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We use photorefractive two-wave mixing for coherent amplification of the object beam in digital holographic recording. Both amplitude and phase reconstruction benefit from the prior amplification as they have an increased SNR. We experimentally verify that the amplification process does not affect the phase of the wavefield. This allows for digital holographic phase analysis after amplification. As the grating formation in photorefractive crystals is just driven by coherent light, the crystal works as a coherence gate. Thus the proposed combination allows for applying digital holography for imaging through scattering media, after the image bearing light is coherence gated and filtered out of scattered background. We show experimental proof-of principle results.
©2011 Optical Society of America
1. Introduction
Holography is a powerful tool for acquisition and reconstruction of 3D optical wave fields. A variety of holographic methods has been proposed since its invention in 1948 [1]. First techniques used holographic plates and films as the recording materials. They had the big disadvantage that the holograms were not erasable and had to be developed, which prohibited their usage in real time imaging [1, 2]. Photorefractive materials like crystals or polymers overcame this problem and allowed erasing the holograms, leading to improved conditions for holographic imaging [3–5]. The reconstruction of the hologram stored in both mentioned recording media, has to be performed optically. Therefore the reference beam illuminates the hologram and is diffracted by the stored grating, so that the object beam is reconstructed.
In digital holography (DH) the hologram is acquired by charge coupled device (CCD) or complementary metal-oxide semiconductor (CMOS) digital cameras [6, 7]. The reconstruction is performed numerically rather than optically. Naturally, the increasing computation possibilities and high quality CCD sensors have favoured the development of digital holography to a powerful field that gains growing attention and has found several applications in a variety of fields. It is for example used in optical metrology [8], shape measurement [9], particle image velocimetry [10], particle tracking [11], particle characterisation [12], non-destructive testing of MEMS [13], cell imaging [14] and tomography of biological specimen [15]. One big advantage of DH is the direct access to the phase of the optical field which opens up unique imaging possibilities [8– 15] like a very good interferometric depth-resolution in the low nm region. Like in classical holography the reconstruction in digital holography contains a DC-term and the twin images, namely the real and the virtual image. In the in-line configuration first proposed by Gabor [1] the mentioned three terms overlap and cannot be distinguished. To overcome this restriction, Leith and Upatnieks [16] introduced the off-axis geometry, where the beams enclose a small angle θ, which leads to a spatial separation of the three components if the angle is chosen properly, i.e. big enough. The finite pixel size of the CCD on the contrary enforces the angle to be kept small enough, so that the reconstruction of the hologram is feasible.
One major limitation of digital holography is that its applicability for imaging through scattering media is restricted, as scattered light forms background noise that saturates conventional intensity-integrating detectors [17] like e.g. CCD cameras. Thus the image quality may suffer substantially from low SNR due to too weak object beams in the presence of the mentioned scattered background noise. Nevertheless digital holography has already been used for imaging through scattering media [18–20].
Photorefractive materials on the other hand have the ability to reject incoherent scattered light, as it does not contribute to the grating formation since the photorefractive medium is sensitive to the spatial derivative of the incident intensity distribution only, rather than the intensity itself [21, 22]. Photorefractive crystals are thus usable as coherence gates [23, 24] that filter the image bearing coherent light out of incoherent background. Another remarkable feature of photorefractive crystals is the ability to couple two beams, named two-wave mixing (TWM). TWM has been widely studied [25, 26], and has found several different applications like image amplification [27], laser-beam cleanup [28], phase-conjugation [29], light-slowing down [30], build-up of oscillators [31], self-organizing laser cavities [32] novelty filters [33], optical interconnects [34], only to name a few. However, besides all advantages, two-wave mixing often suffers from light scattered by defects in the crystal. This scattered light is amplified by the same process that is used for image amplification, as the crystal cannot distinguish between the coherent image-bearing light and the light that is coherently scattered within the crystal. Thus the property of energy coupling leads to amplified noise that creates a fan towards the direction of the energy transfer, named beam-fanning. As this noise may seriously harm the image acquisition several solutions have been proposed to overcome this problem [35–37], mostly based on additional techniques, like rotation of the crystal, a third noise-erasing beam or using a pulsed read-out. An easier way to reduce beam fanning was proposed in [38,39], by only changing the wide-spread forward two-wave mixing geometry to a counterpropagating reflection geometry. When beam fanning is under control the setup for two-wave mixing is rather simple, which allows an easy embedding in other setups and combinations with other techniques. Photorefractive two-wave mixing has, for example, led to an improved performance of acousto-optic imaging [40, 41], which has also been used in combination with digital holography [42]. As two-wave mixing is a powerful discipline that inherently has the ability to amplify light coherently, it seems straightforward to think of a combination of this attribute with digital holography in order to overcome the mentioned limitation of digital holography in imaging through scattering media.
