**1. Introduction**

Incident light with a photon energy ℏ*ω* induces two-photon absorption (TPA) when *Eg* / 2ℏ*ωEg*, where *Eg*is the band gap of the photo-absorption layer of a photodiode (PD). Be‐ cause the absorption coefficient is small, photocurrent generated by TPA is too low to be used in conventional optical signal receivers. However, the nonlinear dependence of the photocurrent on the incident light intensity can be used for optical measurements and opti‐ cal signal processing. It has been used for autocorrelation in pulse shape measurements [1], dispersion measurements [2,3] and optical clock recovery [4]. These applications exploit the dependence of the generated photocurrent on the square of the instantaneous optical inten‐ sity. Measurement systems using TPA in a PD can detect rapidly varying optical phenom‐ ena without using high speed electronics.

This chapter reviews research on TPA and its applications at the optical fiber transmission‐ wavelength. Theory of TPA for semiconductors with diamond and zinc-blende crystal struc‐ tures is reviewed. In contrast to linear absorption for which the photon energy exceeds the band gap, the TPA coefficient depends on the incident lightpolarization. The polarization dependence is described by the nonlinear susceptibility tensor elements.

The polarization dependences of TPA induced by a single optical beam in GaAs- and Si-PDs are compared to evaluate the effect of crystal symmetry. It is found that, in contrast to the GaAs-PD, TPA in the Si-PD is isotropic for linearly polarized light at a wavelength of 1.55 μm. Photocurrents for circularly and elliptically polarized light are also measured. Ratios of the nonlinear susceptibility tensor elements are deduced from these measurements. The dif‐ ferent isotropic properties of GaAs- and Si-PDs are discussed in terms of the crystal and band structures.

Cross-TPA between two optical beams is also studied. The absorption coefficient of cross-TPA strongly depends on the polarizations of the two optical beams. It is shown that the po‐

© 2012 Kagawa; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Kagawa; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

larization dependence of cross-TPA is consistent with the nonlinear susceptibility tensor elements obtained from the self-TPA analysis.

*E* =*Ep p*

wher*e E <sup>p</sup>* and *E <sup>e</sup>* are the electric field strengths and*p*

<sup>4</sup> *<sup>ε</sup>*0(*Ep*

are elements of *p*

3(| *p* **^** <sup>⋅</sup> *<sup>p</sup>*

> | *p* **^** <sup>⋅</sup> *<sup>e</sup>*

zation along the polarization vector *p*

*Pp* (3)= <sup>1</sup>

larization can be written as [7]

<sup>4</sup> *<sup>ε</sup>*0{*Ep*

2(*χxxyy*

and *e <sup>i</sup>*

*Pp* (3)= <sup>1</sup>

+ 2*EpEe*

where*p <sup>i</sup>*

where

process.

the two beams. For circular or elliptical polarization, *p*

<sup>3</sup> ∑ *i*, *j*,*k*,*l pi* \* *pj* \**pk pl*

**^**and*<sup>e</sup>*

**^** <sup>|</sup> <sup>2</sup> <sup>⋅</sup>*χxxyy*

*σ* =

effects, respectively. Terms proportional to the inner product of *p*

*d Ip*

**^**<sup>|</sup> <sup>2</sup> <sup>+</sup> *<sup>χ</sup>xyyx*(1 <sup>+</sup> <sup>|</sup> *<sup>p</sup>*

**^** <sup>+</sup> *Ee<sup>e</sup>*

phase difference between the electric field oscillations along two axes. The nonlinear polari‐

*χijkl* + 2*EpEe*

*\** and *e <sup>i</sup> \**

tively. Because there are only three nonzero independent tensor elements, the nonlinear po‐

<sup>+</sup> <sup>2</sup>*χxyyx* <sup>+</sup> *σχxxxx*∑

**^**\* ⋅ *e* **^**| <sup>2</sup>

*χxxxx* −*χxxyy* −2*χxyxy χxxxx*

The first and second terms are polarization induced by the self- and cross-electric field

rotation of axes and are isotropic. In contrast, terms that are proportional to *σ* vary on the rotation of the axes. Thus, *σ* shows the anisotropy of the third-order nonlinear optical

where *I <sup>p</sup>* and *I <sup>e</sup>* are optical intensity densities of the two beams. The absorption coefficient is

Two optical beams propagate in the crystal under the effect of self- and cross-TPA.

*dz* <sup>=</sup> <sup>−</sup>*βppIp*

proportional to the imaginary part of the nonlinear polarization given in Eq. (4).

**^** is given by

**^**, and*<sup>p</sup> <sup>i</sup>*

**^**and*<sup>e</sup>*

<sup>2</sup> ∑ *i*, *j*,*k*,*l pi* \* *ej* \**pk el χijkl*

*i*

) <sup>+</sup> *σχxxxx*∑


*i*

**^**and *<sup>e</sup>*

**^** (2)

**^**are the polarization unit vectors of

are their complex conjugate, respec‐

<sup>|</sup> *pi* <sup>|</sup> <sup>2</sup> <sup>|</sup>*ei* <sup>|</sup> 2)} (4)

**^**and *<sup>e</sup>*

<sup>2</sup> <sup>−</sup>*βpeIpIe* (6)

(5)

**^**are invariant for

**^** are complex to express the

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 5

) (3)

Cross-TPA can be applied to polarization measurements. Photocurrents generated in the Si-PD by cross-TPA between asignal light under test and a reference light are used to detect the polarization. The light under test is arbitrarily polarized and its Jones vector can be deter‐ mined by photocurrents generated by cross-TPA. This measurement method can detect the instantaneous polarization when the reference light temporally overlaps with the light un‐ der test. Because the time division is limited only by the pulse width of the reference light, it is possible to detect rapid variationsin the polarization. This method can measure not only the linear polarization direction but also the elliptical polarization. Applications to measure‐ ment of the output optical pulse from an optical fiber with birefringence and a semiconduc‐ tor optical amplifier are demonstrated.

#### **2. TPA in semiconductors with diamond and zinc-blende crystals**

#### **2.1. Polarization dependence**

TPA is a third-order nonlinear optical process. Third order nonlinear polarization is induced by the optical electric field according to

$$P\_i^{(\heartsuit)}(\omega\_{i\prime}, \mathbf{k}\_i) = \frac{1}{4} \varepsilon\_0 \sum\_{j,k,l} \chi\_{ijkl} E\_j(\omega\_{j\prime}, \mathbf{k}\_j) E\_k(\omega\_{k\prime}, \mathbf{k}\_k) E\_l(\omega\_{l\prime}, \mathbf{k}\_l) \tag{1}$$

where*ε <sup>0</sup>* is the permittivity of free space, *χ* is the third-order tensor, *ω* is the optical angular frequency, k is the optical wavenumber vector, *E* is the optical electric field [5]. The suffixes *i, j, k,* and *l* denote the orthogonal directions. The relationships between the optical angular frequencies and the wavenumber vectors are determined by energy and momentum conser‐ vation, respectively.

Although the third-order nonlinear susceptibility tensor contains 34 elements, the number of non-zero independent elements is limited by the crystal symmetry and the properties of the incident light. It is apparent that relations *χ xxxx = χ yyyy = χ zzzz*and *χ xxyy = χ xxzz = χ yyzz*, etc. hold for a cubic crystal. Elements like *χ xxxy*and*χ xxyz*will be zero for crystals with 180° rotational symmetry about a crystal axis. For degenerate TPA in which one or two parallel optical beams with the same optical frequency propagate,*ω<sup>i</sup>* = −*ω<sup>j</sup>* =*ω<sup>k</sup>* =*ω<sup>l</sup>* and *χ xyxy=χ xyyx* hold. There are thus only three independent elements, *χ xxxx*, *χ xxyy*, and *χ xyyx,* for degenerate TPA in crystal classes of *m3m* (Si) and 4 ¯3*m* (GaAs) [5,6].

We consider cross- and self-TPA between two optical beams. The electric field is the sum of the electric fields of thetwo incident optical beams.

#### Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 5

$$E = E\_p \stackrel{\Lambda}{p} + E\_e \stackrel{\Lambda}{e} \tag{2}$$

wher*e E <sup>p</sup>* and *E <sup>e</sup>* are the electric field strengths and*p* **^**and*<sup>e</sup>* **^**are the polarization unit vectors of the two beams. For circular or elliptical polarization, *p* **^**and *<sup>e</sup>* **^** are complex to express the phase difference between the electric field oscillations along two axes. The nonlinear polari‐ zation along the polarization vector *p* **^** is given by

$$P\_p(\text{3)} = \frac{1}{4} \varepsilon\_0 \{ E\_p \: ^3 \sum\_{i,j,k,l} p\_i \not^\* p\_j \not^\* p\_k \not^\* p\_l \chi\_{ij\text{kl}} + 2E\_p E\_e \: ^2 \sum\_{i,j,k,l} p\_i \not^\* e\_j \not^\* p\_k e\_l \chi\_{ij\text{kl}} \} \tag{3}$$

where*p <sup>i</sup>* and *e <sup>i</sup>* are elements of *p* **^**and*<sup>e</sup>* **^**, and*<sup>p</sup> <sup>i</sup> \** and *e <sup>i</sup> \** are their complex conjugate, respec‐ tively. Because there are only three nonzero independent tensor elements, the nonlinear po‐ larization can be written as [7]

$$\begin{split} P\_p(\mathbf{3}) &= \frac{1}{4} \varepsilon\_0 \Big| E\_p \mathbf{3} \{ |\bigwedge \mathbf{p} \cdot \bigwedge \mathbf{p}^\*|^2 \cdot \chi\_{\mathbf{x} \mathbf{y} \mathbf{y}} + 2 \chi\_{\mathbf{x} \mathbf{y} \mathbf{x} \mathbf{x}} + \sigma \chi\_{\mathbf{x} \mathbf{x} \mathbf{x} \mathbf{x}} \sum\_i |\ p\_i| \, \mathbf{4} \} \\ &+ 2 E\_p E\_e 2 \langle \chi\_{\mathbf{x} \mathbf{y} \mathbf{y}} \mid \bigwedge \mathbf{\hat{e}} \mid \bigvee \mathbf{2} + \chi\_{\mathbf{x} \mathbf{y} \mathbf{x}} \{ \mathbf{1} + \bigvee \mathbf{\hat{p}} \cdot \mathbf{\hat{e}} \mid \bigvee \mathbf{2} \} + \sigma \chi\_{\mathbf{x} \mathbf{x} \mathbf{x}} \sum\_i |\ p\_i| \, 2 \mid \varrho\_i \mid \mathbf{2} \} \end{split} \tag{4}$$

where

larization dependence of cross-TPA is consistent with the nonlinear susceptibility tensor

Cross-TPA can be applied to polarization measurements. Photocurrents generated in the Si-PD by cross-TPA between asignal light under test and a reference light are used to detect the polarization. The light under test is arbitrarily polarized and its Jones vector can be deter‐ mined by photocurrents generated by cross-TPA. This measurement method can detect the instantaneous polarization when the reference light temporally overlaps with the light un‐ der test. Because the time division is limited only by the pulse width of the reference light, it is possible to detect rapid variationsin the polarization. This method can measure not only the linear polarization direction but also the elliptical polarization. Applications to measure‐ ment of the output optical pulse from an optical fiber with birefringence and a semiconduc‐

**2. TPA in semiconductors with diamond and zinc-blende crystals**

*χijklEj* (*ωj* , *k <sup>j</sup>*

beams with the same optical frequency propagate,*ω<sup>i</sup>* = −*ω<sup>j</sup>* =*ω<sup>k</sup>* =*ω<sup>l</sup>*

TPA is a third-order nonlinear optical process. Third order nonlinear polarization is induced

where*ε <sup>0</sup>* is the permittivity of free space, *χ* is the third-order tensor, *ω* is the optical angular frequency, k is the optical wavenumber vector, *E* is the optical electric field [5]. The suffixes *i, j, k,* and *l* denote the orthogonal directions. The relationships between the optical angular frequencies and the wavenumber vectors are determined by energy and momentum conser‐

Although the third-order nonlinear susceptibility tensor contains 34 elements, the number of non-zero independent elements is limited by the crystal symmetry and the properties of the incident light. It is apparent that relations *χ xxxx = χ yyyy = χ zzzz*and *χ xxyy = χ xxzz = χ yyzz*, etc. hold for a cubic crystal. Elements like *χ xxxy*and*χ xxyz*will be zero for crystals with 180° rotational symmetry about a crystal axis. For degenerate TPA in which one or two parallel optical

There are thus only three independent elements, *χ xxxx*, *χ xxyy*, and *χ xyyx,* for degenerate TPA

We consider cross- and self-TPA between two optical beams. The electric field is the sum of

¯3*m* (GaAs) [5,6].

)*Ek* (*ω<sup>k</sup>* , *k<sup>k</sup>* )*El*

(*ωl* , *k<sup>l</sup>* ) (1)

and *χ xyxy=χ xyyx* hold.

elements obtained from the self-TPA analysis.

4 Photodiodes - From Fundamentals to Applications

tor optical amplifier are demonstrated.

by the optical electric field according to

*Pi* (3)(*ω<sup>i</sup>* , *k<sup>i</sup>* )= <sup>1</sup> <sup>4</sup> *<sup>ε</sup>*0∑ *j*,*k*,*l*

in crystal classes of *m3m* (Si) and 4

the electric fields of thetwo incident optical beams.

vation, respectively.

**2.1. Polarization dependence**

$$\sigma = \frac{\chi\_{\text{xxxx}} - \chi\_{\text{xxyy}} - 2\chi\_{\text{xyxy}}}{\chi\_{\text{xxxx}}} \tag{5}$$

The first and second terms are polarization induced by the self- and cross-electric field effects, respectively. Terms proportional to the inner product of *p* **^**and *<sup>e</sup>* **^**are invariant for rotation of axes and are isotropic. In contrast, terms that are proportional to *σ* vary on the rotation of the axes. Thus, *σ* shows the anisotropy of the third-order nonlinear optical process.

Two optical beams propagate in the crystal under the effect of self- and cross-TPA.

$$\frac{dI\_p}{dz} = -\beta\_{pp} I\_p \, ^2 - \beta\_{pe} I\_p I\_e \, \tag{6}$$

where *I <sup>p</sup>* and *I <sup>e</sup>* are optical intensity densities of the two beams. The absorption coefficient is proportional to the imaginary part of the nonlinear polarization given in Eq. (4).

$$\beta\_{pp} = \frac{\omega}{2n^2 c^2 \varepsilon\_0} (\chi'' \llcorner \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \mid^2 + 2\chi'' \llcorner \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \llcorner \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \llcorner \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \llcorner \hat{\mathbf{p}} \cdot \hat{\mathbf{p}} \tag{7}$$

the light intensity of 107 W/cm2

**3. Experimental setup**

than continuous wave light.

is 2 μm, and the internal efficiency is 1.

final pulse width was compressed toabout 1 ps.

cal beam is also amplified by an EDFA.

independent of the polarization.

tion, that is, cross TPA.

is illuminated on a spot with adiameter of 10 μm. We as‐

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 7

sume that the pulse width is 1 ps, the repetition rate is 100 MHz, absorption layer thickness

Because the photocurrent of PD is proportional to the square of the instantaneous light pow‐ er density, it is necessary to concentrate the optical power into a narrow spatial region and a short time period. Thus, a short pulsed light beam is more suitable for TPAmeasurements‐

Figure 1 shows the experimental setup. A gain-switched laser diode (LD) generated optical pulses with a wavelength of 1.55 μm, a pulse width of 50 ps and a repetition rate of 100 MHz. Light pulse from the gain-switched LD exhibit large wavelength chirping. The pulse was compressed to about 10 ps by an optical fiber with positive wavelength dispersion. Its peak power was then amplified using an Er-doped fiber amplifier (EDFA) to further com‐ press the pulse width through the nonlinear soliton effect in a normal-dispersion fiber. The

To measure cross-TPA between two optical beams, a second gain switched LD with a wave‐ length of 1.55 μm was prepared. Noise due to interference between the two beams does not affect the measurement because the optical frequency difference between the two beams is greater than the bandwidth of the measurement system. Pulse with a repetition rate of 100MHz are completely synchronized with those of the first optical beam. The second opti‐

Both the two beams were made linearly polarized by polarization controllers. After they were launched into free space, they passed through polarizing beam splitters to ensure that they were completely linearly polarized. Half-wave or quarter-wave plates were inserted if it is necessary to control the polarization of the beams. The two beams were spatially over‐ lapped by a polarization-independent beam splitter and they were focused on a PD. It was confirmed that the polarization did not change on reflection at the polarization-independent beam splitter by monitoring the polarization before and after reflection. An optical power meter was placed at the location of the PD and it was used to check if the optical power was

When two optical beams are illuminated on a PD, photocurrents due to self-TPA and cross-TPAare simultaneously generated. It is necessary to detect only the photocurrent generated by the cross-TPA. Optical pulse streams were mechanically chopped at frequencies of 1.0 and 1.4 kHz. Electrical pulsesthat had been synchronized with mechanical choppers were fed into a mixer circuit that generated a sumfrequency of 2.4 kHz. These generated electrical pulses with the sum frequency were used as the reference signal for the lock-in amplifier. Thus, the lock-in amplifier detected only the photocurrent generated by two-beam absorp‐

and

$$\boldsymbol{\beta}\_{pe} = \frac{\omega}{m^2 \boldsymbol{\sigma}^2 \boldsymbol{\varepsilon}\_0} (\boldsymbol{\chi''}\_{\ \boldsymbol{\varepsilon}\_{\ \ \ \mu y}} \mid \boldsymbol{\hat{p}} \cdot \boldsymbol{\hat{e}} \mid \boldsymbol{\hat{e}} + \boldsymbol{\chi''}\_{\ \ \mu y} (\boldsymbol{1} + \boldsymbol{\beta} \mid \boldsymbol{\hat{p}}^\* \cdot \boldsymbol{\hat{e}} \mid \boldsymbol{\hat{e}}) + \boldsymbol{\sigma''} \boldsymbol{\chi''}\_{\ \ \mu \nu} \sum\_{i} \mid p\_i \mid \boldsymbol{2} \mid \boldsymbol{e}\_i \mid \boldsymbol{2} \rangle \tag{8}$$

where *n* is the refractive index, and *c* is the speed of light. *χ* ″ *xxxx*etc. are imaginary parts of the nonlinear susceptibility tensor elements. *σ* ″ is the anisotropy parameter for imaginary parts of the nonlinear susceptibility tensor.

