**5. Writing gratings with a single beam**

**4.2. Local microstructure and diffraction behaviour**

grating periods.

86 Holographic Materials and Optical Systems

are 2.66, 3.37, 6.59 µm, respectively.

right of centre) on the VPH cylindrical lens.

**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

The curves are fitted to theoretical curves generated by putting the parameters calculated

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

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

using the geometry into the KCWT equations and setting the wavelength to 633 nm.

Most standard holographic recording is, of course, achieved by interfering two coherent recording beams. The interference pattern they create is recorded in the medium as a dif‐ fractive structure in the medium. However, due to the challenges of stability and size associ‐ ated with splitting and recombining beams, various methods have been employed in order to achieve recording with a single beam.

Early work by Kukhtarev et al. [21] demonstrated holographic recording using only one input beam in a photorefractive BaTiO<sup>3</sup> crystal using the photogalvanic coupling between orthogo‐ nal birefringent modes. Later, Naruse et al. [22] multiplexed ten gratings into a Fe‐doped LiNbO<sup>3</sup> crystal with a single recording beam. They used the crystal edge in a wavefront split‐ ting arrangement. Mitsuhashi and Obara [23], using a similar approach, demonstrated a com‐ pact holographic memory system using a single‐beam geometry in Fe‐doped LiNbO3 . At that time, they estimated a maximum total capacity of 23 GB based on beam diameter of 5 mm in a system using 635 nm light and a 10 mm crystal using both angular and spatial multiplexing. More recently, Yau et al. [24] have proposed another method that uses a single object beam to record images in a photorefractive LiNbO<sup>3</sup> crystal which allows imaging through a dynami‐ cally varying medium. Chiang et al. proposed a method that uses a single object beam to record multiple images in a medium without the need for a reference wave using a lenticular array [25]. This was demonstrated by recording four holograms in a 30 × 30 × 1 mm3 Fe‐doped LiNbO<sup>3</sup> crystal with a single exposure.

Recent work by Kukhtarev and Kukhtareva develops a dynamic version of the early single‐ beam recording system, demonstrating dynamic holographic interferometry [26] and also holographic amplification of weak images without phase distortions [27].

In the commercial arena, Optware, a Japanese‐based company, developed a new method of holographic storage called collinear holography. Instead of separate signal and reference beams to create the interference pattern, Optware are using a collinear approach by aligning the two laser beams into a single beam of coaxial light to create data fringes. This approach significantly simplified the recording set‐up [28]. Optware released a prototype of this record‐ ing system operating at a wavelength of 532 nm with an overall storage capacity of 200 GB on a recording medium with a diameter of 120 mm (HVD Pro Series 1000). They also released a credit card‐sized layer with a storage capacity of 30 GB and demonstrated a recording system with a storage capacity of 1 TB with a data transfer rate of 128 MB/s.

An example of the single‐beam recording method described here was first reported by the authors in 1998 [29]. After first recording, weak diffraction gratings in the photopolymer were illuminated on Bragg, with just one of the recording beams. For gratings with initial diffrac‐ tion efficiencies ranging from less than 1% to 64%, a further increase in diffraction efficiency was observed during the single‐beam exposure. For example, a grating with 7.5% efficiency was observed to increase to 60% efficiency over several minutes of single‐beam exposure. It was suggested at the time that the increase may be caused by either uniform polymerization of unreacted monomer in the grating, as had been observed in other photopolymers [30], or diffraction from the recorded grating. Further work [31] showed that the grating strength only increased significantly when the single writing beam was incident close to the Bragg angle of the pre‐recorded grating.

In this section, this method of writing holographic gratings using weak pre‐recorded gratings is explored further, because of its potential to allow the use of just one beam at the data writ‐ ing stage. New gratings, angularly separated from the pre‐written grating, are written using a single beam and the dependence on grating thickness is demonstrated. We demonstrate that this approach allows the writing of high diffraction efficiency gratings in unstable conditions due to the fact that the second beam is generated from within the photopolymer layer, in a manner similar to beam pumping in photorefractive crystals. We demonstrate that angular multiplexing is also possible, allowing one grating to be amplified without amplifying the other pre‐recorded gratings.

#### **5.1. Holographic recording process**

A two‐step process was used: (1) recording the weak 'seed' gratings with two recording beams and (2) exposure of the seed grating to one beam. There was a short delay between the two steps, during which both beams were blocked.

