**2. Optical interactions in MEMS**

Understanding the coupling that exists between the thermal and optical behavior in laserirradiated MEMS must begin by looking at the optical properties of the irradiated materials and at how the laser light interacts with each material. The primary factor that affects the magnitude of this interaction is the material's complex refractive index, *n n ik ˆ* . A wave

Optical-Thermal Phenomena in Polycrystalline Silicon MEMS During Laser Irradiation 333

the incident light, respectively. The surface reflectivity, that is the magnitude of the fraction

For normal incidence, *<sup>i</sup> 0* , the polarization dependence disappears and Eq. 3 reduces to

*n 1 ˆ n1 k <sup>R</sup> n 1 ˆ n1 k* 

In any absorbing material (i.e., with *k 0* ) the light transmitted through the medium is

where *oI* is the intensity of light entering the surface, *4 k* is the linear attenuation coefficient of the medium at the wavelength , and *z* is the spatial coordinate with its origin at the surface. The inverse of the attenuation coefficient is known as the optical penetration

> *1 opt <sup>d</sup> 4 k*

While the development above for the Fresnel coefficients assumes a semi-infinite medium (i.e., a single interface separating the two media), the significance of Eq. 6 is that any material whose of thickness *d* >> *opt d* can be considered optically thick, in the sense that it will behave the same as a semi-infinite medium. What constitutes an optically thick layer ultimately depends on the value of the complex part of the refractive index, *k* , as described in Eq. 7. For example, the penetration depth of silicon at 0.3 μm is *opt d* 5.8 nm, very similar to that of aluminium at 0.4 μm or gold at 0.7 μm (Schulz, 1954); at longer wavelengths the penetration depth in silicon increases by over three orders of magnitude (on the order of several micrometers) due to the drastic decrease in the value of *k*. As we will show, the distinction between optically thick and optically thin films will have profound implications in

A different approach must be used in instances where the thickness of the irradiated film is comparable to the optical penetration depth. Such conditions are of significant relevance for surface micromachined polysilicon devices, which generally can have layers and gaps with thicknesses on the order of a few μm (Carter et al., 2005; MEMS Technologies Department,

2 As a consequence of the two polarization conditions, there will be two independent values for reflectivity. For unpolarized irradiation, it is common practice to take the average of the two reflectivity

and it is the distance over which the light intensity is attenuated by *1/e* .

the treatment of the optical thermal coupling that exists in laser heated MEMS.

**2.2 Optically thin and multilayered systems** 

values as the resultant reflectivity.

 

*2 2 2*

*2 2*

*<sup>2</sup> p p R r* . (4)

*<sup>o</sup> I z I ex p( z)* (6)

. (5)

, (7)

*<sup>2</sup> s s R r* and

the well-known expression for bulk surface reflectivity (Born & Wolf, 1999):

of reflected energy, is given for either polarization2, by:

attenuated in accordance with the Beer-Lambert law:

depth

incident on the interface between two media of different refractive indices will undergo reflection and refraction, as shown in Fig. 1. The direction of the reflected and refracted beams follow the well-known laws of reflection and refraction:

$$
\theta\_r = \theta\_i \text{ (law of reflection)},
\tag{1}
$$

$$
\hat{n}\_2 \sin \theta\_t = \hat{n}\_1 \sin \theta\_i \text{ (law of refraction; Snelll's law)}\tag{2}
$$

where *<sup>i</sup>* , *r* and *t* are the angles of incidence, reflection, and refraction, respectively1. The following sections will discuss how the rules above are applied to laser-irradiated structures in order to obtain the magnitudes of the reflected, transmitted, and absorbed light, which ultimately dictate how the energy is deposited in an irradiated microsystem.

Fig. 1. Reflection and refraction of a plane wave incident on the interface between two media.

