A novel synthesis approach for birefringent filters having arbitrarily amplitude transmittances

Optics Communications 369 (2016) 12–17

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

A novel synthesis approach for birefringent filters having arbitrarily amplitude transmittances Abde Rezzaq Halassi a, Rachid Hamdi a,b,n, Djalal Falih Bendimerad b, Badr-Eddine Benkelfat b a b

Laboratoire des Télécommunications, Université 8 mai 1945 Guelma, Algeria SAMOVAR, Télécom SudParis, CNRS, Université Paris-Saclay, 9, rue Charles Fourier, 91011 Evry Cedex, France

art ic l e i nf o

a b s t r a c t

Article history: Received 19 September 2015 Received in revised form 5 February 2016 Accepted 8 February 2016

In this paper, we present a novel procedure for the synthesis of a filter having an arbitrarily specified amplitude transmittance. The filter configuration consists of N birefringent stages placed between a polarizer and an analyzer, with each stage containing an identical section and a variable section. An additional variable section is placed in front of the analyzer. The synthesis procedure is based on the resolution of a generalized nonlinear equation system directly deducted from the Jones matrix formalism to determine the angles of each stage, the angle of the analyzer and the phase shifts of the variable sections. A typical example of a 6-stage birefringent filter having an arbitrarily non-symmetric amplitude transmittance is shown and the opto-geometrical parameters are given to demonstrate the efficiency of the proposed synthesis procedure. The results obtained show an excellent agreement with those developed in the literature. & 2016 Elsevier B.V. All rights reserved.

Keywords: Polarization Birefringence Birefringent filters Optical filter synthesis Optical communication networks

1. Introduction Optical filters whose filtering characteristics can be predefined are highly desirable for a large number of applications [1–8]. Among these filters, birefringent filters have attracted particular interest since the early 20th century [9–12] and several synthesis algorithms have been developed in order to design filters having a specified amplitude transmittance. Harris et al. [13] developed an algorithm based on equations relating the input impulses to the output impulses at each equal length birefringent crystal. N birefringent crystals are then required to realize a symmetric amplitude transmittance approximated with a Fourier series containing (N þ1) terms. The algorithm determines the rotation angle of each crystal and the rotation angle of the output polarizer. Amman et al. [14] extended the Harris’ method to a non-symmetric amplitude transmittance by using birefringent crystals associated to compensators introducing variable phase shifts. Chu et al. [15] proposed a method based on the Z-transform. The two orthogonal components of the electric field deducted from the Jones formalism are expressed as z-polynomial series. The digital filter design techniques are then used to calculate the parameters of each birefringent element. Nevertheless, these algorithms are relatively cumbersome since an inverse transform technique is n Corresponding author at: Laboratoire des Télécommunications, Université 8 mai 1945 Guelma, Algeria. E-mail address: [email protected] (R. Hamdi).

http://dx.doi.org/10.1016/j.optcom.2016.02.016 0030-4018/& 2016 Elsevier B.V. All rights reserved.

usually needed to obtain the specified spectral response. Recently, Vitanov et al. [16,17] designed N-stacked composite plates which, when rotated at specific angles, act as either broadband half-wave "retarders or tunable narrowband filters. The N rotation angles are found by solving N nonlinear equations. In this paper, we present a new method for the synthesis of birefringent filters having an arbitrary-shape amplitude transmittance. It is based on the resolution of a generalized nonlinear equation system deducted from the Jones matrix formalism. The transmittance is approximated by a Fourier series containing (N þ1) complex coefficients Ck , where N is the number of the filter stages. On the other hand, the electric field at the output of the filter is expressed as an exponential series with (N þ 1) complex coefficients Ek which are derived from the Jones matrix formalism. Equating the imaginary and real parts of Ck and Ek leads to a generalized nonlinear equation system which can be resolved by optimization methods. The paper is organized as follows. In Section 2, the mathematical formulation and the setting of the equations deducted from the Jones matrix formalism are exposed. Once the generalized system of nonlinear equations is established, we proceed to its resolution to determine the filter’s opto-geometrical parameters. In Section 3, demonstration results followed by discussions are exposed. We expose, as a proof of principle test, the synthesis of filters whose amplitude transmittances are non-symmetric. Finally, some conclusion remarks are given in Section 4.