#### *5.1.1. Two‐beam recording*

The first step was to record a grating, usually with a diffraction efficiency of approximately 1%, in the photopolymer medium. A standard holographic grating recording arrangement was used, with two coherent interfering beams (532 nm) using beam splitters and mirrors, as shown in **Figure 1**. The arrangement is for unslanted gratings. A He‐Ne beam (633 nm) was used to monitor the diffraction efficiency throughout the initial recording and subsequent illumina‐ tion. This was possible because, in the formulation used, the photopolymer is not sensitive to red light. A short exposure time (around 1 sec) was used for the initial recording in order to keep the efficiency of the initial grating low. The spatial frequency was controlled by adjusting the angle between the two interfering beams and is 500 lines/mm in the work reported here.

#### *5.1.2. Single‐beam recording*

In the commercial arena, Optware, a Japanese‐based company, developed a new method of holographic storage called collinear holography. Instead of separate signal and reference beams to create the interference pattern, Optware are using a collinear approach by aligning the two laser beams into a single beam of coaxial light to create data fringes. This approach significantly simplified the recording set‐up [28]. Optware released a prototype of this record‐ ing system operating at a wavelength of 532 nm with an overall storage capacity of 200 GB on a recording medium with a diameter of 120 mm (HVD Pro Series 1000). They also released a credit card‐sized layer with a storage capacity of 30 GB and demonstrated a recording system

An example of the single‐beam recording method described here was first reported by the authors in 1998 [29]. After first recording, weak diffraction gratings in the photopolymer were illuminated on Bragg, with just one of the recording beams. For gratings with initial diffrac‐ tion efficiencies ranging from less than 1% to 64%, a further increase in diffraction efficiency was observed during the single‐beam exposure. For example, a grating with 7.5% efficiency was observed to increase to 60% efficiency over several minutes of single‐beam exposure. It was suggested at the time that the increase may be caused by either uniform polymerization of unreacted monomer in the grating, as had been observed in other photopolymers [30], or diffraction from the recorded grating. Further work [31] showed that the grating strength only increased significantly when the single writing beam was incident close to the Bragg

In this section, this method of writing holographic gratings using weak pre‐recorded gratings is explored further, because of its potential to allow the use of just one beam at the data writ‐ ing stage. New gratings, angularly separated from the pre‐written grating, are written using a single beam and the dependence on grating thickness is demonstrated. We demonstrate that this approach allows the writing of high diffraction efficiency gratings in unstable conditions due to the fact that the second beam is generated from within the photopolymer layer, in a manner similar to beam pumping in photorefractive crystals. We demonstrate that angular multiplexing is also possible, allowing one grating to be amplified without amplifying the

A two‐step process was used: (1) recording the weak 'seed' gratings with two recording beams and (2) exposure of the seed grating to one beam. There was a short delay between the

The first step was to record a grating, usually with a diffraction efficiency of approximately 1%, in the photopolymer medium. A standard holographic grating recording arrangement was used, with two coherent interfering beams (532 nm) using beam splitters and mirrors, as shown in **Figure 1**. The arrangement is for unslanted gratings. A He‐Ne beam (633 nm) was used to monitor the diffraction efficiency throughout the initial recording and subsequent illumina‐ tion. This was possible because, in the formulation used, the photopolymer is not sensitive to

with a storage capacity of 1 TB with a data transfer rate of 128 MB/s.

angle of the pre‐recorded grating.

88 Holographic Materials and Optical Systems

other pre‐recorded gratings.

*5.1.1. Two‐beam recording*

**5.1. Holographic recording process**

two steps, during which both beams were blocked.

The next step was to illuminate the grating on Bragg, with a single beam and observe the change in diffraction efficiency. The simplest way to do this was to use an additional shutter to block one of the two recording beams. Usually a short interval with no writing beam illumination was allowed after the initial grating recording in order to allow the system to record any spontaneous change that may be occurring in the absence of illumination. Exposure times and writing beam illumination were controlled using Uniblitz electronic shutters. The photopolymer grating was mounted on a rotation stage so that angles of illumination could also be varied.

