Femtosecond micromachining is attractive for the fabrication of photonic devices embedded in glass such as waveguides, lenses, and computer-generated volume holograms [1–4]. Recently, a new effect, femtosecond laser induced birefringence in glass, was observed [5–9]. Birefringence in an otherwise isotropic glass results from nanogratings created through the interaction of the incident light field and the bulk electron plasma wave. The corrugation of these nanogratings is orthogonal to the polarization of the writing laser beam and can exhibit long range periodicity [6, 9]. From a macroscopic viewpoint, the material is anisotropic and its properties can be explained based on the effect of form birefringence. In effect, because the periodicity of the nanogratings is smaller than the wavelength of visible light, these nanogratings act as micro-waveplates with the fast axis normal to the grating lines.

In this Letter we present the design, fabrication, and characterization of polarization selective computer-generated holograms (PSCGH) based on femtosecond laser induced birefringence. A PSCGH performs different diffractive optical functions for different orthogonal polarizations by implementing independent phase or amplitude profiles for each polarization [10–19]. Such capability has application, for example, in optical interconnects and front end optical sensing. In the past PSCGH’s have been realized by etching birefringent crystal substrates to form surface relief profiles [10–13]. The birefringence of the substrates forms different wavefronts for orthogonal polarizations. In general, two surface relief substrates are needed to accurately encode the two polarization dependent wavefronts. Therefore, the fabrication and packaging efforts are high. These methods rely on lithographic processing of birefringent crystals, which is not suited for integrated photonics. Another approach for polarization selective devices is based on form birefringence [14–19]. However, optical form birefringence requires high-resolution lithography that typically limits the range of application to the near infrared [14, 16, 19]. Consequently, femtosecond laser induced birefringence provides an attractive alternative for three-dimensional fabrication and integration of PSCGH’s that operate with visible light.

Two independent wavefronts for each of two orthogonal polarizations can be realized by controlling a two-dimensional array of micro-waveplates. In our case, we control the phase difference between the fast and slow axes of each micro-waveplate by varying the energy of the writing beam and the number of birefringent layers written at different depths. We control the orientation of the principal axis by rotating the polarization of the writing beam.

In our design, the PSCGH implements two phase-only functions, φ ^∥ (x_i,y_j) and φ ^⊥(x_i, y_j), one for each one of the two linear orthogonal polarizations (see Fig. 1). Here, (x_i,y_j) are the Cartesian coordinates of each sample. The phase functions are designed so that their Fourier transforms (FT) provide the desired polarization dependent reconstructions. φ ^∥ is quantized to a given number of phase levels while φ ^⊥ is designed in such a way that it can only differ from φ ^∥ by either Δφ or -Δφ, where Δφ is a constant number:

φ ^⊥ is thus a pseudo-binary phase function in the sense that its phase at each sampling point can only take one of two values relative to the base phase function φ ^∥.

In practice, Δφ is implemented with micro-waveplates and corresponds to the phase difference between the fast and slow axes. Therefore, Δφ = φ _fast - φ _slow where φ _fast and φ _slow are the phase delays relative to bulk glass corresponding to fast and slow axis polarization respectively. The sign of Δφ in a specific cell is controlled by assuming one of two orthogonal orientations of the micro-waveplate.

φ ^∥ and φ ^⊥ are simultaneously encoded by a detour phase method [20] and the degree of freedom in the sign of Δφ. Accordingly, a cell is defined for each sampling point and a sliding rectangle encodes the correspnding phase. As opposed to typical cell encoding methods, the sliding rectangle is not opaque but birefringent and implemented with a micro-waveplate. Therefore, in addition to the detour phase, φ ^detour (x_i,y_j), light accumulates a phase of either φ _fast or φslow, depending on the polarization and the orientation of the micro-waveplate. Hence, once φ ^∥ and φ ^⊥ are calculated, the required φ ^detour is obtained as follows

Since adding a constant phase does not affect the reconstruction, it follows that

As a consequence, when φ ^∥ is encoded for the ∥ polarization in a compensated detour phase, according to Eqs. (3), φ ^⊥ is simultaneously encoded for the ⊥ polarization according to the right hand side of Eqs. (3). So far this design only requires knowledge of the phase difference (Δφ) between fast and slow axes. However, the absolute phase relative to the bulk glass plays an important role in terms of quality, efficiency, and polarization contrast ratio of the reconstruction. For example, it turns out that the best absolute phases correspond to the symmetric cases (φ _fast = (2π - Δφ)/2 and φ _fast = Δφ/2 .

