**3. Analytical methods and corresponding results**

The performance of volume holograms can be attributed to tiny modulations of optical properties inside the holographic material. As a consequence, direct analytical techniques, such as imaging by means of optical microscopy, are only partially appropriate for volume holograms. This is primarily due to the low optical contrast between the grating planes. But it also applies with regard to the most important three‐dimensional structuring of volume holographic gratings.

#### **3.1. Structure and function**

A comprehensive analytical characterization of volume holographic gratings is possible based on the principle *from structure to function*. The idea of a correlation between structure and function for volume holograms is illustrated in **Figure 5**.

**Figure 5** contrasts structural information and the related functionality for one‐dimensional volume gratings. The structural information on the volume holographic grating is displayed on the left hand side. The holographic structure is described by its thickness d, the grating constant Λ and the refractive index contrast Δn. The (optical) functionality of a holographic structure consists in its diffractive properties, exemplarily displayed as angular resolved transmission on the right hand side of **Figure 5**. The link between structure and function consists in mutual determination: When the hologram is recorded, material response, exposure conditions and recording geometry determine the formation of the grating. For the final grating, the structural parameters {d, Λ, Δn} determine the diffractive properties or rather the angular resolved dif‐ fraction efficiency. In consequence, the diffraction efficiency can be utilized to access the char‐ acteristic parameters {d, Λ, Δn}. Corresponding analytical methods will be described below.

#### **3.2. Real‐time observation of grating formation**

The dynamics of volume holographic grating formation may be accessed with the help of a time‐resolved observation of the diffraction efficiency η(t). However, to model the grating growth and to draw conclusions on the interplay of underlying mechanisms, such as polym‐ erization and diffusion, the time evolution of the refractive index contrast Δn(t) is needed. Potential factors, responsible for the change of the refractive index in the course of polymer‐ ization, are molecular polarizability, density and molar mass [28].

*η*<sup>1</sup> = *sin*² (*ν*) (5)

**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

The performance of volume holograms can be attributed to tiny modulations of optical properties inside the holographic material. As a consequence, direct analytical techniques, such as imaging by means of optical microscopy, are only partially appropriate for volume holograms. This is primarily due to the low optical contrast between the grating planes. But it also applies with regard to the most important three‐dimensional structuring of volume holographic gratings.

A comprehensive analytical characterization of volume holographic gratings is possible based on the principle *from structure to function*. The idea of a correlation between structure

**Figure 5** contrasts structural information and the related functionality for one‐dimensional volume gratings. The structural information on the volume holographic grating is displayed on the left hand side. The holographic structure is described by its thickness d, the grating constant Λ and the refractive index contrast Δn. The (optical) functionality of a holographic structure consists in its diffractive properties, exemplarily displayed as angular resolved transmission on the right hand side of **Figure 5**. The link between structure and function consists in mutual determination: When the hologram is recorded, material response, exposure conditions and recording geometry determine the formation of the grating. For the final grating, the structural parameters {d, Λ, Δn} determine the diffractive properties or rather the angular resolved dif‐ fraction efficiency. In consequence, the diffraction efficiency can be utilized to access the char‐ acteristic parameters {d, Λ, Δn}. Corresponding analytical methods will be described below.

The dynamics of volume holographic grating formation may be accessed with the help of a time‐resolved observation of the diffraction efficiency η(t). However, to model the grating growth and to draw conclusions on the interplay of underlying mechanisms, such as polym‐ erization and diffusion, the time evolution of the refractive index contrast Δn(t) is needed. Potential factors, responsible for the change of the refractive index in the course of polymer‐

is the normalized coupling constant with θP the probe beam angle (depicted

where *<sup>ν</sup>* <sup>=</sup> \_\_\_\_\_ *κd*

in **Figure 7**).

cos *θ<sup>P</sup>*

12 Holographic Materials and Optical Systems

(Δλ) or in terms of angle (Δθ) [27].

**3.1. Structure and function**

**3. Analytical methods and corresponding results**

and function for volume holograms is illustrated in **Figure 5**.

**3.2. Real‐time observation of grating formation**

ization, are molecular polarizability, density and molar mass [28].

**Figure 5.** Relationship of structure and function for a volume holographic grating: The grating is determined by its thickness d, grating constant Λ and refractive index contrast Δn (left hand side). Those parameters determine the diffractive properties, exemplarily displayed as angular‐resolved transmission (right hand side, from Ref. [6]).

Optical methods provide an elegant approach to study the kinetics of polymerization and dif‐ fusion in the course of grating formation [19]. Corresponding analysis setups feature in‐situ parts for real‐time, non‐disturbing observation of the grating formation process and enable monitoring of the time evolution of the diffracted part of a probe beam from the very start of exposure. As a consequence, grating growth curves can be obtained.

