**3. Methodology**

The adopted methodology to implement the SLM flexible platform for optical systems can be subdivided into two main sections: (i) the algorithms employed for the CGH generation and optimization methodology (in Section 3.1) and (ii) the SLM framework setup implementation with application in SDM systems and characterization/testing of PIC (in Section 3.2).

The framework ability to improve the overall alignment process and excite different cores of a MCF, can provide a valuable contribution for the impairment mitigation of the system optical path, which can relax digital signal processing (DSP) equalization requirements of the SDM system [5, 18, 22, 34, 49].

Furthermore, its use as a flexible platform for feeding photonic integrated processors was also explored for the characterization/test of PICs, and results have been presented for its implementation as a parallel implementation of the Haar transform (HT) image compression algorithm [8, 18, 23].

### **3.1 CGH pattern establishment**

Holography is a 3D-based display system that comprises exploiting diffraction and interference for recording and reconstructing optical wave fronts. Moreover, computer-generated holography is an effective technique that is appropriate for a broad variety of displays such as two-dimensional (2D), volumetric, autostereoscopic, stereoscopic, and true 3D imaging. It is remarkable that the CGH is becoming feasible due to the emergence of progressively powerful computers that prevents the conventional interferometric recording step in the formation of hologram [50]. In addition, the CGH can be viewed as a phase mask with spatially variable transmittance or a diffractive optical element that can be readily displayed on the devices such as SLM, which are capable of diffracting light [5, 51]. Also, the information that needs to be transformed is presented to an optical system, through the SLM. This is effected with a suitable phase mask for the concerned input function [25].

From a set of different available techniques for the generation of CGH (e.g., IFTA, linear Fourier transform, simulated annealing, and Gerchberg algorithm, as described in Section 2), in our SLM framework, higher focus was given to the linear Fourier transform principle for the calculus of the numerical interference patterns to generate the holograms (CGH). This decision was mainly due to the intensive computational requests and high power loss (up to 9 dB [26]) associated with the implementation of the simulated annealing and Gerchberg-Saxton algorithms and additional computational cost of IFTA when compared to linear phase mask.

Thus, a simplified approach based on the implementation of a linear phase mask generation (in Section 3.1.1) and the development of a new iterative algorithm experimentally driven for CGH effective optimization (in Section 3.1.2) is proposed and tested.

**121**

**Figure 3.**

*estimation [8].*

*Spatial Light Modulation as a Flexible Platform for Optical Systems*

A linear phase mask can be described as a numerical information transformation (in the Fourier domain) of the input function of interest [25], which can be intro-

*Mlinear* = *M*( *fx*, *fy*) = −2*π*(*cx fx* + *cy fy*) (1)

\_ *y* − *y*<sup>0</sup> *wy log* (√ \_ 2 ) ) 2

*Sout* = *ifft*(*H*( *fft*( *Sin*)) (3)

) (2)

The CGH implemented with a linear phase mask can be expressed in the frequency domain as expressed in Eq. (1) [5, 23], where *fx* and *fy* denote the spatial frequency vector components that correspond to the image to be generated in both *X* and *Y* axes, respectively, and *cx* and *cy* represent the horizontal and vertical tilt

A collimated Gaussian beam with transverse profile *Sin* is imaged onto the SLM via a lens, Eq. (2), where (*x*0, *y*0) offer the horizontal and vertical position, respectively, and (*wx*, *wy*) represent the width and the height of the beam, respectively, as

\_ *x* − *x*<sup>0</sup> *wx log* (√ \_ 2 ) ) 2 − ( 2

With the adoption of Fraunhofer approximation, the Fourier transform is produced on the SLM plane, *fft*(*Sin*). Afterward, the subsequent illumination profile is multiplied with the phase mask, *eiHmask*. The resultant signal is then Fourier transformed via the second lens by means of an inverse Fourier transform to achieve the

A graphical user interface (GUI) was also developed to test different masks to be

Different phase masks can be attained by adjusting the different available parameters from the developed GUI. For the *Input Beam* GUI panel the following

*Cartesian coordinate system description of the parameters (x0, y0) and (wx,wy) employed for the input beam Sin*

*DOI: http://dx.doi.org/10.5772/intechopen.88216*

duced into the optical system through an SLM.

