Broad multilayer antireflection coating by apodized and chirped photonic crystal

Optics Communications 284 (2011) 4124–4128

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Broad multilayer antireflection coating by apodized and chirped photonic crystal A. Mouldi ⁎, M. Kanzari Laboratoire de Photovoltaïque et Matériaux Semi-conducteurs (LPMS), Ecole Nationale d'Ingénieurs de Tunis BP 37 le Belvédère 1002 Tunis, Tunisia

a r t i c l e

i n f o

Article history: Received 26 January 2011 Accepted 2 May 2011 Available online 18 May 2011 Keywords: Multilayer antireflection coating Gaussian apodization Chirping

a b s t r a c t In this work, a new examination had been carried out to create a multilayer which acts as an antireflection coating. It has been shown that applying an apodization to a one dimensional photonic crystal can significantly enhance the total optical transmission in the structure. The transmission characteristics of the apodized grating are analyzed. So, in order to achieve a wideband of total transmission, the number of layers and the apodization profile should be optimized. Then, chirping was introduced to the apodized structure to improve the system transmission. The structure was investigated numerically according to its number of layers. Through accurate control of the design parameters, reflectance can achieve a value less than 0.05% across a large range of wavelengths. The enhancement of optical transmission in this structure is very important since the performance of a number of optical devices could be improved. © 2011 Elsevier B.V. All rights reserved.

1. Introduction It is well known that optical devices require an antireflection structure to their input interface to prevent unwanted reflections that can cause interference and cross talk between devices within integrated optical circuits [1,2]. Anti-reflective surfaces are therefore widely used. Traditional ones mostly consist of coatings of two or three dielectric material layers [3]. But some optical systems require coatings with efficient optical properties such as very low residual reflection loss, wider bandwidth and high efficiency of transmission, in addition to high degree of durability against adverse terrestrial and space conditions [4]. Therefore, multilayer antireflection coatings have attracted much attention to improve transmission properties of optical surfaces [4–7]. But the multilayer must have the suitable design. The first approach is to avoid sharp refractive index variation through the stack [5]. Some works considered multilayer systems based on periodic combination of two materials with low variation of refractive indices [4] but the design can be improved more by adopting other approaches. So, some recent works have developed a broad band antireflection by applying a gradient index profile [5–7]. These are called graded refractive index antireflection coating. In this work, we have proposed another approach which has permitted us to reduce the reflection and to enlarge the bandwidth; the proposed approach is based on apodization. Apodization theory is frequently used for astronomy and for optics, for example in quantitative spectroscopy [8] or in fiber Bragg grating in which the profile of gratings varies with the propagation distance [9–13]. In

⁎ Corresponding author. Tel.: + 216 71 874 700; fax: + 216 71 872 729. E-mail addresses: [email protected] (A. Mouldi), [email protected] (M. Kanzari). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.05.005

photonics, apodization have been generally required to suppress the side lobes of the total reflection band and to widen it [9,11–13]. Through the present work, apodization has been used to create a band of total transmission and to enlarge it. A specific function applied to the refraction index of layers and having the form of a Gaussian apodization has been exploited to enhance the transmission. So, each layer has a refractive index obeying to the Gaussian apodization according to its order through the grating. The transmission characteristics of the apodized grating are analyzed. So, in order to achieve a wideband of total transmission, the number of layers and the apodization profile should be optimized. We define this band as the wavelength range when R ≤ 0.05%. After that, the chirping was introduced by applying a global deformation to the structure in accordance with the following rule: y = x 1 + k where x is the coordinates of the initial object, y is the coordinates of the transformed object and k is the coefficient defining the deformation degree [14]. The numerical method employed to obtain the reflection (or transmission) response of the structure is the transfer matrix method.

2. Transfer matrix method For the calculation of system reflection and transmission, we employed the Transfer Matrix Method (TMM). This technique is a finite difference method particularly well suited to the study of the scattering (transmission, reflection, and absorption) spectrum [15]. It is based on Abeles method in terms of forward and backward propagating electric field, that is, E +and E −which were introduced to calculate the reflection and transmission. Abeles showed that the relation between the amplitudes [16] of the electric fields of the incident wave E0+, reflected wave E0−, and transmitted wave after m

A. Mouldi, M. Kanzari / Optics Communications 284 (2011) 4124–4128 + layers, Em + 1, is expressed as the following matrix for stratified films within m layers:

!

E0þ

=

E0−

C1 C2 C3 …Cm + 1 t1 t2 t3 …tm + 1

þ Em +1

!

