coming directly from the object is given by [12–14,18] I 0 ðx; yÞ ¼ I 0;T ðx; yÞ þ I 0;D ðx; yÞ½1 eðt=tÞ 2
(2)
The term I0,T(x,y) is the transmitted object wave and I0,D(x,y) is the holographic reconstruction of the object wave. The latter is written as [12,13,18] I 0;D ðx; yÞ ¼ I 0;O ðx; yÞZ þ I 0;R ðx; yÞ½1 Z þ 2gY cos F (3) In the relation above, I0,O(x,y) and I0,R(x,y) are the object beam and the reference beam intensities, respectively, g is a parameter expressing the polarization coupling of both beams, Y ½Zð1 ZÞI 0;O ðx; yÞI 0;R ðx; yÞ1=2 is the interference term and F is the phase shift on the object beam. The interferogram phase map can be determined by capturing a sequence of four frames. Between each pair of frames a p/2-phase change is introduced in the reference beam, thus changing the interferogram intensity. The intensity of the nth interferogram I0n(x, y) at the point (x, y) is written as[6,12] ðn 1Þp n ¼ 1; 2; 3; 4 I 0n ðx; yÞ ¼ I 0 ðx; yÞ cos2 Fðx; yÞ þ 2 (4) By employing trigonometric relations and combining the intensities one obtains the wrapped phase F(x, y) [6,12–14]: I 04 ðx; yÞ I 02 ðx; yÞ (5) Fðx; yÞ ¼ arctan I 01 ðx; yÞ I 03 ðx; yÞ As the frames are acquired the hologram starts erasing, leading to a temporal decrease of the interferogram visibility according to exp(t/t). In the experiments this can be overcome by shortening the acquisition time and thus making this variation negligible. For this reason the variation of the interferogram visibility in Eqs. (4) and (5) was neglected. In reference the measurement error due to the hologram erasure time was quantitatively analyzed and the optimal value of t, which does minimize such error, was determined. By calculating the phase F(x, y), a 2-D phase map is built and the phase of a given point is represented in a gray level diagram. The 256 gray levels are a measure of
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values between p(black) and p(white) distributed in an image system of 8 bits. The resulting wrapped pattern looks like a border extending from white to black or from black to white. This phase modulation can be removed through several so-called unwrapping methods. In this work the color unwrapping phase map was obtained through the Cellular Automata Method [12,20].
2.2. Wave-optics analysis In the wave-optics analysis for lens evaluation the interferometric methods are usually employed. The Twyman–Green and Mach–Zehnder interferometers are the most popular in classical wave aberration measurements. In our case, we use holographic interferometry combined to the phase-shifting technique for detecting phase variations in the wave front across a plane immediately behind the lens. This analysis is in accordance with geometrical optics, with the additional advantage that in the wave optics approach diffraction effects are completely accounted for. The lens simply delays an incident wave front by an amount proportional to its thickness at each wave front point. The total phase delay suffered by the wave at coordinates (x, y) by passing through a lens may be represented as a multiplicative phase transformation of the form tL(x, y). The complex field U0 L(x, y) across a plane immediately behind the lens is then related to the complex field UL(x, y) incident on a plane immediately at the lens input by [1] U 0L ðx; yÞ ¼ tL ðx; yÞU L ðx; yÞ
(6)
The physical meaning of the lens transformation can be best understood by considering the effect of the spherical lens on a normally incident, unit-amplitude plane wave. Hence, the lens transformation tL(x, y) is described by [1,18] p (7) tL ðx; yÞ ¼ exp½iFðx; yÞ ¼ exp i ðx2 þ y2 Þ lf Considering the input field distribution as unitary, the field after the lens can be written as [1,18] p (8) U 0L ðx; yÞ ¼ exp i ðx2 þ y2 Þ lf where f is the focal length of the lens
3. Optical setup Fig. 1 shows the PS-RTHI experimental setup applied in the wave-optics analysis of optical components. The holographic interferogram is recording in a BSO crystal cut in the transverse electro-optic configuration with dimensions 10 mm 10 mm 3 mm. Using the aniso-
Fig. 1. The phase-shifting real-time holographic interferometry (PS-RTHI) with photorefractive BSO crystals setup, where the light source is an argon ion laser (l ¼ 514,5 nm); M1, M2, M3, M4 and M5 are plane mirrors; BS1 and BS2 are beamsplitters; SF1 and SF2 are spatial filters; L1 and L2 are lenses; LT is the analyzed lens; Po1 and Po2 are polarizers; PZLT/M is a piezoelectric system; BSO is the photorefractive crystal; CCD is a camera; and a computer for digital acquisition and processing data.