*<sup>σ</sup>* ″ <sup>=</sup> *<sup>χ</sup>* ″ *xxxx* −*χ* ″ *xxyy* −2*χ* ″ *xyyx χ* ″ *xxxx* (9)

#### **2.2. Estimate of photocurrent induced by TPA in PDs**

Commercially available PDs are usually designed to be used for photon energies greater than the band gap of the photoabsorption layer. As the absorption coefficient is about 105 cm-1, absorption layer is several micrometers thick. On the other hand, the absorption coeffi‐ cient is much smaller for TPA. If we consider only self-TPA, Eq. (6) is solved as

$$I\_p(z) = \frac{I\_0}{\beta\_{pp} I\_0 z + 1} \approx I\_0 (1 - \beta\_{pp} I\_0 z) \tag{10}$$

where *I <sup>0</sup>* is the initial light intensity density. Using a typical value of 10-18 m2 /V2 for the imaginary parts of the nonlinear susceptibility tensor elements [7], the TPA coefficient is es‐ timated to be about 6×10−<sup>11</sup> m/W. When the incident light density is 107 W/cm2 , *β pp I <sup>0</sup>*is esti‐ mated to be6×10−<sup>6</sup> μm-1. Because only a very small fraction of the incident light is absorbed in PD with a photo-absorption layer that is several micrometers thick, the induced photocur‐ rent is proportional to the absorption coefficient *β*.

When optical pulses with an intensity density *I <sup>0</sup>* and pulse width *T <sup>p</sup>*are irradiated at a repe‐ tition rate of *R*, the induced photocurrent will be

$$J = \eta \beta\_{pp} I\_0 2S \text{ d'} T\_p R \frac{q}{\hbar \omega} \tag{11}$$

where *η* is the internal efficiency of the PD, *d* is the absorption layer thickness, and *S* is the area of the incident beam. The photocurrent is estimated to be about 10-8 A assuming that the light intensity of 107 W/cm2 is illuminated on a spot with adiameter of 10 μm. We as‐ sume that the pulse width is 1 ps, the repetition rate is 100 MHz, absorption layer thickness is 2 μm, and the internal efficiency is 1.

## **3. Experimental setup**

*<sup>β</sup>pp* <sup>=</sup> *<sup>ω</sup>* 2*n* <sup>2</sup> *c* 2 *ε*0 (*χ* ″ *xxyy* | *p* **^** <sup>⋅</sup> *<sup>p</sup>*

6 Photodiodes - From Fundamentals to Applications

and

*<sup>β</sup>pe* <sup>=</sup> *<sup>ω</sup> n* 2 *c* 2 *ε*0 (*χ* ″ *xxyy* | *p* **^** <sup>⋅</sup> *<sup>e</sup>*

mated to be6×10−<sup>6</sup>

**^** <sup>|</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>χ</sup>* ″

*xyyx*(1 + | *p* **^**\* ⋅ *e* **^**| <sup>2</sup> ) + *σ* ″ *χ* ″ *xxxx*∑ *i*

*xxxx* −*χ* ″

cient is much smaller for TPA. If we consider only self-TPA, Eq. (6) is solved as

where *I <sup>0</sup>* is the initial light intensity density. Using a typical value of 10-18 m2

imaginary parts of the nonlinear susceptibility tensor elements [7], the TPA coefficient is es‐

in PD with a photo-absorption layer that is several micrometers thick, the induced photocur‐

When optical pulses with an intensity density *I <sup>0</sup>* and pulse width *T <sup>p</sup>*are irradiated at a repe‐

<sup>2</sup>*<sup>S</sup> <sup>d</sup> TpR <sup>q</sup>*

where *η* is the internal efficiency of the PD, *d* is the absorption layer thickness, and *S* is the area of the incident beam. The photocurrent is estimated to be about 10-8 A assuming that

μm-1. Because only a very small fraction of the incident light is absorbed

*χ* ″ *xxxx*

Commercially available PDs are usually designed to be used for photon energies greater than the band gap of the photoabsorption layer. As the absorption coefficient is about 105 cm-1, absorption layer is several micrometers thick. On the other hand, the absorption coeffi‐

*xxyy* −2*χ* ″

*xyyx*

**^**<sup>|</sup> <sup>2</sup> <sup>+</sup> *<sup>χ</sup>* ″

where *n* is the refractive index, and *c* is the speed of light. *χ* ″

*<sup>σ</sup>* ″ <sup>=</sup> *<sup>χ</sup>* ″

*Ip*(*z*)= *<sup>I</sup>*<sup>0</sup>

timated to be about 6×10−<sup>11</sup> m/W. When the incident light density is 107

*J* =*ηβppI*<sup>0</sup>

the nonlinear susceptibility tensor elements. *σ* ″

**2.2. Estimate of photocurrent induced by TPA in PDs**

rent is proportional to the absorption coefficient *β*.

tition rate of *R*, the induced photocurrent will be

parts of the nonlinear susceptibility tensor.

*xyyx* + *σ* ″

*χ* ″ *xxxx*∑ *i*



*xxxx*etc. are imaginary parts of

(9)

/V2

, *β pp I <sup>0</sup>*is esti‐

W/cm2

<sup>ℏ</sup>*<sup>ω</sup>* (11)

for the

is the anisotropy parameter for imaginary

*<sup>β</sup>ppI*0*<sup>z</sup>* <sup>+</sup> <sup>1</sup> <sup>≈</sup> *<sup>I</sup>*0(1−*βppI*0*z*) (10)

Because the photocurrent of PD is proportional to the square of the instantaneous light pow‐ er density, it is necessary to concentrate the optical power into a narrow spatial region and a short time period. Thus, a short pulsed light beam is more suitable for TPAmeasurements‐ than continuous wave light.

Figure 1 shows the experimental setup. A gain-switched laser diode (LD) generated optical pulses with a wavelength of 1.55 μm, a pulse width of 50 ps and a repetition rate of 100 MHz. Light pulse from the gain-switched LD exhibit large wavelength chirping. The pulse was compressed to about 10 ps by an optical fiber with positive wavelength dispersion. Its peak power was then amplified using an Er-doped fiber amplifier (EDFA) to further com‐ press the pulse width through the nonlinear soliton effect in a normal-dispersion fiber. The final pulse width was compressed toabout 1 ps.

To measure cross-TPA between two optical beams, a second gain switched LD with a wave‐ length of 1.55 μm was prepared. Noise due to interference between the two beams does not affect the measurement because the optical frequency difference between the two beams is greater than the bandwidth of the measurement system. Pulse with a repetition rate of 100MHz are completely synchronized with those of the first optical beam. The second opti‐ cal beam is also amplified by an EDFA.

Both the two beams were made linearly polarized by polarization controllers. After they were launched into free space, they passed through polarizing beam splitters to ensure that they were completely linearly polarized. Half-wave or quarter-wave plates were inserted if it is necessary to control the polarization of the beams. The two beams were spatially over‐ lapped by a polarization-independent beam splitter and they were focused on a PD. It was confirmed that the polarization did not change on reflection at the polarization-independent beam splitter by monitoring the polarization before and after reflection. An optical power meter was placed at the location of the PD and it was used to check if the optical power was independent of the polarization.

When two optical beams are illuminated on a PD, photocurrents due to self-TPA and cross-TPAare simultaneously generated. It is necessary to detect only the photocurrent generated by the cross-TPA. Optical pulse streams were mechanically chopped at frequencies of 1.0 and 1.4 kHz. Electrical pulsesthat had been synchronized with mechanical choppers were fed into a mixer circuit that generated a sumfrequency of 2.4 kHz. These generated electrical pulses with the sum frequency were used as the reference signal for the lock-in amplifier. Thus, the lock-in amplifier detected only the photocurrent generated by two-beam absorp‐ tion, that is, cross TPA.

**Figure 1.** Measurement setup (LD: laser diode; NDF: normal dispersion fiber; ADF: abnormal dispersion fiber; PBS: po‐ larization beam splitter; PIBS: polarization independent beam splitter). The inset shows the rotation of the wave plate. Light from the PBS is linearly polarized along the *x* axis,which is parallel tothe [1] axis of the PD.

