# Volume Holograms in Photopolymers: Comparison between Analytical and Rigorous Theories

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

**K**is the wave vector and can be expressed as function of the grating period; Λ, or as function of the frequency, f:

_{1}is the refractive index modulation; λ

_{0}is the wavelength in vacuum; and ϑ is a parameter that measures the Bragg angle deviation.

**ρ**

_{i}=

**ρ**

_{0}+i

**K**i = 0, ±1, ±2,...

_{i}are the called obliquity factors and are the cosine of the angles that the propagation vectors of the different orders form with the z-axis. In the particular case of non-slanted diffraction gratings, C

_{i}= cosθ

_{0}, where θ

_{0}is the angle of reconstruction inside the medium.

_{1}<< n

_{0}. The cases analyze in this work are consistent with condition.

## 3. Gratings Recorded in Photopolymers

#### 3.1. Theoretical Simulations

_{1}= 0.004 was used [36]. In the second place we performed the analysis with a value of the refractive index modulation n

_{1}= 0.011, a value which is near of maximum for materials based in PVA/AA and is achieved using a crosslinker monomer, NN’,methylene-bis-acrylamide (BMA) [35,36].

_{1}= 0.004, at first Bragg’s condition (Ω takes the value 8.3). The value of Ω shows that we are in Bragg’s regime, and as can be expected the predictions of the three theories are similar (Kogelnik, CW and RCW) in the range of thickness studied.

**Table 1.**Values of the parameters of the phase transmission gratings for different spatial frequencies.

Refractive index modulation | f = 350 lines/mm | f = 500 lines/mm | f = 750 lines/mm |
---|---|---|---|

n_{1} = 0.004 | Ω ~ 8 | Ω ~ 17 | Ω ~ 38 |

n_{1} = 0.011 | Ω ~ 3 | Ω ~ 6 | Ω ~ 14 |

**Figure 1.**Diffraction efficiency of order +1 (first order diffracted), at first Bragg’s angle. The spatial frequency is 350 lines/mm and the refractive index modulation is n

_{1}= 0.004.

**Figure 2.**Efficiency of order 0 (zero order diffracted), at first Bragg’s angle. The spatial frequency is 350 lines/mm and the refractive index modulation is n

_{1}= 0.004.

_{1}= 0.011) for the same spatial frequency the curves presented in Figure 3 and Figure 4 can be obtained (first and zero orders versus thickness at first Bragg angle condition), Ω takes a value of 3, so this grating cannot be classified as a volume one. If these figures are analyzed we can see that there exist deviations between the predictions using Kogelnik’s theory and the results obtained using CW and RCW. This fact occurs because the secondary orders cannot be neglected and part of the energy is diffracted in other directions (secondary diffracted orders). It is also interesting to remark that in all the cases the predictions of CW and RCW are very similar, what shows that the second derivates can be neglected in Equation (8). The results obtained by these two theories are also similar for secondary orders. The energy of orders −1 and +2 as a function of the thickness of the grating are represented in Figure 5.

**Figure 3.**Diffraction efficiency of order +1 (first order diffracted), at first Bragg’s angle. The spatial frequency is 350 lines/mm and the refractive index modulation is n

_{1}= 0.011.

**Figure 4.**Efficiency of order 0 (zero order diffracted), at first Bragg’s angle. The spatial frequency is 350 lines/mm and the refractive index modulation is n

_{1}= 0.011.

**Figure 5.**Efficiency of the orders −1 and +2, at first Bragg’s angle. The spatial frequency is 350 lines/mm and the refractive index modulation is n

_{1}= 0.011.

_{1}= 0.011). In these figures there is good agreement between the CW and RCW theories, but the results obtained by using Kogelnik Coupled Wave Theory slightly deviate from those of RCW and CW theories. This is due to the existence of multi-order diffraction, although the results are clearly closer than in the case of 350 lines/mm.

**Figure 6.**Diffraction efficiency of order +1 (first order diffracted), at first Bragg’s angle. The spatial frequency is 500 lines/mm and the refractive index modulation is n

_{1}= 0.011.

**Figure 7.**Efficiency of order 0 (zero order diffracted), at first Bragg’s angle. The spatial frequency is 500 lines/mm and the refractive index modulation is n

_{1}= 0.011.

_{1}= 0.011 (Ω = 14). It is evident that in this case the agreement between the three theories is almost perfect, so in this case Kogelnik’s theory is highly applicable to analyze these gratings at first Bragg’s angle.

**Figure 8.**Diffraction efficiency of order +1 (first order diffracted), at first Bragg’s angle. The spatial frequency is 750 lines/mm and the refractive index modulation is n

_{1}= 0.011.

**Figure 9.**Efficiency of order 0 (zero order diffracted), at first Bragg’s angle. The spatial frequency is 750 lines/mm and the refractive index modulation is n

_{1}= 0.011.

#### 3.2. Experimental Results

^{2}.

_{1}= 0.0102, and α = 0.0056 μm

^{−1}).

**Figure 11.**Diffraction efficiency of the order +1 for volume transmission gratings recorded in polyvinyl-alcohol/acrylamide (PVA/AA) with spatial frequency 1125 lines/mm and 23 µm of thickness.

_{1}= 0.00476, d = 69 µm and α = 0.0042 μm

^{−1}.

**Figure 12.**Diffraction efficiency of order +1 for volume transmission gratings recorded in PVA/AA with spatial frequency 545 lines/mm and 69 µm of thickness at first Bragg’s angle.

_{1}= 0.011. In Figure 13 it can be seen the discrepancies in the results provided by these two theories. Therefore the approximations made in the Kogelnik’s analysis cannot be done in this case. When Kogelnik’s theory is used to fit the angular response of gratings with spatial frequency of 350 lines/mm important errors will be made in the values of thickness and refractive modulation.

**Figure 13.**Diffraction efficiency of order +1 for volume transmission gratings with spatial frequency 350 lines/mm, n

_{1}= 0.011 and 25 µm of thickness at first Bragg’s angle.

## 4. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Gallego, S.; Neipp, C.; Estepa, L.A.; Ortuño, M.; Márquez, A.; Francés, J.; Pascual, I.; Beléndez, A.
Volume Holograms in Photopolymers: Comparison between Analytical and Rigorous Theories. *Materials* **2012**, *5*, 1373-1388.
https://doi.org/10.3390/ma5081373

**AMA Style**

Gallego S, Neipp C, Estepa LA, Ortuño M, Márquez A, Francés J, Pascual I, Beléndez A.
Volume Holograms in Photopolymers: Comparison between Analytical and Rigorous Theories. *Materials*. 2012; 5(8):1373-1388.
https://doi.org/10.3390/ma5081373

**Chicago/Turabian Style**

Gallego, Sergi, Cristian Neipp, Luis A. Estepa, Manuel Ortuño, Andrés Márquez, Jorge Francés, Inmaculada Pascual, and Augusto Beléndez.
2012. "Volume Holograms in Photopolymers: Comparison between Analytical and Rigorous Theories" *Materials* 5, no. 8: 1373-1388.
https://doi.org/10.3390/ma5081373