tropic diffraction properties of the crystal, the incident beams are linearly polarized by the polarizer at the crystal input, so that the transmitted object and the diffracted waves are orthogonally polarized at the BSO output in a diffusion recording regime. Then it is possible to cut-off or to control the intensities of the transmitted object wave. We used an argon ion laser to supply the recording laser beam emitting at l ¼ 514.5 nm because of the high sensitivity of the BSO crystal for this wavelength. The recording laser beam (power 300 mW) was divided by a beam splitter (BS1) into reference and object beams. These beams have approximately similar optical paths. The reference beam is reflected by a mirror in the PZLT system and collimated by a spatial filter (SF2) and lens (L2) illuminating the recording crystal. The object beam across the variable filter for controlling intensity is collimated by a spatial filter (SF1) and lens (L1), then it is reflected by a beam splitter (BS2) and mirror (M5), so we have the plane wave illuminating the recording crystal (the first hologram), and across the lens test (LT) and illuminating the recording crystal again. Then the interference pattern is formed by diffracted hologram waves and transmitted object waves. We worked in the conditions in which the BSO crystal presented optimal performance for RTHI by diffusion regimen: the angle (a) in the range is from 40 to 501 for response time (t) of 10 s; the ratio between reference and object beam intensities was I0, R(x,y)/I0, O(x,y) ¼ 6.0 and I0, O(x,y)Z/I0,R(x,y)(1Z) ¼ 1, it is in accordance with Ref. [13].
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A phase-shifting PZLT system, a piezoelectrically driven mirror (PZTL) and a step driver, was used to control the phase variation in the reference wave (Df ¼ 0, p/2, p, 3p/2), and then we applied the fourframes technique [5]. The acquisition of digital holographic interferogram images was made with a CCD camera connected to a computer with a frame grabber. These interferograms were analyzed with software developed to calculate the phase and generate a wrapped phase map. The cellular-automata unwrapping technique was used to unwrap the phase map. As listed in references [11–13], the possible errors of the system are due to (1) miscalibration of the phase shifter; (2) spurious reflections and diffraction; (3) quality limitations of the optical components; (4) nonlinearities and resolution of CCD; (5) air turbulence and vibrations, and photorefractive errors, like (6) temporal modulation of holographic interferograms and (7) temporal fluctuation of thermal dependence on the photorefractive effect. Then, a previous calibration of the PZLT system was necessary in order to reduce the errors generated in the phase-shifter system. The effects of air drift and undesired low-frequency vibrations during the holographic recording in a perturbed environment are the most prejudicial error source to the measurement accuracy and to the holographic recording itself. All such effects were minimized by mounting the holographic setup on a table top with pneumatic isolators and by performing the holographic recordings with no ambient air conditioning.
Fig. 2. The details of experimental wave-optics analysis by PS-RTHI, where the wave plane IS crosses a test lens LT with focal length f located at a distance d of recording medium, it shows there is the wave front transformed by (a) a positive lens and (b) a negative lens.
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The wave-optics analysis experimental details by PS-RTHI is shown in Fig. 2, where the wave plane IT crosses a test lens LT with focal length f located at distance d of recording medium. In the figure the wave front was transformed by (a) a positive lens and (b) a negative lens.
4. Experiments and results In Fig. 3(a), we first performed a holographic recording of a plane wave as the object beam. A spherical, positive test lens LT, with nominal focal length fN ¼ 50.0 mm and d ¼ 215.0 mm, was inserted into the object beam path and a second holographic exposure was performed. The holographic reconstruction presented the interference of the plane and the spherical waves with the typically circular, concentric interference fringes as shown in Fig. 3(b). Applying the four-frame phase-shifting procedure we calculated the wrapped phase map of the wave front (Fig. 3(c))
Fig. 3. Diagram of process to obtain the wave front of a optical component, where (a) is the hologram recording plane wave front (150 150 pixels); (b) is the phase-shift holographic interferograms (F ¼ 0, p/2, p, 3p/2, 2p) of the spherical positive lens; (c) is the wrapped phase map by four-frame technique; (d) is the unwrapped phase map; and (e) 3D wave front reconstruction.
