**Diffraction target F.T.-1 Hologram**

In order to implement the CGH, holograms are calculated by using a program based on a variation of the widely adopted simulated annealing optimization algorithm (Dames, Dowling et al., 1991), (Broomfield, Neil et al., 1992) whose cost function to minimize the calculation error is:

$$\mathbf{C} = \sum\_{i} \frac{\left\{I\_i^2 - A^2\right\}^2}{A\_i^2} \tag{2}$$

where *Ii <sup>2</sup>* is the calculated spot intensity for the diffraction order *i*; *Ai <sup>2</sup>* is its defined intensity and *A2* is the average intensity for the diffraction target spots; *t* is the number of process calculations.

There are three steps in a CGH design process:


The CGH designed for this work is a black & white bars pattern implemented onto a Spatial Light Modulator, where there are only two possible states: "1" for white (total transparency or π phase shift) and "0" for black (total darkness or 0 phase shift). Fig. 3 shows the original diffraction target (a), an array of spots with different light intensities (non uniform, as in Fig. 3a), and three consecutive holograms (b, c, d), calculated by the program carrying out the inverse FT according to the algorithm efficiency. A 45% efficiency is an initial calculation value and close to 90% efficiency is practically the best result in the optimization process.

During the calculation of the hologram, the program can find out different holograms which match the diffraction target. It is possible to change, dynamically, the initial conditions (original diffraction target and efficiency, optimization process parameters), to change the direction for the optimization process allowing the algorithm to escape from local minima and reach the correct hologram.

Taking into account the former considerations and by implementing a hologram on the SLM where its spatial period can be modified in real time, we obtain a programmable diffraction

**Hologram F.T. Diffraction target** 

**Diffraction target F.T.-1 Hologram**  In order to implement the CGH, holograms are calculated by using a program based on a variation of the widely adopted simulated annealing optimization algorithm (Dames, Dowling et al., 1991), (Broomfield, Neil et al., 1992) whose cost function to minimize the

*t*

and *A2* is the average intensity for the diffraction target spots; *t* is the number of process

1. Target definition: the target is the diffraction pattern that is to be obtained from the SLM. Depending on the use: filter, switch or others, this target is usually an array or a

2. Fourier transform calculation: the program calculates the inverse Fourier transform (F.T.)-1 of the target. The optimization algorithm compares the FT of the hologram with the defined target improving the efficiency at each calculation time. Hologram pixels are flipped between the amplitude values 0, (or phase 0, π) to reduce an error function, (2), specifying the difference between the desired target in the Fourier plane and the reconstruction obtained from the current state of the hologram, improving the efficiency at each calculation (Efficiency defined as*: η = Σ m orders diffracted light /total* 

3. Finally, CGH implementation in an optical substrate, using a photographic film or

The CGH designed for this work is a black & white bars pattern implemented onto a Spatial Light Modulator, where there are only two possible states: "1" for white (total transparency or π phase shift) and "0" for black (total darkness or 0 phase shift). Fig. 3 shows the original diffraction target (a), an array of spots with different light intensities (non uniform, as in Fig. 3a), and three consecutive holograms (b, c, d), calculated by the program carrying out the inverse FT according to the algorithm efficiency. A 45% efficiency is an initial calculation value and close to 90% efficiency is practically the best result in the optimization process. During the calculation of the hologram, the program can find out different holograms which match the diffraction target. It is possible to change, dynamically, the initial conditions (original diffraction target and efficiency, optimization process parameters), to change the direction for the optimization process allowing the algorithm to escape from local minima

*<sup>2</sup>* is the calculated spot intensity for the diffraction order *i*; *Ai*

*I A <sup>C</sup>*

2 22 2 ( ) *<sup>i</sup>*

(2)

*<sup>2</sup>* is its defined intensity

*i*

*A*

The relationship between the hologram and its Fourier Transform function are:

**2.2 Computer generated hologram design** 

There are three steps in a CGH design process:

matrix of spots. This is the input for the program.

grating.

where *Ii*

calculations.

*incident light*).

and reach the correct hologram.

SLM.

calculation error is:

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 original pattern is shifted

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" instead of a singular one.

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 would impact the efficiency up to a 40%.