#### **4. Pulse width measurement by cross-TPA**

Cross-TPA was used to measure the pulse width generated by the pulse compression proc‐ ess described in the previous section. After the compressed optical pulse was divided into two branches by an optical fiber beam splitter, the timing between them was controlled by a variable delay line. They were then irradiated on the Si-PD. The two beams were made or‐ thogonally linearly polarized to suppress noise due to interference. The photocurrent gener‐ ated by cross-TPA between the divided two optical beams is

$$J(\tau) = \beta \left| h\left(t\right)h\left(t - \tau\right)dt \right. \tag{12}$$

sorbed is quite small, the generated photocurrent is directly proportional to the absorption coefficient*β pp*as shown by Eq. (10). The photocurrents generated in Si- and GaAs-PDs were

compared to discuss the polarization dependence of TPA in Si and GaAscrystals[8].

**Figure 2.** Self-correlation trace of the compressed pulse measured by TPA of Si-PD.

to the *x*-axis. The polarization of the transformed light is expressedby

cos*θ* −sin*θ* sin*<sup>θ</sup>* cos*<sup>θ</sup>* )(

*p* **^** =( *px py* ) =(

delayed by *ϕ* relative to that along the *X*-axis.

In the self-TPA measurement, only one optical beam is illuminated on a PD. The optical beam with a pulse width of 0.9 ps in the measurement setup described in section 3was used in the self-TPAmeasurement. The x- and y- axes are fixed in the laboratory frame. We con‐ sider the case when light that is linearly polarized alongthe *x*-axis is transformed by a halfor quarter-wave plate. The principal axis of the wave plate is rotated at an angle of *θ* relative

1 0

where *ϕ*=π and π/2 for half- and quarter-wave plates, respectively. The inset of Fig. 1 shows the definition of the rotation angle. The principal axes of the quarter-wave plate are repre‐ sented by the *X-* and *Y*-axes. The phase of the polarization component along the *Y*-axis is

The anisotropy of self-TPA for linearly polarized light was measured for Si- and GaAs-PDs. The crystal axis [001] is made parallel to the x-axis. The linear polarization is rotated by a half-wave plate (i.e., *ϕ*=π in Eq. (13)). When the X-axis is tilted by an angle of *θ* relative to the *x*-axis, the polarization direction of the output light from the half-wave plate is tilted by

<sup>0</sup> *<sup>e</sup>iφ*)( cos*<sup>θ</sup>* sin*<sup>θ</sup>* −sin*θ* cos*θ*

)( 1 0

) (13)

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 9

where *h(t)* is the pulse shape, and *τ* is the time delay between the two pulses. The pulse width can be estimated by this self-correlation trace.

Figure 2 shows the self-correlation trace of the compressed optical pulse. The photocurrent due to the cross-TPA is generated only when the two optical pulses temporally overlap on the PD. It disappears when the time delay is larger than the pulse width. The self-correlation trace has a full-width at half-maximum (FWHM) of 1.3 ps. The FWHM of the pulse is esti‐ mated to be about 0.9 ps assuming a Gaussian pulse shape.

#### **5. Polarization dependence of self-TPA in Si- and GaAs-PDs**

Measuring the photocurrent generated in PDs is the easiest way to study the polarization dependence of self-TPA coefficient. Because the fraction of the incident photons that are ab‐ sorbed is quite small, the generated photocurrent is directly proportional to the absorption coefficient*β pp*as shown by Eq. (10). The photocurrents generated in Si- and GaAs-PDs were compared to discuss the polarization dependence of TPA in Si and GaAscrystals[8].

**Figure 2.** Self-correlation trace of the compressed pulse measured by TPA of Si-PD.

**Figure 1.** Measurement setup (LD: laser diode; NDF: normal dispersion fiber; ADF: abnormal dispersion fiber; PBS: po‐ larization beam splitter; PIBS: polarization independent beam splitter). The inset shows the rotation of the wave plate.

Cross-TPA was used to measure the pulse width generated by the pulse compression proc‐ ess described in the previous section. After the compressed optical pulse was divided into two branches by an optical fiber beam splitter, the timing between them was controlled by a variable delay line. They were then irradiated on the Si-PD. The two beams were made or‐ thogonally linearly polarized to suppress noise due to interference. The photocurrent gener‐

where *h(t)* is the pulse shape, and *τ* is the time delay between the two pulses. The pulse

Figure 2 shows the self-correlation trace of the compressed optical pulse. The photocurrent due to the cross-TPA is generated only when the two optical pulses temporally overlap on the PD. It disappears when the time delay is larger than the pulse width. The self-correlation trace has a full-width at half-maximum (FWHM) of 1.3 ps. The FWHM of the pulse is esti‐

Measuring the photocurrent generated in PDs is the easiest way to study the polarization dependence of self-TPA coefficient. Because the fraction of the incident photons that are ab‐

*J*(*τ*)=*β∫h* (*t*)*h* (*t* −*τ*)*dt* (12)

Light from the PBS is linearly polarized along the *x* axis,which is parallel tothe [1] axis of the PD.

**4. Pulse width measurement by cross-TPA**

8 Photodiodes - From Fundamentals to Applications

ated by cross-TPA between the divided two optical beams is

width can be estimated by this self-correlation trace.

mated to be about 0.9 ps assuming a Gaussian pulse shape.

**5. Polarization dependence of self-TPA in Si- and GaAs-PDs**

In the self-TPA measurement, only one optical beam is illuminated on a PD. The optical beam with a pulse width of 0.9 ps in the measurement setup described in section 3was used in the self-TPAmeasurement. The x- and y- axes are fixed in the laboratory frame. We con‐ sider the case when light that is linearly polarized alongthe *x*-axis is transformed by a halfor quarter-wave plate. The principal axis of the wave plate is rotated at an angle of *θ* relative to the *x*-axis. The polarization of the transformed light is expressedby

$$\begin{aligned} \stackrel{\Delta}{p} = \begin{pmatrix} p\_x \\ p\_y \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & e^{i\varphi} \end{pmatrix} \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{aligned} \tag{13}$$

where *ϕ*=π and π/2 for half- and quarter-wave plates, respectively. The inset of Fig. 1 shows the definition of the rotation angle. The principal axes of the quarter-wave plate are repre‐ sented by the *X-* and *Y*-axes. The phase of the polarization component along the *Y*-axis is delayed by *ϕ* relative to that along the *X*-axis.

The anisotropy of self-TPA for linearly polarized light was measured for Si- and GaAs-PDs. The crystal axis [001] is made parallel to the x-axis. The linear polarization is rotated by a half-wave plate (i.e., *ϕ*=π in Eq. (13)). When the X-axis is tilted by an angle of *θ* relative to the *x*-axis, the polarization direction of the output light from the half-wave plate is tilted by 2*θ.* Thus, the polarization is parallel to the [001] and [011] directions when the rotation angle of the half-wave plate is *θ* = 0 and 22.5° , respectively. Using Eq. (7), the anisotropy parame‐ ter *σ"* defined by Eq. (9) can be written as

*L* 011

45° , respectively. The difference in the self-TPA coefficients for linear and circular polariza‐

be 0.1 and 0.39 from the measured photocurrents for linearly and circularly polarized light

**Figure 4.** Photocurrent obtained when elliptically polarized light is incident on (a) GaAs and (b) Si PDs. Linearly polar‐ ized light along the [001] axis is transformed by a quarter-wave plate rotated at an angle of θ.The solid lines indicate

and dichroism parameters. Table 1 lists the obtained ratios for the nonlinear susceptibility

From Eqs. (7), (9), (13) and (16), the dependence of the TPA coefficient on the quarter-wave

*χ*"*xxyy χ*"*xxxx*

1

*xxxx* can be estimated from measured anisotropic

cos8*θ* + 8*δ*cos4*θ*) (17)

*<sup>σ</sup>* ″ (18)

*xyyx* / *χ* ''

*xxxx*(16−*σ* ″ −8*δ* + *σ* ″

2*θ* = −

The absorption coefficient for this elliptically polarized light is greater than*β <sup>L</sup>* 001 .

cos<sup>2</sup>

*χ*"*xxxx* + *χ*"*xxyy* −2*χ*"*xyyx* 2*χ*"*xxxx*

*C* is the TPA coefficient for circularly polarized light. This parameter is estimated to

(16)

11

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491

tion isexpressed by the dichroism parameter

*<sup>L</sup>* 001 −*βpp*

*<sup>L</sup>* <sup>001</sup> <sup>=</sup>

*βpp*

the results calculated using Eq. (17) and the parameters in Table 1.