In this paper we analyze whether such a combination has the potential to improve the performance of DH. We demonstrate that under certain conditions the usage of photorefractive two-wave mixing in digital holography enhances the SNR of digital holographic amplitude and phase reconstructions remarkably and leads to an improved performance of digital holography. We also show first results that prove that this new technique has the potential to enable digital holographic analysis of samples in scattering environment as the coherent part of the light is gated and amplified.
2. Theory
2.1. Photorefractive two-beam coupling
The formation of photorefractive gratings follows four steps. An interference pattern incident on the crystal will photoexcite charge carriers (step I) at points of constructive interference. These carriers will move (II), driven by diffusion or drift processes and will be trapped at empty sites. A non-uniform space-charge distribution is the result. The redistributed charges create an electric field (III) that causes a local change of the refractive index Δn by means of the Pockels effect (IV) according to Eq. (1):
where n is the refractive index in the dark, reff is the effective electro-optic coefficient and ESC is the induced space charge field. The incident interference pattern is thus stored within a refractive index pattern in the photorefractive medium. Depending on the driving process for the carrier movement, there is a shift between those two patterns (Fig. 1 ). A purely diffusion driven process leads to a shift of . Drift dominated movement leads to.
Fig. 1 Recording of the index grating in transmission geometry. Object beam I1 (red) and pump beam I2 (purple) interfere within the crystal. The interference pattern creates a shifted index grating with the grating spacing Λ. + c denotes the positive c-axis of the crystal. The shift between the intensity and index grating leads to a coupling of the two beams. The weak object beam with the intensity I1 is amplified by a factor γ’.
In this paper we use contradirectional degenerate two-wave mixing. The coupled wave equations for this case can be expressed as follows [25]:
with the absorption coefficient α, the intensities of the interacting beams I1 and I2, the refractive index change due to the Pockels effect Δn, the Bragg-angle θ, the wavelength λ and the intensity coupling coefficient γ’. Equation (2-4) show that depending on the coupling coefficient γ’ there is an energy exchange between the two beams leading to an amplification of one beam and concurrently to an attenuation of the other. The coupling coefficient γ’ has a maximum for (Eq. (4). This proves that diffusion based crystals that inherently provide this shift, should allow for maximal coupling. The behaviour of the phases of the two beams during their interaction can be described using following Eq. (5-7) [25]: where β is the phase coupling coefficient. As it can be seen, the phase shift of , that leads to a maximum energy coupling, is accompanied with a phase coupling constant of β = 0, i.e. there is no change of the phase by the coupling process. This characteristic is very important for the present paper as it suggests that it is possible to use phase information of the reconstructed digital hologram for analysis after an amplification process. However, this assumption has to be verified experimentally.2.2. Numerical Reconstruction using the angular spectrum method
Now we want to introduce the procedure which we used for reconstruction of the digital holograms that are shown in the course of the paper. The reconstruction method is based on the angular spectrum method. If h(x,y,0) is the complex object wave field distribution at the hologram plane, the corresponding angular spectrum of the object can be calculated using the Fourier transform (Eq. (8):
where kx and ky are the spatial frequencies corresponding to x and y directions respectively.We use an off-axis geometry as introduced by Leith and Upatnieks [16] where the beams enclose a small angle φ, which leads to a spatial separation of the DC term, the real and the virtual image.
One first order component of the angular spectrum can be filtered by a numerical bandpass-filter [43]. The filtered field can be rewritten as the inverse Fourier-Transform of its filtered angular spectrum (Eq. (9):
From Fourier Optics it is known that the field can now be propagated towards any plane of interest using Eq. (10):
with [44].The reconstructed complex wavefield at any plane at the distance d perpendicular to the propagation axis is given by Eq. (11):
The phase of is distorted due to the aberrations caused by the optical components of the setup. This consequently leads to a distorted reconstruction. By recording a digital hologram of a plane reference surface we can take those distortions into account. In the experiments presented here we use a mirror for this purpose. We reconstruct the reference phase and subtract it from the original object phase. This leads to the phase of the object that may be tilted, as the tilting of the mirror and the object may differ. As the tilting may be too strong, leading to an increased sample depth, we use unwrapping techniques to reconstruct the wavefield.