#### **5.2. Single‐beam holographic recording results**

**Figure 13** shows the diffraction efficiency changing during a typical exposure starting with a standard two‐beam holographic recording of 2 sec followed by a 25 sec delay, during which there is no illumination. Then, at 27 sec, illumination with one of the writing beams commences. The diffraction efficiency increase obtained during the single‐beam exposure is significant and diffraction efficiency of the final grating is much higher than at the point when single‐beam exposure commences. In this case, the grating spatial frequency is 500 lines/mm, and the layer thickness is 135 µm. Each exposing beam has an intensity of 2.5 mW/cm<sup>2</sup> . **Figure 13** shows typical diffraction efficiency increase observed upon exposure to a single on‐Bragg recording beam. No dependence was found on the delay time between the two beams and single‐beam exposure and weak gratings could be enhanced by single‐beam exposure after 12 weeks, as long as the photopolymer was still sensitive.

**Figure 13.** Diffraction efficiency versus exposure time. A standard two‐beam holographic recording of 2 sec is followed by a 25 sec delay (no illumination) and then illumination with just one of the writing beams. Exposure intensity is 2.5 mW/cm2 in each beam.

The increase in diffraction efficiency consistently observed under single‐beam illumination in this photopolymer is due to a new grating formed by the interference between the single writ‐ ing beam and the first‐order beam generated by diffraction at the pre‐recorded weak grating (see discussion below).

#### *5.2.1. Single‐beam recording with Bragg mismatch*

The need for near‐Bragg matching of the single writing beam rules out any bulk photochemi‐ cal effect as the cause of the diffraction efficiency increase, supports the idea that diffraction is the main contributor. In order to study this further, and also assess the potential for multi‐ plexing, the angle of incidence of the single writing beam was varied around the Bragg angle of the pre‐recorded grating. The results are shown in **Figure 14**. The original seed grating dif‐ fraction efficiency was close to 1% in each case and the photopolymer thickness was 200 µm. The spatial frequency of the seed grating was 500 lines/mm. It can be seen from **Figure 14** that there is an optimum angle, close to the Bragg angle of the pre‐recorded grating, that maxi‐ mizes the strength of the grating recorded with the single‐beam writing process. As the illu‐ minating beam is moved further away from the optimal angle, the final diffraction efficiency is reduced under the same exposure conditions. This is probably due to the reduced coupling

**Figure 14.** Bragg curves (the variation of diffraction efficiency with reading beam angle of incidence) for a series of gratings formed using the single beam process, using different angles of incidence of the single writing beam. The layer thickness is 200 µm. The arrows indicate the offset (in degrees) from the Bragg angle of the seed grating (0°).

between the single writing beam and the seed grating. In this example, the optimal angle is about 0.5° from the Bragg angle for the original grating. The asymmetry of the sidelobes is also a consistent feature. Both of these are thought to be due to fringe bending during the formation of the grating under single‐beam exposure as discussed below.

The increase in diffraction efficiency consistently observed under single‐beam illumination in this photopolymer is due to a new grating formed by the interference between the single writ‐ ing beam and the first‐order beam generated by diffraction at the pre‐recorded weak grating

The need for near‐Bragg matching of the single writing beam rules out any bulk photochemi‐ cal effect as the cause of the diffraction efficiency increase, supports the idea that diffraction is the main contributor. In order to study this further, and also assess the potential for multi‐ plexing, the angle of incidence of the single writing beam was varied around the Bragg angle of the pre‐recorded grating. The results are shown in **Figure 14**. The original seed grating dif‐ fraction efficiency was close to 1% in each case and the photopolymer thickness was 200 µm. The spatial frequency of the seed grating was 500 lines/mm. It can be seen from **Figure 14** that there is an optimum angle, close to the Bragg angle of the pre‐recorded grating, that maxi‐ mizes the strength of the grating recorded with the single‐beam writing process. As the illu‐ minating beam is moved further away from the optimal angle, the final diffraction efficiency is reduced under the same exposure conditions. This is probably due to the reduced coupling

**Figure 14.** Bragg curves (the variation of diffraction efficiency with reading beam angle of incidence) for a series of gratings formed using the single beam process, using different angles of incidence of the single writing beam. The layer

thickness is 200 µm. The arrows indicate the offset (in degrees) from the Bragg angle of the seed grating (0°).

(see discussion below).

90 Holographic Materials and Optical Systems

*5.2.1. Single‐beam recording with Bragg mismatch*

This work demonstrates that the grating strength only increases significantly under single‐ beam illumination when the single writing beam is incident close to the Bragg angle of the pre‐recorded grating.

