**4. Analysis of the photonic structure of low spatial frequency lens elements**

Volume phase holographic (VPH) gratings can be recorded on different types of materials such as silver halide, dichromatic gelatin (DCG), photoresists, photopolymer, etc. However, photopolymer provides certain advantages [19] such as higher efficiency, self‐development and cost effectiveness. In general, VPH gratings follow Bragg's law [20] for the propagation of light inside a volume in which periodic modulation of the refractive index forms the grating structure. In certain circumstances, for example thicker media, VPH gratings recorded at low spatial frequency can also be considered as Bragg gratings and they increase the angular and spectral range of VPH gratings [18]. In this section, we examine the photonic structure of the recorded lenses using microscopy in order to verify the model used to design the lens and calculate the spatial frequency and slant angle of the grating pattern at specific points on the recorded lens. We also use Kogelnik's coupled wave theory (KCWT) to predict the expected Bragg curves and compare them to the curves obtained experimentally measured at these locations.

The lens elements are recorded as described above by including a focusing lens into one arm of a standard two‐beam optical arrangement for holographic recording at 532 nm. In this section, the analysis of the local diffraction behaviour using an unexpanded 633 nm laser beam and measuring the Bragg curve (intensity variation in the diffracted beam as the grating is rotated though a range of angles) is complemented by microscopic imaging at the same location.

#### **4.1. Modelling the recorded holographic optical element**

The focusing elements studied were 5 cm focal length off‐axis cylindrical lenses, so the micro‐ structure was expected to vary significantly laterally across the recorded element. **Figure 9** shows a schematic of the recording arrangement. Clearly the angles at which the interfering beams meet varies from left to right and we would expect both slant angle and spatial frequency of the grating pattern to change.

Simple geometry allows us to calculate the inter‐beam angle (which determines the spatial frequency) and the slant angle, i.e. the angle between the bisector of the angle between the two beams (inside the medium) and the normal to the recording plane). Since the lens is not spher‐ ical in this example there is no variation vertically (out of the plane of the page in **Figure 9**). The Bragg angle *θ<sup>o</sup>* is related to the fringe spacing (*Λ*) recorded in the hologram by the relation

$$
\sin \theta\_o = \frac{\lambda}{2\Lambda} \tag{7}
$$

In this work, we use equation (7) to predict the grating structure and verify it experimentally at three specific locations on the element, the centre and points 3 mm either side of the centre. For any local position across the element, the slant could be calculated using the recording geometry and compared to the experimentally observed position of the peak in the Bragg curve. Equally, the local grating period was calculated from knowledge of the inter‐beam angle at that specific location during recording and this was compared with microscopic imaging results.

The shape of the Bragg curve was then modelled using those parameters in KCWT, and fitted to the experimental data (normalised to the measured diffraction efficiency value). **Figure 10(a)** shows the calculated spatial period across the lens element. The inverted red tri‐ angles show the measured values obtained from the microscopy images, which are 2.66, 3.37, 6.59 µm at left, centre and right, respectively (left, right are 3 mm away from centre). **Figure 10(b)** shows the calculated slant angle at positions across the element.

**4. Analysis of the photonic structure of low spatial frequency lens** 

Volume phase holographic (VPH) gratings can be recorded on different types of materials such as silver halide, dichromatic gelatin (DCG), photoresists, photopolymer, etc. However, photopolymer provides certain advantages [19] such as higher efficiency, self‐development and cost effectiveness. In general, VPH gratings follow Bragg's law [20] for the propagation of light inside a volume in which periodic modulation of the refractive index forms the grating structure. In certain circumstances, for example thicker media, VPH gratings recorded at low spatial frequency can also be considered as Bragg gratings and they increase the angular and spectral range of VPH gratings [18]. In this section, we examine the photonic structure of the recorded lenses using microscopy in order to verify the model used to design the lens and calculate the spatial frequency and slant angle of the grating pattern at specific points on the recorded lens. We also use Kogelnik's coupled wave theory (KCWT) to predict the expected Bragg curves and compare them to the curves obtained experimentally measured at these

The lens elements are recorded as described above by including a focusing lens into one arm of a standard two‐beam optical arrangement for holographic recording at 532 nm. In this section, the analysis of the local diffraction behaviour using an unexpanded 633 nm laser beam and measuring the Bragg curve (intensity variation in the diffracted beam as the grating is rotated though a range of angles) is complemented by microscopic imaging at the same location.