*<sup>β</sup>pp* <sup>∝</sup> <sup>1</sup>

This self-TPA coefficient is maximized when

*xxxx* and *χ* ''

<sup>16</sup> *<sup>χ</sup>* ″

*xxyy* / *χ* ''

tensor elements for GaAs and Si.

plate rotation angle *θ* is given by

*C*

*δ* = *βpp*

in the GaAs- and Si-PD, respectively.

where*βpp*

The ratios *χ* ''

*<sup>L</sup>* 001 −*βpp*

*βpp*

**Figure 3.** Photo currents generated when linearly polarized lightirradiated on (a) GaAs PD and (b) Si PD. The linear polarization direction is rotated using a half-wave plate. The horizontal axis is the tilt angle of the half wave plate.The solid lines in (a) and (b) show the values calculated using Eq. (15) with σ" = –0.45 and 0, respectively.

where*βpp <sup>L</sup>* 001 and *βpp L* 011 are the TPA coefficients for linearly polarized light polarized along the [001] and [011] directions, respectively.This parameter can be experimentallydeter‐ mined by the ratio of photocurrents.

Figures 3(a) and (b) respectively show the photocurrents generated in GaAs- and Si-PD sas a function of the rotation angle of the half-wave plate. For the GaAs-PD, the photocurrent var‐ ies with the polarization direction indicating that the TPA is anisotropic. The anisotropy pa‐ rameter *σ''* is estimated to be –0.45. From Eqs. (7), (9) and (13), the dependence of the TPA probability on the rotation angle *θ*of the half-wave plate can be written as

$$
\beta\_{pp}{}^{L} \propto \frac{1}{4} \chi^{''}{}\_{\text{....}} \langle 4 - \sigma^{''} + \sigma^{''} \cos \Theta \rangle \tag{15}
$$

The solid line in Fig. 3 (a) shows the value calculated using Eq. (15) and *σ''*= –0.45. In con‐ trast, the Si-PD exhibits negligibly small dependence on the polarization direction and the TPA coefficient is almost isotropic; |*σ* ″ |is estimated to be less than 0.04.

Figure 4(a) and (b) respectively shows the dependence of the photocurrents generated in the GaAs- and Si-PDs on the rotation angle of a quarter-wave plate (*ϕ=π/2*in Eq. (13)). The inci‐ dent light is linearly polarized along the [001] direction and circularly polarized at *θ* = 0 and 45° , respectively. The difference in the self-TPA coefficients for linear and circular polariza‐ tion isexpressed by the dichroism parameter

$$\delta = \frac{\beta\_{pp}\,^L\,\mathsf{[001]} - \beta\_{pp}\,^C}{\beta\_{pp}\,^L\,\mathsf{[001]}} = \frac{\chi\,^{\mathsf{"}}\_{\mathsf{xxxx}} + \chi\,^{\mathsf{"}}\_{\mathsf{xxyy}} - 2\chi\,^{\mathsf{"}}\_{\mathsf{xyyx}}}{2\chi\,^{\mathsf{"}}\_{\mathsf{xxxx}}}\tag{16}$$

where*βpp C* is the TPA coefficient for circularly polarized light. This parameter is estimated to be 0.1 and 0.39 from the measured photocurrents for linearly and circularly polarized light in the GaAs- and Si-PD, respectively.

**Figure 4.** Photocurrent obtained when elliptically polarized light is incident on (a) GaAs and (b) Si PDs. Linearly polar‐ ized light along the [001] axis is transformed by a quarter-wave plate rotated at an angle of θ.The solid lines indicate the results calculated using Eq. (17) and the parameters in Table 1.

The ratios *χ* '' *xxyy* / *χ* '' *xxxx* and *χ* '' *xyyx* / *χ* '' *xxxx* can be estimated from measured anisotropic and dichroism parameters. Table 1 lists the obtained ratios for the nonlinear susceptibility tensor elements for GaAs and Si.

From Eqs. (7), (9), (13) and (16), the dependence of the TPA coefficient on the quarter-wave plate rotation angle *θ* is given by

$$
\beta\_{pp} \propto \frac{1}{16} \chi''\_{\ \mu\nu} (16 - \sigma'' - 8\delta + \sigma'' \cos 8\theta + 8\delta \cos 4\theta) \tag{17}
$$

This self-TPA coefficient is maximized when

2*θ.* Thus, the polarization is parallel to the [001] and [011] directions when the rotation angle of the half-wave plate is *θ* = 0 and 22.5° , respectively. Using Eq. (7), the anisotropy parame‐

**Figure 3.** Photo currents generated when linearly polarized lightirradiated on (a) GaAs PD and (b) Si PD. The linear polarization direction is rotated using a half-wave plate. The horizontal axis is the tilt angle of the half wave plate.The

along the [001] and [011] directions, respectively.This parameter can be experimentallydeter‐

Figures 3(a) and (b) respectively show the photocurrents generated in GaAs- and Si-PD sas a function of the rotation angle of the half-wave plate. For the GaAs-PD, the photocurrent var‐ ies with the polarization direction indicating that the TPA is anisotropic. The anisotropy pa‐ rameter *σ''* is estimated to be –0.45. From Eqs. (7), (9) and (13), the dependence of the TPA

*xxxx*(4−*σ* ″ + *σ* ″

The solid line in Fig. 3 (a) shows the value calculated using Eq. (15) and *σ''*= –0.45. In con‐ trast, the Si-PD exhibits negligibly small dependence on the polarization direction and the

Figure 4(a) and (b) respectively shows the dependence of the photocurrents generated in the GaAs- and Si-PDs on the rotation angle of a quarter-wave plate (*ϕ=π/2*in Eq. (13)). The inci‐ dent light is linearly polarized along the [001] direction and circularly polarized at *θ* = 0 and

*L* 011 are the TPA coefficients for linearly polarized light polarized

cos8*θ*) (15)

solid lines in (a) and (b) show the values calculated using Eq. (15) with σ" = –0.45 and 0, respectively.

probability on the rotation angle *θ*of the half-wave plate can be written as

TPA coefficient is almost isotropic; |*σ* ″ |is estimated to be less than 0.04.

*βpp*

*<sup>L</sup>* <sup>∝</sup> <sup>1</sup> <sup>4</sup> *<sup>χ</sup>* ″ *L* 011

*<sup>L</sup>* <sup>001</sup> (14)

*<sup>L</sup>* 001 −*βpp*

*βpp*

ter *σ"* defined by Eq. (9) can be written as

10 Photodiodes - From Fundamentals to Applications

where*βpp*

*<sup>L</sup>* 001 and *βpp*

mined by the ratio of photocurrents.

*σ* ″ =2

*βpp*

$$\cos^2 2\theta = -\frac{\boldsymbol{\chi}^{\text{"}}\!\!\!\!\!\!\!\!\/\!\!\/ (\text{x}\!\!\!\/)}{\boldsymbol{\chi}^{\text{"}\!\!\/\!\/\!\/}\!\!\/\!\/\!\!\/ (\text{x}\!\!\/)}\tag{18}$$

The absorption coefficient for this elliptically polarized light is greater than*β <sup>L</sup>* 001 .

The solid lines in Figs. 4(a) and (b) show the results calculated using Eq. (17) for GaAs and Si, respectively. The photocurrent shown in Fig. 4(a) reaches a maximum at *θ* = 15° , which indicates that Eq. (18) holds at this angle. The factor *χ*"*xxyy* / (*<sup>σ</sup>* ″ *χ*"*xxxx* )in Eq. (18) is estimat‐ ed to be –0.75 for GaAs. This value is consistent with the values of *σ* ″ and*χ* ″ *xxyy* / *χ* ″ *xxxx* in Ta‐ ble 1, indicating that thepolarization dependence of the GaAs-PD is consistent with the analysis based on the nonlinear susceptibility.

and s-like states. The polarization directions that induce the first and second transitions must be different from each other. For example, transitions | *pz*(*Γ*15*v*) → | *px*(*Γ*15*c*) and | *px*(*Γ*15*c*) → |*s*(*Γ*1*c*) are induced by dipole moments polarized along the*y*- and *x*-axes, re‐ spectively.| *pz*(*Γ*15*v*) , | *px*(*Γ*15*c*) ,and |*s*(*Γ*1*c*) are wave functions of each band [11]. This proc‐

parameter σ*''* to be non-zero [7]. The matrix element of the optical dipole moment between

On the other hand, Si has the indirect transition type band structure. Figure 5(b) schemati‐ cally shows the band structure and the irreducible representation of this space group [11,12]. A photon energy of 0.8 eV is too small to induce a direct TPA transition without phonon absorption or emission at any point in the first Brillouin zone of Si. The final sate of the TPA transition is Δ1, which has the minimum energy of the conduction band. Many complicated transitionsequences that include optical and phonon transitions exist to reach the final point

When both optical transitions occur at Γ point, an electron is scattered to Δ1in the conduction band. However, two step optical transitions in Si are quite different from that in GaAs. Si crystal has a point group of *O <sup>h</sup>*(*m3m*) that has space inversion symmetry and the wavefunc‐ tion is an eigenstate of the parity at the Γ point. The matrix elements of the dipole moment between the conduction bands of Γ 15, Γ2', and Γ12' vanish because they all have the same pari‐ ty. The only possible virtual final state of the two-step optical transition sequence in Γ point is Γ1 in the higher conduction band. AsΓ1 has a much greater energy than Γ25' andΔ1, the tran‐

**Figure 5.** a) Schematic band structure and allowed–allowed transition in GaAs. (b) Schematic band structure of Si

*xxyy*, but it contributes to *χ* ″

*xyyx*causing the anisotropy

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 13

*xxxx*and*χ* ″

Γ15*<sup>v</sup>* and Γ 15*<sup>c</sup>* is non-zero because *T <sup>d</sup>* lacks space inversion symmetry.

ess does not contribute to *χ* ″

sition probability is thought quite small.