#### **2.1 Optically thick systems**

For monochromatic laser light, incident from vacuum ( *vacuum n 1.0 ˆ* ) at an angle *i* upon a homogeneous, semi-infinite, non-magnetic medium of index *nˆ* , the reflectivity of the interface is dictated by the Fresnel coefficients (Born & Wolf, 1999):

$$r^s = \frac{\cos \Theta\_i - \sqrt{\hat{\mathbf{n}}^2 - \sin^2 \Theta\_i}}{\cos \Theta\_i + \sqrt{\hat{\mathbf{n}}^2 - \sin^2 \Theta\_i}} \; \; r^p = \frac{\hat{\mathbf{n}}^2 \cos \Theta\_i - \sqrt{\hat{\mathbf{n}}^2 - \sin^2 \Theta\_i}}{\hat{\mathbf{n}}^2 \cos \Theta\_i + \sqrt{\hat{\mathbf{n}}^2 - \sin^2 \Theta\_i}} \; \tag{3}$$

where the law of refraction was used to rewrite the expressions in terms of the incident angle and medium refractive index only, and the subscripts *s* and *p* above refer to the *spolarized* or *transverse-electric* (TE) and *p-polarized* or *transverse-magnetic* (TM) polarizations of

<sup>1</sup> For instances where the indices are complex, the quantity θt is also complex-valued and no longer has the same meaning as an angle of refraction.

incident on the interface between two media of different refractive indices will undergo reflection and refraction, as shown in Fig. 1. The direction of the reflected and refracted

where *<sup>i</sup>* , *r* and *t* are the angles of incidence, reflection, and refraction, respectively1. The following sections will discuss how the rules above are applied to laser-irradiated structures in order to obtain the magnitudes of the reflected, transmitted, and absorbed light, which ultimately dictate how the energy is deposited in an irradiated microsystem.

Fig. 1. Reflection and refraction of a plane wave incident on the interface between two

interface is dictated by the Fresnel coefficients (Born & Wolf, 1999):

*s i i*

*cos n sin <sup>ˆ</sup> <sup>r</sup> cos n sin ˆ* 

*2 2*

*2 2 i i*

For monochromatic laser light, incident from vacuum ( *vacuum n 1.0 ˆ* ) at an angle *i* upon a homogeneous, semi-infinite, non-magnetic medium of index *nˆ* , the reflectivity of the

> *2 22 pi i 2 22 i i*

' (3)

*n cos n sin ˆ ˆ <sup>r</sup> n cos n sin ˆ ˆ* 

,

where the law of refraction was used to rewrite the expressions in terms of the incident angle and medium refractive index only, and the subscripts *s* and *p* above refer to the *spolarized* or *transverse-electric* (TE) and *p-polarized* or *transverse-magnetic* (TM) polarizations of

1 For instances where the indices are complex, the quantity θt is also complex-valued and no longer has

media.

**2.1 Optically thick systems** 

the same meaning as an angle of refraction.

*r i* (law of reflection), (1)

*2 t1 i n sin n sin ˆ ˆ* (law of refraction; Snell's law) (2)

beams follow the well-known laws of reflection and refraction:

the incident light, respectively. The surface reflectivity, that is the magnitude of the fraction of reflected energy, is given for either polarization2, by:

$$R^s = \left| r^s \right|^2 \text{ and } \left. R^p = \left| r^p \right|^2 \right. \tag{4}$$

For normal incidence, *<sup>i</sup> 0* , the polarization dependence disappears and Eq. 3 reduces to the well-known expression for bulk surface reflectivity (Born & Wolf, 1999):

$$R = \left| \frac{\hat{n} - 1}{\hat{n} + 1} \right|^2 = \frac{\left(n - 1\right)^2 + k^2}{\left(n + 1\right)^2 + k^2} \,. \tag{5}$$

In any absorbing material (i.e., with *k 0* ) the light transmitted through the medium is attenuated in accordance with the Beer-Lambert law:

$$I(z) = I\_o \exp(-\alpha z) \tag{6}$$

where *oI* is the intensity of light entering the surface, *4 k* is the linear attenuation coefficient of the medium at the wavelength , and *z* is the spatial coordinate with its origin at the surface. The inverse of the attenuation coefficient is known as the optical penetration depth

$$\mathbf{u}\_{opt} = \mathbf{u}^{-1} = \bigvee\_{4 \le k'} \mathbf{4}\_{7} \mathbf{x}' \tag{7}$$

and it is the distance over which the light intensity is attenuated by *1/e* .