**Figure 6** shows recording and analysis setup plus corresponding grating growth curve. In this case, the growth curve reveals a transition of the refractive index contrast [29].

**Figure 6** shows holographic exposure, performed by two freely propagating, s‐polarized recording beams, 2 mm in diameter. Symmetric recording geometry results in unslanted grat‐ ings with periodicity of Λ ≈ 2 μm.

To ensure non‐disturbing observation of the grating formation process, the wavelength for in‐situ observation must be chosen outside of the absorption spectrum of the photosensitizer dye. In the above example, a fiber‐guided 633 nm HeNe laser was used in combination with an adjustable collimator, allowing to probe with a slightly focused beam. This enables to steadily ensure a stable on‐Bragg condition according to Eq. 1. A position sensitive device (PSD) was implemented to detect the diffracted light. The PSD provides time‐resolved information on the diffraction efficiency as well as on the Bragg angle. The time‐resolved information on the grating constant is derived from the position of the diffracted beam on the detector. This also enables to draw conclusions on time‐resolved optical shrinkage [30, 31].

**Figure 6.** Recording and analysis setup for real‐time observation of volume holographic grating formation: Holographic exposure is performed by two freely propagating, s‐polarized recording beams with λ = 405 nm (left). The corresponding grating growth curve shows a two‐step grating growth (right).

The grating growth curve shown in **Figure 6** belongs to a one‐dimensional, plane‐wave vol‐ ume hologram of transmission type, recorded in epoxy‐based polymer [29]. The characteristic two‐step growth can be attributed to a transition of the refractive index contrast from positive to negative values, as a result of competing effects, taking place on overlapping time scales.

Investigations on the dynamics of volume holographic grating formation may also be applied to study the influence of important factors. This applies to material parameters, such as com‐ position or viscosity, to grating parameters such as grating constant or geometry as well as to recording parameters, such as exposure duration or recording intensity [30].

#### **3.3. Angular‐resolved analysis**

In the context of the leading idea *from structure to function*, the angular resolved analysis of volume holograms is of particular importance. Analysis of diffracted light provides basic access to the characteristic properties of the patterns causing the diffraction. Particularly, the angular‐resolved diffraction efficiency provides information on the key features, such as grating constant, grating slant and, by comparison with coupled‐wave‐theory (RCWA) calculations, also about layer thickness, refractive index contrast and refractive index profile.

Analysis of the final holograms is usually accomplished in a rotation scan setup with colli‐ mated probe beam [32]. **Figure 7** shows a rotation scan setup plus corresponding transmission curve, that is, the angular response of a volume hologram (dots) and comparison with RCWA calculations (solid line).

**Figure 7.** Rotation‐scan setup (left) and corresponding angular response of a volume holographic grating, from which conclusions are drawn on grating parameters and material response (right).

The transmitted signal of a 543 nm HeNe laser is detected while the hologram under test is rotated. From the angularly resolved transmission, the following information is derived. First, the maximum diffraction efficiency can be obtained and can be correlated with the exposure energy density E to yield the material response [6]. Second, the grating constant Λ can be derived with the help of Eq. 1a. Finally, RCWA calculations can be used to derive values for the layer thickness d and the refractive index contrast Δn.

#### **3.4. Microscopic techniques**

the diffraction efficiency as well as on the Bragg angle. The time‐resolved information on the grating constant is derived from the position of the diffracted beam on the detector. This also

The grating growth curve shown in **Figure 6** belongs to a one‐dimensional, plane‐wave vol‐ ume hologram of transmission type, recorded in epoxy‐based polymer [29]. The characteristic two‐step growth can be attributed to a transition of the refractive index contrast from positive to negative values, as a result of competing effects, taking place on overlapping time scales.

**Figure 6.** Recording and analysis setup for real‐time observation of volume holographic grating formation: Holographic exposure is performed by two freely propagating, s‐polarized recording beams with λ = 405 nm (left). The corresponding

Investigations on the dynamics of volume holographic grating formation may also be applied to study the influence of important factors. This applies to material parameters, such as com‐ position or viscosity, to grating parameters such as grating constant or geometry as well as to

In the context of the leading idea *from structure to function*, the angular resolved analysis of volume holograms is of particular importance. Analysis of diffracted light provides basic access to the characteristic properties of the patterns causing the diffraction. Particularly, the angular‐resolved diffraction efficiency provides information on the key features, such as grating constant, grating slant and, by comparison with coupled‐wave‐theory (RCWA) calculations, also about layer thickness, refractive index contrast and refractive index

Analysis of the final holograms is usually accomplished in a rotation scan setup with colli‐ mated probe beam [32]. **Figure 7** shows a rotation scan setup plus corresponding transmission curve, that is, the angular response of a volume hologram (dots) and comparison with RCWA

recording parameters, such as exposure duration or recording intensity [30].

