**2. Volume hologram formation in photosensitive materials**

The mechanism of volume hologram formation in photosensitive materials, a complex pro‐ cess where several components are involved, is starting from the interference exposure. Holographic recording induces a generally three‐dimensional spatial modulation in the optical properties. The final grating features optical functionality that consists in specific diffraction of light, to be characterized by means of its diffraction efficiency (defined as the ratio of the input readout power to the diffracted power), as well as angular response and frequency response. Different kinds of gratings are formed, according to the specific record‐ ing conditions. While the geometry clearly determines dimensionality and size of the grating, many factors influence how the material responds to light during the holographic exposure. The material response strongly depends on intrinsic material parameters, such as material composition or viscosity as well as on recording parameters, such as exposure duration and recording intensity [6].

#### **2.1. Grating formation**

In turn, the principle *from structure to function* also comprises the possibility to under‐ stand function through structure. Taking advantage of this relation between structure and function culminates in the attempt of mimicking nature. Notable examples are functional surfaces with hierarchical structures based on the lotus effect to induce superhydropho‐ bicity, the gecko effect for controlled adhesion or the moth's eye effect for anti‐reflection

While the examples mentioned above remain limited to surface phenomena, the third dimen‐ sion opens up entirely new possibilities with major relevance for many applications. Structures with a periodic modulation of the refractive index are of interest wherever light must be manipulated. Volume holographic gratings can be considered as such three‐dimensional (3D)

Volume gratings made by nature can be found in the form of crystals wherever atoms are regularly arranged [1]. The dimensions of atomic and molecular structures usually result in interaction rather with a non‐visible range of the electromagnetic spectrum, enabling access by means of X‐ray crystallography. However, light‐based photonic crystals (PCs), with func‐

While holography allows three‐dimensional imaging, the holographic structure itself extends not necessarily in three dimensions. Depending on the hologram formation tech‐ nique as well as on the recording medium, the hologram itself takes shape as a surface pat‐ tern or rather emerges as a three‐dimensional structure. A volume hologram or photonic crystal may only be formed if recording technique and recording medium allow modifica‐ tion of the optical properties in all three dimensions. The performance of such a grating with thickness in the range of 100 μm differ significantly from thin gratings or surface gratings: Volume Bragg gratings stand out due to their high diffraction efficiency, rigorous wavelength selectivity and the ability that multiple holograms may be superimposed by

There are many ways to create optical surface patterns by photolithography, self‐assem‐ bly or other nano‐ and microfabrication methods with both, bottom‐up and top‐down approaches. However, entering the third dimension in optical structuring is accompanied with considerable challenges. Among existing techniques for three‐dimensional optical structuring, such as direct laser writing [3] or self‐assembly [4], volume holography pro‐ vides the unique possibilities to create optical structures through the entire volume beyond a point‐by‐point, line‐by‐line or plane‐by‐plane fabrication, with high resolution and accu‐

At the same time, the analysis of volume holographic structures emerges as a challenging task. Optical structures inside a volume may not readily be mapped by means of common microscopic methods [5]. This is where the mutuality of function and structure opens up new possibilities. In fact, the diffraction efficiency represents the only accessible parameter to entirely characterize a volume grating. Based on the optical functionality, conclusions may be drawn on grating parameters as well as on material parameters such as material response

coatings.

optical structures with diffractive properties.

4 Holographic Materials and Optical Systems

means of multiplexing.

racy in a single step.

and energetic sensitivity.

tionality in the visible range, can be created artificially [2].

Volume holographic grating formation can be attributed to different physico‐chemical mate‐ rial transformations, depending on the type of photosensitive material. In case of polymers, an interplay of polymerization and diffusion, induced by the spatially modulated exposure, is responsible for hologram formation [7]. A light pattern is projected into the photosensi‐ tive medium, inducing local polymerization, proportional to the light intensity. Thereupon, a chemical gradient is induced, resulting in monomer diffusion and subsequent polymeriza‐ tion. As a consequence, the hologram is formed as a periodic modulation of optical proper‐ ties, according to the recording light pattern. This grating formation mechanism is illustrated in **Figure 1**.

A special characteristic of volume gratings is that their optical functionality can be attributed to very small modulations of optical properties inside the holographic material. In case of phase gratings with a layer thickness of 200 μm, a refractive index contrast in the order of only 10‐3 already results in diffraction efficiency close to 100%.

