**Holographically Recorded Low Spatial Frequency Volume Bragg Gratings and Holographic Optical Elements Provisional chapter Holographically Recorded Low Spatial Frequency Volume Bragg Gratings and Holographic Optical**

Suzanne Martin, Hoda Akbari, Sanjay Keshri,

Dennis Bade, Izabela Naydenova, Kevin Murphy and Suzanne Martin, Hoda Akbari, Sanjay Keshri, Dennis Bade, Izabela Naydenova, Kevin

Vincent Toal Murphy and Vincent Toal

**Elements**

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67296

#### **Abstract**

Low spatial frequency volume gratings (a few hundred lines per millimetre) are near the borderline of what can be considered Bragg gratings. Nevertheless, in some applications, their very low selectivity can be a benefit because it increases the angular and spectral work‐ ing range of the holographic optical element. This chapter presents work carried out using an instantaneously selfdeveloping photopolymer recording material and examines holo‐ graphic optical elements with spatial frequencies below 500 lines/mm. The advantages of volume photopolymer holographic gratings are discussed in the context of existing research. Specific examples explored include a combination of off‐axis cylindrical lenses used to direct light from a solar simulator onto a c‐Si solar cell, producing increases of up to 60% in the energy collected. A study of the microstructure of such elements is also presented. A good fit is obtained between the experimental and theoretical Bragg curves and the microstructure of the element is examined directly using microscopy. This is followed by a discussion of an unusual holographic recording approach that uses the nonlinearities inherent in low spatial frequency grating profiles to record gratings using a single beam. In conclusion, the prop‐ erties of low spatial frequency volume gratings are summarized and future development discussed.

**Keywords:** Bragg gratings, low spatial frequency, photopolymer, holography

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

#### **1. Introduction**

Holographic optical elements (HOEs) are generally considered as either thin or thick gratings when their diffraction behaviour is modelled. Examples of thin gratings are commonplace because most commercially produced holograms, such as those on credit cards or banknotes, are surface relief holograms produced by a stamping process, and so the diffracting part of the structure is only a fraction of micron in thickness.

When a light beam passes through the grating, it obeys the classical grating equation given below, and orders are observed at the predicted angles

$$m\lambda = n\Lambda(\sin\alpha + \sin\beta)\tag{1}$$

where *λ* is the wavelength of light, *Λ* is the spatial period of grating, *m* is the order of diffrac‐ tion, *n* is the refractive index of medium and *α*, *β* are the angles of incidence and diffraction, respectively.

Thick gratings, on the other hand, are usually volume gratings, distinct from surface grat‐ ings because the structures causing the diffraction are distributed through the full thickness of the material. Photopolymer gratings are one example. Some commercial photopolymer holograms have begun to appear on mobile phone batteries and other high value products. In these holograms, the refractive index variation is recorded through the full thickness, typically tens of microns, of the photopolymer layer. Light diffracted from this 'thick' Bragg structure also obeys the grating equation, but the greatest proportion of the light will be diffracted into the first order. This is because the light is effectively reflected from the planes of the fringes. This reflection combined with the grating equation gives the well‐known Bragg condition which can be written as [1]

$$2\imath\lambda\sin\theta = m\lambda\tag{2}$$

where *θ* is the angle of incidence.

The important factor determining whether 'thin' or 'thick' diffraction behaviour is observed is the size of the diffraction features in comparison to the thickness. Usually if the thickness of the recording material is smaller than the average spacing of the interference fringes, the holo‐ grams are considered to be thin holograms. The *Q* parameter, defined by Eq. (3), is derived from Kogelnik's equations for diffraction from volume gratings (discussed in the next section) and can be used in order to determine whether a hologram is 'thin' or 'thick' in its diffraction behaviour.

$$Q = \frac{2\pi\lambda d}{\mathfrak{n}\,\Lambda^2} \tag{3}$$

where *λ* is the wavelength, *d* is the thickness of the recording medium, *n* is the refractive index of the recording material and *Λ* is the fringe spacing.

Generally, gratings with the *Q* < 1 are considered thin gratings while those with values of *Q* > 10 are considered thick [2]. As can be seen from Eq. (3), for any given thick‐ ness and wavelength this depends on fringe spacing or spatial frequency. Low spatial frequency volume Bragg gratings, having grating spatial frequency of typically a few hundred lines/mm, are on the borderline of what can be considered Bragg gratings, but at typical photopolymer thicknesses 30–50 µm, they can be considered Bragg gratings for visible wavelengths. In some applications, their low selectivity can be a significant benefit because it increases the angular and spectral working range of the HOE. In mate‐ rials that perform well at low spatial frequencies, it is interesting to examine the charac‐ teristics and applications of HOEs with spatial frequencies below 500 lines/mm.