Application of Holograms in WDM Components for Optical Fiber Systems 263

which relationship with *D*, the size of the pixel, and *N*, the number of pixels in one

*ND <sup>N</sup> H n*

where *n* is the integer number of black & white bar pairs and depends on the type of

*x f n*

**lens 1 lens 2**

When the other holographic device parameters are fixed, *λ* only depends on *n*, as can be shown from (5). According to fixed or variable values for *n* or/and *x*, different applications for our device can be considered giving an idea of the device's versatility (see Table 2). In the following sections, we design a generic multipurpose device based on the experimental scheme explained above that can operate as a tunable wavelength filter, wavelength multiplexer and wavelength router, by simply modifying in real time the CGH loaded on the SLM. The PC-based interface used to load the CGH on to the SLM also serves to calculate the different patterns needed. The electronic interface allows an automatic

In all cases, the central wavelength channel, *λ0*, is obtained for *n = N/4* in (5), and the

Fig. 6. "4f "dynamic holographic device with a transmissive SLM and fixed grating.

program to be developed for loading different patterns when they are needed.

**n x Application** 

Table 2. Different device applications

operating wavelength range *Δλf* is given by:

fix fix Holographic band-pass filter

fix variable Demultiplexer 1x M variable variable λ router 1x M

variable fix Tunable holographic band-pass filter

**ND**

hologram (pattern). For small angles, equation (3) can be simplified as follows:

0

2

1

*ND d* 

**SLM fixed grating**

**f f x**

**f**

**f**

*<sup>n</sup>* (4)

(5)

**output fibers**

dimension of the SLM is given in (4):

**input fiber**

<sup>1</sup>

For the operation of holographic devices after the generation of holograms, it is necessary to configure with them the SLM. To perform the switching operation a closed control between the holographic component (SLM) and the computer is needed to assign the correspondent hologram from a local database. This procedure is represented in Fig. 5, where a switching control acts over the PC-SLMs interface.

Fig. 5. Tunable holographic device: switching operation

#### **3. Dynamic holographic device design**

In order to design a holographic optical device a "4f" structure is chosen using a transmissive SLM and fixed grating. Fig. 6 illustrates the device used in the present work. The previously calculated CGH (black and white bars) is loaded onto the SLM via a PCbased interface. The SLM-FLC and fixed grating are illuminated by light coming from a singlemode optical fiber collimated by means of a lens. A second lens produces the replicated array of spots explained above on the back focal plane of the lens.

In our experiments, we are interested only in the array of spots corresponding to the first order of diffraction. Therefore, the output optical fibers array is placed in the back focal plane of the lens at a certain angle in order to optimize the coupling. Because of the small size of the singlemode fiber radius, it acts as a spatial light filter.

Output fibers F1,..,F10, must be located at the Fourier lens plane in order to receive the maximum light intensity of the diffracted beams. The relationship between the system diffraction angles (Parker et al., 1998) is in agreement with the expression:

$$
\arctan\left(\frac{\mathbf{x}}{f}\right) = \arcsin\left(\frac{\lambda}{d}\right) + \arcsin\left(\frac{\lambda}{H}\right) \tag{3}
$$

where *x* is the distance of the output optical fiber from the optical axis, *f* is the focal length of the lens, *d* is the spatial period of the fixed grating and *H* is the hologram spatial period, <sup>1</sup>

262 Holograms – Recording Materials and Applications

For the operation of holographic devices after the generation of holograms, it is necessary to configure with them the SLM. To perform the switching operation a closed control between the holographic component (SLM) and the computer is needed to assign the correspondent hologram from a local database. This procedure is represented in Fig. 5, where a switching

PC-SLMs Interface

(Hi assigment to the SLMj

according to nij

Switching control

In order to design a holographic optical device a "4f" structure is chosen using a transmissive SLM and fixed grating. Fig. 6 illustrates the device used in the present work. The previously calculated CGH (black and white bars) is loaded onto the SLM via a PCbased interface. The SLM-FLC and fixed grating are illuminated by light coming from a singlemode optical fiber collimated by means of a lens. A second lens produces the

In our experiments, we are interested only in the array of spots corresponding to the first order of diffraction. Therefore, the output optical fibers array is placed in the back focal plane of the lens at a certain angle in order to optimize the coupling. Because of the small

Output fibers F1,..,F10, must be located at the Fourier lens plane in order to receive the maximum light intensity of the diffracted beams. The relationship between the system

where *x* is the distance of the output optical fiber from the optical axis, *f* is the focal length of the lens, *d* is the spatial period of the fixed grating and *H* is the hologram spatial period,

*f d H*

(3)

replicated array of spots explained above on the back focal plane of the lens.

diffraction angles (Parker et al., 1998) is in agreement with the expression:

size of the singlemode fiber radius, it acts as a spatial light filter.

arctan arcsin arcsin *<sup>x</sup>*

SLMs

nij

Holographic device

1

M

)

control acts over the PC-SLMs interface.