The contrast of such optical structures can be further enhanced with additives such as nanoparticles or quantum dots [8, 9]. It can also be combined with other mechanisms such as photochemical isomerization of optically anisotropic components or with polymer‐dispersed liquid crystals (PDLCs) for switchable or tuneable optical devices [10, 11]. Furthermore, the light‐induced mass transport may result in the formation of additional surface‐relief gratings [12].

**Figure 1.** Schematic illustration of grating formation in photosensitive polymers: Unreacted polymer chains, cross‐ linker, dopant (e.g., monomers) and photoinitiator are needed for the material composition and layer formation (top). Interference exposure induces local polymerization (middle) and subsequent diffusion of components (either into the bright regions, as illustrated above, or into the dark regions) to form the permanent grating (below).

#### *2.1.1. Specific material requirements*

First of all, a photosensitive medium for volume holographic recording must be capable to undergo molecular or structural transformations with the result of a local, and in most cases, permanent change of the optical properties, as described in the previous section. In addi‐ tion, and with focus on the functionality of volume holographic gratings, the medium must comply with high material standards. In this context, a number of material parameters must be optimized, namely sensitivity and dynamic range, resolution, transparency and stability. The following table gives an overview of important material parameters for volume holo‐ graphic recording.


**Table 1.** Overview on volume holographic material parameters.

Indispensable prerequisite for the material composition is the ability to form stable layers with *thickness* d in the range of at least 50 μm where a *refractive index contrast* Δn in the order of 10‐3 can be induced.

The material *resolution* U refers to the precision within the ability of a photosensitive mate‐ rial to transfer the interference pattern of exposure into a permanent modulation of opti‐ cal properties during the recording process, depending on the smallest structure size of the exposure pattern. It is therefore related to the precision of the final recorded structure. With the objective to fully exploit the interference pattern of the exposure beams, high material resolution is required. With regard to the maximum spatial frequency response, that is, the highest frequency inducing a permanent refractive index modulation, a value in the range of 5000–10,000 lines per mm is aspired [13].

The *sensitivity* S provides information on how the input energy is converted into a certain holographic contrast. High sensitivity is required with respect to the possible application of low‐power laser sources but also to ensure high recording speed [14]. In this context, the *dynamic range* M# refers to the total response of a medium, when divided up to n holograms. It determines how many holograms can be multiplexed in a single volume [15].

Highest *transparency* T of the material at the operating wavelength is required to achieve high diffraction efficiency. Low attenuation becomes particularly important in case of thick layers, desired for volume holographic applications. In general, losses arising from absorp‐ tion and scattering should not exceed 30%. Although much lower values might be required, depending on the layer thickness, for certain specific applications [16].

Next to a high sensitivity, high spatial resolution and low losses, a high thermal stability and long‐time stability of samples and systems are required [14].

Altogether, the diversity of material requirements is accompanied by a diversity of approaches to meet these needs.

#### *2.1.2. Photosensitive media for volume holography*

*2.1.1. Specific material requirements*

6 Holographic Materials and Optical Systems

graphic recording.

First of all, a photosensitive medium for volume holographic recording must be capable to undergo molecular or structural transformations with the result of a local, and in most cases, permanent change of the optical properties, as described in the previous section. In addi‐ tion, and with focus on the functionality of volume holographic gratings, the medium must comply with high material standards. In this context, a number of material parameters must be optimized, namely sensitivity and dynamic range, resolution, transparency and stability. The following table gives an overview of important material parameters for volume holo‐

**Figure 1.** Schematic illustration of grating formation in photosensitive polymers: Unreacted polymer chains, cross‐ linker, dopant (e.g., monomers) and photoinitiator are needed for the material composition and layer formation (top). Interference exposure induces local polymerization (middle) and subsequent diffusion of components (either into the

bright regions, as illustrated above, or into the dark regions) to form the permanent grating (below).

**Parameter Symbol Unit Target value**

Layer thickness d [μm] >50 Refractive index contrast Δn [‐] >0.001 Resolution U [lines/mm] >5000 Sensitivity S [cm²/J] >1 Transparency T [%] >0.7

**Table 1.** Overview on volume holographic material parameters.

The range of photosensitive media used for volume holographic recording is as diverse as the spectrum of potential applications.

Although photographic emulsion, the original holographic recording material, is capable to form amplitude as well as phase holograms with good sensitivity and stability as well as high spatial resolution, it nevertheless appears inappropriate for volume holography [1]. This is due to the fact that the sample thickness is limited to only a few microns, a serious disadvan‐ tage in view of the aspired recording of thick gratings for volume holography.