**3.3. Angular‐resolved analysis**

14 Holographic Materials and Optical Systems

grating growth curve shows a two‐step grating growth (right).

profile.

calculations (solid line).

enables to draw conclusions on time‐resolved optical shrinkage [30, 31].

As outlined above, low contrast is the main problem in imaging of volume holographic grat‐ ings. It might be proposed to apply fluorescent media or dyes as contrast agents to improve the image contrast. In this case, agglomeration of the contrast media along the grating planes would be prerequisite to achieve the desired effect, which cannot always be ensured. To achieve con‐ trast in electron microscopy, conductive species are necessarily required. This is the case where nanoparticles are incorporated [33, 34]. Thus, transmission electron microscopy (TEM) is used to evaluate the degree of nanoparticle assembly [33, 35]. Scanning force microscopy (SFM) may be applied additionally, to map surface modulations [33, 36]. Nanocomposite materials are also the subject of investigations by luminescence microscopy [37]. However, mapping of nanopar‐ ticles yields only a limited description of lattice structures, not necessarily identical with the grating of interest, which is linked to the diffractive properties. In fact, it has been demonstrated that photoinsensitive nanoparticles experience counterdiffusion during grating buildup [38].

The imaging task is becoming increasingly complex without conductive species and with regard to the three dimensionality of volume phase gratings. However, it is the third dimen‐ sion in particular to which the specific features of volume holograms can be assigned.

All the limitations notwithstanding, optical microscopy might nevertheless be applied to ana‐ lyze volume holograms. Corresponding images are shown in **Figure 8**.

A one‐dimensional volume phase grating is shown on the left side of **Figure 8**. In case of higher dimensional gratings, optical microscopy may only be applied to picture single planes of the structure. As an example, the middle and right hand side of **Figure 8** show two planes of a three‐dimensional holographic grating with hexagonal close packing crystal structure. It was produced by four mutually coherent exposure beams. The lateral distance of neighboring crystal units is 2 μm, to be read from **Figure 8**. The grating constant perpen‐ dicular to the image plane amounts to 22 μm (not shown) [39].

**Figure 8.** Optical microscopic imaging of volume phase gratings in photosensitive polymer: one‐dimensional grating (left hand side) and 3D photonic grating under variation of the microscopic focal plane (middle and right hand side).

#### **3.5. Spatially resolved diffraction analysis**

Optical microscopy provides local information on the grating in the context of geometry and dimensionality. No information on the optical functionality, such as on the Bragg selectivity, is provided. However, this information is accessible according to the relation between structure and function and can be derived from spatial‐resolved analysis. In case of transmission type gratings, this can be accomplished by means of scanning the lateral plane to obtain local val‐ ues of the grating parameters. Local values of thickness d, refractive index contrast Δn, as well as grating period Λ and the grating slant Φ can be obtained. The lateral scan method is keeping track of the hologram shape, which is determined by the material response to the Gaussian intensity distribution of the recording beams [30].

Probing only a fraction of the exposed area is primarily for the purpose of measuring precision [29, 40]. However, it also enables scanning of the grating by moving the sample perpendicular to the optical axis. A sequence of rotation scans through the grating diam‐ eter constitutes a lateral scan. This analytical method allows the determination of the holo‐ gram characteristics along the sample surface. Thereupon, it is possible to compare and track respective properties from the center of the grating to the edges, corresponding to the areas of highest and lowest recording intensity. As a consequence, spatial sequences of the grating parameters are derived, providing insight into the local material characteristics.

**Figure 9** illustrates the principle of lateral scanning. The ratio of probe beam to exposure beam diameter was 1:6. The local diffraction efficiency is displayed along the lateral posi‐ tion of five different volume holograms. The gratings were recorded with different exposure dose. The respective energy density of exposure (E) is displayed in **Figure 9**. The lateral scan reveals the material response, resulting in overmodulation (*ν* > \_\_ *π* <sup>2</sup> ) in case of E > 330 mJ/cm2 .

The results from spatially resolved investigations of the grating constant reveal the influence of the recording intensity and exposure duration on the Bragg selectivity [30]. The advantage of lateral scanning consists in the exploitation of the Gaussian intensity distribution of the recording beams. Thus, lateral scanning provides direct access to characterize the material response, based on one single exposure. As a result, other influential factors on the material response, such as pre‐exposure, are eliminated.

**Figure 9.** Principle of lateral scanning: Holograms were recorded with exposure beam diameter of 3mm and scanned step‐by‐step with a probe beam diameter of 0.5 mm. Five lateral scans are shown. The corresponding gratings were recorded with different exposure energy densities.