The focusing elements studied were 5 cm focal length off‐axis cylindrical lenses, so the micro‐ structure was expected to vary significantly laterally across the recorded element. **Figure 9** shows a schematic of the recording arrangement. Clearly the angles at which the interfering beams meet varies from left to right and we would expect both slant angle and spatial frequency

Simple geometry allows us to calculate the inter‐beam angle (which determines the spatial frequency) and the slant angle, i.e. the angle between the bisector of the angle between the two beams (inside the medium) and the normal to the recording plane). Since the lens is not spher‐ ical in this example there is no variation vertically (out of the plane of the page in **Figure 9**).

In this work, we use equation (7) to predict the grating structure and verify it experimentally at three specific locations on the element, the centre and points 3 mm either side of the centre. For any local position across the element, the slant could be calculated using the recording geometry and compared to the experimentally observed position of the peak in the Bragg curve. Equally, the local grating period was calculated from knowledge of the inter‐beam angle at that specific

location during recording and this was compared with microscopic imaging results.

is related to the fringe spacing (*Λ*) recorded in the hologram by the relation

<sup>2</sup>*<sup>Λ</sup>* (7)

**4.1. Modelling the recorded holographic optical element**

sin *<sup>θ</sup><sup>o</sup>* <sup>=</sup> \_\_\_*<sup>λ</sup>*

of the grating pattern to change.

The Bragg angle *θ<sup>o</sup>*

**elements**

84 Holographic Materials and Optical Systems

locations.

**Figure 9.** Schematic illustration of the geometry of the recording set‐up showing the angles at which the interfering beams meet at the photopolymer plate of width *d*, at a slant angle *φ*.

**Figure 10.** (a) Spatial period across the VPH lens: 0 corresponds to the right side and 14 corresponds to the left side of the lens and (b) slant angle across the VPH lens.

#### **4.2. Local microstructure and diffraction behaviour**

**Figure 11** shows the experimental intensity data obtained by illuminating the three positions with an unexpanded 633 nm He‐Ne beam and varying the angle of incidence. As expected, the three curves are shifted relative to one another, verifying that each location has a different slant angle and inter‐beam angle. Each curve has a different FWHM because of the different grating periods.

The curves are fitted to theoretical curves generated by putting the parameters calculated using the geometry into the KCWT equations and setting the wavelength to 633 nm.

As can be seen from the figure very good agreement is obtained. The small broadening of the Bragg peaks can be accounted for by the fact that the incident beam had a finite width, and so in reality the result is not for a specific location but for an average over the beam width. The thick‐ ness of the photopolymer used was 70 ± 5 µm and was measured by white light interferometry.

In order to verify the theoretical calculation of spatial period of grating, the microscopic image of the gratings has been taken using a phase contrast microscope (Olympus DP72) at three different positions, centre and 3 mm away from the centre (right and left). These are shown in **Figure 12**. The spatial periods measured using the microscope are 2.6 ± 0.1, 3.9 ± 0.1 and 7.0 ± 0.1 µm at left, centre and right of the VPH cylindrical lens, respectively. The theoretical values for these positions can be seen from the graph of **Figure 12(a)** and are 2.66, 3.37, 6.59 µm, respectively.

It can be observed that the holographic recording has produced the predicted diffraction grat‐ ing patterns. The experimental results for the diffraction characteristics of the local pattern fit well with the theoretical predictions for these spatial frequencies made with KCWT. A small

**Figure 11.** Theoretical and experimental angular selectivity curves for three different positions (centre and 3 mm left and right of centre) on the VPH cylindrical lens.

Holographically Recorded Low Spatial Frequency Volume Bragg Gratings and Holographic Optical Elements http://dx.doi.org/10.5772/67296 87

**Figure 12.** Microscopic image of the surface structure of the holographic lens at the (a) left position (3 mm from centre), (b) centre and (c) right position (3 mm from centre).

amount of broadening is observed in the experimental curves, but this is probably due to the size of the probe beam, beam walk off or scattering. The spatial periods obtained from the microscopic images agree with the theory, and it can be seen from the microscopic images that the spatial period is increasing from left to right of the VPH cylindrical lens. Future work will involve fabricating and modelling lens elements at different wavelengths.