Photosensitive polymers have been used as holographic media since 1969 [17]. Polymers com‐ bine many advantages, namely low cost, ease of fabrication, flexibility and the ability to be integrated in more complex systems, such as optical circuits. They fulfill the requirements for volume holographic recording with no need for solvent processing, good dimensional stability, variable thickness, high energetic sensitivity, large dynamic range and sharp angu‐ lar selectivity [6, 15, 18, 19].

Among polymers for volume holographic recording, there are two material classes, differing in the mechanism of polymerization. The performances of free‐radical (FRP) and cationic ring‐opening polymerization (CROP) systems differ in many respects. Ranking among well‐known FRP systems, glass‐like polymer based on poly‐(methyl methacrylate) (PMMA) with distributed phenanthrenequinone (PQ) is known as effective and thermally stable holo‐ graphic recording material. Results on volume gratings within this chapter are based on investigations on gratings in free‐surface epoxy‐based polymer samples, prepared by micro‐ resist technology GmbH. High performance is achieved in volumetrically stable, free‐surface samples with variable layer thickness [6]. The corresponding mechanism of polymerization is a cationic ring‐opening polymerization. Similar to polyvinylalcohol/acrylamide (PVA/AA) material, grating formation occurs primarily as a consequence of photopolymerization and mass transport processes [20].

Photorefractive materials, such as lithium niobate, barium titanate or gallium arsenide, are capable to form temporary, erasable holograms as a result of a nonlinear optical effect [21]. Recorded data may be erased by flooding the crystal with uniform illumination. Improved properties with respect to the dynamic range, sensitivity and signal‐to‐noise‐ratio have been demonstrated [22]. Drawbacks are partially slow response time and low stability.

#### **2.2. Grating types**

Volume holographic gratings can be categorized according to different criteria. The following section gives an overview on the different types of gratings, and how they can be distinguished with regard to their optical functionality such as diffraction efficiency and angular response.

#### *2.2.1. Modulation*

With regard to the modulated optical property, phase gratings can be differentiated from absorption gratings. Depending on the physico‐chemical processes involved in the grating for‐ mation, the diffraction efficiency can be attributed to a refractive index contrast and/or to a mod‐ ulation of the absorption, respectively. However, both can also be observed together [23]. In this case, the modulation of the refractive index yields a part of the total diffraction efficiency while the absorption modulation induces additional diffraction. As a consequence, it is not possible to distinguish between phase and absorption gratings, based on the diffraction efficiency. Nor can such information be derived from a microscopic image. However, it may be provided from local analysis of the optical properties, namely refractive index and absorption, respectively.

#### *2.2.2. Dimensionality*

The dimensionality of a volume holographic grating indicates in how many spatial directions the modulation spreads. This is determined by the recording interference pattern. **Figure 2** schematically illustrates how the dimensionality of a volume grating relates to the number and orientation of corresponding recording beams.

In case of a two‐beam exposure, a one‐dimensional volume grating is formed, illustrated by the grating planes on the left side of **Figure 2**. The grating vector *K* <sup>→</sup> is defined by the two wave vectors of the recording beams (this is also illustrated in **Figure 3**). At least three record‐ ing beams are needed to build a two‐dimensional grating or rather optical pillars (center of **Figure 2**). A three‐dimensional grating, with a modulation in all three directions, results from at least four exposure beams (right side in **Figure 2**).

**Figure 2.** Schematic view of one‐ , two‐ and three‐dimensional volume gratings formed by two, three and four recording beams, respectively.

Higher dimensional gratings can also be obtained by means of superposition. The same applies to the functionality of more complex structures. The diffraction pattern provides respec‐ tive information according to the correlation of function and structure. A one‐dimensional grating shows diffraction with only one rotational degree of freedom (left side of **Figure 2**). Each additional recording direction adds a spatial direction to the modulation of the grating with the result of a higher dimension (middle and right side of **Figure 2**).

Microscopic images of one‐dimensional and three‐dimensional volume phase gratings are shown in **Figure 8**.

#### *2.2.3. Geometry*

Among polymers for volume holographic recording, there are two material classes, differing in the mechanism of polymerization. The performances of free‐radical (FRP) and cationic ring‐opening polymerization (CROP) systems differ in many respects. Ranking among well‐known FRP systems, glass‐like polymer based on poly‐(methyl methacrylate) (PMMA) with distributed phenanthrenequinone (PQ) is known as effective and thermally stable holo‐ graphic recording material. Results on volume gratings within this chapter are based on investigations on gratings in free‐surface epoxy‐based polymer samples, prepared by micro‐ resist technology GmbH. High performance is achieved in volumetrically stable, free‐surface samples with variable layer thickness [6]. The corresponding mechanism of polymerization is a cationic ring‐opening polymerization. Similar to polyvinylalcohol/acrylamide (PVA/AA) material, grating formation occurs primarily as a consequence of photopolymerization and

Photorefractive materials, such as lithium niobate, barium titanate or gallium arsenide, are capable to form temporary, erasable holograms as a result of a nonlinear optical effect [21]. Recorded data may be erased by flooding the crystal with uniform illumination. Improved properties with respect to the dynamic range, sensitivity and signal‐to‐noise‐ratio have been

Volume holographic gratings can be categorized according to different criteria. The following section gives an overview on the different types of gratings, and how they can be distinguished with regard to their optical functionality such as diffraction efficiency and angular response.

