Synthesis of controlled semi-reflecting multilayer broad band mirrors

Optik 127 (2016) 6673–6681

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Original research article

Synthesis of controlled semi-reflecting multilayer broad band mirrors S.S. Farag A member of the Optics group of the Physics Department, Faculty of Science, Ain Shams University, Abbasia, Cairo, Egypt

a r t i c l e

i n f o

Article history: Received 29 February 2016 Accepted 19 April 2016 Keywords: Thin films Broad-band mirrors design Light propagation Interferometry

a b s t r a c t Broad-band dielectric multilayer mirrors formed by staggering achromatic periods are developed here to reach multilayer systems of controlled reflectance as well as band width. The basic theory and the complete synthesis procedure are presented for the design of multilayer mirrors with controlled reflectance. Six designs with different reflectance levels, ranging from 44% up to 90%, average reflectance over the whole visible region, are presented. These broad mirrors are useful in many research experiments of interferometry and spectroscopy. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction Periodic structures of high- and low-index multilayer thin films, besides their great impact on the industry of optics, are always assumed a vital topic for many authors [1–8] involved in the development of modern optics. As a branch of high reflectance coatings, semi, or partial reflecting coatings are achieved by either metallic or all-dielectric layers. In the present work, multilayer all-dielectric reflectors are concerned, apart from metallic ones. Mirrored interferometers are one of the most popular applications of partial reflectors, by which beam splitting is achieved. Many authors approached the subject in previous works [9–11]. In the present work a different approach, considering the group propagation of waves is presented, for the design of broad band multilayer partial (semi) reflectors at normal incidence of light waves. This approach is based on a previous study of a double layer interferometer DLI [12,13]. What concerns us presently, about that study, is the comparison between the path lengths of the two different dispersive media comprising the DLI. The optical thicknesses of those media filling the interferometer gaps, are of integral proportionality within the range of spectrum studied, and they exactly match each other at a certain wavelength o , where the waves are travelling with their group velocities. Thus, achromatization takes place at o . The idea of matching two dispersive media in an interferometer, is reformulated here to apply to periods of thin films. Each period is formed of two dispersive high- and low-index materials. These periods are treated as double layer interferometers which are, in our case, considered as the basic unit for multilayer systems built up of periods of alternative high- and low-index materials. Also, design techniques for broad band mirrors are developed here to fit our approach. 2. Method As well known, the classical stack of quarter-wave thickness dielectric layers of alternate high- and low-index is the basic way to obtain all dielectric high reflectance mirrors. The high- and low-index layers form periods of equal phase thicknesses,

E-mail address: samy [email protected] http://dx.doi.org/10.1016/j.ijleo.2016.04.104 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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Fig. 1. The typical performance of a classical stack of 15 alternating high- (Ti2 O3 ) and low-index (SiO2 ) layers each of thickness o /4 where o = 550 nm.

which are repeated to reach the required reflectance level. Fig. 1 shows the characteristic curve for the reflectance of a typical classical stack of multilayer. The high- and low-index layers are matched according to the equation, nH hH = nL hL

(1)

where n and h are the refractive index and the geometrical thickness, respectively, the suffices H and L symbolize the highand low-index, respectively. A serious defect in such systems is that the high reflectance is obtained over a limited range of wavelengths. This defect was a real challenge for many pioneers [14–16] who made a number of attempts to extend the range of reflectance by altering the design of the basic classical stack. Most of these attempts have involved the staggering of the successive layers throughout the stack to form a regular progression, so that at any wavelength in a wide range of spectrum, enough layers in the stack have optical thicknesses near a quarter-wave to give high reflectance. As an example of how this staggering method was achieved, Heavens and Liddel [17] computed the thicknesses of successive layers so as to be in either arithmetic or geometric progression. The method presented here is also a staggering method which differs from before in that it is actually a staggering of periods [18–20] of high- and low-index layers, rather than of successive layers, building up the stack. Also, the essentials of the achromatization condition, states that at o nH hH GH = nL hL GL

(2)

That is the optical thicknesses are modulated with the group propagation factors G, where GH = 1 − and GL = 1 −

    dn  H nH

d

    dn  L

nL

d

(3)

o

o

(4)

Therefore, the two layers forming a unit period are not just matched by their optical thicknesses, by the way of (1), but also by their dispersion functions by the way of (2), within the range d. Thus, the matching Eq. (2) can take the form



(nHhH − nL hL )o = o hH

 dn  H

d

− hL

 dn  L

d

o

(5)

which may be restated as (nH hH − nL hL )o =1 o [hH (dnH /d) − hL (dnL /d)]o

(6)

at the specified reference wavelength o . At other wavelengths, for the same matched period, Eq. (6) differs from unity. Therefore, (6) is used here as the staggering tool, by which we can change the reference wavelength o for each period, in the progression pattern, forming the complete design. Now we can summarize our method as follows: First, the starting wavelength o is chosen, and the first basic period is calculated by the way of (2), at the starting wavelength.

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Fig. 2. The dispersion of Ti2 O3 in the visible.

Fig. 3. The dispersion of SiO2 in the visible.