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Fig. 4. Wave front analysis for spherical lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for different nominal focal length fN: (a) 25.0 mm, (b) 50.0 mm, (c) 100.0 mm and (d) 150.0 mm, for d ¼ 215.0 mm.
and the unwrapped phase map (Fig. 3(d)). Fig. 3(e) shows the 3D wave front reconstruction for a 50 mm focal length spherical lens. Fig. 4 shows the phase distributions of the positive lenses for different nominal focal lengths fN: (a) 25.0 mm, (b) 50.0 mm, (c) 100.0 mm and (d) 150.0 mm, for d ¼ 215.0 mm. It is possible to see that the thickness of the fringes decreases when the value of fN increases. This is due to the reduction of the distance from the focal point to the crystal, see Fig. 2. Therefore, it is possible to compare two lenses and to see which has the biggest focal distance. And we can explain the big size of the spot at the center of the phase map to be due to the small size of the phase map, approximately 2 mm 2 mm on the axis of the lens.
Fig. 5 presents the phase distributions of the negative lenses with different nominal focal lengths fN: (a) 79.0 mm and (b) 139.0 mm, for d ¼ 215.0 mm. In this case the thickness of the fringes increases when fN increases. This behavior was expected for this kind of lenses. In Fig. 6, the test lens is a positive cylindrical lens, with nominal focal length fN ¼ 100.0 mm, for d ¼ 215.0 mm, where we obtained the phase distributions. Fig. 7 presents the phase distributions of two positive lens with same nominal focal lengths fN ¼ 50.0 mm, for d ¼ 215.0 mm. But in case (a) the test lens is perfect and in (b) the test lens presents scratches on its surface. The presence of the scratches on the lens introduces noises in the phase map. It is possible to analyze the optical quality of thin lenses.
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Fig. 5. The PS-RTHI with a spherical negative lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for two nominal focal length fN: (a) 79.0 and (b) 139.0 mm.
Fig. 6. The PS-RTHI with a cylindrical positive lens, where column (I) is the wrapped phase map, column (II) is the unwrapped phase map and column (III) is the 3D wave front reconstruction by test lens fN ¼ 100 mm, for d ¼ 215.0 mm, (150 150 pixels).
Fig. 8 presents the phase distributions of two positive lens with same nominal focal length fN ¼ 50.0 mm, but in case (a) d ¼ 115.0 mm and for (b) d ¼ 215.0 mm. Again the thickness of the fringes increases when the value of d increases. This case is similar to the decrease in the value of fN because there is an increase in the distance from the focal point to the crystal, then the thickness of fringes increases. Finally, the light-induced lens effects in non-linear optical materials may cause self-focusing or selfdefocusing of laser beam depending on the properties of each material. The known technique to study lightinduced lens effect analyses the transmittance in far field, see references [21–23]. Some works study the lightinduced lens by changes in diameter of the transmitted beam intensity using a CCD camera [24]. In this work,
we introduce the possibilities of the PS-RTHI for wave optics analysis in light-induced lens formed in a non-linear optical material, where the transmitted beam intensity and the phase shifts in wave front can be measured directly with monitoring, visualization and real-time analysis. Analysis from the intensity and phase changes in transmitted beam because of the light-induced lens effect in non-linear medium is very important in studies of the non-linear optical advanced materials [21]. Some measurements in two non-linear optical crystals using PS-RTHI are showing here. Fig. 9(a) shows the PS-RTHI analysis of a positive lens with the focal length fN ¼ 50.0 mm located at d ¼ 215.0 mm. In Fig. 9(b), a Bi12TiO20 sample with transverse electro-optic
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Fig. 7. Wave front analysis of two spherical lens with fN ¼ 50 mm, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction: (a) perfect lens and (b) scratches lens, for d ¼ 215.0 mm.
Fig. 8. The PS-RTHI with a spherical positive lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for nominal focal length fN ¼ 50.0 mm: (a) for d ¼ 115.0 mm and (b) for d ¼ 215.0 mm.
configuration and thickness 3.0 mm was added with test lens sample. Consequently, the self-focusing effect due to thermo-optical effects, and, self-defocusing effect due to photorefractive and photo-chromic effects (without external electric field) to produce anisotropic phase change DF(x, y) provide the astigmatic wave front in light-induced astigmatic lens generated by focused laser beam. Fig. 9(c), the LiNbO3:Fe(0.025 ppm Fe2+)
presents the self-defocusing effect due to photovoltaic effect [21–23], without external electric field, and the other light-induced effects are neglected in this case. Thus, light-induced lens effect by wave-optics analysis, the self-defocusing of beam waist to produce increases of phase change DF(x, y), and spherical wave front in light-induced spherical lens are generated by laser beam.