*Sin* = *exp* (− (

output signal *Sout*, which can be defined as Eq. (3) [5, 8]:

applied to the SLM device [18] (see **Figure 4**).

2

*3.1.1 Linear phase mask CGH*

parameters, respectively.

depicted in **Figure 3** [8].

All algorithms were developed and implemented in MATLAB© [52].

#### *3.1.1 Linear phase mask CGH*

*Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies*

and their coexistence on a shared infrastructure. They can also be densely and efficiently multiplexed which aids the optical networks, not only to extend the reach but also the per channel bit rate. It has also been envisaged that implementation of WSS and SDM will significantly help further in extending the network reach and

The adopted methodology to implement the SLM flexible platform for optical systems can be subdivided into two main sections: (i) the algorithms employed for the CGH generation and optimization methodology (in Section 3.1) and (ii) the SLM framework setup implementation with application in SDM systems and

The framework ability to improve the overall alignment process and excite different cores of a MCF, can provide a valuable contribution for the impairment mitigation of the system optical path, which can relax digital signal processing

Furthermore, its use as a flexible platform for feeding photonic integrated processors was also explored for the characterization/test of PICs, and results have been presented for its implementation as a parallel implementation of the Haar

Holography is a 3D-based display system that comprises exploiting diffraction and interference for recording and reconstructing optical wave fronts. Moreover, computer-generated holography is an effective technique that is appropriate for a broad variety of displays such as two-dimensional (2D), volumetric, autostereoscopic, stereoscopic, and true 3D imaging. It is remarkable that the CGH is becoming feasible due to the emergence of progressively powerful computers that prevents the conventional interferometric recording step in the formation of hologram [50]. In addition, the CGH can be viewed as a phase mask with spatially variable transmittance or a diffractive optical element that can be readily displayed on the devices such as SLM, which are capable of diffracting light [5, 51]. Also, the information that needs to be transformed is presented to an optical system, through the SLM. This is effected with a suitable phase mask for the concerned

From a set of different available techniques for the generation of CGH (e.g., IFTA, linear Fourier transform, simulated annealing, and Gerchberg algorithm, as described in Section 2), in our SLM framework, higher focus was given to the linear Fourier transform principle for the calculus of the numerical interference patterns to generate the holograms (CGH). This decision was mainly due to the intensive computational requests and high power loss (up to 9 dB [26]) associated with the implementation of the simulated annealing and Gerchberg-Saxton algorithms and additional computational cost of IFTA when compared to linear

Thus, a simplified approach based on the implementation of a linear phase mask

generation (in Section 3.1.1) and the development of a new iterative algorithm experimentally driven for CGH effective optimization (in Section 3.1.2) is proposed

All algorithms were developed and implemented in MATLAB© [52].

(DSP) equalization requirements of the SDM system [5, 18, 22, 34, 49].

transform (HT) image compression algorithm [8, 18, 23].

**120**

capacity [40].

**3. Methodology**

characterization/testing of PIC (in Section 3.2).

**3.1 CGH pattern establishment**

input function [25].

phase mask.

and tested.

A linear phase mask can be described as a numerical information transformation (in the Fourier domain) of the input function of interest [25], which can be introduced into the optical system through an SLM.

The CGH implemented with a linear phase mask can be expressed in the frequency domain as expressed in Eq. (1) [5, 23], where *fx* and *fy* denote the spatial frequency vector components that correspond to the image to be generated in both *X* and *Y* axes, respectively, and *cx* and *cy* represent the horizontal and vertical tilt parameters, respectively.

$$M\_{linear} = M\left(f\_x, f\_y\right) = -2\pi \left(c\_x f\_x + c\_y f\_y\right) \tag{1}$$

A collimated Gaussian beam with transverse profile *Sin* is imaged onto the SLM via a lens, Eq. (2), where (*x*0, *y*0) offer the horizontal and vertical position, respectively, and (*wx*, *wy*) represent the width and the height of the beam, respectively, as depicted in **Figure 3** [8]. \_ \_

$$\text{LS}\_{\text{in}} = \exp\left(-\left(2\frac{\mathbf{x} - \mathbf{x}\_0}{w\_{\text{x}}\log\left(\sqrt{2}\right)}\right)^2 - \left(2\frac{\mathbf{y} - \mathbf{y}\_0}{w\_{\text{y}}\log\left(\sqrt{2}\right)}\right)^2\right) \tag{2}$$

