# Holographic Spectroscopy: Wavelength-Dependent Analysis of Photosensitive Materials by Means of Holographic Techniques

^{*}

## Abstract

**:**

**PACS**42.40.-i; 42.40.Lx; 42.40.Pa; 42.40.My; 42.65.-k; 42.70.Ln

## 1. Introduction

## 2. Recording a Test Hologram and the Properties of Its Permittivity

#### 2.1. Recording of a Test Hologram

**Figure 1.**Sketch of an optical setup for the recording of a test hologram: reference (R) and signal (S) waves that are mutually coherent and feature a flat-top intensity distribution and planar wavefronts, superimpose in a photosensitive medium of thickness l and generate a sinusoidal intensity distribution. M1–M6: mirrors; L1–L4: lenses; BS1–BS3: beam splitters, F1–F4: filters, P: polarizers; $\lambda /2$: half-wave retarder plates; SH1–SH3: computer-controlled beam shutters; PD1-PD4: Si-PIN diodes; BD1,BD2: beam dumps; SFF: spatial frequency filter, PH: pinhole. The direction of the corresponding wavevectors of the recording beams (${\mathbf{k}}_{\mathrm{R}}$, ${\mathbf{k}}_{\mathrm{S}}$) and the intensity distribution with respect to cartesian coordinates $I\left(x\right)$ is depicted in the inset.

#### 2.2. Properties of the Permittivity of the Test Hologram

**Table 1.**Characteristics of the optical setup and their relation to simplifications made in the theoretical analysis.

Characteristic of the optical setup | Simplification for the theoretical analysis |
---|---|

hologram recording is performed with equal wavelengths of R- and S-waves degenerate wave-mixing) | ${\lambda}_{\mathrm{R}}={\lambda}_{\mathrm{S}}$, i.e., $|{\mathbf{k}}_{\mathrm{R}}|=|{\mathbf{k}}_{\mathrm{S}}|=2\pi /\lambda $ |

hologram recording is performed by the superposition of planar wave fronts | sinusoidal permittivity modulation * |

R- and S-waves feature flat-top intensity profile | the electric field amplitudes are constant within the beam paths and the beam is assumed to have infinite diameter |

hologram recording is performed with equal directions of the electric field vectors of R- and S-wave and with equal intensities | $|{R}_{0}{|}^{2}={\left|{S}_{0}\right|}^{2}$, i.e., the modulation depth becomes unity ($m=1$), ${\mathbf{e}}_{R}\xb7{\mathbf{e}}_{S}=1$ |

hologram recording is homogeneously over the entire volume, i.e., exponential decrease of the grating parameters in z-direction is excluded (cf. [9]) ** | permittivity modulation is not z-dependent |

wave vector of the hologram is directed perpendicularly to the samples’ normal | permittivity modulation is aligned parallel to x-axis (unslanted hologram) |

## 3. Dispersion of First Order Diffracted Waves

#### 3.1. Signal Wave Reconstruction

**Figure 2.**Sketch of an optical setup for the reconstruction of a test hologram: Closing of shutter SH1 stops the hologram recording process. Then, reconstruction with a wavelength different from the recording wavelength is enabled by opening shutter SH4. The direction of the probe beam, particularly the angle of incidence with respect to the samples’ normal, is adjusted according to momentum conservation depicted in the inset figure.

**K**. However, in any practical situation, this condition is softened by at least two effects, leading to the well known rocking curve (i.e., the shape of the signal wave’s intensity profile upon angular or wavelength detuning from the Bragg condition): 1. Every existing laser system has a certain wavelength bandwidth. Consequently, there is not only one writing wavelength λ but a range $[\lambda -\delta \lambda ,\lambda +\delta \lambda ]$, which is resembled either by the range of possible reconstruction wavelengths under a certain angle of incidence or the range of angles for a given wavelength; 2. The sharpness of a grating is inversely proportional to the dimensions of the grating (cf. Figure 24.5a,b in [25]). In z-direction, the uncertainty of the Bragg condition of a grating with thickness d is given by $2\pi /d$. Consequently, the Bragg condition is a property of thick holograms. In x-direction, the dependency on the dimensions is even more obvious, as for an extension of the illumination that is smaller than Λ, the modulation would lose its grating characteristics. As commonly used beam diameters typically illuminate several hundreds or thousands of grating periods, finiteness in x-direction can be neglected. The finiteness of the grating in z-direction, however, which is governed either by the thickness of the crystal or the effective penetration depth of the recording beam, is covered. The z-dependency will be considerable for common samples and may be used to determine the effective grating thickness ${d}_{\text{eff}}$.

#### 3.2. Coupled-Wave Theory

#### 3.3. Diffraction Efficiency of Out-of-Phase Mixed Gratings

#### 3.4. Pure Refractive Index and Absorption Gratings

#### 3.5. In-Bragg Cases

#### 3.6. Transmission Efficiency

## 4. Examples

#### 4.1. Spectroscopic in-Bragg Analysis of the Diffraction Efficiency

**Figure 3.**Full absorption coefficient (thick black), base absorption (grey black), modulated absorption band (dashed black) and refractive index change emerging from Kramers–Kronig relation (blue) for Gaussian absorption band related to Fe-doping (

**a**) and polaronic absorption (

**b**).

**Figure 4.**Diffraction efficiency for a Gaussian absorption band (

**a**) and a polaronic absorption band (

**b**). Each figure shows the overall diffraction efficiency (black) and the efficiency from the pure absorption (blue) and the pure refractive index grating (red).