H1, H2,..Hi

the PC

Holograms (Hi

,.. Hn

) stored in

**3. Dynamic holographic device design** 

Fig. 5. Tunable holographic device: switching operation

which relationship with *D*, the size of the pixel, and *N*, the number of pixels in one dimension of the SLM is given in (4):

$$H = \frac{\text{ND}}{n} \qquad 0 < n < \frac{N}{2} \tag{4}$$

(5)

where *n* is the integer number of black & white bar pairs and depends on the type of hologram (pattern). For small angles, equation (3) can be simplified as follows:

> *x f n*

1

*ND d* 

Fig. 6. "4f "dynamic holographic device with a transmissive SLM and fixed grating.

When the other holographic device parameters are fixed, *λ* only depends on *n*, as can be shown from (5). According to fixed or variable values for *n* or/and *x*, different applications for our device can be considered giving an idea of the device's versatility (see Table 2).

In the following sections, we design a generic multipurpose device based on the experimental scheme explained above that can operate as a tunable wavelength filter, wavelength multiplexer and wavelength router, by simply modifying in real time the CGH loaded on the SLM. The PC-based interface used to load the CGH on to the SLM also serves to calculate the different patterns needed. The electronic interface allows an automatic program to be developed for loading different patterns when they are needed.


Table 2. Different device applications

In all cases, the central wavelength channel, *λ0*, is obtained for *n = N/4* in (5), and the operating wavelength range *Δλf* is given by:

Application of Holograms in WDM Components for Optical Fiber Systems 265

The operation as a tunable CWDM/DWDM filter is obtained by changing the hologram period, *n*. From the output fibers (*F1 to F10*), a CWDM tunable filter uses F4 and a DWDM

In Table 4 tuning between 1311 < λ < 1591 nm (CWDM, output fiber F4) is obtained for 17 < *n* < 328 and between 1531 < λ < 1591 nm (DWDM, output fiber F8) is reached for 138 < *n* <201.

**λ (nm)** 1311 1351 1391 **1431** 1471 1511 1551 1591

**λ (nm)** 1531 **1551** 1571 1591

For CWDM applications the holographic filter has a tuning range of *Δλf* =1591 -1311 = 280 nm with a -3dB passband of 1 nm. In Fig. 7 the transmission response is shown, according to (Parker et al., 1998). For wavelengths very close to the centre, the shape is Gaussian (*λ < λ<sup>0</sup>* +/- 1,5 nm); from these wavelengths the shape is like a Bessel function and the zero

Table 5 shows, in case of CWDM systems, the different values of *n* and corresponding central wavelengths separated by 40 nm, from 1311 to 1591 nm. In this case, an adjacent channel isolation > 50 dB is achieved and the complete filter tuning range is covered

**a)** *n* **328 276 227 180 136 94 55 17** 

**b)** *n* **201 180 159 138** 

convergence is slower ( > 20 dB for *λ > λ0* +/- 1,5 nm ; > 40 dB for *λ > λ0* +/- 5 nm).


the shape is Gaussian and like a Bessel function for wavelengths λ > λ0 +/-1.5 nm

Fig. 7. Holographic filter shape. For wavelengths close to the central value, λ < λ0 +/-1.5 nm,

**n 180 136 94 55 λ0 (nm)** 1431 1471 1511 1551 **BW (nm)** 1 1 1 1 **Δλ (-3 dB)** 1430.5-1431.5 1470.5 -1471.5 1510.5-1511.5 1550.5-1551.5 **Δλ (>-50 dB)** 1421-1441 1461-1481 1501-1521 1541-1561

**Filter Bandwidth, nm**

Table 4. a) Filter operation for CWDM (F4) and b) for DWDM (F8)

**Transmission Loss , dB**

tunable filter uses F8 (see Fig. 6).

**4.1 Wavelength response** 

according to the type of hologram.