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Fig. 9. Analysis in light-induced lens in two non-linear optical crystals by PS-RTHI, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map, column (III) is the 3D wave front reconstruction and column (IV) is detailed 3D wave front reconstruction in the x-plane and y-plane: in (a) a positive lens; in (b) a positive lens added a Bi12TiO20 sample, the astigmatic wave front in light-induced astigmatic lens generated by laser beam; and in (c) a positive lens added a LiNbO3:Fe(0.025 ppmFe2+) sample presents increases of phase change DF(x, y), and spherical wave front in light-induced spherical lens generated by laser beam.
5. Conclusions
Acknowledgments
In this paper, we present an investigation of the potentialities of the PS-RTHI technique for wave-optics analysis. This technique was used to obtain quantitative measurements from the phase distributions of the wave front for different types and locations of lenses along the propagation direction. The lens quality can be evaluated by examining the distortion of the wave front with in situ visualization, monitoring and analysis in real-time. Also, we presented the potentialities of the study of light-induced wave front changes by selffocusing and self-defocusing effects in samples of Bi12TiO20 and LiNbO3:Fe crystals. The PS-RTHI presents some advantages related to the conventional techniques for studies in non-linear optical materials: measurements direct from the phase distributions; the diffraction effect is considerate; the phase depth analysis is doing with visualization, monitoring and in real-time.
This work was supported by the Conselho Nacional de Desenvolvimento Cientı´ fico e Tecnolo´gico (CNPq) and Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´ vel Superior (Capes).
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Optik
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Optik 121 (2010) 80–88 www.elsevier.de/ijleo
Wave optics analysis by phase-shifting real-time holographic interferometry M.R.R. Gesualdia,, M. Muramatsub, D. Sogab, R.D. Paiva Jrb a Centro de Engenharia, Modelagem e Cieˆncias Sociais Aplicadas, Universidade Federal do ABC, Rua Santa Ade´lia 166, CEP 09.210-170, Bairro Bangu, CEP 09210-170, Santo Andre´ – SP, Brazil b Instituto de Fı´sica, Universidade de Sa˜o Paulo, Rua do Mata˜o, Travessa R 187, CEP 05508-900, Cidade Universita´ria, Sa˜o Paulo, SP, Brazil
Received 18 December 2007; accepted 26 May 2008
Abstract The purpose of this work is to study the potentialities in the phase-shifting real-time holographic interferometry using photorefractive crystals as the recording medium for wave-optics analysis in optical elements and non-linear optical materials. This technique was used for obtaining quantitative measurements from the phase distributions of the wave front of lens and lens systems along the propagation direction with in situ visualization, monitoring and analysis in real time. r 2008 Elsevier GmbH. All rights reserved. Keywords: Wave optics; Lens testing; Phase shifting; Real-time holography; Photorefractive crystals
1. Introduction The most important components of optical imaging and optical data processing systems are the lens or lens systems. A wave-optics analysis of the lens is in accordance with the geometrical optics theory, with the additional advantage that in the wave-optics approach is completely accorded for diffraction effects [1–2]. The interferometric methods are typically used in wave front analysis for lens and imaging quality evaluation. Two-beam interferometers have been used for the testing of optical components and optical systems. They present many advantages over other test methods: it is usually non-contact and does not damage fragile optical surfaces; in most cases, the interference Corresponding author. Tel.:+55 11 4437 8457.