As discussed in Ref. [31], the angular position of the Bragg peak for the gratings recorded in this way is linearly dependent on the angle of incidence of the single writing beam. This shows that the formation of a new grating formed by diffraction is responsible for the observed increases.

We propose that the weak diffracted beam interferes with the undiffracted beam to produce a low‐contrast interference pattern which is immediately recorded in the material. If the new grating is in phase with the original grating, more light will then be diffracted into the weaker beam, reducing the beam ratio and increasing the contrast in the interference pattern, in turn producing an even stronger diffracted beam. In this way, quite weak gratings can rapidly 'seed' the growth of relatively high diffraction efficiency gratings. This growth of a new grating is anal‐ ogous to the energy transfer between the strong beam and the weak beam in 'beam pumping' in photorefractive crystals, except that the refractive index modulation created is permanent.

A potential difficulty with the above explanation is the phase mismatch between the 'seed' grating and any grating created by diffraction at the 'seed' grating. This is due to the fact that there will be a phase difference (*π*/2) between the incident beam and the beam diffracted by the phase grating, which would cause any new grating to be out of phase with the original. Beam pumping in photorefractives, which is also initiated by diffraction at a weak grating, occurs only because the recorded grating in photorefractive crystals is laterally displaced with respect to the interference fringes that create it.

Unlike photorefractive crystals, photopolymers are usually considered to produce gratings that are not laterally shifted from the interference pattern that creates them. Thus we might not expect the interference pattern created by the incident beam and its diffracted beam to produce a grating that is in phase with the one that created it. However, such shifts and non‐linear recording profiles have been observed in holographic recording materials such as acrylamide photopolymers [32], nanoparticle‐doped photopolymers [33] and silver halide emulsions [34]. Murciano et al. [35] reported that effects such as beam bending and two wave mixing have been observed even with very small phase shifts, by them and other authors in similar materials. They analysed the origin and effects of fringe bending and Bragg detuning in holographic gratings recorded in rigid media such as photopolymerizable inorganic silica glass materials and proposed that they occurred as a result of the non‐sinusoidal nature of the recorded pattern. Using an acrylamide photopolymer‐doped sol gel, Murciano et al. observed two‐wave mixing during two‐beam recording and used a two‐wave mixing model to explain the asymmetry and angular shift (fringe bending) observed in their angular selectivity (Bragg) curves. The reconstruction model used by Murciano et al. to analyse the gratings used cou‐ pled wave theory taking into account two wave mixing occurring during recording. Fringe bending becomes larger as thickness and refractive index modulation increase and, of course, depends greatly on the initial beam ratio. Good agreement was obtained between experimen‐ tal and theoretical results of simulations with a shift of just 2.6° between the recorded grating and the light pattern, demonstrating that very small shifts can cause such effects.

It seems likely that our results described above would have a similar origin. That is the non‐ sinusoidal refractive index profile of the recorded grating provides enough of a phase shift to allow coupling from the strong beam to the weak diffracted beam during single‐beam recording. It should also be borne in mind that the beam ratio is very large (typically 99:1) at the start of the single‐beam recording step, so small amounts of energy transfer from the strong beam will have a significant effect and the fringe period is very much larger than the wavelength for these low spatial frequency gratings. In the case of these acrylamide‐based photopolymers, gratings are recorded via photopolymerization and diffusion. Either of these processes can dominate depending on the recording intensity, spatial frequency and photo‐ polymer formulation [36], and non‐sinusoidal grating profiles are also common especially at low spatial frequencies [37]. It has been observed that the process of enhancing diffraction efficiency reported here is stronger at lower spatial frequencies.

#### *5.2.2. Single‐beam recording with different layer thickness*

Experiments with different layer thicknesses ranging from 60 to 240 µm showed that the enhance‐ ment of the seed grating occurs much more in thicker layers. **Figure 15** shows the diffraction efficiency increasing under single‐beam illumination from an initial diffraction efficiency of approximately 1% for thicknesses of 60, 130, 190 and 240 µm. These curves are each obtained in the same way as that in **Figure 13** but the delay between the two exposures was 110 sec in these experiments. The gratings were exposed to a single beam at the Bragg angle for 150 sec. The spatial frequency is 500 lines/mm and each exposing beam has an intensity of 2.5 mW/cm2 . It is observed that the increase in efficiency when recording with a single beam is much more pronounced in thicker layers and practically non‐existent in the layer with thickness 60 µm It is probable that this dependence is due the increased distance over which energy can be transferred from the weaker beam to the stronger beam, as described above.