With regard to the modulated optical property, phase gratings can be differentiated from absorption gratings. Depending on the physico‐chemical processes involved in the grating for‐ mation, the diffraction efficiency can be attributed to a refractive index contrast and/or to a mod‐ ulation of the absorption, respectively. However, both can also be observed together [23]. In this case, the modulation of the refractive index yields a part of the total diffraction efficiency while the absorption modulation induces additional diffraction. As a consequence, it is not possible to distinguish between phase and absorption gratings, based on the diffraction efficiency. Nor can such information be derived from a microscopic image. However, it may be provided from local analysis of the optical properties, namely refractive index and absorption, respectively.

The dimensionality of a volume holographic grating indicates in how many spatial directions the modulation spreads. This is determined by the recording interference pattern. **Figure 2** schematically illustrates how the dimensionality of a volume grating relates to the number

In case of a two‐beam exposure, a one‐dimensional volume grating is formed, illustrated by

<sup>→</sup> is defined by the two

demonstrated [22]. Drawbacks are partially slow response time and low stability.

mass transport processes [20].

8 Holographic Materials and Optical Systems

**2.2. Grating types**

*2.2.1. Modulation*

*2.2.2. Dimensionality*

and orientation of corresponding recording beams.

the grating planes on the left side of **Figure 2**. The grating vector *K*

With regard to the geometry, transmission gratings can be distinguished from reflection gratings. The geometry of a grating is determined by the recording geometry, as illustrated in **Figure 3**.

The transmission and reflection curves, respectively, are strongly peaked at the Bragg angle, which is defined by Bragg's law. In case of unslanted transmission type gratings, the Bragg angle is equiva‐ lent to half the angle between reference and signal beam (Θ/2). The grating period Λ is (where n is the refractive index of the recording medium and λ is the free‐space recording wavelength):

$$
\Lambda\_t = \frac{\lambda}{2n\sin\frac{\theta}{2}}\tag{1a}
$$

$$
\Lambda\_{\prime} = \frac{\lambda}{2n \cos \frac{\theta}{2}} \tag{1b}
$$

for transmission (*Λ<sup>t</sup>* ) and reflection gratings (*Λ<sup>r</sup>* ), respectively.

A grating is transmission type if the angle between incoming light wave vector *k* <sup>→</sup> and grating vector *K* <sup>→</sup> is less than 90 degrees, that is, ∠( → *k*, *K* → ) <sup>&</sup>lt; \_*<sup>π</sup>* <sup>2</sup> . This is the case if both recording beams approach the sample from the same side (see left hand side of **Figure 3**). In contrast, the grating is reflection type if the recording beams come from both sides of the sample (see center of **Figure 3**). Again, transmission and reflection gratings may also be observed simultaneously. Beyond the possibilities to overlap gratings by means of multiplexing, superimposed holograms may also be formed due to the reflection of recording beams at the sample‐substrate interface. This case of secondary gratings is illustrated on the right hand side of **Figure 3**.

**Figure 3.** Recording geometries for transmission grating (TG) and reflection grating (RG). Wave vectors of recording beams *k* <sup>→</sup> as well as grating vectors *<sup>K</sup>* → are displayed. The reflected wave forms a secondary grating (RG') [5].

#### *2.2.4. Selectivity*

The selectivity of a volume phase grating serves as a criterion to classify the hologram with regard to the optical functionality. Therefore, it is indicated to define an important parameter with respect to the diffractive properties—the coupling constant κ:

$$
\kappa = \frac{\pi \Delta \mathfrak{m}}{2\lambda} \tag{2}
$$

where Δn is the refractive index contrast and λ is the recording wavelength. The coupling constant κ serves as a measure for the strength of a grating.

Holograms can be categorized into Raman‐Nath type and Bragg type, respectively. A Raman‐ Nath hologram causes multiple diffraction orders, leading to low diffraction efficiency. A Bragg hologram shows single diffraction, enabling high diffraction efficiency and good selectivity. While Raman‐Nath holograms may be recorded in a thin film, thick films are required to obtain Bragg holograms with good optical functionality [24].