3. Experiments
The aim of the paper is to analyze whether a combination of photorefractive two-wave mixing and digital holography can benefit of the capability of photorefractive crystals to coherence-gate coherent light out of scattered background. We will start the experimental part of this paper with two-wave mixing where we will analyze the capabilities of our setup for two-wave mixing amplification. Afterwards we will switch to digital holography and will give examples for the performance and quality of digital holographic reconstructions and will consider their limitations. In the last chapter we will show the results of the combined setups, where the amplified object beam is analyzed with digital holography. We will show the influence of the amplification process on amplitude and phase reconstructions and will present the results of the main experiment, where we use coherence gating and TWM before digital holographic reconstruction, at the end.
3.1. Two-wave mixing
We start the experiments with the implementation of a two-wave mixing setup. A sketch of the setup is shown in Fig. 2 . The beam of a HeNe-Laser is split into an object beam and a pump beam that is used for two-wave mixing amplification. The photorefractive medium we use is a 0°-cut Rh:BaTiO3 crystal with dimensions of 7.0 mm x 3.0 mm x 7.0 mm (c axis parallel to one of the long dimensions) and 1000 ppm Rhodium in the melt [45]. The object beam is perpendicularly incident onto the sample where it is reflected. The reflected light passes a 4f configuration with f = 150 mm. The sample used is a 1951 USAF test chart. A CCD chip which consists of 1392 x 1040 pixels with a pixel size of 6.45 μm x 6.45 μm is used to detect the object beam. The crystal is placed in the Fourier-plane between the two lenses. This geometry is advantageous for weak signal amplification [46], but may also cause problems, as the crystal has to be tilted. This leads to a decrease of the effective interaction region, as the crystal acts as a filter that may cut off high frequencies.
To overcome this restriction the usage of a 45°-cut crystal could be helpful as the desired angle between the crystal c-axis and the grating vector can be obtained without the need to tilt the crystal that much. The angle between the writing beams in our case is about 160° in air, which leads to a grating period of Λ=0.4 μm in the crystal. All beams are p-polarized, in order to make use of the largest electro-optic coefficient r42 of the BaTiO3 crystal which is oriented such that its c-axis lies in the plane of incidence. The crystal is used without external field, as the diffusion based configuration has the essential π/2-phase shift between intensity and index grating, needed for maximum energy coupling and minimum phase coupling. The usage of neutral density filters allows for setting a certain beam ratio r between the amplifying pump beam and the object beam. At first we choose a ratio of about r=40000 as the achievable gain depends strongly on the ratio and gets higher with higher ratios [3].
The photorefractive gain factor γ is defined as the intensity at the output of the crystal in the presence of the pump beam, divided by the intensity without the pump light [3]. In order to measure γ the CCD is replaced by a detector. The left part of Fig. 3 shows the pure object beam. The intensity of the object beam is measured to be approximately 2 nW. This is that weak that no image can be seen on the camera. The object beam is then blocked and the pump beam is directed towards the crystal. The pump light that is scattered towards the detector has a power of 10 nW. Now pump and object beams are unblocked and interfere within the crystal, writing a photorefractive reflection grating. This grating leads to energy coupling between the beams as described in the last chapter. It has to be noted that we illuminate the crystal for 10 minutes with the pump beam to generate carrier so that when the object beam is unblocked the resulting grating just needs to resort the carrier. This speeds up the grating formation and it takes between 5 and 10 seconds until the maximum gain value is reached. The object beam is amplified at the expense of the amplification beam. The resulting object beam has now a power of 1.7 µW. This corresponds to a gain factor of .