While the development above for the Fresnel coefficients assumes a semi-infinite medium (i.e., a single interface separating the two media), the significance of Eq. 6 is that any material whose of thickness *d* >> *opt d* can be considered optically thick, in the sense that it will behave the same as a semi-infinite medium. What constitutes an optically thick layer ultimately depends on the value of the complex part of the refractive index, *k* , as described in Eq. 7. For example, the penetration depth of silicon at 0.3 μm is *opt d* 5.8 nm, very similar to that of aluminium at 0.4 μm or gold at 0.7 μm (Schulz, 1954); at longer wavelengths the penetration depth in silicon increases by over three orders of magnitude (on the order of several micrometers) due to the drastic decrease in the value of *k*. As we will show, the distinction between optically thick and optically thin films will have profound implications in the treatment of the optical thermal coupling that exists in laser heated MEMS.

#### **2.2 Optically thin and multilayered systems**

A different approach must be used in instances where the thickness of the irradiated film is comparable to the optical penetration depth. Such conditions are of significant relevance for surface micromachined polysilicon devices, which generally can have layers and gaps with thicknesses on the order of a few μm (Carter et al., 2005; MEMS Technologies Department,

<sup>2</sup> As a consequence of the two polarization conditions, there will be two independent values for reflectivity. For unpolarized irradiation, it is common practice to take the average of the two reflectivity values as the resultant reflectivity.

Optical-Thermal Phenomena in Polycrystalline Silicon MEMS During Laser Irradiation 335

Fig. 5. Schematic representation of the LTR method. A multilayer stack is represented by an

This technique is similar to the transfer matrix method in that each layer is assigned a mathematical entity made up of the reflection and transmission coefficients for the layer. However, unlike the matrix method where the layer matrix depends on the properties of the media surrounding the layer, the coefficients are referenced with respect to vacuum (i.e., a wave is considered to be travelling into or from vacuum), simplifying the calculations and giving the technique its modularity. Thus, a three-element LTR vector containing the leftand right-side reflection coefficients, as well as the transmission coefficient, is defined as:

**X** . (8)

, (9)

*1 p <sup>r</sup> 1 pr <sup>L</sup>*

*1 p <sup>r</sup> 1 pr*

 *<sup>2</sup> 2 2 <sup>i</sup> p exp i d n sin ˆ*

*1 r T p 1 pr <sup>R</sup>*

For a wave incident at an angle *i* upon a layer of thickness *d* and refractive index *nˆ* , the

whereas the coefficient *r* considers reflection from the interfaces and for the two possible polarization conditions3 is given by Eq. 3. The Fresnel coefficients above assume the wave

<sup>3</sup> As discussed in footnote 2, this method will yield polarization-dependent results for reflectance, transmittance, and absorptances. For unpolarized irradiation, the accepted value is the average of the

coefficient *p* in Eq. 8 considers propagation in the medium and is defined as:

two polarization cases.

LTR element, where each layer is also made up of an LTR element.

2008)—comparable to optical penetration depths at visible-to-near infrared wavelengths (Phinney & Serrano, 2007; Serrano & Phinney, 2009; Serrano et al., 2009). In theses cases, depicted in Fig. 3, light transmitted across the first interface will encounter a second interface and undergo reflection and refraction. The process of reflection and refraction at both interfaces can repeat itself numerous times, as shown in Fig. 3, and with each reflection, the wave can undergo a phase change of 180°. If the incident light is monochromatic, with sufficiently large coherence length (i.e., laser light), then the multiple reflections will interfere with each other constructively and destructively. This thin film interference will yield deviations from the values obtained with Eqs. 3 and 4 for the optical response of the irradiated surface.

Fig. 3. Reflection and refraction in a multilayered system showing the multiple reflections from the two interfaces.