A.R. Halassi et al. / Optics Communications 369 (2016) 12–17

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Fig. 1. The generic diagram of the birefringent filter having arbitrarily amplitude transmittances. Γ : Phase shift introduced by the identical section, bk : Phase shift introduced by the variable section of the kth stage, φk : Rotation angle of the kth stage.

2. Synthesis method and filter structure Fig. 1 depicts the physical structure of the birefringent filter whose amplitude transmittance has an arbitrary shape. The filter consists of a stack of N stages. Each stage is composed of an identical section associated to a variable section whose slow and fast axes are parallel to each other. An additional variable section introducing a phase shift bp is placed in front of the output analyzer with its slow axis parallel to the transmission axis of this analyzer. The identical sections, oriented each at an angle φk (k¼1, 2,…N) with respect to the x axis, have the same geometrical thickness and introduce the same phase shift Γ . The variable sections introduce variable phase shifts bk. For the sake of simplicity, we assume that the transmission axis of the polarizer P is chosen parallel to the laboratory x axis while the transmission axis of the analyzer A is parallel to the u axis as shown in the inset of Fig. 1. We also assume that the input electric vector has an amplitude unity and is totally polarized parallel to the transmission axis of the polarizer. The synthesis method adopted is then based on the Jones formalism to express the electric field vector at the output of the filter as a function of the opto-geometrical parameters of the constituent sections. Referring to Fig. 1, the output electric vector Eu _ trans emerging parallel to the transmission axis of the analyzer is found by multiplying the input electric vector by the overall Jones matrix representing the birefringent filter as, ⎛ E u _ trans ⎞ ⎟ = PA. R (φP ) . {R ( − φP ) . MP . (bP ) . R (φP )}. {R ( − φN ) . MN (ΓN ) . ⎜ ⎠ ⎝ 0 R (φN )}...... {R ( − φ 2) . M2(Γ2) . R (φ 2)}. {R ( − φ1) . ⎛ 1⎞ M1(Γ1) . R (φ1)}. Pp. ⎜ ⎟ ⎝ 0⎠

⎛ −ibP 0 ⎞ MP (bP ) = ⎜ e ⎟ is the Jones matrix of the last variable section ⎝ 0 1⎠ expressed in its own reference frame, ⎛ 1 0⎞ ⎟, is the Jones matrix of the polarizer, expressed in the Pp = ⎜ ⎝0 0⎠ x–y laboratory reference frame, ⎛ 1 0⎞ ⎟, is the Jones matrix of the analyzer, expressed in its PA = ⎜ ⎝0 0⎠ own u–v reference frame. This field can also be written as: ⎛E _ ⎞ ⎛ 1⎞ ⎜ u trans ⎟ = PA. MP (bP ). {R(θP ). MN (ΓN )}...... {R(θ 2). M1(Γ1)} . R(θ1). ⎜ ⎟ ⎝ 0⎠ ⎝ 0 ⎠

(2)

where ⎛ cos δ sin δ ⎞ ⎟ R (δ ) = ⎜ ⎝− sin δ cos δ ⎠

is the rotation matrix expressed in the x–y

laboratory reference frame with δ = φp , θp, φk , θk where θk represents the relative angle such as

θ1 = φ1 θ2 = φ2 − φ1 ..... θN = φN − φN − 1 θP = φP − φN Eq. (2) can be written in the form,

(1)

where φk represents the rotation angle of the kth stage, with respect to the x axis, and Γk = Γ + bk is the phase shift introduced by the stage, where Γ is the phase shift introduced by the identical section, and bk represents the phase shift introduced by the variable section of the stage.

⎛ −iΓk 0 ⎞ Mk = ⎜ e k = 1, 2, .... N , is the matrix of the kth section ⎟ ⎝ 0 1⎠ expressed in its own reference frame,

⎛ ⎞ cos θ Pe−iΓN sin θ P ⎟ . .⎜ ⎜ ⎟ −iΓN cos θ P ⎠ ⎝ − sin θ Pe ⎞ N ⎛ cos θ ke−iΓk − 1 sin θ k ⎟ ⎛ cos θ 1 ⎞ ⎟ .⎜ ∏ .⎜⎜ ⎟ −iΓk − 1 cos θ k ⎠ ⎝ − sin θ 1⎠ k = 2 ⎝ − sin θ ke

