**6. Results and discussions**

#### **6.1. Characterization of a volume holographic spherical lens**

The volume holographic in-line spherical lens was characterized, and its efficiency *η* was calculated as the ratio between the power focused by the holographic lens (*Pf*\_*Holo*\_*lens*) and the power focused by a commercial Fresnel lens (*Pf*\_*Fresnel*) with the same focusing features:

$$
\eta = \mathcal{P}\_{f\_{\\_H \text{hole\\_lens}}} / \mathcal{P}\_{f\_{\\_F \text{Fresand}}} \tag{7}
$$

In particular, both holographic and Fresnel lenses have been illuminated by a beam with a diameter comparable with that of the lenses (≈4.5 cm), and a power metre was positioned at the focal length. The evaluated efficiency was 42%.

The angular selectivity was assessed by measuring the diffracted intensity from the holographic spherical lens as a function of the angle of incidence. Two different laser sources, emitting at 532 and 633 nm, were used in order to consider different behaviours of the lens at different wavelengths. Experimental results of the angular scan at different incident wavelengths are showed in **Figure 7(a)**. As expected, the angle at which the diffraction intensity is maximum increases as the wavelength increases [14]. Additionally, the lens chromatic aberration was investigated. Ideally, an optical lens should focus all of the component colours of white light to a single point. This means that the lens should refract all of the colours of white light in the same way, so they all intersect each other at the same location (or focus). The measurement of axial or longitudinal chromatic aberration is given by the difference of focal length between blue (442 nm), green (532 nm) and red (633 nm), caused by chromatic dispersion.

**5.2. Recording set-up**

40 Holographic Materials and Optical Systems

state of technology.

The dimensions of an individual HOE range from 1 cm × 1 cm to 10 cm × 10 cm. In a step-bystep exposure process by coherent and monochromatic light (laser), the holograms are produced in patterns on a film, which can have a maximum size of 1 m × 2 m at the present

The experimental set-up used to record holographic in-line spherical lenses was a typical Michelson interferometer with a concave mirror with a focal length of 5 cm placed on the object beam. A recording light source emitting at 532 nm (green) with a maximum power of 750 mW in CW and a coherence length up to 100 m was used. The diameter of the hologram was about 4 cm. To record an off-axis holographic cylindrical lens, the experimental set-up was modified, and two beams of equal intensity interfere with an angle α at the surface of the recording medium. A commercial cylindrical lens, with a focal length of 5.08 cm, is placed on the object beam. Finally, to record a volume holographic grating (VHG), two collimated beams interfere

The volume holographic in-line spherical lens was characterized, and its efficiency *η* was calculated as the ratio between the power focused by the holographic lens (*Pf*\_*Holo*\_*lens*) and the power focused by a commercial Fresnel lens (*Pf*\_*Fresnel*) with the same focusing features:

In particular, both holographic and Fresnel lenses have been illuminated by a beam with a diameter comparable with that of the lenses (≈4.5 cm), and a power metre was positioned at

The angular selectivity was assessed by measuring the diffracted intensity from the holographic spherical lens as a function of the angle of incidence. Two different laser sources, emitting at 532 and 633 nm, were used in order to consider different behaviours of the lens at different wavelengths. Experimental results of the angular scan at different incident wavelengths are showed in **Figure 7(a)**. As expected, the angle at which the diffraction intensity is maximum increases as the wavelength increases [14]. Additionally, the lens chromatic aberration was investigated. Ideally, an optical lens should focus all of the component colours of white light to a single point. This means that the lens should refract all of the colours of white light in the same way, so they all intersect each other at the same location (or focus). The measurement of axial or longitudinal chromatic aberration is given by the difference of focal length between blue (442 nm), green (532 nm) and red (633 nm), caused by chromatic disper-

*f Holo lens f Fresnel ηP P* \_\_ \_ = / (7)

with an angle α at the surface of the recording medium.

**6.1. Characterization of a volume holographic spherical lens**

the focal length. The evaluated efficiency was 42%.

**6. Results and discussions**

sion.

**Figure 7.** (a) Angular scan of the volume holographic lens at two different wavelengths and (b) focal length for different incident wavelengths both for conventional plano-convex lens and for holographic spherical lens.