Volume phase holograms have advantages over surface relief holograms because of their high efficiency and resistance to surface contaminants. Applications include spectroscopy, astronomy and ultrafast lasers [3], as well as solar concentrators [4]. A number of different materials are available for recording them including silver halide, dichromatic gelatin (DCG), photoresists and photopolymers. The primary advantages of thick volume holograms over thin surface holograms are the possible high efficiencies (theoretically 100%) and the fact that most of the energy is diffracted into a single direction (order). For most applications where the hologram is expected to perform some or all of the functions of a conventional optical ele‐ ment such as a lens, maximum efficiency and minimum additional beams are advantageous. Beam splitting applications are an exception, of course. Photopolymers are useful materi‐ als because of their ease of use and potential for very high diffraction efficiency, but their key advantage in optical device applications is their self‐developing capability, because it removes the need for chemical or physical processing after exposure. Assuming shrinkage is not a significant problem in the photopolymer used, the avoidance of chemical and physical processing is very important in maintaining the original photonic structure written in the volume of the material during the exposure/recording step. The longer term shelf life of the recorded devices varies according to the photopolymer used. In the acrylamide‐based photo‐ polymer used in the examples in this chapter, long‐term stability of recorded gratings is good when the photopolymer layer is laminated with a protective plastic layer. Recent studies of the shelf life of such gratings sealed in plastic showed varied results however [5] pointing to the need for improved protection and alternative more robust formulations [6].

For thick gratings and elements, Kogelnik's coupled wave theory [1] is a widely accepted model that relates diffraction efficiency and angular selectivity of gratings to the grating's physical characteristics (thickness, spatial frequency and refractive index modulation).

According to Kogelnik, the diffraction efficiency (*η*) can be calculated using Eq. (4), allowing us to model how the diffraction efficiency varies with the angle of incidence, near the Bragg angle. This allows us to observe how grating thickness and spatial frequency affect the angu‐ lar selectivity of an individual grating:

$$\eta = \frac{\sin 2\sqrt{(\xi^2 + \upsilon^2)}}{\left(1 + \frac{\xi^2}{\upsilon^2}\right)} \tag{4}$$

The parameters *ξ* and *υ* are defined as:

**1. Introduction**

74 Holographic Materials and Optical Systems

respectively.

behaviour.

which can be written as [1]

where *θ* is the angle of incidence.

*Q* = \_\_\_\_\_ <sup>2</sup>πλ*<sup>d</sup>*

of the recording material and *Λ* is the fringe spacing.

the structure is only a fraction of micron in thickness.

below, and orders are observed at the predicted angles

Holographic optical elements (HOEs) are generally considered as either thin or thick gratings when their diffraction behaviour is modelled. Examples of thin gratings are commonplace because most commercially produced holograms, such as those on credit cards or banknotes, are surface relief holograms produced by a stamping process, and so the diffracting part of

When a light beam passes through the grating, it obeys the classical grating equation given

 *mλ* = *nΛ*(sin*α* + sin*β*) (1) where *λ* is the wavelength of light, *Λ* is the spatial period of grating, *m* is the order of diffrac‐ tion, *n* is the refractive index of medium and *α*, *β* are the angles of incidence and diffraction,

Thick gratings, on the other hand, are usually volume gratings, distinct from surface grat‐ ings because the structures causing the diffraction are distributed through the full thickness of the material. Photopolymer gratings are one example. Some commercial photopolymer holograms have begun to appear on mobile phone batteries and other high value products. In these holograms, the refractive index variation is recorded through the full thickness, typically tens of microns, of the photopolymer layer. Light diffracted from this 'thick' Bragg structure also obeys the grating equation, but the greatest proportion of the light will be diffracted into the first order. This is because the light is effectively reflected from the planes of the fringes. This reflection combined with the grating equation gives the well‐known Bragg condition

2*nΛ*sin*θ* = *mλ* (2)

The important factor determining whether 'thin' or 'thick' diffraction behaviour is observed is the size of the diffraction features in comparison to the thickness. Usually if the thickness of the recording material is smaller than the average spacing of the interference fringes, the holo‐ grams are considered to be thin holograms. The *Q* parameter, defined by Eq. (3), is derived from Kogelnik's equations for diffraction from volume gratings (discussed in the next section) and can be used in order to determine whether a hologram is 'thin' or 'thick' in its diffraction

where *λ* is the wavelength, *d* is the thickness of the recording medium, *n* is the refractive index

Generally, gratings with the *Q* < 1 are considered thin gratings while those with values of *Q* > 10 are considered thick [2]. As can be seen from Eq. (3), for any given thick‐ ness and wavelength this depends on fringe spacing or spatial frequency. Low spatial

*<sup>n</sup> <sup>Λ</sup>*<sup>2</sup> (3)

$$
\xi = \ \Delta\theta \frac{kd}{2} \tag{5}
$$

$$\upsilon = \frac{\pi \, n\_i \, d}{\lambda \cos \theta} \tag{6}$$

where *d* is the thickness of the grating, *n*<sup>1</sup> is the refractive index modulation, *λ* is the wavelength of the reconstructing beam, *Δθ* is the deviation from the Bragg angle and *k* is interference fringe vector, normal to the fringes with a magnitude *K* = 2*π*/spatial period. It can be seen from Eq. (4) that increasing the spatial period (reducing spatial frequency) is effective in controlling the angular selectivity. In materials where the refractive index modulation is small, significant thickness is required for high efficiency. Decreasing the spatial frequency (increasing the period) can be the better approach to control the angular selectivity as long as the gratings still behave as thick volume gratings [7].