Table 5. Tunable pass band filter (CWDM)

$$
\Delta\lambda\_{\gamma'} = \lambda \left( n = 0 \right) - \lambda \left( n = \frac{N}{2} \right) \tag{6}
$$

The -3dB passband width, *BW*, for each wavelength channel tuned, is limited by the output fiber characteristics and the wavelengths coupled inside the core diameter *φcore*. Taking this into account, and from (3), the following expression relates the bandwidth *BW* for every wavelength channel tuned in the filter and the focal distance *f* of the lens according the optical power coupled into the output optical fiber (Parker et al., 1998):

$$f > \phi\_{\text{surr}} \frac{d}{BW} \times \left(1 - \frac{\lambda\_o^2}{d^2}\right)^{\gg 2} \tag{7}$$

In order to obtain minimum losses, the collimated light through the SLM has to illuminate the maximum quantity of pixels. As its intensity distribution has a Gaussian profile, it is sufficient that *1/e2* beam bandwidth illuminates the SLM aperture. According to optical Gaussian laws, the following condition is reached:

$$ND = 4\lambda\_0 \frac{f}{\pi \phi\_{core}}\tag{8}$$

For commercial FLC-SLMs, available pixel size *D* is > 5 µm and the number of pixels, *N*, usually is from 250 to 1000. From expressions (5) and (7), it is possible to calculate the *x*  value and *λmax* and *λmin* for the operating range of tuning.

#### **4. Tunable holographic filter application**

In order to design a tunable holographic filter with a -3dB passband width, *BW*, of 1 nm (125 GHz), for each wavelength channel tuned, we take *d* = 3.5 µm for the spatial period of the fixed grating. To use the same device for CWDM/DWDM, a SLM with a value of *N* = 720 and *D* = 7 µm for the spatial period, is chosen. The output singlemode fibers used in our device have a core diameter, *φcore*, of 9 µm. Then, from (7), *f* must be greater than 23.9 mm. As a practical value we assume *f* = 25 mm.

Table 3 summarizes the filter figures for CWDM systems applications where channels are allocated between *λmin*= 1290 nm and *λmax*= 1590 nm, with central wavelength *λ0* = 1431 nm, and for DWDM systems (*λmin*= 1530 nm and *λmax*= 1590 nm, *λ0* = 1551 nm).


Table 3. Device parameters for CWDM(DWDM) systems

The operation as a tunable CWDM/DWDM filter is obtained by changing the hologram period, *n*. From the output fibers (*F1 to F10*), a CWDM tunable filter uses F4 and a DWDM tunable filter uses F8 (see Fig. 6).

In Table 4 tuning between 1311 < λ < 1591 nm (CWDM, output fiber F4) is obtained for 17 < *n* < 328 and between 1531 < λ < 1591 nm (DWDM, output fiber F8) is reached for 138 < *n* <201.


Table 4. a) Filter operation for CWDM (F4) and b) for DWDM (F8)

#### **4.1 Wavelength response**

264 Holograms – Recording Materials and Applications

2 *<sup>f</sup>*

The -3dB passband width, *BW*, for each wavelength channel tuned, is limited by the output fiber characteristics and the wavelengths coupled inside the core diameter *φcore*. Taking this into account, and from (3), the following expression relates the bandwidth *BW* for every wavelength channel tuned in the filter and the focal distance *f* of the lens according the

> <sup>2</sup> 1 - *core <sup>d</sup> <sup>f</sup> BW d*

In order to obtain minimum losses, the collimated light through the SLM has to illuminate the maximum quantity of pixels. As its intensity distribution has a Gaussian profile, it is sufficient that *1/e2* beam bandwidth illuminates the SLM aperture. According to optical

*<sup>f</sup> ND*

For commercial FLC-SLMs, available pixel size *D* is > 5 µm and the number of pixels, *N*, usually is from 250 to 1000. From expressions (5) and (7), it is possible to calculate the *x* 

In order to design a tunable holographic filter with a -3dB passband width, *BW*, of 1 nm (125 GHz), for each wavelength channel tuned, we take *d* = 3.5 µm for the spatial period of the fixed grating. To use the same device for CWDM/DWDM, a SLM with a value of *N* = 720 and *D* = 7 µm for the spatial period, is chosen. The output singlemode fibers used in our device have a core diameter, *φcore*, of 9 µm. Then, from (7), *f* must be greater than 23.9 mm.