The resulting amplified image is shown on the right part of Fig. 3. The intensity is clearly enhanced and the sample is now visible. But as can be seen, the amplification does not take place homogeneously. This is due to the intensity profiles of the used laser beams that play an important role for the homogeneity of the amplification as was pointed out in [47]. The necessity to use the crystal in a tilted geometry in the Fourier plane also has a negative effect. Imperfections of the amplification process lead to a decreased resolution and a blur in the amplified images as can be seen in Fig. 4 . Figure 4a) shows elements 3 to 6 of the third group of our test chart. We use filters to create the weak intensity depicted in Fig. 4b). The pump beam at first is slightly misaligned from the ideal case, leading to the blurred amplified object beam with a strongly inhomogeneous amplitude distribution (Fig. 4c). Though in this example a good alignment leads to very good results shown in 4d), this blurring is present in nearly all images, as optimizing one part of the image leads to degraded quality of other parts.
Fig. 4 First row: a) Image, b) same image with decreased intensity, c+d) amplified images (γ=100) with slightly misaligned (c) and aligned (d) pump beam. Second row: The ‘4’ can be resolved without (e) but gets blurred with amplification (f).
Another example is shown in the bottom of Fig. 4. The character ‘4’ of element four of group four before amplification is shown in 4e). In this example the object beam was set to a sufficient power even without amplification. The ‘4’ is clearly resolved corresponding to a lateral resolution of about 25 µm. After amplification the opening of the amplified ‘4’ is not distinguishable (Fig. 4f) but the whole ‘4’ is blurred. Optimization of the alignment, such that the whole sample is completely homogeneously amplified is a challenge under these conditions. Nevertheless it should be possible to mitigate these problems and optimize the image quality by using the mentioned 45°-cut crystal and by adjusting and optimizing the beam profiles, the setup and the used equipment. However, for our proof of principle experiments the performance of the setup is sufficiently good even without these improvements.
To analyze the interferometric performance of two-wave mixing we record fringes, replacing the object by a mirror and again amplifying the object beam. The CCD records the resulting fringes. Exemplary results are shown in Fig. 5 . Here the amplification is just γ ≈10, as the power of the pure object beam is increased compared to the first example because we use a different sample. As it can be seen, the fringe visibility and thus signal to noise ratio is considerably increased by the amplification, resulting in a better interference contrast. It is also notable that the noise is clearly reduced. This proves that the amplification process also leads to improved interferometric conditions that should be advantageous for digital holography. This will be analyzed in more detail in section 3.3.
3.2. Digital holography
In the last chapter we introduced our setup and results for two-wave mixing. As the main topic of this paper is the combination of two-wave mixing with digital holography in the following chapter we will at first consider the performance and limits of “normal” digital holographic recording. For the experiments we use the Mach-Zehnder configuration sketched in Fig. 6 .
The beam of the HeNe-laser is again split into object and reference beams. The object beam is perpendicularly incident onto the sample where it is reflected. The test sample now consists of a metallic island which is deposited on Si-substrate. The island has lateral dimensions of about 1.0 mm x 1.3 mm and a height of about 200 nm. Comparable samples have already been used in [13]. The reflected light passes a 2f1 −2f2 configuration with focal lengths of f1 = 100 mm and f2 = 250 mm which corresponds to a magnification of the sample by about m = f2/f1≈2.5. The object beam interferes with the reference beam on the CCD. The angle between the reference beam and the object beam is approximately φ = 1° forming an off-axis geometry. The digital holograms are computed using the angular spectrum method introduced in section 2.2. In this configuration our sample covers 560x550 pixel of the CCD. At first, the diffuser plate shown in Fig. 6 is not placed in the setup. The reconstructed holograms are shown in Fig. 7 .
Fig. 7 Reconstructed digital holograms of the metallic island. Amplitude reconstruction (left), phase (middle) and unwrapped phase (right)
The left part of the figure shows the reconstructed amplitude of the metallic island. The corresponding wrapped phase is depicted in the middle part of the image. In order to get the quantitative height, we record a reference hologram of a mirror and subtract the reconstructed wrapped phases. Afterwards we apply an unwrapping algorithm that uses a phase derivative variance path [48] and obtain the unwrapped image on the right. The small tilt that can be seen in the unwrapped image can be removed numerically. We compare a linescan of the obtained surface profile to a profile measured with a profilometer (Ambios XP2). In order to analyze the mismatch we plot the difference of both profiles in Fig. 8 . As can be seen the mismatch is small, amounting to less then 0.1 π.