#### *5.2.3. Single‐beam recording at different spatial frequencies*

**Figure 16** shows the Bragg curves of a number of single‐beam gratings recorded at different spatial frequencies. Although the diffraction efficiency of the 'seed' grating was approximately 1% in each case and the single‐beam recording time is 30 sec for all gratings, a significant dependence on grating spatial frequency is observed. In this instance, the recording intensity is 1.8 mW/cm2 . The initial exposure time with two beams was 1 sec for 2000 and 1500 lines/mm and 0.75 sec for 1000–250 lines/mm in these examples but the diffraction efficiency was close to 1% in each case.

The pronounced increase observed for lower spatial frequencies indicates that fringe period may be a crucial factor. It was suggested above that the non‐sinusoidal refractive index profile of the recorded grating could provide enough of a phase shift to allow coupling from the strong Holographically Recorded Low Spatial Frequency Volume Bragg Gratings and Holographic Optical Elements http://dx.doi.org/10.5772/67296 93

bending becomes larger as thickness and refractive index modulation increase and, of course, depends greatly on the initial beam ratio. Good agreement was obtained between experimen‐ tal and theoretical results of simulations with a shift of just 2.6° between the recorded grating

It seems likely that our results described above would have a similar origin. That is the non‐ sinusoidal refractive index profile of the recorded grating provides enough of a phase shift to allow coupling from the strong beam to the weak diffracted beam during single‐beam recording. It should also be borne in mind that the beam ratio is very large (typically 99:1) at the start of the single‐beam recording step, so small amounts of energy transfer from the strong beam will have a significant effect and the fringe period is very much larger than the wavelength for these low spatial frequency gratings. In the case of these acrylamide‐based photopolymers, gratings are recorded via photopolymerization and diffusion. Either of these processes can dominate depending on the recording intensity, spatial frequency and photo‐ polymer formulation [36], and non‐sinusoidal grating profiles are also common especially at low spatial frequencies [37]. It has been observed that the process of enhancing diffraction

Experiments with different layer thicknesses ranging from 60 to 240 µm showed that the enhance‐ ment of the seed grating occurs much more in thicker layers. **Figure 15** shows the diffraction efficiency increasing under single‐beam illumination from an initial diffraction efficiency of approximately 1% for thicknesses of 60, 130, 190 and 240 µm. These curves are each obtained in the same way as that in **Figure 13** but the delay between the two exposures was 110 sec in these experiments. The gratings were exposed to a single beam at the Bragg angle for 150 sec. The spatial frequency is 500 lines/mm and each exposing beam has an intensity of 2.5 mW/cm2

It is observed that the increase in efficiency when recording with a single beam is much more pronounced in thicker layers and practically non‐existent in the layer with thickness 60 µm It is probable that this dependence is due the increased distance over which energy can be transferred

**Figure 16** shows the Bragg curves of a number of single‐beam gratings recorded at different spatial frequencies. Although the diffraction efficiency of the 'seed' grating was approximately 1% in each case and the single‐beam recording time is 30 sec for all gratings, a significant dependence on grating spatial frequency is observed. In this instance, the recording intensity

and 0.75 sec for 1000–250 lines/mm in these examples but the diffraction efficiency was close

The pronounced increase observed for lower spatial frequencies indicates that fringe period may be a crucial factor. It was suggested above that the non‐sinusoidal refractive index profile of the recorded grating could provide enough of a phase shift to allow coupling from the strong

. The initial exposure time with two beams was 1 sec for 2000 and 1500 lines/mm

.

and the light pattern, demonstrating that very small shifts can cause such effects.

efficiency reported here is stronger at lower spatial frequencies.

from the weaker beam to the stronger beam, as described above.

*5.2.3. Single‐beam recording at different spatial frequencies*

is 1.8 mW/cm2

to 1% in each case.

