**2. Operating principle**

The working principle of a holographic device design is based on the wavelength dispersion produced in a diffraction grating element (Agrawal, 2002). When a polychromatic light reaches a diffraction grating, there is an angular dispersion (diffraction) according to the incident light wavelength.

Equation (1) expresses the relationship between the diffraction angle and the wavelength of the incident light λ:

$$
\sin \Phi = \frac{m\lambda}{d} \tag{1}
$$

by considering the incident light perpendicular to the grating, Ф is the diffracted light angle, *m* is the diffraction order and *d* the grating spatial period.

The light diffracted, in a far field approximation, follows the Fourier transform distribution and the intensity for the different diffraction orders, *m,* is proportional to *sinc2(Фd/λ)*; the separation between diffraction orders is given by *λR/d*, where *R* is the distance between the binary transmissive diffraction grating and the Fourier plane (Kashnow, 1973).

Most diffraction grating elements are not practically useful for changing the spatial period or the wavelength. A way to allow these variations is, by using a Spatial Light Modulator (SML), to implement on it a Computer Generated Hologram (CGH). The pixelated structure of the SLM produces the effect of a two-dimensional diffraction grating when the device is illuminated with a coherent light. In the SLM every ferro-electric liquid crystal (FLC) pixel can be electro-optically configured to provide a phase modulation to the incident light. Therefore, by managing the hologram on the SLM and its spatial period a programmable diffraction grating is obtained.

In optical fiber communications, wavelengths around 0.8 – 1.6 µm are used. Thus, an SLM pixel pitch close to these wavelength values is required. Unfortunately, current commercial SLMs do not have enough resolution. Therefore, to solve this limitation, a fixed diffraction grating with a low spatial period, together with the SLM giving a high resolution filter, is used (Parker et al., 1998).


Table 1. Relationship between phases and contrast

#### **2.1 2 and 4-phases holograms**

Different types of holograms can be used (Horche & Alarcón, 2004) in the SLM. In order to optimize losses, phase holograms are preferred instead of amplitude holograms due to its intrinsic 3 dB of loss and 4-phase holograms are used instead of 2-phase (binary) holograms because of its greater efficiency (40.5% 81%), which is proportional to sinc2(π/M), where M is the number of phases. Table 1 summarizes the relationships between phase and contrast for 2 and 4 phase holograms.

Fig. 1. two/four-phases bars holograms

The working principle of a holographic device design is based on the wavelength dispersion produced in a diffraction grating element (Agrawal, 2002). When a polychromatic light reaches a diffraction grating, there is an angular dispersion (diffraction) according to the

Equation (1) expresses the relationship between the diffraction angle and the wavelength of

sin *<sup>m</sup>*

by considering the incident light perpendicular to the grating, Ф is the diffracted light angle,

The light diffracted, in a far field approximation, follows the Fourier transform distribution and the intensity for the different diffraction orders, *m,* is proportional to *sinc2(Фd/λ)*; the separation between diffraction orders is given by *λR/d*, where *R* is the distance between the

Most diffraction grating elements are not practically useful for changing the spatial period or the wavelength. A way to allow these variations is, by using a Spatial Light Modulator (SML), to implement on it a Computer Generated Hologram (CGH). The pixelated structure of the SLM produces the effect of a two-dimensional diffraction grating when the device is illuminated with a coherent light. In the SLM every ferro-electric liquid crystal (FLC) pixel can be electro-optically configured to provide a phase modulation to the incident light. Therefore, by managing the hologram on the SLM and its spatial period a programmable

In optical fiber communications, wavelengths around 0.8 – 1.6 µm are used. Thus, an SLM pixel pitch close to these wavelength values is required. Unfortunately, current commercial SLMs do not have enough resolution. Therefore, to solve this limitation, a fixed diffraction grating with a low spatial period, together with the SLM giving a high resolution filter, is

% darkness PHASE (rad) % darkness PHASE (rad)

binary transmissive diffraction grating and the Fourier plane (Kashnow, 1973).

2-Phases 4-Phases

Table 1. Relationship between phases and contrast

black 100 0 100 π/4 grey1 - - 66 3π/4 grey2 - - 33 -3π/4 white 0 π 0 -π/4

Different types of holograms can be used (Horche & Alarcón, 2004) in the SLM. In order to optimize losses, phase holograms are preferred instead of amplitude holograms due to its intrinsic 3 dB of loss and 4-phase holograms are used instead of 2-phase (binary) holograms because of its greater efficiency (40.5% 81%), which is proportional to sinc2(π/M), where M is the number of phases. Table 1 summarizes the relationships between phase and

*m* is the diffraction order and *d* the grating spatial period.

*d*

(1)

**2. Operating principle** 

incident light wavelength.

diffraction grating is obtained.

used (Parker et al., 1998).

**2.1 2 and 4-phases holograms** 

contrast for 2 and 4 phase holograms.

the incident light λ:

Fig 1 shows a bars hologram for 2 and 4-phases and their diffraction target in a far field approach. As we can see, the main difference in the holograms is the grey bars in the 4 phases holograms; in this case there is a white bar, a black bar and two different grey bars for addressing the 4-phases (π/4, 3π/4, -3π/4, -π/4); with regard to the diffraction target. Another characteristic is the loss of the symmetry for the diffraction orders.

Fig. 2. Examples of 2/4-phases holograms and diffraction targets

In Fig. 2 examples of calculated holograms are shown. The program calculates the inverse Fourier transform (F.T.)-1 of the diffraction target (result) by an annealing optimization algorithm. In this case both holograms have a calculated efficiency of 85% and the grey bars are clearly visible in the figure. In the following Section some guidelines about design of holograms by computer are given.

Application of Holograms in WDM Components for Optical Fiber Systems 261

a) Diffraction target b) η= 45% eff hologram

c) η =7 0% eff hologram d) η=90% eff hologram Fig. 3. Hologram calculation process according to the algorithm efficiency η. a) diffraction target; b), c) and d) are calculated holograms with η = 45%, 70% and 90%, respectively

Fig. 4. a) "Zoom" of the original diffraction target, b) original shifted diffraction pattern along the y axis, c) calculated diffraction target and d) corresponding hologram when the

Computer calculations are very sensitive to the geometrical distribution of the original diffraction target. A very slight misalignment on it (centre*: x = 0, y = 0*) can produce a hologram completely different from the correct one. This effect is shown in Fig. 4 when the original array of spots (Fig. 4a) is shifted by 30% of spot separation *δ* (Fig. 4b), along the vertical axis *y*; the calculated target (Fig. 4c) is an array of spots "duplicated" and "shifted"

To avoid small misalignments, along the *x* axis, of the output fibers array positions , with impact on the efficiency, we can optimize the hologram pattern, introducing an offset in the bar positions to correct them (Crossland et al., 2000) An offset of 5% of the hologram period

original pattern is shifted

instead of a singular one.

would impact the efficiency up to a 40%.