There are various ways to obtain a numerical description of the overall optical performance of such a multilayered system. The most common method is the transfer matrix method (Born & Wolf, 1999; Katsidis & Siapkas, 2002) whereby each individual layer is assigned a matrix of Fresnel coefficients, which capture the interaction of the incident wave with the layer. This method, while useful for obtaining the net response of the stratified structure, does not easily permit extracting information on how the energy is deposited within the layers, a detail of paramount importance when analyzing laser-irradiated MEMS. To obtain interlayer absorptance values, we turn to a similar analysis called the LTR method (Mazilu et al., 2001), which combines the layer responses in a modular form. This modularity then permits the extraction of the absorptances for the layers in the structure.

#### **2.2.1 LTR method**

The LTR method (Mazilu et al., 2001), which is stands for **L**eft-side reflectance, **T**ransmittance, and **R**ight-side reflectance, considers a stack of material irradiated from the left and right sides, as shown in Fig. 4. The technique leverages the fact that for an irradiated layered system only three terms are needed to fully describe its optical response—the reflectances of either side and a transmittance term. While most typically utilized for obtaining the net response of a stratified system, the modular nature of the LTR method facilitates the extraction of absorptance values for each individual layer, making it particularly useful for laser-irradiated MEMS (Serrano & Phinney, 2009; Serrano et al., 2009).

2008)—comparable to optical penetration depths at visible-to-near infrared wavelengths (Phinney & Serrano, 2007; Serrano & Phinney, 2009; Serrano et al., 2009). In theses cases, depicted in Fig. 3, light transmitted across the first interface will encounter a second interface and undergo reflection and refraction. The process of reflection and refraction at both interfaces can repeat itself numerous times, as shown in Fig. 3, and with each reflection, the wave can undergo a phase change of 180°. If the incident light is monochromatic, with sufficiently large coherence length (i.e., laser light), then the multiple reflections will interfere with each other constructively and destructively. This thin film interference will yield deviations from the values obtained with Eqs. 3 and 4 for the optical response of the

Fig. 3. Reflection and refraction in a multilayered system showing the multiple reflections

permits the extraction of the absorptances for the layers in the structure.

There are various ways to obtain a numerical description of the overall optical performance of such a multilayered system. The most common method is the transfer matrix method (Born & Wolf, 1999; Katsidis & Siapkas, 2002) whereby each individual layer is assigned a matrix of Fresnel coefficients, which capture the interaction of the incident wave with the layer. This method, while useful for obtaining the net response of the stratified structure, does not easily permit extracting information on how the energy is deposited within the layers, a detail of paramount importance when analyzing laser-irradiated MEMS. To obtain interlayer absorptance values, we turn to a similar analysis called the LTR method (Mazilu et al., 2001), which combines the layer responses in a modular form. This modularity then

The LTR method (Mazilu et al., 2001), which is stands for **L**eft-side reflectance, **T**ransmittance, and **R**ight-side reflectance, considers a stack of material irradiated from the left and right sides, as shown in Fig. 4. The technique leverages the fact that for an irradiated layered system only three terms are needed to fully describe its optical response—the reflectances of either side and a transmittance term. While most typically utilized for obtaining the net response of a stratified system, the modular nature of the LTR method facilitates the extraction of absorptance values for each individual layer, making it particularly useful for laser-irradiated MEMS (Serrano & Phinney, 2009; Serrano

irradiated surface.

from the two interfaces.

**2.2.1 LTR method** 

et al., 2009).

Fig. 5. Schematic representation of the LTR method. A multilayer stack is represented by an LTR element, where each layer is also made up of an LTR element.

This technique is similar to the transfer matrix method in that each layer is assigned a mathematical entity made up of the reflection and transmission coefficients for the layer. However, unlike the matrix method where the layer matrix depends on the properties of the media surrounding the layer, the coefficients are referenced with respect to vacuum (i.e., a wave is considered to be travelling into or from vacuum), simplifying the calculations and giving the technique its modularity. Thus, a three-element LTR vector containing the leftand right-side reflection coefficients, as well as the transmission coefficient, is defined as:

$$\mathbf{X} = \begin{pmatrix} L \\ T \\ R \end{pmatrix} = \begin{pmatrix} r \frac{1 - p^2}{1 - p^2 r^2} \\ r \frac{1 - r^2}{1 - p^2 r^2} \\ r \frac{1 - p^2}{1 - p^2 r^2} \end{pmatrix}. \tag{8}$$