⎛ E u _ trans ⎞ ⎛ 1 0 ⎞ ⎛ −ibP ⎞ 0 ⎟. ⎟. ⎜ e ⎜ ⎟=⎜ ⎝ ⎠ ⎝ 0 0⎠ ⎝ 0 0 1⎠

(3)

Carrying out the matrix product in Eq. (3) leads to the expression of the electric field Eu _ transm in the form of an exponential series such as, N

Eu _ transm =

∑ Eke−ikΓ = E0 + E1e−iΓ + … … + ENe−iNΓ k=0

(4)

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A.R. Halassi et al. / Optics Communications 369 (2016) 12–17

where Ek (k = 0, 1 … … , N ) are complex coefficients. It is obvious that the coefficient EN of the term e−iNΓ is of the form,

EN (θ1, … θN , θP ; b1, … bN , bP ) = cos(θ1). cos(θ2) … .. cos(θN − 1). cos(θN ). cos(θP ). e−i(b1+ b2 +… . bN + bP ) The other coefficients are obtained in the same way. In order to synthesize a filter that will have an ideal arbitrarily specified amplitude transmittance G(w) at its output, we first approximate G(w) with the Fourier series C(w) on a finite number of (N þ 1) coefficients such as: N

C (w ) =

∑ Cke−ikwΔt = C0 + C1e−iwΔt + … .. + CNe−iNwΔt k=0

N

∑ Cke−ikΓ = C0 + C1e−iΓ + … … + CNe−iNΓ k=0

(6)

By identification of Eqs. (4) and (6), we obtain the following generalized system,

⎧ E (θ , ... θ , θ ; b , ... b , b ) = C N p 1 N P 0 ⎪ 0 1 ⎪ E1(θ1, ... θN , θp; b1, ... bN , bP ) = C1 ⎪ ⎪ E2(θ1, ... θN , θp; b1, ... bN , bP ) = C2 ⎨ ⎪ E3(θ1, ... θN , θp; b1, ... bN , bP ) = C3 ⎪ ⎪⋮ ⎪ E (θ , ... θ , θ ; b , ... b , b ) = C N p 1 N P N ⎩ N 1

(7)

The coefficients Ek and Ck are complex. This equality implies the equality of the imaginary (Im) and real (Re) parts of both coefficients. The equation system (7) generates a new system of (2N þ2) nonlinear equations with (2N þ 2) unknown variables such as:

⎧ Re(E (θ , ... θ , θ , b , ... b , b )) 0 1 N P 1 N P ⎪ ⎪ Re(E1(θ1, ... θN , θP , b1, ... bN , bP )) ⎪⋮ ⎪ ⎪ Re(EN (θ1, ... θN , θP , b1, ... bN , bP )) ⎨ ⎪ Im(E0(θ1, ... θN , θP , b1, ... bN , bP )) ⎪ ⎪ Im(E1(θ1, ... θN , θP , b1, ... bN , bP )) ⎪⋮ ⎪ ⎩ Im(EN (θ1, ... θN , θP , b1, ... bN , bP ))

= Re(C0)

=0 =0 =0 =0 =0

(9)

The system Eq. (9) is a system of (2N þ2) nonlinear equations with (2N þ2) unknown variables that can be solved by optimization methods such as, for instance, PSO algorithm [18,19].

3. Results and discussions In this section, we will illustrate the new synthesis procedure through an example. Let's choose an ideal non-symmetrical amplitude transmittance G(w) to be synthesized as shown in Fig. 2. It is interesting to notice here that the filter structure proposed in Fig. 1 can be implemented in different technologies such as the birefringent plates [14], the birefringent fiber sections [15], the equivalent optical delay-line circuit [20], etc. In this example, we use a birefringent plate (BP) as the identical section and a liquidcrystal cell (LCC) [10] as the variable section as shown in Fig. 3. The LCCs introduce variable phase shifts bk and the additional LCC introduces a phase shift bP . The ideal amplitude transmittance is defined as: ⎧ wΔ t + 1, 0 ≤ wΔt ≤ π ⎪− π ⎪ ⎪ 2wΔt 3π G(w ) = ⎨ − 2, π < wΔt ≤ 2 ⎪ π ⎪ 3π 1, < wΔt ≤ 2π ⎪ ⎩ 2

