**3. Achromatization of HPM with surface diffraction gratings**

As discussed, HPMs can successfully imprint their phase pattern as long as the wavelength satisfies the Bragg condition but to achieve this, the HPM needs to be angle tuned which cannot be considered pure achromatization. Such achromatization of HPMs can be accomplished with the concept of pairing the Bragg grating with two surface gratings [30]. According to the grating dispersion equation (Eq. (5)), a surface grating with a given period (*Λ*SG) will diffract normally incident light at an angle (*θ*) as a function of its wavelength (*λ*):

will be created that can simultaneously convert multiple beams into different modes while combining them to a single beam. For example, fiber lasers that operate at higher order mode are of interest because they are considered to overcome the power limitations of fiber lasers operating at the fundamental mode. Therefore, beam combining several lasers operating at higher order modes into one high-power fundamental mode beam will be very beneficial and

As an example, **Figure 15** shows how a double multiplexed HPM can convert individually two TEM11 modes (**Figure 15a** and **15b**) to TEM00 modes (**Figure 15c** and **15d**) while also spectrally beam combining the beams into one beam (**Figure 15e**) [27]. The lasers were operating at 1061 and 1064 nm and the HPM consisted two four-sector phase masks integrated in two TBGs that had a generate output. The authors attributed the difference between the far-field profiles of the two laser beams after their conversion to different collimations. In the final combined beam, there are wings present but these were credited to the generation of the initial TEM11 modes which was done by a set of HPMs. This brought some alignment challenges as shown in **Figure 15(c)** and. Nevertheless, it is evident that the integration of VBGs and phase plates could open new optical design spaces in areas such as high-power beam combining,

of interest as a power scaling approach.

66 Holographic Materials and Optical Systems

mode multiplexing in communication systems, and others.

**3. Achromatization of HPM with surface diffraction gratings**

b, d); a multiplexed four-sector HPM spectrally combines the two beams and converts them to TEM00 (e).

As discussed, HPMs can successfully imprint their phase pattern as long as the wavelength satisfies the Bragg condition but to achieve this, the HPM needs to be angle tuned which can-

**Figure 15.** Demonstration of conversion from the TEM11 mode to TEM00 for 1061 and 1064 nm lasers separately (a, c and

$$
\Lambda\_{\rm sc} \sin \theta = m\lambda \tag{5}
$$

Based on coupled wave theory [2], a VBG will diffract light if the Bragg condition (Eq. (6)) is met and can reach diffraction efficiencies as high as ≈100% [3]:

$$2\,\Lambda\_{\text{vac}}\sin\theta\_{\text{s}} = \lambda\tag{6}$$

Since both of these diffraction angles are dependent on the corresponding grating periods, if the surface grating period is double the period of the volume Bragg grating (Eq. (7)), then any first-order diffraction by normally incident light will be at the corresponding Bragg condition of the volume Bragg grating and that will hold for any wavelength [30]:

$$2\,\Lambda\_{\text{vac}} = \Lambda\_{\text{sc}}\tag{7}$$

Therefore, a surface grating with twice the period of a TBG can make different wavelengths get diffracted by the TBG at the same time as long as they have the same incident angle. In order to recollimate the diffracted beams, an identical surface grating needs to be added in a mirror orientation to the transmitting volume Bragg grating, as shown in **Figure 16**. This grating completely cancels out the dispersion of the first surface grating and recollimates the outgoing beam. Applying this concept to an HPM will eliminate the need for angle tuning in order to meet the Bragg condition for different wavelengths, making, therefore, the device a fully achromatic phase element.

**Figure 16.** Concept of using surface grating pairs to meet the Bragg condition for various wavelengths regardless of angle tuning [30].

The experimental proof was carried out by using two surface gratings with a grove spacing of 150 lines/mm (a period of 6.66 µm) aligned to an HPM with a period of 3.4µm in setup shown in **Figure 17** [31]. The goal of the experiment was to achieve successful broadband mode conversion from a Gaussian to a TEM11 mode without the need to angularly tune the HPM. Three different TEM00 tunable diode laser sources were used in order to get a wavelength range of over 300 nm (765–1071 nm).

**Figure 17.** Experimental setup for observing Gaussian to TEM11 conversion of the HPM surface grating system with three different diode sources [31].

**Figure 18** shows as an example, the far-field profiles of three different wavelengths ((a) 765, (b) 978, and (c) 1071 nm) that were converted to the TEM11 mode without any angular adjustment of the HPM. This successfully demonstrates that full achromatization of a holographic phase mask can be achieved with the combination of surface gratings and phase-encoded transmitting volume Bragg grating.

**Figure 18.** Far-field profile of the diffracted beam after propagating through a holographic four-sector mode converting mask aligned to two surface gratings at (a) 765 nm, (b) 978 nm, and (c) 1071 nm. The sizes shown are not to scale.

In conclusion, this is a demonstration of a way to make phase masks fully achromatic—something not possible until recently. This is achieved by the combination of surface gratings and phase-encoded transmitting volume Bragg grating.