With the adoption of Fraunhofer approximation, the Fourier transform is produced on the SLM plane, *fft*(*Sin*). Afterward, the subsequent illumination profile is multiplied with the phase mask, *eiHmask*. The resultant signal is then Fourier transformed via the second lens by means of an inverse Fourier transform to achieve the output signal *Sout*, which can be defined as Eq. (3) [5, 8]:

$$\mathbf{S}\_{out} = \mathbf{i}\hat{g}\mathbf{f}\mathbf{t}\left(H\left(\hat{f}\mathbf{f}\mathbf{t}\left(\mathbf{S}\_{in}\right)\right)\right) \tag{3}$$

A graphical user interface (GUI) was also developed to test different masks to be applied to the SLM device [18] (see **Figure 4**).

Different phase masks can be attained by adjusting the different available parameters from the developed GUI. For the *Input Beam* GUI panel the following

#### **Figure 3.**

*Cartesian coordinate system description of the parameters (x0, y0) and (wx,wy) employed for the input beam Sin estimation [8].*

**Figure 4.** *GUI SLM\_mask to generate the different phase masks applied to the SLM [18].*

input parameters are available: (i) horizontal position (*x*0); (ii) vertical position (*y*0); (iii) width of the beam (*wx*); and (iv) height of the beam (*wy*) (see GUI panel *Input Beam* in the **Figure 4**).

$$\mathcal{S}\_{\text{out}} = \hat{\
iff}\left(H\{\hat{\
iff}\{\mathcal{S}\_{\text{in}}\}\}\right) \tag{4}$$

**123**

**Figure 5.**

*Spatial Light Modulation as a Flexible Platform for Optical Systems*

steps of the algorithm are enumerated as follows [23]:

ii.Initially set the four values *a*1−4 to 1, from *H* = ∠(*a*<sup>1</sup>

camera, and feed this data to the algorithm.

expected, defined as error factor: δ = |*ISLM* − *I*<sup>1</sup>

from the optical chip anticipated output [8, 23].

ing the values of *a*1−4 to compensate for the error factor.

superimposition of four independent holograms generated by Eq. (1). Then, the corresponding linear transformations in the Fourier domain provided in Eq. (5) and

*H* = ∠(*eiH*<sup>1</sup> + *eiH*<sup>2</sup> + *eiH*<sup>3</sup> + *eiH*4) (5)

The block diagram of the employed algorithm is given in **Figure 5**, and the major

i.Generate a first linear phase mask to produce the expected initial field based

iii.Acquire the replay field from the hologram generated by SLM (*I SLM*) with a

iv.Calculate the difference between the hologram generated and the initial field

v.If the condition δ ≤ 0.1 is not satisfied, repeat steps ii–iv by iteratively adjust-

The developed algorithm in MATLAB® is capable of controlling both SLM and camera hardware, providing a dynamic experimentally driven algorithm for effec-

The error factor (δ) is defined to quantify the generated hologram deviation

The employed SLM is a reflective LCoS phase-only type, and its model

is PLUTO-TELCO-012. It can operate within the wavelength range of

*Block diagram of the algorithm applied for the optimization of the linear phase mask CGH [23].*

*H*1 = *exp* (*i*2*π*(*cx*1 *fx* + *cy*1 *fy*)) (6)

 *e iH*<sup>1</sup>


+ *a*2 *e iH*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88216*

Eq. (6) were applied [8, 23]:

on Eq. (5).

tive CGH optimization.

**3.2 CGH generation setup**

+ *a*3 *e iH*<sup>3</sup> + *a*4 *e iH*<sup>4</sup> ).

The Phase Mask GUI panel offers the correspondent input parameters: (i) horizontal translation (*dx*); (ii) vertical translation (*dy*); (iii) horizontal frequency delay (*cx*); (iv) vertical frequency delay (*cy*); (v) percentage of zoom (%); (vi) rotation in degrees (°); and (vii) selection of three possible input functions, i.e., sinusoidal Eq. (4), linear Eq. (1), or defined by the user (user-defined). The option to save or replace the phase mask file is also made available, as depicted in the Phase Mask GUI panel from **Figure 4**.