The focal length was measured as function of wavelength both for a conventional lens and for a holographic spherical lens. Results are reported in **Figure 7(b)**. The conventional lens was a 2" plano-convex lens with a focus distance of 6 cm. Its theoretical focal length was also evaluated by using a simplified thick lens equation:

$$f = \mathbb{R} / \left(n - 1\right) \tag{8}$$

where *n* is the index of refraction and *R* is the radius of curvature of the lens surface. The lens used in our experiments had *R* = 30.9 mm, and it was fabricated from N-BK7, so we used the index of refraction for N-BK7 at the wavelength of interest to approximate the wavelengthdependent focal length. Seeing **Figure 7(b)**, it is evident that, while for the conventional lens the focal length slightly increases by increasing the incident wavelength, the holographic lens shows a marked decrease of the focal length by increasing the incident wavelength. Also, this behaviour can be explained by considering the Bragg condition. Indeed, the Bragg angle *θB* increases when wavelength increases. In particular, the focal length is related to *θB*, and so to

$$
\lambda \text{, by the geometrical relationship } f = \frac{\nu\_{/2}}{\tan \left( \theta\_B \right)} \text{, here } D \text{ is the lens diameter.}
$$

Chromatic aberration of the holographic lenses can be reduced in the visible range designing an achromatic doublet by using two holographic elements: a holographic lens and a holographic grating, as proposed by Udupa et al. [67]. Therefore, the combined two holographic element systems behave like a single element holographic achromatic lens.

The beam profile in the focal point of the holographic optical lens was also characterized, and a comparison with the beam profiles both of a conventional lens and of a Fresnel lens was carried out. In **Table 1** the evaluated widths of the beam as four times the standard deviation (4-sigma) for the three different lenses characterized are summarized.


**Table 1.** Beam width 4-sigma evaluated for three different lenses.

It is clear that the holographic spherical lens shows a reduced ability to concentrate light at the focal distance with respect to the other two lenses considered. This result can depend from the chromatic aberration that affects the holographic lens.

#### **6.2. Characterization of a volume holographic cylindrical lens**

The volume holographic off-axis cylindrical lens was characterized, and its efficiency *η* was evaluated as the ratio between the power focused by the holographic lens (*Pf*\_*Holo*\_*lens*) and the power focused by a commercial in-line cylindrical lens (*Pf*\_*Cyl*) with the same focusing features. A beam with a diameter comparable with that of the lenses (≈1.5 cm) was used, and the focused power was measured by a power metre positioned at the focal length. The obtained value for the efficiency was 25%.

Considering that both holographic spherical and holographic cylindrical lenses follow the same theoretical laws; also in the case of cylindrical holographic lenses, it is expected that the focal length as a function of the incident wavelengths follows the same behaviour of that observed for the holographic in-line spherical lens described in the previous section (**Figure 7(b)**). Therefore, a decrease in the focal length of the holographic cylindrical lens is expected by increasing the incident wavelength.

In **Table 2** the evaluated widths of the beam as 4-sigma obtained by the intensity profile of the wavefront acquired at the focal plane of the off-axis holographic cylindrical lens with focal length f = 5.08 cm are summarized.


**Table 2.** Beam width 4-sigma evaluated for the off-axis cylindrical lens.

As can be noted from **Table 2**, in the case of the off-axis cylindrical lens, the beam width (4 sigma) along the y-axis is greatly reduced in the focus of the lens, thereby drastically reducing the width along the y-dimension. For that reason, cylindrical lenses are the most commonly suggested to avoid tracking in one direction; indeed, if the incidence direction varies in the perpendicular plane, the angular selectivity is lower, so it is possible to eliminate tracking in this direction [24].

#### **6.3. Solar concentrators in the space**

**Beam width (4-sigma) X [μm] Y [μm]** Holographic spherical lens 4754.62 5279.20 Conventional optical lens 4171.23 4155.35 Fresnel lens 4007.91 4018.47

It is clear that the holographic spherical lens shows a reduced ability to concentrate light at the focal distance with respect to the other two lenses considered. This result can depend from the

The volume holographic off-axis cylindrical lens was characterized, and its efficiency *η* was evaluated as the ratio between the power focused by the holographic lens (*Pf*\_*Holo*\_*lens*) and the power focused by a commercial in-line cylindrical lens (*Pf*\_*Cyl*) with the same focusing features. A beam with a diameter comparable with that of the lenses (≈1.5 cm) was used, and the focused power was measured by a power metre positioned at the focal length. The obtained value for

Considering that both holographic spherical and holographic cylindrical lenses follow the same theoretical laws; also in the case of cylindrical holographic lenses, it is expected that the focal length as a function of the incident wavelengths follows the same behaviour of that observed for the holographic in-line spherical lens described in the previous section (**Figure 7(b)**). Therefore, a decrease in the focal length of the holographic cylindrical lens is