E-mail address: [email protected] (M.R.R. Gesualdi). 0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.05.019
fringe patterns are diagnostic of the errors in the test piece and are easily interpreted; the quantitative and qualitative analysis in wave front aberrations; further, can be null tests. The Twyman–Green and Mach–Zehnder are the most popular interferometers in classical wave aberration measurements [2–6]. On the other hand, the holographic interferometry techniques are powerful optical methods for wave front analysis in the field of non-destructive testing [5–7]. In recent years, the use of the photorefractive sillenite crystals [8,9] as holographic recording medium has added a new dimension to holographic interferometry since the writing–reading hologram process is made in real time [10]. In this direction, the phase-shifting real-time holographic interferometry (PS-RTHI) technique [11–18] using sillenite crystal serves to capture holographic interferograms in real time and in the interferogram analysis by spatial phase measurement:
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the phase-shifting technique. In this technique, a variation of the wave front is observed directly in real time, with sequential reading of holographic interferograms and the phase map calculation. It can be used on quantitative measurements in static and dynamic processes in surfaces of opaque [11–14,18] and transparent objects [15–18]. So, the possibilities in optical testing can be interesting applications of this technique, where it is possible to obtain the intensity and phase distribution from the wave front in any arbitrary plane located between the optical element and the recording plane. To the best of our knowledge, possibility of this technique for reconstructing wave front optical phase has not been fully explored in the wave front sensing area for optical testing. Quantitative determination of the wave front phase allows investigation of the modifications suffered by the wave front through phase-distorting media: optical elements and systems [16–17], lens with aberrations or ground-glass screen [7–18], non-linear optical materials [19] and other cases. In this paper, we describe a phase-shifting real-time holographic interferometer using photorefractive sillenite crystal as the holographic recording medium, which is suitable for performing quantitative measurements in lens from the intensity and phase distributions of the wave front for different locations along the propagation direction in real time. Also, we introduced, for the first time to our knowledge, the potentialities of this technique to study light-induced lens effect in non-linear optical materials with in situ visualization, monitoring and analysis in real time.
2. Theoretical background 2.1. Real-time holography using photorefractive sillenite crystals In real-time holographic interferometry (RTHI) the hologram is recorded, and during the readout process the object is also illuminated. Since the hologram and the object remain in their original positions, the resulting image is the superposition of the holographically reconstructed object wave front and the wave front coming directly from the object. Each perturbation on the object produces fringes on its image due to the interference between the diffracted wave front and the modified object wave front. Let us consider the incidence of two coherent monochromatic waves onto a sillenite crystal. For holographic recording by diffusion, the hologram diffraction efficiency is given by [10–14] Z¼
pDn sin rL l cos y rL
2
m2
(1)
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where l is the recording wavelength, r is the crystal rotatory power, L is the crystal thickness, m is the modulation of the incident interference pattern and 2y is the angle between the interfering beams. The refractive index modulation of the hologram is written as Dn ¼ n3o r41 E sc =2, where n0 is the refractive index, r41 is the linear electro-optic coefficient of the sillenite crystal and Esc is the electric field generated by the redistributed space charges in the crystal. If t is the hologram response (build-up or erasure) time, the light intensity I0 at a point (x, y) resulting from the overlap of the wave diffracted by the hologram with the transmitted wave coming directly from the object is given by [12–14,18] I 0 ðx; yÞ ¼ I 0;T ðx; yÞ þ I 0;D ðx; yÞ½1 eðt=tÞ 2
(2)
The term I0,T(x,y) is the transmitted object wave and I0,D(x,y) is the holographic reconstruction of the object wave. The latter is written as [12,13,18] I 0;D ðx; yÞ ¼ I 0;O ðx; yÞZ þ I 0;R ðx; yÞ½1 Z þ 2gY cos F (3) In the relation above, I0,O(x,y) and I0,R(x,y) are the object beam and the reference beam intensities, respectively, g is a parameter expressing the polarization coupling of both beams, Y ½Zð1 ZÞI 0;O ðx; yÞI 0;R ðx; yÞ1=2 is the interference term and F is the phase shift on the object beam. The interferogram phase map can be determined by capturing a sequence of four frames. Between each pair of frames a p/2-phase change is introduced in the reference beam, thus changing the interferogram intensity. The intensity of the nth interferogram I0n(x, y) at the point (x, y) is written as[6,12] ðn 1Þp n ¼ 1; 2; 3; 4 I 0n ðx; yÞ ¼ I 0 ðx; yÞ cos2 Fðx; yÞ þ 2 (4) By employing trigonometric relations and combining the intensities one obtains the wrapped phase F(x, y) [6,12–14]: I 04 ðx; yÞ I 02 ðx; yÞ (5) Fðx; yÞ ¼ arctan I 01 ðx; yÞ I 03 ðx; yÞ As the frames are acquired the hologram starts erasing, leading to a temporal decrease of the interferogram visibility according to exp(t/t). In the experiments this can be overcome by shortening the acquisition time and thus making this variation negligible. For this reason the variation of the interferogram visibility in Eqs. (4) and (5) was neglected. In reference the measurement error due to the hologram erasure time was quantitatively analyzed and the optimal value of t, which does minimize such error, was determined. By calculating the phase F(x, y), a 2-D phase map is built and the phase of a given point is represented in a gray level diagram. The 256 gray levels are a measure of
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values between p(black) and p(white) distributed in an image system of 8 bits. The resulting wrapped pattern looks like a border extending from white to black or from black to white. This phase modulation can be removed through several so-called unwrapping methods. In this work the color unwrapping phase map was obtained through the Cellular Automata Method [12,20].