Δ1 for electron.

On the other hand, the photocurrent generated in the Si-PD is maximized when the an‐ gle is 0 and the incident light is linearly polarized, which contrasts the situation for the GaAs PD. Because the anisotropy parameter is small, Eq. (18) does not hold at any rota‐ tion angle *θ*.

#### **6. Discussion of self-TPA polarization dependence**

The polarization dependence of self-TPA is strongly dependent on the crystal symmetry and the band structure. Hutchings and Wherettcalculated nonlinear susceptibility tensor ele‐ ments based on kp perturbation [9]. The ratios listed in Table 1 are consistent with their re‐ sults. Murayamaand Nakayama[10] have performed *ab initio* calculations.Their calculated values for the ratios*χ*"*xxyy* / *χ*"*xxxx* and*χ*"*xyyx* / *χ*"*xxxx* depend on the photon energy. The val‐ ues of ratios shown in Table 1 are very similar to those calculated for a photon energy of 1 eV. The small discrepancy between the photon energies is probably due to the parameters used in the calculation.


**Table 1.** Parameters obtained from the polarization dependence of the photocurrents of GaAs and Si PDs at a wavelength of 1.55 μm.

It is very reasonable that GaAs and Si were observed to have quite different anisotropies be‐ cause of their different crystal symmetries and band structures. As GaAs has a direct transi‐ tion type band structure, an optical transition occurs at around the Γ point. The anisotropy for GaAs is due to the allowed–allowed transition [7,9] (see Fig. 5(a)), which is the two-step optical transition of*Γ*15*<sup>v</sup>* →*Γ*15*<sup>c</sup>* →*Γ*1*c*. Γ15*<sup>v</sup>*, Γ1*<sup>c</sup>*, and Γ15*<sup>c</sup>*are irreducible representations of the point group *T <sup>d</sup>*(4 ¯3*m*) of the GaAs crystal for the highest valence band, the lowest conduction band, and the higher conduction band at the Γ point, respectively [11]. The first transition *Γ*15*<sup>v</sup>* →*Γ*15*<sup>c</sup>* occurs between p-like states, the second transition *Γ*15*<sup>c</sup>* →*Γ*1*c* occurs between p-like

and s-like states. The polarization directions that induce the first and second transitions must be different from each other. For example, transitions | *pz*(*Γ*15*v*) → | *px*(*Γ*15*c*) and | *px*(*Γ*15*c*) → |*s*(*Γ*1*c*) are induced by dipole moments polarized along the*y*- and *x*-axes, re‐ spectively.| *pz*(*Γ*15*v*) , | *px*(*Γ*15*c*) ,and |*s*(*Γ*1*c*) are wave functions of each band [11]. This proc‐ ess does not contribute to *χ* ″ *xxxx*and*χ* ″ *xxyy*, but it contributes to *χ* ″ *xyyx*causing the anisotropy parameter σ*''* to be non-zero [7]. The matrix element of the optical dipole moment between Γ15*<sup>v</sup>* and Γ 15*<sup>c</sup>* is non-zero because *T <sup>d</sup>* lacks space inversion symmetry.

The solid lines in Figs. 4(a) and (b) show the results calculated using Eq. (17) for GaAs and Si, respectively. The photocurrent shown in Fig. 4(a) reaches a maximum at *θ* = 15° , which

ble 1, indicating that thepolarization dependence of the GaAs-PD is consistent with the

On the other hand, the photocurrent generated in the Si-PD is maximized when the an‐ gle is 0 and the incident light is linearly polarized, which contrasts the situation for the GaAs PD. Because the anisotropy parameter is small, Eq. (18) does not hold at any rota‐

The polarization dependence of self-TPA is strongly dependent on the crystal symmetry and the band structure. Hutchings and Wherettcalculated nonlinear susceptibility tensor ele‐ ments based on kp perturbation [9]. The ratios listed in Table 1 are consistent with their re‐ sults. Murayamaand Nakayama[10] have performed *ab initio* calculations.Their calculated values for the ratios*χ*"*xxyy* / *χ*"*xxxx* and*χ*"*xyyx* / *χ*"*xxxx* depend on the photon energy. The val‐ ues of ratios shown in Table 1 are very similar to those calculated for a photon energy of 1 eV. The small discrepancy between the photon energies is probably due to the parameters

> Anisotropy parameter σ" -0.45 |σ ″ | <0.04 Dichroism parameter δ 0.1 0.39 χ"*xxyy* / χ"*xxxx* 0.34 0.39 χ"*xyyx* / χ"*xxxx* 0.56 0.31

**Table 1.** Parameters obtained from the polarization dependence of the photocurrents of GaAs and Si PDs at a

It is very reasonable that GaAs and Si were observed to have quite different anisotropies be‐ cause of their different crystal symmetries and band structures. As GaAs has a direct transi‐ tion type band structure, an optical transition occurs at around the Γ point. The anisotropy for GaAs is due to the allowed–allowed transition [7,9] (see Fig. 5(a)), which is the two-step optical transition of*Γ*15*<sup>v</sup>* →*Γ*15*<sup>c</sup>* →*Γ*1*c*. Γ15*<sup>v</sup>*, Γ1*<sup>c</sup>*, and Γ15*<sup>c</sup>*are irreducible representations of the

band, and the higher conduction band at the Γ point, respectively [11]. The first transition *Γ*15*<sup>v</sup>* →*Γ*15*<sup>c</sup>* occurs between p-like states, the second transition *Γ*15*<sup>c</sup>* →*Γ*1*c* occurs between p-like

¯3*m*) of the GaAs crystal for the highest valence band, the lowest conduction

GaAs Si

*χ*"*xxxx* )in Eq. (18) is estimat‐

*xxyy* / *χ* ″

*xxxx* in Ta‐

and*χ* ″

indicates that Eq. (18) holds at this angle. The factor *χ*"*xxyy* / (*<sup>σ</sup>* ″

**6. Discussion of self-TPA polarization dependence**

analysis based on the nonlinear susceptibility.

12 Photodiodes - From Fundamentals to Applications

tion angle *θ*.

used in the calculation.

wavelength of 1.55 μm.

point group *T <sup>d</sup>*(4

ed to be –0.75 for GaAs. This value is consistent with the values of *σ* ″

On the other hand, Si has the indirect transition type band structure. Figure 5(b) schemati‐ cally shows the band structure and the irreducible representation of this space group [11,12]. A photon energy of 0.8 eV is too small to induce a direct TPA transition without phonon absorption or emission at any point in the first Brillouin zone of Si. The final sate of the TPA transition is Δ1, which has the minimum energy of the conduction band. Many complicated transitionsequences that include optical and phonon transitions exist to reach the final point Δ1 for electron.

When both optical transitions occur at Γ point, an electron is scattered to Δ1in the conduction band. However, two step optical transitions in Si are quite different from that in GaAs. Si crystal has a point group of *O <sup>h</sup>*(*m3m*) that has space inversion symmetry and the wavefunc‐ tion is an eigenstate of the parity at the Γ point. The matrix elements of the dipole moment between the conduction bands of Γ 15, Γ2', and Γ12' vanish because they all have the same pari‐ ty. The only possible virtual final state of the two-step optical transition sequence in Γ point is Γ1 in the higher conduction band. AsΓ1 has a much greater energy than Γ25' andΔ1, the tran‐ sition probability is thought quite small.

**Figure 5.** a) Schematic band structure and allowed–allowed transition in GaAs. (b) Schematic band structure of Si

When a phonon process occurs after the first optical transition, the polarization effect of the first optical transition on the intermediate state of TPA can be destroyed by the phonon process. The anisotropy is thus considered to be reduced by this process.

normalized using the minimum photocurrent. The photocurrent is strongly dependent on the orientation of the two linear polarization axes and has a maximum and minimum values when the polarization axes of the two optical pulses are parallel and perpendicu‐

*xyyx*) cos4*θ* +

The solid line in Fig.6 shows the result calculated using Eq. (19) and the parameters in Table 1.

**^**) =2 <sup>+</sup> *<sup>χ</sup>* ″

Using the parameters in Table 1 which were obtained from the self-TPA of Si, this ratio is

**Figure 7.** Photocurrent due to cross-TPA between linear polarized and elliptical polarized lights. Solid line is the calcu‐

*xxyy χ* ″ *xyyx*

1 <sup>2</sup> (*<sup>χ</sup>* ″

*xxyy* + 3*χ* ″

*xyyx*) (19)

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491

**^**are parallel and or‐

(20)

15

**^**and *<sup>e</sup>*

lar, respectively.

