## Abstract

We derive a simple equation to predict the center-wavelength shift of a Fabry-Perot type narrow-bandpass filter by using the conventional characteristic matrix method and the elastic strain model as the temperature varies. We determine the thermal expansion coefficient of substrate from the zero-shift condition of the center wavelength of the filter. The calculated shifts are in a good agreement with the experimental ones, in which the narrow-bandpass filters are prepared by plasma ion-assisted deposition on four substrates with different thermal expansion coefficients.

© 2004 Optical Society of America

## 1. Introduction

For the narrow-bandpass filter, it is well-known that the variation of the optical thickness in each layer due to atmospheric moisture and temperature deviation plays a significant part in the change of transmission characteristics including the center-wavelength shift. An effect by moisture adsorption results from the low packing density of the layer, which is deposited often by a conventional thermal evaporation method, while an effect by temperature deviation originates in the thermal properties of a layer such as Poisson’s ratio, thermal expansion coefficient, and so on.

Recently, ion- and plasma-based deposition techniques, such as ion-assisted deposition, plasma ion-assisted deposition, ion-beam sputtering, and ion-beam-assisted sputtering, have been developed so as to make it possible to increase the packing density to near unity [1–4]. If the packing density of the layer is equal to unity, then the undesirable wavelength shift of the narrow-bandpass filter owing to atmospheric moisture can be removed, so that the center-wavelength shift can be induced by the temperature level of the environment only. Takashashi reported a theoretical model for the thermal stability of single-cavity narrow-bandpass filters prepared by ion-assisted deposition and in his paper the elastic strain model was applied to an effective refractive index and total thickness of an equivalent layer representing the whole narrow-bandpass filter [4].

In this paper, we apply the elastic strain model to the characteristic matrix of each layer of Fabry-Perot type narrow-bandpass filters. We derive a simple and analytic equation to calculate the center-wavelength shift of the filters as the temperature changes, taking into account the fact that a variation of the optical phase thickness at each layer is very small and the minimum reflectance is determined by admittances of the incident and exit media. From the equation the thermal expansion coefficient of substrates to obtain the zero-shift of the center wavelength is calculated. Also, we deposit single-cavity Fabry-Perot filters on glass substrates with different thermal expansion coefficients by means of plasma ion-assisted deposition and compare the calculated spectral shifts of the center wavelength for various substrates with the experimental results.

## 2. Basic theory

#### 2.1 Refractive index and physical thickness of a film under temperature variation

At temperature *T*
_{0} we assume that the packing density of a film is equal to unity and the refractive index and the physical thickness of the film at *T*
_{0} are ${n}_{f}^{0}$ and ${d}_{f}^{0}$, respectively. As the temperature changes from *T*
_{0} to *T*, following the elastic strain model [4], the refractive index and the physical thickness of the film at *T* can be written as

and

respectively, where

and (*dn/dT*)
_{f}
is the thermal coefficient of the refractive index of the film material, *α*_{f}
and *α*_{S}
are thermal expansion coefficients of film and substrate, respectively, *ν*_{f}
is the Poisson ratio of film, and Δ*T*=*T*-*T*
_{0}.

#### 2.2 Derivation of the center-wavelength shift from the matrix method

Let us consider a Fabry-Perot type narrow-bandpass filter with a single cavity such as [(*LH*)^{p} (2*L*)^{q} (*HL*)^{p}] or [(*HL*)^{p} (2*H*)^{q} (*LH*)^{p}] structure, where *H* and *L* denote the quarterwave optical thickness of high- and low-index films, respectively. If the refractive index and the physical thickness of a layer are represented as *n*_{f}
and *d*_{f}
, respectively, the characteristic matrix of a layer is given by [5]

where *f*=*H* or *L*, and the optical phase thickness of the layer at normal incidence, *δ*_{f}
, is given by

When the temperature changes from *T*
_{0} to *T*, the optical phase thickness at ${T\mathit{,}\delta}_{f}^{T}$
, can be written as

where ${\delta}_{f}^{0}=\frac{\pi}{2}$ is the quarter-wave optical phase thickness at *T*_{0}
and *ε*_{f}
is a small variation in the optical phase thickness due to the temperature deviation. The resultant characteristic matrix [** M**]

^{T}for the filter at

*T*can be given by

where subscripts *C* and *D* correspond to *H* and *L* or *L* and *H*. Then each component of the matrix, [* M*]

^{T}, can be expressed in terms of the first order of

*ε*

_{H}and

*ε*

_{L}in Eq. (6).