In **Table 2** the evaluated widths of the beam as 4-sigma obtained by the intensity profile of the wavefront acquired at the focal plane of the off-axis holographic cylindrical lens with focal

As can be noted from **Table 2**, in the case of the off-axis cylindrical lens, the beam width (4 sigma) along the y-axis is greatly reduced in the focus of the lens, thereby drastically reducing the width along the y-dimension. For that reason, cylindrical lenses are the most commonly suggested to avoid tracking in one direction; indeed, if the incidence direction varies in the perpendicular plane, the angular selectivity is lower, so it is possible to eliminate tracking in

**Beam width (4-sigma) X [μm] Y [μm]** Holographic cylindrical lens off-axis 995.84 237.78

**Table 1.** Beam width 4-sigma evaluated for three different lenses.

42 Holographic Materials and Optical Systems

chromatic aberration that affects the holographic lens.

expected by increasing the incident wavelength.

**Table 2.** Beam width 4-sigma evaluated for the off-axis cylindrical lens.

length f = 5.08 cm are summarized.

this direction [24].

the efficiency was 25%.

**6.2. Characterization of a volume holographic cylindrical lens**

In order to utilize a material in space environment, an appropriate characterization of this material for these applications is required. The first problem is the large temperature range of operation. For this reason, a characterization in temperature was made to verify the possibility to use HOEs described before in the aerospace industry.

In order to characterize the behaviour of the new proposed photopolymer as a function of the temperature, a VHG was recorded with a diffraction angle α of 30° leading a VHG with 1000 lines per millimetre. Experimentally, the diffraction efficiency was evaluated by using the following relationship:

$$\eta = \bigvee\_{\bullet}^{P\_1} \left( P\_0 + P\_1 \right) \tag{9}$$

where P1 is the measured power of the first diffraction order and P0 is the measured power of the zero diffraction order. With this aim, a He-Ne laser emitting at 633 nm, a motorized goniometer and a power metre were used. Measures were carried out at room temperature (TR = 24°) in a given point of the VHG that was rotated at the Bragg angle. Afterward, the VHG was exposed to a stress in temperature, to verify its behaviour in terms of diffraction efficiency. With this aim, the temperature of the photopolymer, and so of the grating, was increased at 150° for 2 h. Then, the temperature was reported at TR, and the diffraction efficiency at the same previous point of the VHG was measured again. Finally, the temperature was lowered at −80° for 23 h, and the diffraction efficiency was measured again in the same previous point of the grating when the temperature was reported to TR. In **Table 3** the efficiencies measured at TR in initial condition and after two different thermal stresses (increased and lowered of temperature of material) are summarized.


**Table 3.** Diffraction efficiency measured as a function of temperature.

Preliminary results confirm that the diffraction efficiency of the VHG subjected to thermal stress does not change significantly. The little variations observed in the diffraction efficiency maybe are ascribed to errors in positioning the red laser for measurement in the same point. Therefore, we can conclude that VHGs do not lose efficiency after a single cycle of thermal stress, and so this very promising material could be successful used for aerospace applications. Of course, further study has to be performed to demonstrate that no changes in the VHG performance are observed even after several cycles of thermal stress.

To consolidate the previous results, a test of outgassing of the photopolymer was carried out. This characterization is very important to evaluate the behaviour of the photopolymer in the absence of pressure (space conditions). In this test, the grating was firstly cleaned with isopropanol and then was inserted in a chamber connected with a turbo pump variation. The chamber was also connected to a residual gas analyzer (RGA) consisting of a mass spectrometer. Two tests were performed at different pressures. The first one had a base pressure of 2 × 10−8 mbar after 6 days of pumping at room temperature. The results, acquired with a mass spectrometer, reported the presence only of main gas species known that are the contaminants based on the chamber.

The second test had a base pressure 7 × 10−8 mbar after 3 days of pumping at room temperature. Results acquired with the mass spectrometer highlighted the presence of unknown main gas species, with atomic mass units (AMU) up to 100. However, the intensity of the unknown gas is very small and comparable with the contaminants of the same chamber of the test.

Finally, the behaviour of the focal length for different incident wavelengths reported in **Figure 7** can be useful in aerospace application. Indeed, considering that in the infrared region the focal length is very far by the focal length in the visible region (where the PV cell will be placed), the thermal overheating of the photovoltaic cell due to the absorption of infrared radiation is avoided, reducing cooling requirements. All these results confirmed the possibility to use this photopolymer in spatial applications.