2.2. Wave-optics analysis In the wave-optics analysis for lens evaluation the interferometric methods are usually employed. The Twyman–Green and Mach–Zehnder interferometers are the most popular in classical wave aberration measurements. In our case, we use holographic interferometry combined to the phase-shifting technique for detecting phase variations in the wave front across a plane immediately behind the lens. This analysis is in accordance with geometrical optics, with the additional advantage that in the wave optics approach diffraction effects are completely accounted for. The lens simply delays an incident wave front by an amount proportional to its thickness at each wave front point. The total phase delay suffered by the wave at coordinates (x, y) by passing through a lens may be represented as a multiplicative phase transformation of the form tL(x, y). The complex field U0 L(x, y) across a plane immediately behind the lens is then related to the complex field UL(x, y) incident on a plane immediately at the lens input by [1] U 0L ðx; yÞ ¼ tL ðx; yÞU L ðx; yÞ
(6)
The physical meaning of the lens transformation can be best understood by considering the effect of the spherical lens on a normally incident, unit-amplitude plane wave. Hence, the lens transformation tL(x, y) is described by [1,18] p (7) tL ðx; yÞ ¼ exp½iFðx; yÞ ¼ exp i ðx2 þ y2 Þ lf Considering the input field distribution as unitary, the field after the lens can be written as [1,18] p (8) U 0L ðx; yÞ ¼ exp i ðx2 þ y2 Þ lf where f is the focal length of the lens
3. Optical setup Fig. 1 shows the PS-RTHI experimental setup applied in the wave-optics analysis of optical components. The holographic interferogram is recording in a BSO crystal cut in the transverse electro-optic configuration with dimensions 10 mm 10 mm 3 mm. Using the aniso-
Fig. 1. The phase-shifting real-time holographic interferometry (PS-RTHI) with photorefractive BSO crystals setup, where the light source is an argon ion laser (l ¼ 514,5 nm); M1, M2, M3, M4 and M5 are plane mirrors; BS1 and BS2 are beamsplitters; SF1 and SF2 are spatial filters; L1 and L2 are lenses; LT is the analyzed lens; Po1 and Po2 are polarizers; PZLT/M is a piezoelectric system; BSO is the photorefractive crystal; CCD is a camera; and a computer for digital acquisition and processing data.
tropic diffraction properties of the crystal, the incident beams are linearly polarized by the polarizer at the crystal input, so that the transmitted object and the diffracted waves are orthogonally polarized at the BSO output in a diffusion recording regime. Then it is possible to cut-off or to control the intensities of the transmitted object wave. We used an argon ion laser to supply the recording laser beam emitting at l ¼ 514.5 nm because of the high sensitivity of the BSO crystal for this wavelength. The recording laser beam (power 300 mW) was divided by a beam splitter (BS1) into reference and object beams. These beams have approximately similar optical paths. The reference beam is reflected by a mirror in the PZLT system and collimated by a spatial filter (SF2) and lens (L2) illuminating the recording crystal. The object beam across the variable filter for controlling intensity is collimated by a spatial filter (SF1) and lens (L1), then it is reflected by a beam splitter (BS2) and mirror (M5), so we have the plane wave illuminating the recording crystal (the first hologram), and across the lens test (LT) and illuminating the recording crystal again. Then the interference pattern is formed by diffracted hologram waves and transmitted object waves. We worked in the conditions in which the BSO crystal presented optimal performance for RTHI by diffusion regimen: the angle (a) in the range is from 40 to 501 for response time (t) of 10 s; the ratio between reference and object beam intensities was I0, R(x,y)/I0, O(x,y) ¼ 6.0 and I0, O(x,y)Z/I0,R(x,y)(1Z) ¼ 1, it is in accordance with Ref. [13].