 

Fig. 1. Schematic of the PSCGH cell design according to Eq. (2). Two phase functions are encoded by shifting a square birefringent window within each cell (φ ^detour) and by rotating the micro-waveplate axis.

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We now show that this encoding provides an asymmetric reconstruction for +1 and -1 orders instead of a conjugate symmetric reconstruction as in the case of regular CGHs that use classical detour cell encoding. Moreover, we show that by switching the polarization of the incident light between the two states, ∥ and ⊥, the +1 and -1 orders are interchanged. Indeed, when FT{φ ^∥} is realized at +1 order, FT{-φ ^⊥(x_i, y_j)} is realized at -1 order and vice versa. This property is explained if we consider that the reconstruction from the detour phase coding produces a π change in phase from the +1 order to the -1 order while the phase difference introduced by the micro-waveplate is the same for both orders. From Eq. 3 if φ ^∥ generates the +1 order, the phase is

then the phase for the -1 order is

which is essentially - φ ^⊥.

We use a projection algorithm to design the two phase functions [21, 22]. First, we calculate φ ^∥ using a phase-only constraint. In a second step, we enforce the constraint of Eq. (1) for the calculation of the pseudo-binary phase φ ^⊥ during a second iterative process. It is not surprising to find out that Δφ has to be π/2 rad in order to have the best annihilation of the reconstruction of φ ^∥ because it provides a π rad swing among different samples.

In the experiments we used a regenerative Ti: Sapphire amplifier (λ=800 nm, 100 kHz, 60 fs, 100mW) . Attenuators and polarizers controlled the pulse energy, a half wave plate after the last polarizer rotated the polarization of the writing beam, and a shutter controlled the pulse output. The laser beam was focused 40 μm below the surface of a fused silica slab with an objective lens of numerical aperture NA=0.65. The sample was mounted on a three-dimensional computer controlled motorized stage that moved at a speed of 300 μm/s.

 

Fig. 2. (a) Microscope image of a 2 × 9 array of 5 μm side micro-waveplates placed between two crossed polarizers. The principal axes of neighboring squares are rotated by 20 degrees. b) Measured retardance between fast and slow axes, Δφ, for double layers as a function of layer separation in depth (black). The retardance for single layers fabricated at the same depths, 40 μm (red) and 45–65 μm (green), are shown for reference. Given all parameters, like pulse energy, duration, and translation speed, the graph shows that the total retardance can be controlled by the separation between the two layers. Measurements were performed at 514 nm.

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First, to characterize the femtosecond writing process, we fabricated a set of sample micro-waveplates and the phase difference between the fast and slow axes was obtained by measuring transmission through the waveplate sandwiched between two crossed polarizers as a function of its rotation angle [8]. It was found that under our experimental conditions a π/2 phase difference could not be achieved in a single layer. In order to achieve the required π/2 phase difference, two layers of micro-waveplates were stacked along the propagation direction of the beam. Figure 2(a) shows a microscope image of a series of 5 μm side laser-induced waveplates located between crossed polarizers. Each waveplate has different orientation of its principal axis from 0° to 360°, so the right column reproduces the same result as the left column. Each micro-waveplate was realized by writing multiple lines with a line separation of 0.5 μm. The graph of Fig. 2(b) presents data of the measured phase difference, Δφ, as a function of the separation between layers. With separations above 15 m the phase difference for two layers of waveplates was equal to the sum of the phase difference of the individual layers. However, when the separation was smaller, the value deviated from the sum of the individual layers, which was explained by the overlapping of the two layers.