ð

Þ

ð2Þ

rjp =

ˆ j−1 cosθj −n ˆ j cosθj−1 n ˆ j−1 cosθj + n ˆ j cosθj−1 n

ð3Þ

tjp =

ˆ j−1 cosθj−1 2n : ˆ j−1 cosθj + n ˆ j cosθj−1 n

ð4Þ

Moreover, for perpendicular (S) polarization: ˆ j−1 cosθj−1 −n ˆ j cosθj n ˆnj−1 cosθj−1 + n ˆ j cosθj

ˆ j−1 cosθj−1 n tjp = 2 : ˆ j−1 cosθj−1 + n ˆ j cosθj n

ϕj−1 =

ð5Þ

r=

t=

Em + E0þ

þ

=

1

c a

=

ð10Þ

The quantities a and c are the matrix elements of all the product Cj matrix: 2

C1 C 2 C 3 …Cm + 1 =



 a b : c d

2

  ˆ n cosθm + 1 2 = Re m + 1 jtP j ˆ 0 cosθ0 n

ð14Þ

for S and P polarization, respectively, where Re indicates the real part. 3. Model and formalism In this paper, apodized and chirped structures were introduced in the photonic crystal to enhance transmission. Apodized structure was to change the refractive indices of the dielectric layers, and chirped structure was to change the optical thicknesses of the dielectric layers. With different chirped coefficients and apodized functions, the total transmission band gap appeared in different wavelength ranges.

The structure considered in this paper is formed by p layers and the refraction index of each layer is chosen to have the following form ð15Þ

Where j is the order of the layer in the grating, a and b are the apodization parameters. In the first study the optical thicknesses of the layers were taken quarter wavelengths. Therefore, geometrical thicknesses of the layers have the following form dð jÞ =

λ0 : 4nð jÞ

ð16Þ

In mathematics, Gaussian function takes the form: "

ð8Þ

ð9Þ t1 t2 …tm + 1 : a

j

ð6Þ

Except for j = 1, λ is the wavelength of the incident light in vacuum − and dj − 1 is the thickness of the (j − 1) th layer. By putting Em + 1 = 1, because there is no reflection from the final phase, Abeles obtained a convenient formula for the total reflection and transmission coefficients, which correspond to the amplitude reflectance r and transmittance t, respectively, as follows: E0− E0þ

j

h i 2 nð jÞ = 1 + exp −ð j−aÞ = b :

ð7Þ 2π ˆ d n cosθj−1 : λ j−1 j−1

  ˆm + 1 n cosθm + 1 = n ˆm + 1 TP = Re t ˆ0 cosθ0 = n ˆ0 S n

ð13Þ

3.1. Apodization

The complex refractive indices and the complex angles of ˆ j sinθj (j = 1, ˆ j−1 sinθj−1 = n incidence obviously follow Snell's law: n 2… m + 1). The values ϕj − 1 in Eq. (2) indicate the change in the phase of the wave between (j − 1) th and j th boundaries and are expressed by the equation: ϕ0 = 0

ð12Þ

  ˆ n cosθm + 1 2 TS = Re m + 1 jtS j ˆ 0 cosθ0 n

where tj and rj are the Fresnel transmission and reflection coefficients, respectively, between the (j − 1) th and j th layer. The Fresnel coefficients tj and rj can be expressed as follows by using the complex ˆ j = nj + ikj and the complex refractive angle θj . For refractive index n parallel (P) polarization

rjp =

R = jrj :

For (S) and (P) polarizations, and the energy transmittance T as:

Here, Cj is the propagation matrix with the matrix elements.

Cj =

By using Eqs. (9) and (10), we can easily obtain the energy reflectance R as: 2

ð1Þ

− Em +1

    rj exp −i ϕj−1 exp i ϕj−1     exp −i ϕ j−1 rj exp i ϕj−1

4125

ð11Þ

2

ðx−βÞ f ðxÞ = αexp − 2γ 2

# :

ð17Þ

The graph of a Gaussian is a characteristic symmetric “bell curve” shape that quickly falls off towards plus/minus infinity. The parameter α is the height of the curve's peak, β is the position of the center of the peak, and γ controls the width of the “bell”. The form used in this work is "

# ðx−aÞ2 f ðxÞ = 1 + exp − : b

ð18Þ

If we compare this form to the general one of the Gaussian function, we can say that the function is composed of two parts, one is constant and equal to one, the second has a Gaussian profile rffiffiffi b . So, it is a profile which tends to 1 at the with α = 1 β = a γ = 2 edges of studied interval. Thus, refraction index is taken in the range 1–2. Dielectric materials with refractive indices involving in this range are for example SiO2 (1.45), MgF2 (1.38), MgO (1.78), Si3N4 (1.9) etc.