The implemented scripts and GUI were written in MATLAB© [52].

### *3.1.2 Optimization of the linear phase mask CGH*

In an effort to realize the hologram that can suitably replicate the output signal, we estimated the hologram of the beam through the image phase-only information of the generated hologram. Thus, a first linear phase mask is generated to produce the expected initial field, i.e., the input function of interest.

Since a phase-only SLM does not permit the inverse Fourier of the desired pattern to be addressed into the far field and replicated into the resultant distribution of amplitude and phase on the SLM directly. It is quite demanding to generate a CGH with guarantees for the light to be spatially modulated with the required accuracy and resolution. To address these challenges and obtain the desired hologram with an error factor δ ≤ 10%, we implemented an iterative algorithm to optimize the generation of the linear phase mask. Also, the error factor threshold was set so as to prevent an infinite loop in the adopted optimization algorithm, while guaranteeing that the output result has an accuracy ≥90%.

The algorithm was implemented to generate a hologram that replicates the output of the four waveguides (WG) of an optical chip for data compression proposes [8, 23, 53]. A hologram of four beams was calculated by a phase-only

input parameters are available: (i) horizontal position (*x*0); (ii) vertical position (*y*0); (iii) width of the beam (*wx*); and (iv) height of the beam (*wy*) (see GUI panel

*GUI SLM\_mask to generate the different phase masks applied to the SLM [18].*

The Phase Mask GUI panel offers the correspondent input parameters: (i) horizontal translation (*dx*); (ii) vertical translation (*dy*); (iii) horizontal frequency delay (*cx*); (iv) vertical frequency delay (*cy*); (v) percentage of zoom (%); (vi) rotation in degrees (°); and (vii) selection of three possible input functions, i.e., sinusoidal Eq. (4), linear Eq. (1), or defined by the user (user-defined). The option to save or replace the phase mask file is also made available, as depicted in the Phase Mask

In an effort to realize the hologram that can suitably replicate the output signal, we estimated the hologram of the beam through the image phase-only information of the generated hologram. Thus, a first linear phase mask is generated to produce

Since a phase-only SLM does not permit the inverse Fourier of the desired

The algorithm was implemented to generate a hologram that replicates the output of the four waveguides (WG) of an optical chip for data compression proposes [8, 23, 53]. A hologram of four beams was calculated by a phase-only

pattern to be addressed into the far field and replicated into the resultant distribution of amplitude and phase on the SLM directly. It is quite demanding to generate a CGH with guarantees for the light to be spatially modulated with the required accuracy and resolution. To address these challenges and obtain the desired hologram with an error factor δ ≤ 10%, we implemented an iterative algorithm to optimize the generation of the linear phase mask. Also, the error factor threshold was set so as to prevent an infinite loop in the adopted optimization algorithm, while guaranteeing that the output result has

The implemented scripts and GUI were written in MATLAB© [52].

*Sout* = *ifft*(*H*( *fft*( *Sin*)) (4)

*Input Beam* in the **Figure 4**).

**Figure 4.**

GUI panel from **Figure 4**.

*3.1.2 Optimization of the linear phase mask CGH*

the expected initial field, i.e., the input function of interest.

**122**

an accuracy ≥90%.

superimposition of four independent holograms generated by Eq. (1). Then, the corresponding linear transformations in the Fourier domain provided in Eq. (5) and Eq. (6) were applied [8, 23]:

$$H = \angle \left( e^{iH\_1} + e^{iH\_2} + e^{iH\_3} + e^{iH\_4} \right) \tag{5}$$

$$H\_1 = \exp\left(i2\pi \left(c\_{x1}f\_x + c\_{y1}f\_y\right)\right) \tag{6}$$

The block diagram of the employed algorithm is given in **Figure 5**, and the major steps of the algorithm are enumerated as follows [23]:


The developed algorithm in MATLAB® is capable of controlling both SLM and camera hardware, providing a dynamic experimentally driven algorithm for effective CGH optimization.

The error factor (δ) is defined to quantify the generated hologram deviation from the optical chip anticipated output [8, 23].