For a wave incident at an angle *i* upon a layer of thickness *d* and refractive index *nˆ* , the coefficient *p* in Eq. 8 considers propagation in the medium and is defined as:

$$p = \exp\left(i\frac{2\pi}{\lambda}d\sqrt{\hat{n}^2 - \sin^2\Theta\_i}\right),\tag{9}$$

whereas the coefficient *r* considers reflection from the interfaces and for the two possible polarization conditions3 is given by Eq. 3. The Fresnel coefficients above assume the wave

<sup>3</sup> As discussed in footnote 2, this method will yield polarization-dependent results for reflectance, transmittance, and absorptances. For unpolarized irradiation, the accepted value is the average of the two polarization cases.

Optical-Thermal Phenomena in Polycrystalline Silicon MEMS During Laser Irradiation 337

*10 L 1 2 n n ˆ ˆ N 1 N 01 R* 

Since the result of the composition is another LTR vector, if the fields *E1* and *E2* incident on the stack are known, then the remaining fields, *F1* and *F2* , can be easily found using Eqs. 10

While the LTR construct is useful for capturing the response of a multilayered structure irradiated from the front and the back (left and right in Fig. 3), only front-side illumination is considered here, as that is the most common configuration encountered in MEMS applications. For single-sided illumination the structure is assumed to be illuminated only

> *2 R R*

*Re n cos*

*Re n cos* 

If the incident medium on the left is vacuum or air, Eq. 22 can be rewritten fully in terms of

*Re n sin ˆ*

With the fields on the left- and right-most layers defined, the fields entering and exiting each layer can be obtained by recursively applying Eqs. 10 and 11 to each layer. Once these fields are defined, the individual layer absorptances can be easily obtained by noting that each layer is referenced to vacuum and the absorptance is simply the difference between the

where the left-most fields of the first layer and the right-most fields for the last layer are

*cos*

*L L*

 *2 2 sub i 2*

*i*

L T R

L *F E 1 2* (left side reflection); (18)

T *F E 2 2* (transmission); (19)

R *0* (right side reflection). (20)

*<sup>2</sup> R* L , (21)

T , and (22)

*A1RT* . (23)

T . (24)

*2 222 iiii AE E F F i1 2 1 2* , (25)

(17)

**LTR S X X X X S**

The total reflected, transmitted, and absorbed intensities are then:

*T*

*T*

the angle of incidence and the substrate index, *sub nˆ* , as

entering and exiting field magnitudes:

obtained from Eqs. 18 and 19.

and 11.

from the left (i.e., *E 0 <sup>1</sup>* in Fig. 5), and

travels from vacuum through the layer and out into vacuum once again. If the amplitudes of the fields incident on the layer from the right and the left are given, as shown in Fig. 5, by *E1* and *E2* , respectively, the elements of **X** can be used to describe the amplitudes of the fields, *F1* and *F2* , exiting the layer as:

$$F\_1 = TE\_1 + LE\_2 \text{ (and } \tag{10}$$

$$F\_2 = RE\_1 + TE\_2 \,. \tag{11}$$

The LTR method additionally defines a vector for a single interface: one for an interface with a wave travelling from vacuum into a medium of index *nˆ* ( **S***<sup>01</sup>* ) and another for a wave travelling from the medium into vacuum ( **S***<sup>10</sup>* ):

$$\mathbf{S}\_{01}\left(\hat{n}\right) = \begin{pmatrix} L \\ T \\ R \end{pmatrix} = \begin{pmatrix} r \\ t\_{01} \\ -r \end{pmatrix}, \text{and} \tag{12}$$

$$\mathbf{S}\_{10}\left(\hat{n}\right) = \begin{pmatrix} L \\ T \\ R \end{pmatrix} = \begin{pmatrix} -r \\ t\_{10} \\ r \end{pmatrix},\tag{13}$$

where the coefficient *r* is given in Eq. 3 for the two polarization conditions, and