Second, the parameters calculated in the first step is then substituted in the left hand side of (6), then a value other than unity is chosen for the right hand side to match the left side at a new wavelength. This chosen value is fixed as constant throughout the synthesis procedure, therefore it is more convenient to be designated as  where =

(nH hH − nL hL )o o [hH (dnH /d) − hL (dnL /d)]o

(7)

Third, the wavelength in the RHS of (7) is then changed in an iterative like routine, whereas the other quantities change dependently, until a certain wavelength is reached at which  in (7) is verified. Fourth, the new wavelength is considered as the reference wave for the second period in the stack. This period is calculated by the way of (2) at the new reached wavelength. Again, if the new resulting optical thickness is substituted in (6) it gives unity. Steps 1–4 are repeated every time a new period is added to the stack. 3. Application Our aim here is the control of the reflectance level, and bandwidth, by way of the above procedure, for the design of broad partial reflectors. Ti2 O3 and SiO2 are chosen for the high- and low-index films, respectively. The dispersion curves for the two materials are shown in Figs. 2 and 3 as a guide for their matching by way of (2). The design is started by choosing a starting wavelength, at which the first period in the stack is calculated, by the way of (2). The optical thickness of either the high or the low-index layer, in (2), is taken equal quarter the starting wavelength and the other layer, in the period, is calculated accordingly. From Figs. 2 and 3, the starting wavelength is chosen first in an intermediate position in the region of maximum dispersion. We start from 475 nm and then proceed towards longer wavelengths, in the staggering procedure. The other following wavelengths in the staggering process are determined by choosing a fixed constant value for , defined by way of (7).

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Fig. 4. The performance of the design listed in Table 1. Table 1 The details of the design with resulting reflectance shown in Fig. 4. The starting wave (475 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from short waves towards the longer. The listed thicknesses are calculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3 Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

Start wave: 475 nm L-material: SiO2 Period

Ph. Thk (nm)

1 0.67 2 01.45 3 122.61 4 140.34 5 151.98 6 157.82 7 160.18 8

40.06 8626 9.22 .0848 76.71 1.3111 90.18 1.5007 98.61 1.6252 102.75 1.6876 104.4 1.7129 105.

: 3 Wave (nm)

Multp of /4

75

6747

96

9974

720

1.292

823

1.5189

891

1.6609

925

1.7305

939

1.7584

944

1.768

Substrate ns = 1.52.

Then find the wavelength verifying that value by changing o , in the right hand side, until the chosen  is verified. The bigger is the value chosen for  the wider are the steps between the wavelengths, building up the design. The second period in the design is then calculated, by way of (2), at this new wavelength, and so on. Fig. 4 shows the resulting reflectance for a 15-layer system, built as above, with  taken equals to 3, which gives an average reflectance of 66% over the whole range. The irregularity in reflectance is noticed at the region of maximum dispersion of materials. The design details are listed in Table 1. In order to overcome the effect of this region of maximum dispersion, a shorter wavelength is chosen, as the starting wavelength, in addition to narrowing the steps of changing , in the staggering process, by way of . Obviously, from (7),  is directly proportional to dn/d and thus a smaller  reduces the effect of dispersion. Fig. 5 shows the reflectance of a multilayer system starting from 450 nm with a progression factor  equals to 1.5. The reflectance appears maintained at a regular level, almost around 90%, over the whole visible range. The design details are listed in Table 2. Now, a higher control for the reflectance level, and band width, is a requirement. It is found that a better control could be achieved at wavelengths far enough from the region of maximum dispersion. Since our working range is still in the visible band, the staggering process is inverted by starting from long wavelengths, in the near infrared, and proceeds towards shorter wavelengths, in the visible. Figs. 6–8 show the resulting reflectance for designs starting from 850 nm but differ in the progression factor . Fig. 6 gives an average reflection around 63% with  equals to 3, Fig. 7 gives 56% average with  equals to 4 and Fig. 8 gives an average of 53% with  equals to 5. Design details for Figs. 6–8, are listed in Tables 3–5. Obviously, increasing the

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Fig. 5. The performance of the design listed in Table 2.

Table 2 The details of the design with resulting reflectance shown in Fig. 5. The starting wave (450 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from a short wave towards the longer. The listed thicknesses are calculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3 Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

Start wave: 450 nm L-material: SiO2

: 1.5

Period

Ph. Thk (nm)

Wave (nm)

1 76.39 2 82.93 3 90.1 4 97.64 5 105.34 6 113. 7 120.47 8

35.75 .8169 42.26 .8868 49.05 .9635 55.88 1.0442 62.57 1.1264 68.98 1.2084 75.02 1.2882 80.37

450

.6022

488

.7118

530

.8261

574

.9411

619

1.0539

664

1.1619

707

1.2635

749

1.354

Substrate ns = 1.52.

Fig. 6. The performance of the design listed in Table 3.

Multp of /4

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S.S. Farag / Optik 127 (2016) 6673–6681

Fig. 7. The performance of the design listed in Table 4.

Fig. 8. The performance of the design listed in Table 5.

Table 3 The details of the design with resulting reflectance shown in Fig. 6. The starting wave (850 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from a long wave towards the shorter. The listed thicknesses are calculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3 Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Substrate ns = 1.52.