As an example of PSCGH design, we encoded in 100×100 cells the University of Colorado’s CU logo as the polarization selective target image, with the letter C as the FT of the base phase function φ ^∥, and the letter U as the FT of the pseudo-binary phase φ ^⊥. The designed PSCGH was fabricated considering the measured phase retardance and writing two layers with a depth separation of 7.5 μm. Each layer was realized by writing first the micro-waveplates with one orientation and then the micro-waveplates with the orthogonal orientation. The cell size was 20 × 20 μm while the whole PSCGH size was 2 × 2mm. φ ^∥ was quantized to ten phase levels to provide an accurate representation and to accommodate ten orthogonal positions of the sliding window within the cell. Each micro-waveplate implemented a birefringent sliding rectangular aperture whose size was 4 × 20 μm. A large sliding aperture size helps increase diffraction efficiency and suppresses higher orders. Moreover, it also assures that diffraction effects are not significant when light propagates through each micro-waveplate.

Figure 3 shows a microscope image of a detail of the fabricated PSCGH and the far field reconstruction. The reconstruction was realized by illuminating the PSCGH with a 514 nm laser source. A polarizer and a half wave plate were used to control the polarization of the incident beam, while a digital camera was used to capture the far field images. As expected, the far field reconstruction is asymmetric as a consequence of the encoding.

Figure 4 shows the simulated and experimental results of the reconstruction for different linear polarization states. The bottom part of Fig. 4 shows the reconstruction of the PSCGH as the polarization of the illuminating beam is rotated. As expected, the relative intensity of each reconstruction depends on the angle of polarization of the incident beam. The experimental images were captured with a CCD camera at the Fourier plane of a lens of 200 mm focal length. The measured diffraction efficiency was 1.8% for the U and 2.8% for the C, which are typical values for detour phase encoding. This difference in the maximum intensities results from the nature of the two encoded phase functions, one pseudo-binary and one with ten-level phase encoding. For comparison, depending on the value of the absolute phase, the simulated efficiency values for the letter C ranged from 1.6% to 6.8% and for the letter U from 1.2% to 9%. Unfortunately, it is not possible to arbitrarily control the absolute phase simultaneously with the phase retardance to achieve the higher efficiencies. The polarization for the best reconstruction of C and the polarization for the best reconstruction of U were indeed orthogonal as expected. The reconstruction of the letter C essentially disappears when the reconstruction of the letter U is maximized and vice versa. Here, we define the contrast ratio as the ratio of the integrated intensity of the desired reconstruction and the integrated intensity of the undesired reconstruction. The polarization contrast ratio for the reconstruction of the letters C and U were 12 and 4.6 respectively.

 

Fig. 3. Left: Microscope image of a detail of the fabricated PSCGH. Right: far field reconstruction. Note that the reconstruction is asymmetric. When the polarization is changed between two orthogonal states, the reconstruction switches the +1 and -1 orders.

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These PSCGH’s are created in the bulk of fused silica glass without affecting the optical characteristics of the surface. Therefore, they can easily be integrated with other free-space and guided wave devices, created in glass with the same ultrafast laser, to assemble three-dimensional photonic circuits. Another advantage of the technique presented here is the flexible control of the birefringence that enables rapid and inexpensive prototyping of polarization devices. Further research is still required to study the creation of nanogratings at high speeds. For instance, present estimates show that by optimizing the laser power and writing speed, a 1 mm² hologram could be written in a few minutes. In order to realize the above-mentioned applications, current work is directed at increasing the contrast ratio by better controlling the micro-waveplate fabrication to achieve tighter retardance tolerances. Furthermore, we are investigating alternative coding techniques that can provide higher efficiency by nano-patterning the whole area of the PSCGH.

In conclusion, we demonstrated the first PSCGH fabricated with femtosecond laser induced birefringence in fused silica. In the design process we interleaved a phase-only encoding and a pseudo-binary phase encoding. These phases were further implemented in a cell-oriented technique combined with the freedom to rotate the micro-waveplates composing each cell. The reconstruction was consistent with the design and the simulations, showing the expected polarization selectivity.

 

Fig. 4. Top-simulation of the reconstruction at +1 order for each of two orthogonal linear polarizations. Bottom-experimental reconstruction at +1 order for different polarizations. From left to right, the polarization angles are rotated clockwise 0, 30, 60, and 90 degrees. See 576KB movie of this reconstruction when the polarization is gradually rotated.

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