$$t\_{01}^s = \frac{2\cos\Theta\_i}{\cos\Theta\_i + \sqrt{\hat{n}^2 - \sin^2\Theta\_i}} \quad t\_{01}^p = \frac{2\hat{n}\cos\Theta\_i}{\hat{n}^2\cos\Theta\_i + \sqrt{\hat{n}^2 - \sin^2\Theta\_i}} \tag{14}$$

$$t\_{10}^s = \frac{2\hat{n}\cos\Theta\_i}{\cos\Theta\_i + \sqrt{\hat{n}^2 - \sin^2\Theta\_i}}, \text{ and } \ t\_{10}^p = \frac{2\hat{n}^2\cos\Theta\_i}{\hat{n}\cos\Theta\_i + \sqrt{\hat{n}^2 - \sin^2\Theta\_i}}. \tag{15}$$

Combination of multiple layers is implemented by the use of a composition rule, as shown below for two layers. Under the LTR scheme, each layer is considered a separate entity, separated from adjacent layers by a zero- thickness vacuum layer, such that the wave exits one layer into vacuum and enters the next layer from vacuum.

$$\mathbf{LTR} = \mathbf{X}\_1 \oplus \mathbf{X}\_2 = \begin{pmatrix} L\_1 \\ T\_1 \\ R\_1 \end{pmatrix} \oplus \begin{pmatrix} L\_2 \\ T\_2 \\ R\_2 \end{pmatrix} = \begin{pmatrix} L\_1 + \frac{L\_2 T\_1^2}{1 - R\_1 L\_2} \\ \frac{T\_1 T\_2}{1 - R\_1 L\_2} \\ \frac{T\_1 T\_2}{1 - R\_1 L\_2} \\ R\_2 + \frac{R\_1 T\_2^2}{1 - R\_1 L\_2} \end{pmatrix} = \begin{pmatrix} \mathcal{L} \\ \mathcal{T} \\ \mathcal{R} \end{pmatrix}.\tag{16}$$

This rule enables modeling of a multilayer structure by sequential application of the composition rule to all the layers in the stack including the media on the left and right side of the multilayer structure.

travels from vacuum through the layer and out into vacuum once again. If the amplitudes of the fields incident on the layer from the right and the left are given, as shown in Fig. 5, by *E1* and *E2* , respectively, the elements of **X** can be used to describe the amplitudes of the

The LTR method additionally defines a vector for a single interface: one for an interface with a wave travelling from vacuum into a medium of index *nˆ* ( **S***<sup>01</sup>* ) and another for a wave

> *L r nT t ˆ*

 

*R r*

*L r*

 

*R r*

, *<sup>p</sup> <sup>i</sup> <sup>01</sup> 2 22*

*1*

*2*

*1 2 <sup>2</sup>*

*1 RL L L T T T T*

This rule enables modeling of a multilayer structure by sequential application of the composition rule to all the layers in the stack including the media on the left and right side

*2ncos <sup>ˆ</sup> <sup>t</sup>*

*<sup>10</sup> <sup>10</sup>*

, and

Combination of multiple layers is implemented by the use of a composition rule, as shown below for two layers. Under the LTR scheme, each layer is considered a separate entity, separated from adjacent layers by a zero- thickness vacuum layer, such that the wave exits

*1 2*

*R R*

*12 1 2*

where the coefficient *r* is given in Eq. 3 for the two polarization conditions, and

*i i*

*i i*

*cos n sin ˆ* 

*cos n sin ˆ* 

one layer into vacuum and enters the next layer from vacuum.

*nT t ˆ*

*<sup>01</sup> <sup>01</sup>*

*F TE LE <sup>112</sup>* , and (10)

*F RE TE 2 12* . (11)

**S** , and (12)

**S** , (13)

*i i*

*i i*

, (14)

. (15)

. (16)

*2 p i <sup>10</sup> 2 2*

*ncos n sin ˆ ˆ* 

> *2 2 1*

> > *1 2*

L T R

*1 2*

*1 RL*

*R T <sup>R</sup> 1 RL*

*L T <sup>L</sup>*

*1 2*

*1 2*

*1 2*

*n cos n sin ˆ ˆ* 

*2n cos <sup>ˆ</sup> <sup>t</sup>*

fields, *F1* and *F2* , exiting the layer as:

travelling from the medium into vacuum ( **S***<sup>10</sup>* ):