Fig. 8 Comparison of profiles obtained by digital holography and by a profilometer scan (left). Height mismatch of both profiles (right).
Next, we introduce a ground glass diffuser plate (Thorlabs DG10-600) into the object beam path as a scattering medium as shown in Fig. 6. The diffusing angle of the diffuser is ±10°. As is shown in the figure the diffuser is placed between the first lens and the beamsplitter in the object beam path. In this configuration the object beam passes the diffuser once. The diffuser is slightly tilted, so that the object beam path encloses an angle of approximately 5° to the perpendicular incidence. The variable attenuator is removed and thus the intensity of the object beam is increased. Again digital holograms are recorded and the recording procedure described above is repeated leading to the results shown in Fig. 9 . It is obvious that the digital holographic reconstruction is severely disturbed by the light that is scattered by the diffuser. Thus the image cannot be reconstructed.
Fig. 9 Reconstructed digital holograms of the metallic island with a diffuser plate in the beam path. Amplitude reconstruction (left), wrapped phase (middle) and unwrapped phase (right)
3.3. Amplified digital holography
In chapters 3.1 and 3.2 we verified that two-wave mixing amplification has a strong potential for amplifying weak beams. We also showed results that verified that our digital holography setup allows for analyzing the phase information of wavefields, leading to very good depth resolutions in the nm region. But we also showed a limitation of digital holography, as we introduced a diffuser into the setup which scattered the object beam very severely and prohibited the digital holographic reconstruction. In the following chapter we show that a combination of two-wave mixing and digital holography allows for improving the performance of digital holography. At first we will show the influence of the amplification on the amplitude- and phase reconstructions, before we will present the results of the main experiments.
The used setup is a combination of the setups shown in Figs. 2 and 6 as is depicted in Fig. 10 . We again use the 2f1-2f2 configuration introduced in chapter 3.2. Again, at first the diffuser is not placed in the beam path. We use the pure object beam and the amplified one and record digital holograms, in order to compare the results. It should be noted that the intensity ratio r in this example is set to be approximately 1200. Figure 11 shows the reconstructed amplitude images of the USAF test chart. It is evident that the reconstruction of the amplified beam (γ = 20) has a much better signal to noise ratio compared to the reconstruction of the weak beam without amplification. The reference beam power in this example is kept constant for both measurements at approximately 5 µW which is in the magnitude of the amplified object beam power. This naturally leads to significantly different ratios between the power of reference and object beams for the cases with and without amplification. With amplification the ratio p between reference and object beam is approximately p≈1, whereas for the case without amplification the ratio amounts to approximately p≈20. This difference has influence on the signal to noise ratio of the digital holographic reconstruction as it depends on the mentioned ratio. Weak object beams can benefit from very strong reference beams as they are coherently amplified [49]. In the example presented here this difference should lead to better SNR-values for the weak object beam than a ratio of p=1 would do. Nevertheless in the following considerations this circumstance is not further investigated as it is not within the main scope of this paper.
Fig. 10 Sketch of the setup for amplified digital holography. The inset marked with b) shows components that replace the marked part of the setup in later experiments (path lengths not to scale).
Fig. 11 Reconstructed amplitude without amplification (left), and with amplification (γ = 20) (right). The SNR of the reconstructed amplitude is clearly enhanced.
In order to quantify the increase in visibility and signal to noise ratio in our example, Fig. 12 shows an intensity plot of vertical line-scans, across the lines marked in Fig. 11. Thus the intensities of the reconstructed amplitude image with and without amplification can directly be compared, and an exemplary value of the SNR enhancement can be given. In the example plotted here, the SNR is improved by a factor of roughly 20. This leads to a better visibility of the reconstructed object and thus to improved reconstructions.
Fig. 12 Intensity of the line plot marked in the reconstructed Fig. 11 without amplification (black), and with amplification (red). The SNR of the reconstructed amplitude is clearly enhanced.
The example shows that the amplification of the weak object beam increases the SNR in the digital holographic reconstruction thus two-wave mixing is capable of supporting digital holography and enhancing its performance. Nevertheless the analysis so far just concentrated on the amplitude reconstruction. As mentioned, the extraordinary feature of DH is the access to the phase. As introduced, the usage of the crystal in a diffusion based regime is expected to lead to a conservation of the object beam phase, as the phase coupling coefficient becomes β=0. This should enable analyzing the amplified version of the reconstructed phase. This assumption is to be verified in the following.