Equation (8) can be written as

lated results using parameters in Table 1.

*<sup>β</sup>pe* <sup>∝</sup> <sup>1</sup>

<sup>2</sup> (*<sup>χ</sup>* ″

*xxyy* + *χ* ″

The absorption coefficient hasa maximum and minimum when*p*

thogonal, respectively. The ratio of the maximum to minimum values is

*βpe*(*p* **^** / / *<sup>e</sup>* **^**)

*βpe*(*p* **^** <sup>⊥</sup>*<sup>e</sup>*

3.26. This value is consistent with the measured cross-TPA shown in Fig 6..

#### **7. Cross-TPA in Si-APD**

As shown in the previous section, TPA in Si is isotropic. Thus, TPA in Si-PD is simpler than that in GaAs-PD. In addition, a Si avalanche photodiode (APD) with the multiplication gain is commercially available. Consequently, we concentrate on cross-TPA in Si-APD.

Cross-TPA depends on the relationship between the polarization vectors of the two beams. We measure three cases: when both beams are linearly polarized, when one optical beam is linearly polarized and the other is varied between linear, elliptical, and circular polarization by a quarter-wave plate, and when one beam is circularly polarized and the other is varied between linear, elliptical, and circular polarization [13].

**Figure 6.** Photocurrent due to cross- TPA between two linearly polarized beams. Solid line is the calculated results using parameters in Table 1.

Figure 6 shows the photocurrent when both beams are linearly polarized. The horizon‐ tal axis of the figure is the rotation angle of the half- wave plate. The photocurrent was normalized using the minimum photocurrent. The photocurrent is strongly dependent on the orientation of the two linear polarization axes and has a maximum and minimum values when the polarization axes of the two optical pulses are parallel and perpendicu‐ lar, respectively.

Equation (8) can be written as

When a phonon process occurs after the first optical transition, the polarization effect of the first optical transition on the intermediate state of TPA can be destroyed by the phonon

As shown in the previous section, TPA in Si is isotropic. Thus, TPA in Si-PD is simpler than that in GaAs-PD. In addition, a Si avalanche photodiode (APD) with the multiplication gain

Cross-TPA depends on the relationship between the polarization vectors of the two beams. We measure three cases: when both beams are linearly polarized, when one optical beam is linearly polarized and the other is varied between linear, elliptical, and circular polarization by a quarter-wave plate, and when one beam is circularly polarized and the other is varied

**Figure 6.** Photocurrent due to cross- TPA between two linearly polarized beams. Solid line is the calculated results

Figure 6 shows the photocurrent when both beams are linearly polarized. The horizon‐ tal axis of the figure is the rotation angle of the half- wave plate. The photocurrent was

is commercially available. Consequently, we concentrate on cross-TPA in Si-APD.

between linear, elliptical, and circular polarization [13].

process. The anisotropy is thus considered to be reduced by this process.

**7. Cross-TPA in Si-APD**

14 Photodiodes - From Fundamentals to Applications

using parameters in Table 1.

$$\beta\_{p\epsilon} \propto \frac{1}{2} (\chi''\_{\ \!\!\!\/ \!\!\/ \!\/ \!\!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/) \cos 4\Theta + \frac{1}{2} (\chi''\_{\ \!\!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/ \!\/) \tag{19}$$

The solid line in Fig.6 shows the result calculated using Eq. (19) and the parameters in Table 1. The absorption coefficient hasa maximum and minimum when*p* **^**and *<sup>e</sup>* **^**are parallel and or‐ thogonal, respectively. The ratio of the maximum to minimum values is

$$\frac{\beta\_{pe}(\stackrel{\frown}{\mathbf{p}} \mid / \stackrel{\frown}{\mathbf{e}})}{\beta\_{pe}(\mathbf{p} \perp \mathbf{e})} = 2 + \frac{\chi''}{\chi''}\_{\text{array}}\tag{20}$$

Using the parameters in Table 1 which were obtained from the self-TPA of Si, this ratio is 3.26. This value is consistent with the measured cross-TPA shown in Fig 6..

**Figure 7.** Photocurrent due to cross-TPA between linear polarized and elliptical polarized lights. Solid line is the calcu‐ lated results using parameters in Table 1.

Figure 7 shows the photocurrent when one beam (*e* **^**) was linearly polarized and the polari‐ zation of the other beam (*p* **^**) was varied using a quarter-wave plate. The horizontal axis is the rotation angle of the quarter-wave plate. The polarization of the second beam varied be‐ tween linear, elliptical, and circular in this case. The solid line shows the calculated value using the parameters in Table 1. The photocurrent had maximum and minimum values when the second beam was linearly and circularly polarized, respectively. The ratios are theoretically written as

$$\frac{\beta\_{p\text{\tiny p\text{\tiny \text{\textdegree}}}}(\stackrel{\text{\textdegree}}{\text{p\text{\textdegree}}}/\stackrel{\text{\textdegree}}{\text{\textdegree}})}{\beta\_{pe}(\text{p\text{\textdegree}};circular)} = \frac{4 + 2\chi\stackrel{\text{\textdegree}}{\chi\text{\textdegree}}\_{xyy}/\chi\stackrel{\text{\textdegree}}{\text{\textdegree}}\_{xyy}}{3 + \chi\stackrel{\text{\textdegree}}{\text{\textdegree}}\_{xyy}/\chi\stackrel{\text{\textdegree}}{\text{\textdegree}}\_{xyy}}\tag{21}$$

using Eq (8). This ratio is calculated to be 1.53 from the parameters in Table 1, and is consis‐ tent with the measurement.

Figure 8 shows the photocurrent when one beam was circularly polarized while the polari‐ zation of the other beam was varied using a quarter-wave plate between linear, elliptical, and circular polarization. The unit vectors for circular polarization are *σ* **^** <sup>+</sup> <sup>=</sup> <sup>1</sup> 2 (*x* **^** <sup>+</sup> *<sup>i</sup> <sup>y</sup>* **^**)and *σ* **^** <sup>−</sup> <sup>=</sup> <sup>1</sup> 2 (*x* **^** <sup>−</sup>*<sup>i</sup> <sup>y</sup>* **^**). An arbitrary polarization vector *<sup>p</sup>* **^** can be written as a linear combination of

these unit vectors.

$$
\stackrel{\wedge}{p} = p\_+ \stackrel{\wedge}{\sigma}\_+ + p\_- \stackrel{\wedge}{\sigma}\_- \tag{22}
$$

**Figure 8.** Photocurrent due to cross TPA when one optical beam is circularly polarized.

er optical beam because values of *χ* ″

principle, they are not exactly equal for Si.

analysis based on the nonlinear susceptibility tensor.

**8. Polarization measurement by cross-TPA**

The solid line in Fig. 8 is the calculated results using Eq. (24). When one optical beam is cir‐ cularly polarized, cross-TPA exhibits very weak dependences on the polarization of the oth‐

The calculated values shown by the solid lines in Figs. 6, 7, and 8 are obtained by nonlinear susceptibility tensor elements that were deduced from the polarization dependence of self-TPA. There is relatively good agreement with the measured cross-TPA. This demonstrates that the polarization dependences of self- and cross-TPA of Si are consistent with theoretical

The polarization dependence of the cross-TPA in Si-APD can be used to measure the polari‐ zation. In this method, a Si-APD is irradiated by the arbitrarily polarized light to be meas‐ ured (signal light) and a linearly polarized referencebeam. The photocurrents generated by cross-TPA between the signal light and the linearly polarized reference light are measured. Polarization direction of the reference beam was varied in four ways. Polarization of the arbitrarily polarized light can be determined from the four photocurrents of the APD [14]. Several applications require the ability to detect rapid variations in the polarization of an optical signal. In all conventional polarization measurement methods, the temporal resolu‐

*xyyx* are very close to each other: however, in

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 17

*xxyy* and *χ* ″

When*e* **^**=*<sup>σ</sup>* **^** <sup>−</sup>, Eq (8) can be written as

*<sup>β</sup>pe* <sup>∝</sup>*<sup>χ</sup>* ″ *xxyy* <sup>|</sup> *<sup>p</sup>*<sup>+</sup> <sup>|</sup> <sup>2</sup> <sup>+</sup> *<sup>χ</sup>* ″ *xyyx* (1 + | *p*<sup>−</sup> | 2). (23)

We used the relations *σ* **^** <sup>+</sup> ⋅*σ* **^** <sup>−</sup> =1and*σ* **^** <sup>+</sup> ⋅*σ* **^** <sup>+</sup> =*σ* **^** <sup>−</sup> ⋅*σ* **^** <sup>−</sup> =0. *β pe* is independent of *p* **^** when *χ*"*xxyy* =*χ*"*xyyx* because| *p*<sup>+</sup> | <sup>2</sup> + | *p*<sup>−</sup> | <sup>2</sup> =1.The photocurrent depends on the polarization, as Fig. 8 shows, but this dependence is relatively small.