*s i <sup>01</sup> 2 2*

*2cos <sup>t</sup>*

*s i <sup>10</sup> 2 2*

*2ncos <sup>ˆ</sup> <sup>t</sup>*

**LTR X X**

of the multilayer structure.

$$\mathbf{LTR} = \mathbf{S}\_{10} \left( \hat{n}\_1 \right) \oplus \mathbf{X}\_1 \oplus \mathbf{X}\_2 \oplus \dots \oplus \mathbf{X}\_{N-1} \oplus \mathbf{X}\_N \oplus \mathbf{S}\_{01} \left( \hat{n}\_R \right) = \begin{pmatrix} \mathcal{L} \\ \mathcal{T} \\ \mathcal{R} \end{pmatrix} \tag{17}$$

Since the result of the composition is another LTR vector, if the fields *E1* and *E2* incident on the stack are known, then the remaining fields, *F1* and *F2* , can be easily found using Eqs. 10 and 11.

While the LTR construct is useful for capturing the response of a multilayered structure irradiated from the front and the back (left and right in Fig. 3), only front-side illumination is considered here, as that is the most common configuration encountered in MEMS applications. For single-sided illumination the structure is assumed to be illuminated only from the left (i.e., *E 0 <sup>1</sup>* in Fig. 5), and

$$\mathcal{L} = \mathcal{F}\_1 / \mathcal{E}\_2 \text{ (left side reflection)}; \tag{18}$$

$$\mathcal{T}' = \mathcal{F}\_2 / \mathcal{E}\_2 \tag{19}$$

$$\mathcal{R} = 0 \quad \text{(right side reflection)}.\tag{20}$$

The total reflected, transmitted, and absorbed intensities are then:

$$R = \left| \mathcal{L} \right|^2,\tag{21}$$

$$T = \left| \mathcal{T} \right|^2 \frac{\text{Re}\left(n\_R \cos \Theta\_R\right)}{\text{Re}\left(n\_L \cos \Theta\_L\right)}, \text{and} \tag{22}$$

$$A = \mathbf{1} - \mathbf{R} - T \tag{23}$$

If the incident medium on the left is vacuum or air, Eq. 22 can be rewritten fully in terms of the angle of incidence and the substrate index, *sub nˆ* , as

$$T = \left| \mathcal{T} \right|^2 \frac{\text{Re}\left(\sqrt{\hat{n}\_{sub}^2 - \sin^2 \Theta\_i} \right)}{\cos \Theta\_i}. \tag{24}$$

With the fields on the left- and right-most layers defined, the fields entering and exiting each layer can be obtained by recursively applying Eqs. 10 and 11 to each layer. Once these fields are defined, the individual layer absorptances can be easily obtained by noting that each layer is referenced to vacuum and the absorptance is simply the difference between the entering and exiting field magnitudes:

$$A\_i = \left| E\_1^i \right|^2 + \left| E\_2^i \right|^2 - \left| F\_1^i \right|^2 - \left| F\_2^i \right|^2,\tag{25}$$

where the left-most fields of the first layer and the right-most fields for the last layer are obtained from Eqs. 18 and 19.

Optical-Thermal Phenomena in Polycrystalline Silicon MEMS During Laser Irradiation 339

The previous section detailed the response of MEMS optical systems in strictly athermal terms. However, in laser-irradiated MEMS or MEMS exposed to extreme thermal environments the consequences of a changing thermal environment could be significant, especially in regards to the optical response. For simplicity, we shall consider cases where the incident laser energy is responsible for any temperature fluctuation in the irradiated structure, although the same principles are valid for structures subject to bulk external

Laser irradiation of an absorbing structure, such as micromachined polysilicon MEMS, will lead to a corresponding temperature increase. The magnitude of the induced temperature rise will depend on several factors, including the geometry, and thermal and optical properties of the irradiated materials. Because all of the parameters that play a role in determining the energy deposition exhibit some temperature dependence, the laser-induced heating of the structure will be dynamic in nature as the properties change during the