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A. Mouldi, M. Kanzari / Optics Communications 284 (2011) 4124–4128

a

3.2. Chirping

2

The deformation was introduced by applying a power law, so that the coordinates y which represent the transformed object were determined using the coordinates x of the initial object in accordance with the following rule:

1.8

ð19Þ

where k is the deformation degree. The initial optical phase thickness when we apply the y function is: ϕ=

2π λ0 cosθ: λ 4

1.7

Refraction index

k + 1

y=x

1.9

1.6 1.5 1.4

ð20Þ

1.2 1.1

The optical phase thickness of the j th layer is: i 2π λ0 h k + 1 k + 1 cosðθð jÞÞ: j −ð j−1Þ ϕð jÞ = λ 4

1

b

i λ0 h k + 1 k + 1 : j −ð j−1Þ  4 nð jÞ

3

4

5

ð22Þ

ð23Þ

8

9

10

11

12

13

(j,a,b)=(6,4,6) (j,a,b)=(7,6,7) (j,a,b)=(9,8,9) (j,a,b)=(13,11,19)

15

10

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Wavelength λ/λ0

4. Results and discussions

When the apodized structures are introduced in the standard photonic crystal, the structure will not be periodic any more. Fig. 1 shows the reflection response of the apodized grating as a function of normalized wavelength λ/λ0. It is worth noting that transmission is enhanced and instead of achieving a total reflection band like in periodic structure, through apodized structure, we got a total transmission band. We have purpose to minimize the number of layers with realizing a broad high transmission band. So, we have studied the optical reflection by varying the layer number. We have optimized the apodization parameters a and b for each number of layers. Table 1 presents the optimized couples (a,b) according to p where p varies between 6 and 21. These values have been chosen to correspond to a reasonable optical reflection. It was illustrated after numerical simulations of the optical reflection versus the normalized wavelength for some p, a and b values. We note some regularity in a and b for the higher values of p. For p ≥ 11, a generally takes the value p-2 while b takes the values 17, 18 and 19. For p ≥ 14, b keeps the value 18. Fig. 1(a) shows some profiles of refractive indices for some p values. Fig. 1(b) gives the corresponding optical reflection spectra obtained by apodization technique. For p = 6, some narrow bandwidths appeared. For p ≥ 9, by increasing the number of layers the curve flattens and the zero reflection band becomes wider. Table 2 illustrates the band properties for different design (characterized by

7

5

The studied structure is deposited on glass substrate with a refractive index of 1.5.

4.1. Transmission enhancement by apodization technique

6

25

20

Therefore, geometrical thicknesses of the apodized and chirped layers have the following form: dð jÞ =

2

layer number j

Reflection (%)

i λ0 h k + 1 k + 1 j −ð j−1Þ : 4

1

ð21Þ

For the deformed system, the optical thickness of each layer becomes variable and depends on the j th layer and the deformation degree k. So, the optical thickness of each layer after deformation by the y function takes the following form: xð jÞ =

(p,a,b)=(6,4,6) (p,a,b)=(7,6,7) (p,a,b)=(9,8,9) (p,a,b)=(13,11,19)

1.3

Fig. 1. (a) Optimized refractive index profiles for different numbers of layers (b) Reflection spectrum of the structure as a function of the normalized wavelength for different numbers of layers and refractive index profiles.

the triplet (p,a,b)), λshort and λlong are the wavelengths of respectively the lower and the upper band edges. For p ≥ 10, it has been found that p has no regular effect on the bandwidth but the curve becomes flatter by increasing p. The efficiency of apodization becomes maximum when we use 13 layers and the normalized bandwidth attains 0.8628 μm. On the other hand for p ≥ 16, no effect on the normalized bandwidth was observed with regular 0.8296 μm value. So, by comparing these results with the classical Bragg dielectric mirror, the bandwidth increases by increasing the number of the layers. However, the bandwidth of the Bragg mirror becomes stable for a critical p value [3]. It is clear, that the same result was found for the design simulated in the present work to have a total transmission band.