The optical admittance of the filter *Y*^{T}
at *T* can be written as

where

and ${n}_{s}^{T}$
is the refractive index of a substrate at *T*. Then the reflectance at *T* is given by

where ${n}_{0}^{T}$ is the refractive index of incident medium at *T*. Since the minimum reflectance of the filter, i.e. the maximum transmittance at the center wavelength, is the value at the interface between the incident medium and the substrate, i.e.,

the admittance at the center wavelength of the filter should be written as

If we neglect the terms of (${n}_{L}^{T}$${\mathit{/}n}_{H}^{T}$
)^{2p} for large *p*, Eq. (12) can be written as a simple form,

where

for low-index cavity, and

for high-index cavity. From Eq. (6), the deviation of the optical phase thickness *ε*_{f}
of the layer is given by

and by inserting Eq. (16) into Eq. (13), the new center-wavelength of the filter at *T* can be written as

and the amount of center-wavelength shift of a single-cavity filter from temperature *T*
_{0} to *T* can be written as

From Eqs. (1)~(3), (14), (15) and (18), it is noted that the center-wavelength shift of the narrow-bandpass filter is dependent on the thermal properties of films as well as the thermal expansion coefficient of the substrate, but independent of the refractive index of the substrate. Also Eq. (18) is independent of *p*, the period of high- and low-index pairs in the reflectors, for large *p*, and can be applied for the multiple cavity filters regardless of the number of cavities based on [(*LH*)^{p} (2*L*)^{q} (*HL*)^{p}] or [(*HL*)^{p} (2*H*)^{q} (*LH*)^{p}] structures.

## 3. Simulations and Experiments

#### 3.1 Simulations

In order to simulate the center-wavelength shift of the Fabry-Perot type narrow-bandpass filters, SiO_{2} and Ta_{2}O_{5} are chosen as the low- and high-index materials, respectively, because they are widely used for the DWDM narrow-bandpass filters. A transmission spectrum of a narrow-bandpass filter of [air | (*LH*)^{10} (2*L*)^{4} (*HL*)^{10} | fused silica] structure is calculated by substituting Eqs. (1) and (2) into the characteristic matrix of each layer and multiplying all the matrices of the filter and shown in Fig. 1 as a function of wavelength and temperature deviation. When the amount of the temperature deviation increases, the center wavelength of the filter shifts to the longer wavelength region, and it turns out that the amount of the center-wavelength shift of the filters is 1.20 nm. On the other hand, it is calculated by 1.20 nm at Δ*T*=100 °C from Eq. (18). Therefore, it indicates that Eq. (18) is a simple equation to calculate the spectral shift of the center wavelength of a narrow-bandpass filter without using the characteristic matrix multiplication process.

In order to verify the simple formula for the center-wavelength shift of the narrow-bandpass filter, Eq. (18) was applied to Takashashi’s experimental results [4]. In Takashashi’s paper [4], Fabry-Perot filters with a high-index cavity of [air | (*HL*)^{9} (2*H*) (*LH*)^{9} | substrate] were prepared by ion-assisted deposition on several substrates with different thermal expansion coefficients. The experimental results for Δ*λ*/Δ*T* in reference 4 show a good agreement with the calculated ones from Eq. (18) as shown in Table 1.

It is noted that thermal expansion coefficient of a substrate for zero-shift of the center-wavelength of the filter can be calculated by putting (Δ*λ/λ*
_{0})=0 in Eq. (18) and is shown in Fig. 2 as a function of *q* for low- and high-index cavity filters. It shows that different substrates of *α*_{s}
=10.487 and 11.343 ppm/°C should be used at *q*=4 for low- and high-index cavity filters, respectively.