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A phase-shifting PZLT system, a piezoelectrically driven mirror (PZTL) and a step driver, was used to control the phase variation in the reference wave (Df ¼ 0, p/2, p, 3p/2), and then we applied the fourframes technique [5]. The acquisition of digital holographic interferogram images was made with a CCD camera connected to a computer with a frame grabber. These interferograms were analyzed with software developed to calculate the phase and generate a wrapped phase map. The cellular-automata unwrapping technique was used to unwrap the phase map. As listed in references [11–13], the possible errors of the system are due to (1) miscalibration of the phase shifter; (2) spurious reflections and diffraction; (3) quality limitations of the optical components; (4) nonlinearities and resolution of CCD; (5) air turbulence and vibrations, and photorefractive errors, like (6) temporal modulation of holographic interferograms and (7) temporal fluctuation of thermal dependence on the photorefractive effect. Then, a previous calibration of the PZLT system was necessary in order to reduce the errors generated in the phase-shifter system. The effects of air drift and undesired low-frequency vibrations during the holographic recording in a perturbed environment are the most prejudicial error source to the measurement accuracy and to the holographic recording itself. All such effects were minimized by mounting the holographic setup on a table top with pneumatic isolators and by performing the holographic recordings with no ambient air conditioning.
Fig. 2. The details of experimental wave-optics analysis by PS-RTHI, where the wave plane IS crosses a test lens LT with focal length f located at a distance d of recording medium, it shows there is the wave front transformed by (a) a positive lens and (b) a negative lens.
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The wave-optics analysis experimental details by PS-RTHI is shown in Fig. 2, where the wave plane IT crosses a test lens LT with focal length f located at distance d of recording medium. In the figure the wave front was transformed by (a) a positive lens and (b) a negative lens.
4. Experiments and results In Fig. 3(a), we first performed a holographic recording of a plane wave as the object beam. A spherical, positive test lens LT, with nominal focal length fN ¼ 50.0 mm and d ¼ 215.0 mm, was inserted into the object beam path and a second holographic exposure was performed. The holographic reconstruction presented the interference of the plane and the spherical waves with the typically circular, concentric interference fringes as shown in Fig. 3(b). Applying the four-frame phase-shifting procedure we calculated the wrapped phase map of the wave front (Fig. 3(c))
Fig. 3. Diagram of process to obtain the wave front of a optical component, where (a) is the hologram recording plane wave front (150 150 pixels); (b) is the phase-shift holographic interferograms (F ¼ 0, p/2, p, 3p/2, 2p) of the spherical positive lens; (c) is the wrapped phase map by four-frame technique; (d) is the unwrapped phase map; and (e) 3D wave front reconstruction.
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Fig. 4. Wave front analysis for spherical lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for different nominal focal length fN: (a) 25.0 mm, (b) 50.0 mm, (c) 100.0 mm and (d) 150.0 mm, for d ¼ 215.0 mm.
and the unwrapped phase map (Fig. 3(d)). Fig. 3(e) shows the 3D wave front reconstruction for a 50 mm focal length spherical lens. Fig. 4 shows the phase distributions of the positive lenses for different nominal focal lengths fN: (a) 25.0 mm, (b) 50.0 mm, (c) 100.0 mm and (d) 150.0 mm, for d ¼ 215.0 mm. It is possible to see that the thickness of the fringes decreases when the value of fN increases. This is due to the reduction of the distance from the focal point to the crystal, see Fig. 2. Therefore, it is possible to compare two lenses and to see which has the biggest focal distance. And we can explain the big size of the spot at the center of the phase map to be due to the small size of the phase map, approximately 2 mm 2 mm on the axis of the lens.
Fig. 5 presents the phase distributions of the negative lenses with different nominal focal lengths fN: (a) 79.0 mm and (b) 139.0 mm, for d ¼ 215.0 mm. In this case the thickness of the fringes increases when fN increases. This behavior was expected for this kind of lenses. In Fig. 6, the test lens is a positive cylindrical lens, with nominal focal length fN ¼ 100.0 mm, for d ¼ 215.0 mm, where we obtained the phase distributions. Fig. 7 presents the phase distributions of two positive lens with same nominal focal lengths fN ¼ 50.0 mm, for d ¼ 215.0 mm. But in case (a) the test lens is perfect and in (b) the test lens presents scratches on its surface. The presence of the scratches on the lens introduces noises in the phase map. It is possible to analyze the optical quality of thin lenses.
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Fig. 5. The PS-RTHI with a spherical negative lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for two nominal focal length fN: (a) 79.0 and (b) 139.0 mm.