*5.2.2. Single‐beam recording with different layer thickness*

92 Holographic Materials and Optical Systems

**Figure 15.** Growth in diffraction efficiency, under single beam illumination, of 'seed' gratings in samples prepared with different thicknesses. The grating spatial frequency is 500 lines/mm and the layer thicknesses are approximately 60 µm (solid), 130 µm (dotted‐line), 190 µm (dashed line) and 240 µm (dotted line). Initial diffraction efficiency is approximately 1% for the two‐beam grating.

**Figure 16.** Bragg curves (the variation of diffraction efficiency with reading beam angle of incidence) for a series of gratings formed using the single beam process using different spatial frequency seed gratings. Layer thickness is approximately 140 µm.

beam to the weak diffracted beam during single‐beam recording. It seems reasonable that this process will be more efficient when the fringe period is very much larger than the wavelength.

As well as the clear dependence of final on‐Bragg diffraction efficiency on spatial frequency, the changes in the width of the Bragg curve are also obvious. In addition, we can see that the shift in the position of the Bragg peak increases as the spatial frequency decreases and the magnitude of the effect increases.

#### *5.2.4. Angular multiplexing of data using a one beam system*

In some applications, it may be necessary to obtain a diffraction efficiency increase in one grat‐ ing selected from a number of gratings angularly multiplexed in the same region of the pho‐ topolymer layer, without affecting its neighbours. The selectivity demonstrated in **Figure 14** implies that this should be possible.

In order to investigate if it would be possible to selectively boost one grating from a series, a layer of photopolymer 135 µm in thickness was used and five seed gratings of equal strength were angularly multiplexed into it. One of the gratings was then illuminated at its Bragg angle in order to increase its efficiency. For separations up to 1.8°, illumination caused a sig‐ nificant increase in the diffraction efficiency of gratings on either side of the intended grating. **Figure 17** shows the result for gratings 2° apart. Although there are only five seed gratings in these examples, the principle is demonstrated that one of a set of angularly multiplexed gratings, separated by 2°, can be enhanced by illuminating it at the appropriate angle with a single beam of light. Working with higher spatial frequency and thickness would be likely to allow smaller angular separations between neighbouring gratings.

**Figure 17.** Variation of diffraction efficiency with the reading beam angle of incidence for a series of gratings, from which one individual grating has been enhanced by illuminating with a single on‐Bragg beam of light. The angular separation between neighbouring gratings is 2°.

#### **6. Conclusion**

High efficiency diffractive optical elements have been recorded holographically with low spa‐ tial frequency. Three slanted gratings were successfully stacked using lamination and shown to increase significantly the range of angles from which light could be collected. However, positioning gratings in the path of the detector/cell meant that although light incident at large angles was coupled into the detector, an equivalent amount of light was deflected away from the detector at lower angles. When arranged off‐axis, however, they increased the total light collected. The HOEs were then tested with a solar simulator and shown to improve the energy collected at a Si solar cell by up to 60% when used off‐axis in pairs.

Modelling off‐axis low spatial frequency focusing elements confirms that a range of grat‐ ings spatial frequencies and slant angles exists in the recorded elements and by simple geometry we can predict the grating spacing and slant angle at any point across the ele‐ ment and use coupled wave theory to predict the diffraction behaviour at a particular loca‐ tion. Good agreement is shown between theory and experiment and measurements made using phase contrast microscopic imaging of the photonic structure also agree closely.

A method of writing low spatial frequency holographic gratings with a single beam was also presented and shown to be capable of writing high diffraction efficiency gratings under unstable conditions. The technique is based on the exposure of very weak pre‐recorded grat‐ ings to a single illuminating beam in order to write new high efficiency gratings in the pho‐ topolymer material. Strong spatial frequency dependence was shown. Diffraction efficiencies of 60% were obtained in just 30 sec with a 250 line/mm grating. With longer exposures up to 80% was achieved. The potential for angular multiplexing was shown by illuminating a single grating from among five multiplexed weak gratings and increasing its efficiency eightfold with a negligible effect on the other gratings.

In conclusion, these studies of low spatial frequency holographic optical elements have shown their potential for solar applications, their capacity to function as thick volume gratings and the success with which their microstructure can be controlled and modelled. Their particu‐ lar potential for recording self‐interference has also been exploited as a vibration‐immune holographic recording method. The challenges that remain include modelling of the non‐lin‐ ear effects that occur during low spatial frequency holographic recording, development and modelling of more complex low spatial frequency elements and combinations of elements. Future work will focus on developing new diffractive elements and exploitation of these in practical applications.