Table 1 Optimized apodization parameters for different numbers of layers. p

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

a b

4 6

6 7

7 9

8 9

9 9

9 17

10 18

11 19

12 18

13 18

14 18

15 18

16 18

17 18

18 18

19 18

A. Mouldi, M. Kanzari / Optics Communications 284 (2011) 4124–4128

p

Δλ/λ0

λshort/λ0

λlong/λ0

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.072 0.4924 0.6226 0.5446 0.1882 0.8504 0.8628 0.8504 0.8188 0.8296 0.8296 0.8296 0.8296 0.8296 0.8296

0.965 0.8128 0.7788 0.7974 0.915 0.732 0.732 0.732 0.7368 0.7352 0.7352 0.7352 0.7352 0.7352 0.7352

1.037 1.3054 1.4014 1.3414 1.1032 1.5824 1.5916 1.5824 1.5556 1.5648 1.5648 1.5648 1.5648 1.5648 1.5648

a

20

Reflection (%)

Table 2 The zero reflection band properties of apodized structure for different numbers of layers.

4127

15 10 5

0 0.2 0.15

deformation degree k 0.1 0.05 0

b It can be summarized that design containing 13 layers and based on apodization with (a,b) = (11,19) offers better optical properties, particularly wider spectral coverage.

0

2

1

4

3

6

5

8

7

Wavelength (μm)

3

2.5

λ

/λ

λ

long 0

4.2. Transmission enhancement by apodization and chirping techniques

center

/λ

0

We display in Fig. 2 the reflection spectra of the apodized and chirped grating for different values of k with p = 13, a = 11 and b = 19. Compared with the band realized with only apodized structure (k = 0), the width of the band was extended. With k = 0.1, the range for R b 0.05% is enlarged with an increase of 26%. In order to study the effect of k variation on the bandwidth, we display in Fig. 3 the reflection spectra as a function of the normalized wavelength and the deformation degree which is chosen to vary between 0 and 0.2. We note that as the parameter k increases, the total transmission band broadens and shifts to the higher wavelength region. To have the suitable band, we have chosen k which gives the broadest band without causing a great shifting of the lower band edge (λshort). For this, we have chosen a specific condition where λshort/λ0 ≈ 1.5. With p = 13, the value of k optimum is about 0.18. We propose now to study the effect of layer number on the apodized and chirped grating response. We give in Table 3 the band properties for k = 0.18 and different layer numbers.

Reflection (%)

18 16

k=0

14

k=0.05 k=0.2

10 8 6 4 2 0 0.5

1

1.5

2

2.5

3

3.5

4

1.5

λ 0.5

0

/λ

short 0

0.02 0.04 0.06 0.08

0.1

0.12 0.14 0.16 0.18

0.2

deformation degree k Fig. 3. (a) Reflection spectrum of the apodized and chirped structure with 13 layers as a function of the normalized wavelength and chirping coefficient k. (b) The zero reflection band shift as a function of chirping coefficient k.

It is seen that with apodized and chirped structure, the more layers there are, the larger the zero reflection band. Even for the high values of p, the bandwidth still enlarges when we increase the number of layers. So, we have attained the normalized bandwidth value of about

Table 3 The zero reflection band properties of apodized and chirped structure for different numbers of layers.

k=0.1

12

λ/λ0

2

4.5

5

Normalized Wavelength λ/λ0 Fig. 2. Reflection spectrum of the apodized and chirped structure with 13 layers as a function of the normalized wavelength for different chirping coefficients.

p

Δλ/λ0

λshort/λ0

λlong/λ0

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.0869 0.5522 0.9389 1.0174 0.4456 1.1295 1.2781 1.5415 1.62 1.634 1.648 1.6649 1.6789 1.6957 1.7069

1.4352 1.5109 1.4128 1.4436 1.7351 1.396 1.4721 1.424 1.4268 1.4464 1.4605 1.4745 1.4885 1.4997 1.5137

1.5221 2.0631 2.3518 2.4611 2.18 2.5255 2.7502 2.9656 3.0468 3.0805 3.1085 3.1393 3.1674 3.1954 3.2206

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A. Mouldi, M. Kanzari / Optics Communications 284 (2011) 4124–4128

1.7 when we have 21 layers with a shifting of λshort of almost 1.5 from λ0 value. 5. Conclusions In this work, the reflectance properties of the proposed structure are different comparing with those of the conventional multi-layered structure, because in this case dielectric multilayer gives a band structure with zero reflection under a specific apodized design. The reflectance properties of the grating with apodized structure are investigated in detail by numerical simulations. Chirping has been considered a good technique to enlarge the total reflection band in photonic crystal [14]. Through this work, introducing the chirping into the apodized structure was benefic to improve the flatness of the total transmission band. The suggested models may be applied to design photonic crystal based antireflection coatings. It can also be of military and commercial interest because the design can be used to enhance or better transmit infrared images. The progress in thin layers technology, especially the control of dielectric refractive index, will allow the realization of the suggested model.

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