Therefore we record digital holograms and amplified digital holograms. We process the obtained data as presented above. For reference we again use the mirror, but this time through the crystal with a blocked pump beam, i.e. without amplification. We unwrap the phase for both cases and thus we get the surface profiles corresponding to the amplified and non-amplified case. The height mismatch of a line-scan across the profiles is used to judge whether the phase changes significantly during the amplification process. The results are plotted in Fig. 13 . The intensity ratio r of pump and object beams is varied by tuning the variable attenuator thus changing the gain factor. While for “low” gain the phase mismatch is very small, increasing the intensity ratio and thus the gain, leads to variations of the height mismatch that reach 0.1*π as a worst case. This mismatch can be explained by numerical errors because of the unwrapping procedure. Another reason is that the shift between the index and intensity pattern is ideally , but in reality it has a certain distribution across the recorded grating that will surely differ from this ideal value. This will lead to an inhomogeneous distribution of β across the object beam [47]. As also the distribution of the intensity in the object beam is not homogeneous this will lead to an inhomogeneous amplification and altogether to a certain degree of phase coupling across the image. Nevertheless the results are sufficiently good, when compared to the scans in Fig. 8.
Fig. 13 Height mismatch obtained analyzing the linescans across the sample surface for amplified digital holography and non amplified digital holography.
In the following we want to analyze whether it is possible to apply two-wave mixing during digital holographic recording through the diffuser plate used in chapter 3.2.
As already mentioned the photorefractive effect should allow for coherence-gating the image bearing out of the scattered light of the object beam. Thus we place again the diffuser into the beam path. We started the experiment using the HeNe laser, but the obtained results were not satisfactory. The reason for this is, that the very long coherence length of the HeNe laser disturbs the measurement. Scattered light off the diffuser keeps its coherence to the reference and also interferes with the pump beam leading to degraded results. For the following experiments we change the light source and use a titanium sapphire laser in CW operation-mode instead of the HeNe laser as the coherence length is much smaller. Changing the laser and especially the used bandwidth, naturally also affects the TWM performance as the achievable amplification strongly depends on the used bandwidth [50]. When low coherence lengths are used in interferometric off-axis applications, beam walk-off [51] should always be considered. But in our setup this influence should be negligible. The coherence length is calculated to be about 2 mm.
In the following we want to make some considerations on what influence the amplification has on interferometric measurements in presence of scattering media. We want to show how the bandwidth of the light source affects the measurements and results. Therefore we introduce a basic model that aims at providing beam ratios that can be used for further experimental optimization. The goal of this simple model is the calculation of a rough estimate rather than a detailed quantitative simulation. The intensity Isum that is detected by the CCD amounts to, where Iinterf is the interference-term. Ibal is the intensity of the ballistic light that passes the scattering medium without being scattered. Icohscatt is the coherently scattered light which is slightly scattered but keeps its coherence. Thus it contributes to the grating formation within the crystal and is also amplified leading to undesired amplified speckle-background. Naturally this part of the light strongly depends on the bandwidth, i.e. the coherence length of the used source. Iref is the reference beam intensity and Iscatter is the incoherently scattered light. Iinterf is composed of three terms, as the image bearing object beam, the reference beam and the coherently scattered light interfere mutually. We obtain , with α1, α2 and α3 being the relative phases between object, reference and coherently scattered beams. For simplicity in the following we assume α1=α2=α3=0. We define γcoh1 to be the coherent amplification factor of the object beam and γcoh2 to be the amplification factor of the coherently scattered light. It has to be added that in the following we calculate the amplification just by the beam ratios and do not consider inhomogeneities of the intensity across the beam profile. The scattered fraction of the intensity is large compared to the coherent light but it is proportional and can thus be defined as Iscatter =k (Ibal+Icohscatt). The relation between image bearing and coherent light that has no image information can be written as Ibal = qIcohscatt. We express the reference intensity as a function of the complete intensity coming from the object beam path as .