#### 4.2. Effects of Phase Shifts in Mixed Gratings

**Figure 5.**Zeroth (

**a**) and first order (

**b**) diffraction efficiency for ${\u03f5}_{1}^{\u2033}\left(\lambda \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\u03f5}_{1}^{\prime}\left(\lambda \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1,13\times {10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{F\xb7m}}^{-1}$. Figure (

**a**): $\Phi =0$ (red), $\Phi =\pi /4$ (black), $\Phi =\pi /2$ (blue), $\Phi =3\pi /4$ (green). Figure (

**b**): $\Phi =0$ (red), $\Phi =\pi /4$ and $\Phi =3\pi /4$ (black), $\Phi =\pi /2$ (blue).

## 5. Quantitative Analysis of the Borrmann Effect

**Figure 6.**Proportions ${\eta}_{1}\left(\theta \right)/{\eta}_{1}(-\theta )$ (black) and ${\eta}_{1}(-\theta )/{\eta}_{1}\left(\theta \right)$ (red). In Figure (

**a**), the phase shift Φ-depending ratios are show for ${Q}_{BA}=3$; In Figure (

**b**) the ${Q}_{BA}$-depending ratios are given for $\Phi =\pi /2$.

**Figure 7.**Graph of Equation (44). Figure (

**a**) shows the relation between ${Q}_{\text{BA}}$ and Φ, for ${\eta}_{1}\left(\theta \right)/{\eta}_{1}(-\theta )=4/3$ (green), 2 (black), 3 (red), 6 (blue); Figure (

**b**) shows the complete mathematical solution including ${\eta}_{1}\left(\theta \right)/{\eta}_{1}(-\theta )<1$ , for ${\eta}_{1}\left(\theta \right)/{\eta}_{1}(-\theta )=4/3$ (green), 2 (black), $3/4$ (red), $1/2$ (blue).

## 6. Conclusions

## Nomenclature of Material Properties

ϵ(x, λ) | Complex permittivity |

${\u03f5}_{0}^{\prime}\left(\lambda \right)$ | Mean real permittivity |

${\u03f5}_{0}^{\u2033}\left(\lambda \right)$ | Mean imaginary permittivity |

${\u03f5}_{1}^{\prime}\left(\lambda \right)$ | Amplitude of real permittivity grating |

${\u03f5}_{1}^{\u2033}\left(\lambda \right)$ | Amplitude of imaginary permittivity grating |

Φ_{B} | Phase offset of real permittivity grating compared with interference pattern at z = 0 |

Φ_{A} | Phase offset of imaginary permittivity grating compared with interference pattern at z = 0 |

Φ | Φ _{A} − Φ_{B} (for diffraction experiments with no interference pattern at z = 0) |

n(x, λ) | Complex refractive index |

α_{0}(λ) | Mean absorption coefficient |

n(x, λ) | Real refractive index |

n_{0}(λ) | Mean real refractive index |

n_{1}(λ) | Amplitude of real refractive index grating |

Φ_{n} | Phase offset of real refractive index grating |

κ_{0}(λ) | Mean absorption index |

κ_{1}(λ) | Amplitude of grating |

Φ_{κ} | Phase offset of absorption index grating |

K | Grating vector in material |

## Nomenclature of LightWave Properties

R(z)/S(z) | Electric field amplitude of reference/signal beam inside material |

R_{0}/S_{0} | Intial electric field amplitude of reference/signal beam inside material |

e_{R}(z)/e_{S}(z) | Polarisation unit vector of reference/signal beam |

Φ_{Diff} | Phase difference between reference and signal beam inside the material |

k | Wave vector of reference beam inside material |

Φ_{0} | Phase shift of interference pattern between reference/signal beam |

m | Modulation depth of interference pattern |

I/I_{R}/I_{S} | Intensity of interference pattern/reference beam/signal beam |

η_{0}/η_{1} | Transmission efficiency/diffraction efficiency |

## Acknowledgements

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**MDPI and ACS Style**

Voit, K.-M.; Imlau, M.
Holographic Spectroscopy: Wavelength-Dependent Analysis of Photosensitive Materials by Means of Holographic Techniques. *Materials* **2013**, *6*, 334-358.
https://doi.org/10.3390/ma6010334

**AMA Style**

Voit K-M, Imlau M.
Holographic Spectroscopy: Wavelength-Dependent Analysis of Photosensitive Materials by Means of Holographic Techniques. *Materials*. 2013; 6(1):334-358.
https://doi.org/10.3390/ma6010334

**Chicago/Turabian Style**

Voit, Kay-Michael, and Mirco Imlau.
2013. "Holographic Spectroscopy: Wavelength-Dependent Analysis of Photosensitive Materials by Means of Holographic Techniques" *Materials* 6, no. 1: 334-358.
https://doi.org/10.3390/ma6010334