With reference to the coupled wave analysis [25], the parameter:

*<sup>Λ</sup><sup>t</sup>* <sup>=</sup> \_\_\_\_\_\_ *<sup>λ</sup>*

*<sup>Λ</sup><sup>r</sup>* <sup>=</sup> \_\_\_\_\_\_ *<sup>λ</sup>*

) and reflection gratings (*Λ<sup>r</sup>*

<sup>→</sup> is less than 90 degrees, that is, ∠(

for transmission (*Λ<sup>t</sup>*

10 Holographic Materials and Optical Systems

grating vector *K*

hand side of **Figure 3**.

*2.2.4. Selectivity*

<sup>→</sup> as well as grating vectors *<sup>K</sup>*

beams *k*

2*n*sin \_*<sup>θ</sup>* 2

2*n*cos \_*<sup>θ</sup>* 2

A grating is transmission type if the angle between incoming light wave vector *k*

beams approach the sample from the same side (see left hand side of **Figure 3**). In contrast, the grating is reflection type if the recording beams come from both sides of the sample (see center of **Figure 3**). Again, transmission and reflection gratings may also be observed simultaneously. Beyond the possibilities to overlap gratings by means of multiplexing, superimposed holograms may also be formed due to the reflection of recording beams at the sample‐substrate interface. This case of secondary gratings is illustrated on the right

The selectivity of a volume phase grating serves as a criterion to classify the hologram with regard to the optical functionality. Therefore, it is indicated to define an important parameter

**Figure 3.** Recording geometries for transmission grating (TG) and reflection grating (RG). Wave vectors of recording

→ are displayed. The reflected wave forms a secondary grating (RG') [5].

*<sup>κ</sup>* <sup>=</sup> \_\_\_\_\_\_\_ *πΔ<sup>n</sup>*

where Δn is the refractive index contrast and λ is the recording wavelength. The coupling

Holograms can be categorized into Raman‐Nath type and Bragg type, respectively. A Raman‐ Nath hologram causes multiple diffraction orders, leading to low diffraction efficiency. A Bragg hologram shows single diffraction, enabling high diffraction efficiency and good selectivity. While Raman‐Nath holograms may be recorded in a thin film, thick films are required to

with respect to the diffractive properties—the coupling constant κ:

constant κ serves as a measure for the strength of a grating.

obtain Bragg holograms with good optical functionality [24].

), respectively.

→ *k*, *K* → ) <sup>&</sup>lt; \_*<sup>π</sup>* (1a)

(1b)

<sup>→</sup> and

<sup>2</sup> . This is the case if both recording

<sup>2</sup>*<sup>λ</sup>* (2)

$$
\Omega = \frac{\left| \mathbf{x}^\* \right|}{2k} \tag{3}
$$

with κ the coupling constant, defined by Eq. 2, can serve as an indicator for the presence of volume‐type gratings. For small Ω, multi‐wave diffraction occurs and little selectivity is shown. If *Ω* ≥ 10 is fulfilled, only two diffraction orders are excited [26]. This is the case if the Bragg condition is satisfied (i.e. if the probe angle corresponds to the Bragg angle, defined by Eq. 1).

From a more general perspective, a Bragg grating, also called thick grating [14], satisfies the condition:

$$d \gg \frac{\Lambda^2}{\Lambda} \tag{4}$$

with d the thickness of the grating or rather the layer thickness of the recording material.

The most important two‐wave case or rather the Bragg regime, is illustrated in **Figure 4**.

**Figure 4.** Diffraction efficiency η over normalized thickness ζ and normalized coupling constant ν in Bragg regime. A grating is overmodulated in case of *ν* > \_\_\_ *<sup>π</sup>* 2 .

In Bragg regime, the diffraction efficiency of the first diffraction order (*η*) is as follows:

$$
\eta\_1 = \sin^2 \langle \nu \rangle \tag{5}
$$

where *<sup>ν</sup>* <sup>=</sup> \_\_\_\_\_ *κd* cos *θ<sup>P</sup>* is the normalized coupling constant with θP the probe beam angle (depicted in **Figure 7**).

**Figure 4** shows how the diffraction efficiency distributes over normalized thickness ζ and normalized coupling constant ν. Hereby, the normalized thickness serves as off‐Bragg param‐ eter, accounting for small deviations from the Bragg condition either in terms of wavelength (Δλ) or in terms of angle (Δθ) [27].