(10)

G(w) is approximated by the Fourier series to the seven order as,

C (w ) = (0.0225 − 0.0113i) − 0.0507e−iwΔt

= Re(C1)

+ (0.2027 + 0.1013i)e−i2wΔt + 0.6250e−i3wΔt + … + (0.2027 − 0.1013i)e−i4wΔt − 0.0507e−i5wΔt

= Re(CN )

+ (0.0225 + 0.0113i)e−i6wΔt

= Im(C0) = Im(C1) = Im(CN )

=0

(5)

where Δt represents the time shift between two successive impulses at the output of the identical section. Obviously, Γ = ωΔt , where w is the angular frequency and the amplitude transmittance C(w) can be written in the form:

C (w ) =

The system Eq. (8) can be written in the following form, ⎧ f (θ , ... θ , θ , b , ... b , b ) = Re(E ) − Re(C ) 0 0 N P 1 N P ⎪ 0 1 ⎪ f0 (θ 1, ... θ N , θ P , b1, ... bN , bP ) = Re(E1) − Re(C1) ⎪ ⎪⋮ ⎪ fN (θ 1, ... θ N , θ P , b1, ... bN , bP ) = Re(EN ) − Re(CN ) ⎨ ⎪ g 0 (θ 1, ... θ N , θ P , b1, ... bN , bP ) = Im(E 0 ) − Im(C 0 ) ⎪ = Im(E1) − Im(C1) ⎪ g1(θ 1, ... θ N , θ P , b1, ... bN , bP ) ⎪⋮ ⎪ ⎩ gN (θ 1, ... θ N , θ P , b1, ... bN , bP ) = Im(EN ) − Im(CN )

(8)

(11)

The application of the method described in Section 2 leads to the generation of the system Eq. (8) of (2N þ2) nonlinear equations with (2N þ2) unknown variables where N ¼6 such as,

⎧ − cos(θ )⋅ cos(θ )⋅ cos(θ )⋅ cos(θ )⋅ cos(θ )⋅ sin(θ )⋅ sin(θ )⋅e−ibP = 0.0225 − 0.0113i 2 3 4 5 6 1 p ⎪ ⎪ −i(b6 + bP ) θ θ θ θ θ θ θ − ( )⋅ ( )⋅ ( )⋅ ( )⋅ ( )⋅ ( )⋅ ( +… cos cos cos cos cos sin sin 2 3 4 5 p 1 6)⋅e ⎪ ⎪ −i(b1+ bP ) + … ⎪ − cos(θ1)⋅ cos(θ3)⋅ cos(θ4 )⋅ cos(θ5)⋅ cos(θ6)⋅ sin(θ2)⋅ sin(θp)⋅e ⎪ −i(b3 + bP ) + cos(θ2)⋅ cos(θ5)⋅ cos(θ6)⋅ sin(θ1)⋅ sin(θ3)⋅ sin(θ4 )⋅ sin(θp)⋅e +… ⎪ ⎪ ⎨ + cos(θ )⋅ cos(θ )⋅ cos(θ )⋅ sin(θ )⋅ sin(θ )⋅ sin(θ )⋅ sin(θ )⋅e−i(b4 + bP ) + … 2 3 6 1 4 5 p ⎪ ⎪ −i(b5 + bP ) +… ⎪ + cos(θ2)⋅ cos(θ3)⋅ cos(θ4 )⋅ sin(θ1)⋅ sin(θ5)⋅ sin(θ6)⋅ sin(θp)⋅e ⎪ −i(b2 + bP ) = − 0.0507 ⎪ + cos(θ4 )⋅ cos(θ5)⋅ cos(θ6)⋅ sin(θ1)⋅ sin(θ2)⋅ sin(θ3)⋅ sin(θp)⋅e ⎪ ⋮ ⎪ −i(b1+ b2 + b3 + b4 + b5 + b6 + bP ) ⎪ = 0. 0225 + 0.0113i ⎩ + cos(θ1)⋅ cos(θ2)⋅ cos(θ3)⋅ cos(θ4 )⋅ cos(θ5)⋅ cos(θ6)⋅ cos(θp)⋅e

(12)