Fig. 6. The PS-RTHI with a cylindrical positive lens, where column (I) is the wrapped phase map, column (II) is the unwrapped phase map and column (III) is the 3D wave front reconstruction by test lens fN ¼ 100 mm, for d ¼ 215.0 mm, (150 150 pixels).
Fig. 8 presents the phase distributions of two positive lens with same nominal focal length fN ¼ 50.0 mm, but in case (a) d ¼ 115.0 mm and for (b) d ¼ 215.0 mm. Again the thickness of the fringes increases when the value of d increases. This case is similar to the decrease in the value of fN because there is an increase in the distance from the focal point to the crystal, then the thickness of fringes increases. Finally, the light-induced lens effects in non-linear optical materials may cause self-focusing or selfdefocusing of laser beam depending on the properties of each material. The known technique to study lightinduced lens effect analyses the transmittance in far field, see references [21–23]. Some works study the lightinduced lens by changes in diameter of the transmitted beam intensity using a CCD camera [24]. In this work,
we introduce the possibilities of the PS-RTHI for wave optics analysis in light-induced lens formed in a non-linear optical material, where the transmitted beam intensity and the phase shifts in wave front can be measured directly with monitoring, visualization and real-time analysis. Analysis from the intensity and phase changes in transmitted beam because of the light-induced lens effect in non-linear medium is very important in studies of the non-linear optical advanced materials [21]. Some measurements in two non-linear optical crystals using PS-RTHI are showing here. Fig. 9(a) shows the PS-RTHI analysis of a positive lens with the focal length fN ¼ 50.0 mm located at d ¼ 215.0 mm. In Fig. 9(b), a Bi12TiO20 sample with transverse electro-optic
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Fig. 7. Wave front analysis of two spherical lens with fN ¼ 50 mm, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction: (a) perfect lens and (b) scratches lens, for d ¼ 215.0 mm.
Fig. 8. The PS-RTHI with a spherical positive lens, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map; column (III) is the 3D wave front reconstruction for nominal focal length fN ¼ 50.0 mm: (a) for d ¼ 115.0 mm and (b) for d ¼ 215.0 mm.
configuration and thickness 3.0 mm was added with test lens sample. Consequently, the self-focusing effect due to thermo-optical effects, and, self-defocusing effect due to photorefractive and photo-chromic effects (without external electric field) to produce anisotropic phase change DF(x, y) provide the astigmatic wave front in light-induced astigmatic lens generated by focused laser beam. Fig. 9(c), the LiNbO3:Fe(0.025 ppm Fe2+)
presents the self-defocusing effect due to photovoltaic effect [21–23], without external electric field, and the other light-induced effects are neglected in this case. Thus, light-induced lens effect by wave-optics analysis, the self-defocusing of beam waist to produce increases of phase change DF(x, y), and spherical wave front in light-induced spherical lens are generated by laser beam.
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Fig. 9. Analysis in light-induced lens in two non-linear optical crystals by PS-RTHI, where column (I) is the wrapped phase map (150 150 pixels), column (II) is the unwrapped phase map, column (III) is the 3D wave front reconstruction and column (IV) is detailed 3D wave front reconstruction in the x-plane and y-plane: in (a) a positive lens; in (b) a positive lens added a Bi12TiO20 sample, the astigmatic wave front in light-induced astigmatic lens generated by laser beam; and in (c) a positive lens added a LiNbO3:Fe(0.025 ppmFe2+) sample presents increases of phase change DF(x, y), and spherical wave front in light-induced spherical lens generated by laser beam.
5. Conclusions
Acknowledgments
In this paper, we present an investigation of the potentialities of the PS-RTHI technique for wave-optics analysis. This technique was used to obtain quantitative measurements from the phase distributions of the wave front for different types and locations of lenses along the propagation direction. The lens quality can be evaluated by examining the distortion of the wave front with in situ visualization, monitoring and analysis in real-time. Also, we presented the potentialities of the study of light-induced wave front changes by selffocusing and self-defocusing effects in samples of Bi12TiO20 and LiNbO3:Fe crystals. The PS-RTHI presents some advantages related to the conventional techniques for studies in non-linear optical materials: measurements direct from the phase distributions; the diffraction effect is considerate; the phase depth analysis is doing with visualization, monitoring and in real-time.
This work was supported by the Conselho Nacional de Desenvolvimento Cientı´ fico e Tecnolo´gico (CNPq) and Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´ vel Superior (Capes).
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