Start wave: 850 nm L-material: SiO2 Period

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

1 144.91 2 131.02 3 114.11 4 96.44 5 80.05 6 67.34 7 59.29 8

Ph. Thk (nm) 93.53 1.5496 83.22 1.4011 69.91 1.2203 54.82 1.0313 39.46 .856 25.72 .7201 14.76 .634 8.38

: 3 Wave (nm)

Multp of /4

850

1.5753

769

1.4016

670

1.1774

567

.9234

471

.6647

397

.4332

350

.2486

320

.141

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Table 4 The details of the design with resulting reflectance shown in Fig. 7. The starting wave (850 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from a long wave towards the shorter. The listed thicknesses are calculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3 Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

Start wave: 850 nm L-material: SiO2

: 4

Period

Ph. Thk (nm)

Wave (nm)

Multp of /4

1 144.91 2 127.48 3 106.5 4 85.82 5 69.08 6 58.94 7 53.34 8

93.53 1.5496 80.5 1.3632 63.57 1.1389 45.05 .9177 27.8 .7387 14.23 .6302 5.67 .5704 2.45

850

1.5753

748

1.3559

626

1.0708

505

.7588

407

.4682

347

.2396

314

.0954

296

.041

Substrate ns = 1.52.

Table 5 The details of the design with resulting reflectance shown in Fig. 8. The starting wave (850 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from a long wave towards the shorter. The listed thicknesses are calculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3 Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

Start wave: 850 nm L-material: SiO2

: 5

Period

Ph. Thk (nm)

Wave (nm)

Multp of /4

1 144.91 2 124.92 3 101.28 4 79.13 5 63.44 6 55.09 7 50.88 8

93.53 1.5496 78.53 1.3359 59.08 1.083 38.54 .8462 20.7 .6785 8.29 .5891 2.39 .5441 .86

850

1.5753

733

1.3226

595

.9951

466

.6492

374

.3486

325

.1397

300

.0403

288

.015

Substrate ns = 1.52.

value of , at a fixed starting wavelength, reduces the number of layers contributing to the reflectance, in the visible. This is clear from tables. As a further illustration for the method, both the starting wavelength and , are incremented to larger values, to lower the reflectance. The resulting reflectance is shown in Fig. 9, giving an average of 44% over the visible region. This is achieved by starting wave of 950 nm, and  equals to 10. Table 6 lists the details for this last design. As seen from the listed details, the number of layers are fixed in all the designs presented, so that the role of the above equations is highlighted. All the above results are, additionally, piled up altogether in Table 7, for comparison.

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Fig. 9. The performance of the design listed in Table 6. Table 6 The details of the design with resulting reflectance shown in Fig. 9. The starting wave (950 nm) is listed corresponding to the first period, at which it is calculated. The consequent listed wavelengths are the result of the chosen , and the corresponding period is calculated at this wave. In this design the progression of waves starts from a long wave towards the shorter. The listed thicknesses recalculated at the monitoring wave (550 nm). A high index material is added to the even periods to adjust the reflectance level. No. of periods: 7 Monitoring wave: 550 nm H-material: Ti2 O3

Start wave: 950 nm L-material: SiO2

Lyr Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

h1 L1 h2 L2 h3 L3 h4 L4 h5 L5 h6 L6 h7 L7 h8

: 10

Period

Ph. Thk (nm)

Wave (nm)

Multp of /4

1 162.03 2 155.11 3 136.42 4 107.72 5 79.66 6 60.8 7 52.25 8

105.69 1.7327 100.84 1.6587 87.28 1.4588 64.6 1.1519 39.07 .8518 16.97 .6502 4.13 .5588 1.01

950

1.7802

910

1.6984

800

1.4701

633

1.0881

469

.6581

358

.2859

308

.0696

288

.017

Substrate ns = 1.52. Table 7 Comparison between the different results for all the designs presented. Ti2 O3 and SiO2 are chosen to be the high- and low-index materials, respectively, for all designs Design no. range

No. of periods

No. of layers

Progression direction

Starting wavelength



Average R% over the visible of spectrum

1 2 3 4 5 6

7 7 7 7 7 7

15 15 15 15 15 15

From shorter to longer wavelength From shorter to longer wavelength From longer to shorter wavelength From longer to shorter wavelength From longer to shorter wavelength From longer to shorter wavelength

475 450 850 850 850 950

3 1.5 3 4 5 10

66 63 63 56 53 44

4. Conclusion The concept of matching two dispersive media by the group propagation of waves is applied to the calculation of matched high- and low-index thin films, forming periods in high reflectance multi-layers. The problem of achieving a specific reflection level over a specific range of wavelengths is approached by the way of straightforward fully controlled parameters. The values for these parameters are chosen arbitrarily, according to the required effect, and consequently the related equations

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synthesizes the layer system automatically. Different designs are presented for broad-band mirrors with different reflectance levels over the whole visible spectrum. Acknowledgements I would like to thank Professor Doctor M. Medhat and also Professor Doctor El-Sayed El-Zaiat for their fruitful discussions and encouragement. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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