A.R. Halassi et al. / Optics Communications 369 (2016) 12–17

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Referring to the system Eq. (9), we can now write Eq. (12) as,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

f0 (θ1, ... θ6, θP , b1, ... b6 , bP ) = − cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )..... .sin(θ1).sin(θP ).cos(bP )

− Re(0.0225 − 0.0113i)

=0

f1 (θ1, ... θ6, θP , b1, ... b6 , bP ) = − cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θP ).sin(θ1).sin(θ6 )…. .cos(b6 + bP )− cos(θ1).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )…. .sin(θ 2).sin(θP ).cos(b1 + bP ) + cos(θ 2).cos(θ5).cos(θ6 )…. .sin(θ1).sin(θ 3).sin(θ4 ).sin(θP ). cos(b3 + bP ) + cos(θ 2)…. .cos(θ 3).cos(θ6 ).sin(θ1).sin(θ4 ).sin(θ5).sin(θP ).cos(b4 + bP ) + … + cos(θ 2).cos(θ 3).cos(θ4 ).sin(θ1).sin(θ5).sin(θ6 ).sin(θP )…. .cos(b5 + bP ) + cos(θ4 ).cos(θ5).cos(θ6 ).sin(θ1).sin(θ 2)…. .sin(θ 3).sin(θP ).cos(b 2 + bP ) − Re( − 0.0507) ⋮ f6 (θ1, ... θ6, θP , b1, ... b6 , bP ) = + cos(θ1).cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )….

=0

.cos(θP ).cos(b1 + b 2 + b3 + b4 + b5 + b6 + bP ) − Re(0.0225 + 0.0113i)

=0

g0 (θ1, ... θ6, θP , b1, ... b6 , bP ) = + cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )… .. .sin(θ1).sin(θP ).sin(bP )

− Im(0.0225 − 0.0113i)

= 0

g1(θ1, ... θ6, θP , b1, ... b6 , bP ) = + cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θP ).sin(θ1).sin(θ6 )…. .sin(b6 + bP ) + cos(θ1).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )…. .sin(θ 2).sin(θP ).sin(b1 + bP )− cos(θ 2).cos(θ5).cos(θ6 )…. .sin(θ1).sin(θ 3).sin(θ4 ).sin(θP ). sin(b3 + bP )− cos(θ 2)…. .cos(θ 3).cos(θ6 ).sin(θ1).sin(θ4 ).sin(θ5).sin(θP ).sin(b4 + bP ) + … − cos(θ 2).cos(θ 3).cos(θ4 ).sin(θ1).sin(θ5).sin(θ6 ).sin(θP )…. .sin(b5 + bP )− cos(θ4 ).cos(θ5).cos(θ6 ).sin(θ1).sin(θ 2)…. .sin(θ 3).sin(θP ).sin(b 2 + bP ) − Im(−0.0507) ⋮ g6 (θ1, ... θ6, θP , b1, ... b6 , bP ) = − cos(θ1).cos(θ 2).cos(θ 3).cos(θ4 ).cos(θ5).cos(θ6 )….

=0

.cos(θP ).sin(b1 + b 2 + b3 + b4 + b5 + b6 + bP ) − Im(0.0225 + 0.0113i)

The system Eq. (13) can be solved by the PSO algorithm to find the opto-geometrical parameters of the filter. The settings used for the PSO are: The maximum number of iterations equals to 1500 iterations and the population equals to 150. In Table 1, we show the values of the opto-geometrical filter parameters obtained by the new synthesis method compared to those obtained by Amman's algorithm [14]. Fig. 4 shows the electric field responses of the filter: the ideal

Norm. Transmittance

1

=0

response, AMMAN's algorithm response and the response calculated by the new synthesis procedure. As can be seen, the three curves show a good agreement. Unlike the phase shifts of the AMMAN's compensators which are determined by the thickness of each compensator, the values of the phase shifts b1 to bP calculated by the new method are achieved by carefully adjusting the electrically controlled birefringence of the liquid crystal cells LCC1 to LCCp. This birefringence tunability makes the filter fully reconfigurable and has

Ideal response

0.8 0.6 0.4 0.2 0 0

(13)

100 200 300 Phase in degree

Fig. 2. Ideal non-symmetric amplitude transmittance to be synthesized.