We have already seen the potential effects of different layer thicknesses on the absorptance of an irradiated structure. However, while those fluctuations might arise out of manufacturing variability, the same effect can be observed during the heating of an as-built device. Geometrical and dimensional considerations during the heating result from any temperature-induced displacement and deformation of the MEMS when exposed to elevated temperatures (Knoernschild et al., 2010; Phinney et al., 2006). If the irradiating wavelength is in the optically thick regime for the irradiated material, the dimensional changes do not have a significant effect in the optical response of the structure since the incident energy is fully absorbed within the material. Nevertheless, depending on the structure, small deflections and deformations could have a significant effect on the heat

transfer mechanisms on the heated device (Gallis et al., 2007; Wong & Graham, 2003).

a small change in gap height can lead to as much as a six-fold change in absorptance.

Additionally, due to the phase changes upon reflection, the trends in absorptance repeat for different values of thicknesses and gaps, as seen in Figs 6 and 7. The recurrence period can be estimated from Eq. 9 by finding the thickness increase *d* for which the path length

When the conditions are such that thin film interference becomes important in the optical response, particularly for multilayered systems, the deformation will have a more dramatic effect. Depending on the design and geometry of the irradiated structure, the heating can alter both the thickness of the individual layers (via thermal expansion) and the spacing between them (via thermal expansion, buckling, etc.). Such deformations will produce changes in the absorptance of the laser irradiation, as shown in Fig. 7 for a Poly4 SUMMiT V™ structure similar to the one described by Phinney et al, (Phinney et al., 2006) and shown in Fig. 6a. The cantilevered structure in that reference suffered deflections of over 10 μm during laser irradiation. In Fig. 7, just a variation in the air gap height of ±500 nm suffices to demonstrate the type of deflection-induced changes in absorptance encountered in these tests. Assuming the deflection is caused by the temperature excursion of the structure, then

**3. Optical-thermal coupling in laser-irradiated MEMS** 

heating and laser irradiation (Burns & Bright, 1998).

**3.1 Temperature-induced geometry changes** 

difference is equivalent to an integer multiple of :

heating event.

#### **2.2.2 MEMS**

As discussed in the previous section, the optical response of laser-irradiated materials depends strongly on various parameters. For optically thick materials, the refractive index of the irradiated medium determines the reflectivity of the surface and thus the fraction of the energy that is deposited in the material. When the optical penetration depth is comparable to film thickness, the geometry and composition of the structure becomes as important as refractive index in dictating the optical response. This becomes evident when analyzing the response of sacrificial micromachined MEMS fabricated from polysilicon.

In polysilicon-based MEMS the typical layer thickness is approximately 2 μm, with intermediate gaps of the same order (Carter et al., 2005; MEMS Technologies Department 2008). Such thicknesses are comparable to the penetration depth for both silicon and polysilicon for wavelengths above 550 nm (Jellison Jr & Modine, 1982a, 1982b; Lubberts et al., 1981; Xu & Grigoropoulos, 1993) and therefore the likelihood for thin film interference, as explained above, increases. Indeed, calculations carried out for air-spaced polysilicon structure fabricated from Sandia National Laboratories' SUMMiT-V™ process (MEMS Technologies Department, 2008), as shown in Fig. 6, show that the absorptance of the top-most layer can vary significantly as a function of the layer thickness. The multiple reflections from the various layers in the structure lead to conditions of local maxima and minima for different layer thicknesses. These extrema correspond to thicknesses where the interference between the multiply reflected waves is fully constructive or destructive as will be shown later.

Fig. 6. (a) Schematic of a SUMMiT V™ polysilicon MEMS structure and (b) its optical response at different wavelengths as a function of the thickness of the top-most layer.

The variation in the amplitude and width of the absorptance peaks in this structure is related to the relative reflectivity of the two polysilicon surfaces at the particular wavelength much like a Fabry-Perot cavity (Born & Wolf, 1999) and will ultimately depend on the overall composition of the multilayered structure. For a coupled optical-thermal analysis, the existence of these periodic variations in the absorptance must be taken into account to predict the thermal behavior of laser-irradiated MEMS accurately.