Table 3 summarizes the filter figures for CWDM systems applications where channels are allocated between *λmin*= 1290 nm and *λmax*= 1590 nm, with central wavelength *λ0* = 1431 nm,

> CWDM DWDM 1270 -1590 Δλ (nm) 1510 -1590 1 BW (nm) 1 1431 λ0 (nm) 1551 25.00 *f* (mm) 25.00 11.499 *x* (mm) 12.463 24.71 *Ф* (º) 26.51 1591 λmax (nm) 1591 1311 λmin (nm) 1531

and for DWDM systems (*λmin*= 1530 nm and *λmax*= 1590 nm, *λ0* = 1551 nm).

optical power coupled into the output optical fiber (Parker et al., 1998):

*n n* 

*N*

3/2 <sup>2</sup> 0

*core*

(6)

(7)

(8)

 0

Gaussian laws, the following condition is reached:

0 4

**4. Tunable holographic filter application** 

As a practical value we assume *f* = 25 mm.

value and *λmax* and *λmin* for the operating range of tuning.

Table 3. Device parameters for CWDM(DWDM) systems

For CWDM applications the holographic filter has a tuning range of *Δλf* =1591 -1311 = 280 nm with a -3dB passband of 1 nm. In Fig. 7 the transmission response is shown, according to (Parker et al., 1998). For wavelengths very close to the centre, the shape is Gaussian (*λ < λ<sup>0</sup>* +/- 1,5 nm); from these wavelengths the shape is like a Bessel function and the zero convergence is slower ( > 20 dB for *λ > λ0* +/- 1,5 nm ; > 40 dB for *λ > λ0* +/- 5 nm).

Table 5 shows, in case of CWDM systems, the different values of *n* and corresponding central wavelengths separated by 40 nm, from 1311 to 1591 nm. In this case, an adjacent channel isolation > 50 dB is achieved and the complete filter tuning range is covered according to the type of hologram.

**Filter Bandwidth, nm**

Fig. 7. Holographic filter shape. For wavelengths close to the central value, λ < λ0 +/-1.5 nm, the shape is Gaussian and like a Bessel function for wavelengths λ > λ0 +/-1.5 nm


Table 5. Tunable pass band filter (CWDM)

Application of Holograms in WDM Components for Optical Fiber Systems 267

Table 6 summarizes the fiber positions in order to demultiplex the wavelengths used in the CWDM/DWDM systems. A CWDM system uses *F1, F2, F3, F4, F5, F6, F8* and F10 and a DWDM uses *F7, F8, F9*, and *F10* output fibers (see Fig. 6). It is necessary to emphasize that a better performance as demultiplexer could be implemented if only this function is required. For example, we could design a demultiplexer device with channel separation smaller than 50 GHz (Parker, Cohen et al., 1997). However, the novel idea is to design a compatible

**Output fiber** *λ* **(nm) CWDM** *λ* **(nm) DWDM** *x***(µm)** 

Maintaining output fibers in the same place as shown in Table 6, if *n* value (type of hologram) is properly varied, a certain wavelength coming from the input fiber can be routed to any one of the output fibers. As an example, Table 7 highlights the *n* values for routing *λ0* = 1431nm (CWDM) and *λ<sup>0</sup>* = 1551 (DWDM) towards an output fiber; these values

> *x x* 1 *n ND n ND f d f*

For *Δx* = 161 µm, *Δn* was calculated by using (10) resulting in *Δn* = 21 and for *Δx* = 321 µm, *Δn* is 45. Therefore, the device is a 1x8 λ router in case of CWDM and a 1x4 λ router for DWDM systems. It is necessary to highlight that the positions of the fibers are compatible with all applications and that the crosstalk resulting from high-order diffraction beams (*m* = 2, 3,.) are outside of the locations of the output array fibers (*ΔФ*

(10)

have been calculated from (10), considering the variation of *n* according to *Δx*:

*F1* 1311 10 535 *F2* 1351 10 856 *F3* 1391 11 178 *F4* 1431 11 499 *F5* 1471 11 821 *F6* 1511 12 142 *F7* 1531 12 303 *F8* 1551 1551 12 463 *F9* 1571 12 624 *F10* 1591 1591 12 785

CWDM/DWDM device able to carry out different functions.

Table 6. CWDM/DWDM demultiplexers (n = 180)

**6. Wavelength routing application** 

= 4º), (Horche, 2004).

This feature allows the possibility of a multiple pass band filter in the same optical fiber, but with an increment of losses penalty according to the expression 10 log *C* (dB), where *C* is the number of simultaneously tuned channels (Parker et al., 1998). In this case, *C* = 4 and therefore the increment of losses in the device is: Δ losses = 10 log 4 = 6 dB.