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A.R. Halassi et al. / Optics Communications 369 (2016) 12–17

Fig. 3. Basic configuration of the N-stage liquid-crystal birefringent filter. LCC: Liquid-Crystal Cell, BP:Birefringent Plate, P: Polarizer, A: Analyzer.

Table 1 Opto-geometrical filter parameters obtained by the new method and AMMAN's algorithm.

LCC's phase Relative angles shifts b1 → bp θ1 → θP (degree)

Relative angles θ1 → θP (degree)

(degree)

68.66 60.62 45.85 52.70 45.85 60.62 21.34

18.97 2.35  47.95  47.95 2.35 19.05 26.66

68.66 60.61 45.85 52.69 45.85 60.61 21.33

Compensator's phase shifts b1 → bp (degree)

19.02 2.33  47.90  47.90 2.33 19.02 26.56

the benefit to keep the same filter's physical structure when we want to change the shape of the synthesized amplitude transmittance. It should be noted that these results are not unique. There are in fact 2(N + 1) possible sets of solutions but all are equivalent, where N is the number of the required birefringent sections [14]. It is interesting to notice that the number N of the required birefringent sections depends on the features of the desired output transmission function. A better fit to the target spectral shape needs a bigger N. However, N can take a lower value if a proper spectral weighting function is used to emphasize some spectral regions of the synthesized amplitude transmittance [2,3]. Fig. 5 illustrates the calculated spectral responses corresponding to different values of N. As can be seen, a better fit to the ideal target spectral shape G(w) is achieved as N increases. Moreover, in order to demonstrate the efficiency of the proposed method in synthesizing different spectral shapes, we have plotted in Fig. 6a triangular spectral response and a flat-top spectral response using six birefringent sections. As can be seen, the proposed method is universal and can be used to synthesize arbitrarily transmittance shapes. Fig. 7 depicts the mean square error (MSE) between the calculated responses for three different spectral shapes and the corresponding ideal target spectral responses for different N. As can be noted, the MSE shows a rapid decrease until it reaches a minimum value as N increases beyond 12. For the flat-top shape case, it is expected that for a better fit, N takes a greater value due to the Gibbs phenomenon arising from the rising and falling edges of the target spectral response [21].

Norm. Transmittance

AMMAN's algorithm

0.8

Ideal response Amman's algorithm New method

0.6 0.4 0.2 0 0

100 200 Phase in degree

300

Fig. 4. Electrical field responses of the filter having a non-symmetric amplitude transmittance obtained by the new synthesis procedure and AMMAN's algorithm.

Norm. Transmittance

New method

1

Ideal response N=6 N=12 N=20

1 0.8 0.6 0.4 0.2 0

0

100

200

300

Phase in degree Fig. 5. Comparison of the electrical field responses for different number N of birefringent sections.

4. Conclusion In this work, we have presented a generalized method for the synthesis of birefringent filters whose amplitude transmittances can have any arbitrary shape. The method is based on the resolution of a generalized nonlinear equation system deducted

from the Jones matrix formalism to determine the angles of the identical and variable sections, the angle of the analyzer and the phase shifts of the variable sections. As a proof of principle test, we have calculated the opto-geometrical parameters of a 6-stage

A.R. Halassi et al. / Optics Communications 369 (2016) 12–17

Norm. Transmittance

1

Ideal N=6

0.8 0.6

References

0.2 0

100

200

300

Phase in degree

1 Norm. Transmittance

birefringent filter having a non-symmetric amplitude transmittance. The results obtained show an excellent agreement with those developed in the literature. Further investigations are being carried out in order to study the validity of the present synthesis method to filters having any amplitude and phase transfer functions. These kinds of filters whose amplitude transmittances can have any tailored shapes are a solution of choice for, for instance, short pulse shaping and improving the performance of future optical communication networks.

0.4

0

Ideal N=6

0.8 0.6 0.4 0.2 0

0

100

200

300

Phase in degree Fig. 6. Triangular spectral response (Top) and flat top spectral response (Down) for a filter with six birefringent sections.

10

Mean Square Error

17

10 10 10 10

-1

-2

G(w) Triangular Flat-top

-3

-4

-5

0

6 12 18 24 30 36 42 48 Number N of birefringent sections

Fig. 7. Mean Square Error between the responses for three different spectral shapes calculated by the new synthesis procedure and the corresponding ideal target spectral responses for different N.

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