#### 3.2 Preparation of Fabry-Perot filters

Fabry-Perot filters with a low-index single cavity, [air | (*LH*)^{9} (2*L*)^{4} (*HL*)^{9} | substrate], were prepared on four glass substrates by means of plasma ion-assisted deposition. Because the packing density of the film deposited by this technique approaches unity level at room temperature [3], it is assumed that the center-wavelength shift of the filter is due only to temperature deviation.

Two groups of the glass substrate were prepared: one with the small *α*_{S}
such as Zerodur (Schott) and fused silica (Schott), and the other with the large *α*_{S}
such as F7 (Schott) and WMS-02 (Ohara). The diameter of the substrates was 25 mm and the thickness was 6.35 mm, which is thick enough to avoid a substrate bend due to the residual stress in thin film formed during the deposition. Optical and thermal properties of the four types of glass used in the simulations are listed in Table 2.

Four filters were deposited in a plasma ion-assisted deposition system (APS1104, Leybold) with an advanced plasma source. The transmission spectra of the filters were measured by using a tunable laser source (AQ4321A, Ando Electric Co.) and an optical spectrum analyzer (Model 6317B, Ando Electric Co.) in the range of 1540 to 1550 nm. Filters were placed between GRIN lens collimators at a normal incidence and a thermo-electric module device was used as a heating source. The temperature of the filter was measured by a K-type thermocouple, which was in contact with the filter on an aluminum block. The center-wavelength shift of the filters was measured from room temperature to 70 °C.

#### 3.3 Experiments

Figure 3 shows the center-wavelength shift of the fabricated filters vs. the temperature deviation. The temperature coefficients of the center-wavelength shift (Δ*λ*/Δ*T*) for the filters were obtained from the slopes in Fig. 3, and are compared in Table 3 with the derived ones from transmission spectrum simulated by using Eqs. (1) and (2) and the calculated ones by using Eq. (18). It is found that the experimental results for Δ*λ*/Δ*T* are in a good agreement with those from Eq. (18). Also it is shown that the Δ*λ*/Δ*T* of the filters is very small for F7 and WMS-02 substrates with high *α*_{S}
, while it is large for Zerodur and fused-silica substrates with low *α*_{S}
. It is expected that F7 and WMS-02 substrates with *α*_{S}
of near 10.487 ppm/°C can be chosen for the filters to be insensitive to the temperature deviation, and it is known that they are actually used in the fabrication of the DWDM filters.

## 4. Conclusions

As the temperature varies, we have derived a simple equation for the center-wavelength shift of the Fabry-Perot type narrow-bandpass filter by taking an advantage of the fact that the minimum reflectance of the Fabry-Perot filter is the one between the incident medium and the substrate. It leads to calculate the thermal expansion coefficient of substrates for the zero-shift of the center wavelength of the Fabry-Perot filters. The calculated shifts of the center wavelength from the equation are in a good agreement with the experimental ones for various substrates. The derived equation can be very useful in prediction of the center-wavelength shift and the thermal expansion coefficient of the substrate for zero-shift before depositing the Fabry-Perot filters.

## Acknowledgments

The authors wish to thank the Korea Electric Terminal Co., Ltd in Incheon, Korea, for the transmittance measurements of the filters. This work was supported by the Korea Science and Engineering Foundation through the Quantum Photonic Science Research Center at Hanyang University.

## References and links

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**2. **R.Y. Tsai, C.S. Chang, C.W. Chu, T. Chen, F. Dai, D. Lin, S. Yan, and A. Chang, “Thermally stable narrow-bandpass filter prepared by reactive ion-assisted sputtering,” Appl. Opt. **40**, 1593–1598 (2001). [CrossRef]

**3. **A. Zöller, R. Götzelmann, K. Matl, and D. Cushing, “Temperature-stable bandpass filters deposited with plasma ion-assisted deposition,” Appl. Opt. **35**, 5609–5612 (1996). [CrossRef] [PubMed]

**4. **H. Takashashi, “Temperature stability of thin-film narrow-bandpass filters produced by ion-assisted deposition,” Appl. Opt. **34**, 667–675 (1995). [CrossRef] [PubMed]

**5. **H.A. Macleod, *Thin-Film Optical Filters*, 3rd ed. (IoP, Bristol, 2001). [CrossRef]

**6. **Essential Macleod is commercial thin film software from Thin Film Center Inc, USA.