The coherent amplifications γcoh1 and γcoh2 can be derived from the given formulas and the expression given in [25] to Eq. (12):
We measure γ’ to be ≈-32.9 cm−1. L is the interaction length of the two beams and . When the coherence length of the source is very short, the coherently scattered light can be neglected and the equations are simplified. For this case we can define the fringe modulation depth of the two-beam interference as . This leads to the expression for the modulation depth described in Eq. (13):
Using Eq. (12) and (13) we simulate the influence of the intensity ratios between the two-wave mixing pumping and the reference beam on the modulation depth recorded on the CCD. We estimate k = 50 for our diffuser. The results of the simulations are depicted on the left part of Fig. 14 . The amplification clearly enhances the modulation depth leading to improved conditions for interferometric recording. It has to be noted that an increase of the pump power leads to an increase of z and thus the maximum of the modulation depth can be reached with decreasing c. As mentioned, the coherently scattered light was neglected in this example. Therefore the complete modulation depth is image bearing information.
Fig. 14 Simulated influence of the beam ratios on the modulation-depth with and without amplification for (k = 50 and negligible Icohscatt) (left). Comparison of the influence of a highly coherent source and a low-coherent source on the results(right).
If a source with a longer coherence length is used, the coherently scattered light cannot be neglected. Then the modulation within the interference pattern will consist of the three terms described by Iinterf. The definition of the modulation depth as does not hold for this case of three beam interference as the three interference terms are not linearly independent. Furthermore m now contains image as well as speckle information.
To describe the ratio between image and speckle information in the modulation we define . As an overall quality criterion we use Q = rimg*m as for the image recovery both modulation depth and ratio rimg are important.
To investigate the influence of the bandwidth of the used source, we choose parameters for a highly coherent source describing the HeNe laser and one with shorter coherence length as the Ti:sapphire-laser. Using the HeNe-laser most of the scattered light will be scattered coherently which means that k≈0. It also leads to a small ratio q between image bearing and coherently scattered light. We convert the value for k we assumed in the foregoing case to values of k and q in the case with coherently scattered light and thus we obtain q = 0.02. In this case the amplified coherently scattered light will be the main contributor to the modulation in the interference pattern. The coherence length of the Ti:sapphire-laser is small compared to the HeNe-laser. Here most of the light will be scattered incoherently, leading to bigger k and q values. In this example we estimate k = 45 and q = 10. The results are depicted in the right part of Fig. 14. The plot shows the improvement which is offered by the amplification process. Using the laser source with the short coherence length leads to a clearly improved ΔQ compared to the highly coherent source. The model predicts that the results get better the shorter the coherence length is. Attention has to be paid, though, as a very short coherence length induces beam walk-off as already mentioned. This would lead to a decrease of the interaction region of the beams within the crystal, which would consequently result in a degraded amplification performance.
This indicates that two-wave mixing amplification could prove more advantageous for measurements using low coherence length sources. To validate this prediction we perform the experiment with a Ti:sapphire. Due to the use of the shorter coherence length source the setup has to be slightly changed, as now the path lengths have to be matched more accurately. In order to adjust the pump beam path length to the object beam path length the setup has to be changed as is shown in Fig. 10 when the inset b) replaces the marked part in the figure. The shown configuration that results allows for controlling the length of the pump beam path.
The power of the light from the object beam path is approximately Iobj≈1 mW. The power of the reference beam is Iref≈2 mW, which compensates for the difference in beam diameters. This power dependence is optimized for maximum modulation depth for the case without amplification [17]. The pump beam power is set to Ipump≈10 mW. Thus the intensity ratio between reference and object beam is r≈10. It has to be considered that now using the measured values we obtain r = Ipump/ (Ibal+Iscatter+Icohscatt).
At first we try to coherence gate and amplify the object beam scattered by the diffuser. The result with blocked pumping beam is shown on the left of Fig. 15 . The detected light in this case just consists of scattered light. Using two-wave mixing the coherent part of the light is successfully amplified as shown on the right of the same image. Though the amplification is rather inhomogeneous the whole sample is visible and its shape can be recognized. It is difficult to give values for the net amplification as it is not the complete intensity that is amplified. Following the definition used so far the gain amounts to γ ≈2.
Fig. 15 left: Object beam through diffuser. right: Object beam through diffuser and with two-wave mixing amplification.
In the following we use the beams shown in Fig. 15 for digital holographic recording. A line-scan across the acquired interference patterns is shown in Fig. 16 . We record both the reference beam and the light coming from the object beam path and subtract them from the acquired interference pattern. Thus we obtain just the modulated interference term. It is remarkable that the non-amplified interference pattern has a certain modulation. But as the reconstruction fails the origin seems not to be the object. The modulation depth of the amplified version is clearly enhanced. In this example it is increased by a factor of about 3.
Fig. 16 Line-scan across the acquired hologram with blocked and unblocked two-wave mixing pumping through the diffuser. The modulation depth of the amplified interference is clearly enhanced.
Naturally the digital holograms recorded without two-wave mixing show similar “bad” results as the ones recorded with the HeNe-laser. Without amplification it is not possible to reconstruct the metallic island. The reconstructed amplitude is depicted in Fig. 17 . The left part of the image shows the case without amplification. Here just scattered light is reconstructed. On the contrary, the amplified version clearly allows for reconstructing the sample as can be seen in the middle part of the image. The right side of Fig. 17 illustrates the line-scan across the lines marked in the left and middle-images. Again the amplified reconstruction has an inhomogeneous intensity distribution, similarly to the pure amplified object beam depicted in Fig. 14.
Fig. 17 left: Reconstructed amplitude of the object beam through the diffuser.middle: Reconstructed amplitude of the object beam through the diffuser with two-wave mixing amplification. right: Comparison of reconstructed intensities across the marked red lines.
A comparison to the similar line-scan across Fig. 14 shows the capability of digital holography to coherently amplify images if the reference beam used for DH recording has a much higher intensity than the object beam (Fig. 18 ) [51]. The noise in the reconstructed image is clearly reduced leading to an improved signal-to noise ratio. Thus the TWM image amplification also benefits from the coherent image amplification that digital holography offers.
Fig. 18 Line-scan across pure vs. reconstructed amplitude of amplified object beam. The image that is amplified with two-wave mixing benefits of the digital holographic capability of coherent amplification, when weak object beams are recorded with intense reference beams.
The reconstructed wrapped and unwrapped phases are depicted in Fig. 19 . The results prove that the phase reconstruction through the diffuser is enabled by amplified digital holography. Thus indeed two-wave mixing has the ability to improve the performance of DH. As a last step we want to compare the line-scan of the amplified version through the diffuser and the normal digital holographic recording. In order to do this, one further issue still has to be addressed. Though Fig. 19 shows that the phases contain information of the object after amplification, the unwrapping procedure we have used so far does not work here sufficiently well. Using a reference through the diffuser would be the most straightforward approach. But this was unfortunately not successful and led to results with very high noise. Alternatively one may use the same reference as before, i.e. without diffuser that led to the result shown in the second row of Fig. 19. Unfortunately this phase is wrongly reconstructed because if we subtract the phase of the reference from the object phase we do not remove the system-errors but we obtain a curved phase. The main problem is that the phase of the light that is not coherence gated and amplified, i.e. the scattered light, is completely destroyed due to the diffuser. Finally, to obtain the surface profile we apply a very minimalistic approach. We use the amplitude reconstruction to identify the borders of the sample. We treat the phases of the object and the reference without diffuser with a low-pass filter and subtract them. Afterwards we take a line-scan and manually unwrap the phase, which leads to the profile shown in Fig. 20 . The phase mismatch is plotted on the right side of the figure. The maximum error amounts to 0.2 π. This proves that two-wave mixing has the capability to enhance the performance of digital holography.
Fig. 19 . First row: without amplification. Second row: with amplification. And from left to right: wrapped phase and unwrapped phase.
Fig. 20 . (left) Comparison of profiles obtained by digital holography and amplified digital holography through the diffuser. (right) Height mismatch of both profiles. The unwrapping in this case was performed manually.
4. Conclusions
To conclude, we show that a combination of photorefractive two-wave mixing and digital holography has the potential to increase the performance of digital holography. As the photorefractive crystal works as a coherence gate using two-wave mixing in combination with digital holography allows for imaging through scattering media. The diffused object is filtered and amplified and can afterwards be recorded and processed as usual in DH. We have verified that the phase is not affected using the diffusion based crystal. This is the key for analyzing the phase of the wavefield after reconstruction. We also have shown that the reconstructed amplitude has a significantly improved SNR when two-wave mixing is introduced. The proposed combination of digital and photorefractive holography could open up new applications for digital holography for example for imaging through scattering media in biomedical applications.
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