The dependence of the absorption coefficient on the rotation angle is

*<sup>β</sup>pe* <sup>∝</sup>(*<sup>χ</sup>* ″ *xxyy* −*χ* ″ *xyyx* )sin2*θ* + *χ* ″ *xxyy* + 3*χ* ″ *xyyx* (24)

The ratio of the maximum tominimum values is

$$\frac{\beta\_{\rm pre}(\stackrel{\scriptstyle \alpha}{\mathfrak{p}}=\stackrel{\scriptstyle \alpha}{\widetilde{\sigma}}\_{\rightarrow})}{\beta\_{\rm pre}(\mathfrak{p}=\sigma\_{-})} = \frac{1}{2} (1 + \frac{\chi''}{\chi''}\_{\
u \mu \underline{\chi}}) \tag{25}$$

It is estimated to be 1.13 using the parameters in Table 1.

**Figure 8.** Photocurrent due to cross TPA when one optical beam is circularly polarized.

Figure 7 shows the photocurrent when one beam (*e*

*βpe*(*p* **^** / / *<sup>e</sup>* **^**)

**^**). An arbitrary polarization vector *<sup>p</sup>*

<sup>−</sup>, Eq (8) can be written as

*<sup>β</sup>pe* <sup>∝</sup>*<sup>χ</sup>* ″

**^** <sup>+</sup> ⋅*σ* **^**

Fig. 8 shows, but this dependence is relatively small.

*<sup>β</sup>pe* <sup>∝</sup>(*<sup>χ</sup>* ″

It is estimated to be 1.13 using the parameters in Table 1.

The ratio of the maximum tominimum values is

**^**; *circular*) <sup>=</sup> <sup>4</sup> <sup>+</sup> <sup>2</sup>*<sup>χ</sup>* ″

and circular polarization. The unit vectors for circular polarization are *σ*

*p* **^** <sup>=</sup> *<sup>p</sup>*+*<sup>σ</sup>* **^** <sup>+</sup> + *p*−*σ* **^**

<sup>−</sup> =1and*σ* **^** <sup>+</sup> ⋅*σ* **^** <sup>+</sup> =*σ* **^** <sup>−</sup> ⋅*σ* **^**

The dependence of the absorption coefficient on the rotation angle is

*xxyy* −*χ* ″ *xyyx*

*βpe*(*p* **^** <sup>=</sup>*<sup>σ</sup>* **^** +)

*βpe*(*p* **^** <sup>=</sup>*<sup>σ</sup>* **^** −) = 1

*xxyy* <sup>|</sup> *<sup>p</sup>*<sup>+</sup> <sup>|</sup> <sup>2</sup> <sup>+</sup> *<sup>χ</sup>* ″

*χ*"*xxyy* =*χ*"*xyyx* because| *p*<sup>+</sup> | <sup>2</sup> + | *p*<sup>−</sup> | <sup>2</sup> =1.The photocurrent depends on the polarization, as

)sin2*θ* + *χ* ″

<sup>2</sup> (1 <sup>+</sup> *<sup>χ</sup>* ″

*xxyy χ* ″ *xyyx*

*xxyy* + 3*χ* ″

*βpe*(*p*

zation of the other beam (*p*

16 Photodiodes - From Fundamentals to Applications

theoretically written as

tent with the measurement.

*σ* **^** <sup>−</sup> <sup>=</sup> <sup>1</sup> 2 (*x* **^** <sup>−</sup>*<sup>i</sup> <sup>y</sup>*

When*e* **^**=*<sup>σ</sup>* **^**

these unit vectors.

We used the relations *σ*

**^**) was linearly polarized and the polari‐

**^** <sup>+</sup> <sup>=</sup> <sup>1</sup> 2 (*x* **^** <sup>+</sup> *<sup>i</sup> <sup>y</sup>* **^**)and

**^** can be written as a linear combination of

<sup>−</sup> (22)

*xyyx* (1 + | *p*<sup>−</sup> | 2). (23)

<sup>−</sup> =0. *β pe* is independent of *p*

*xyyx* (24)

) (25)

(21)

**^** when

**^**) was varied using a quarter-wave plate. The horizontal axis is

the rotation angle of the quarter-wave plate. The polarization of the second beam varied be‐ tween linear, elliptical, and circular in this case. The solid line shows the calculated value using the parameters in Table 1. The photocurrent had maximum and minimum values when the second beam was linearly and circularly polarized, respectively. The ratios are

3 + *χ* ″

using Eq (8). This ratio is calculated to be 1.53 from the parameters in Table 1, and is consis‐

Figure 8 shows the photocurrent when one beam was circularly polarized while the polari‐ zation of the other beam was varied using a quarter-wave plate between linear, elliptical,

*xxyy* / *χ* ″ *xyyx*

*xxyy* / *χ* ″ *xyyx*

> The solid line in Fig. 8 is the calculated results using Eq. (24). When one optical beam is cir‐ cularly polarized, cross-TPA exhibits very weak dependences on the polarization of the oth‐ er optical beam because values of *χ* ″ *xxyy* and *χ* ″ *xyyx* are very close to each other: however, in principle, they are not exactly equal for Si.

> The calculated values shown by the solid lines in Figs. 6, 7, and 8 are obtained by nonlinear susceptibility tensor elements that were deduced from the polarization dependence of self-TPA. There is relatively good agreement with the measured cross-TPA. This demonstrates that the polarization dependences of self- and cross-TPA of Si are consistent with theoretical analysis based on the nonlinear susceptibility tensor.

#### **8. Polarization measurement by cross-TPA**

The polarization dependence of the cross-TPA in Si-APD can be used to measure the polari‐ zation. In this method, a Si-APD is irradiated by the arbitrarily polarized light to be meas‐ ured (signal light) and a linearly polarized referencebeam. The photocurrents generated by cross-TPA between the signal light and the linearly polarized reference light are measured. Polarization direction of the reference beam was varied in four ways. Polarization of the arbitrarily polarized light can be determined from the four photocurrents of the APD [14].

Several applications require the ability to detect rapid variations in the polarization of an optical signal. In all conventional polarization measurement methods, the temporal resolu‐ tion is limited by the response speed of the PD and/or electrical devices employed. Measure‐ ments based on TPA can be employed to measure rapidly varying polarization without the need to use high-speed electronics. Since the reference beam can be short pulses, the tempo‐ ral polarization of a short-time period can be measured using this method. The temporal resolution is limited by only the pulse width of the reference light.

#### **8.1. Principle of polarization measurement**

The polarization of thelight to be measured can be generally described by the Jones vector

$$\stackrel{\frown}{\mathcal{P}} = \begin{pmatrix} a\_{\times} \\ a\_{\times} e^{i\alpha} \end{pmatrix} \tag{26}$$

*ax*

that *a <sup>x</sup>*, *a <sup>y</sup>*, and *α*are quite insensitive to *x*.

short reference light pulse.

coefficients as

and

<sup>2</sup> <sup>=</sup> *<sup>x</sup>* <sup>+</sup> <sup>2</sup>−*β*<sup>2</sup> / *<sup>β</sup>*<sup>1</sup>

*ay* <sup>2</sup> =1−*ax*

*x* ≡*χ* ″

*β*5(*e* **^**<sup>=</sup> <sup>1</sup> 2 ( 1 *i*

*β*6(*e* **^**<sup>=</sup> <sup>1</sup> 2 ( 1 −*i*

respectively. The sign of sin*α* is positive when *β <sup>6</sup> >β <sup>5</sup>*and vice versa.

**8.2. Measurement of stationary polarization**

cos*<sup>α</sup>* <sup>=</sup> (*<sup>x</sup>* <sup>+</sup> 3)(1−*β*<sup>4</sup> / *<sup>β</sup>*3)

*xxyy* / *χ* ″

The polarization can be determined from the ratios of the photocurrent *β <sup>2</sup> /β <sup>1</sup>* and *β <sup>4</sup> /β <sup>3</sup>*. *x* is the ratio of the two independent non-diagonal elements of the third-order nonlinear sus‐ ceptibility tensor; it was estimated to be 1.3 using values in Table 1. However, it was found

Let us consider the case when the pulse width of the reference light is much shorter than that of the light to be measured. The measured photocurrent produced by APD due to cross-TPA samples the polarization of the light being measured during the reference light pulse. It is thus possible to measure polarization as a function of time by varying the timing of the

One problem with this measurement method is that the sign of sin*α* cannot be deter‐ mined as long as the reference light is linearly polarized. When it is important to deter‐ mine the sign ofsin*α*, it is necessary to compare the two photocurrents generated by cross-TPA for right and left circularly polarized localized lights; let us define these absorption

Polarization measurements were performed using the same setup as that shown in Fig. 1. The reference light is linearly polarized and its polarization direction *γ* was varied in four ways by a half- wave plate. On the other hand, for the signal light, linear polarization was

(*<sup>x</sup>* <sup>+</sup> 1)(1 <sup>+</sup> *<sup>β</sup>*<sup>2</sup> / *<sup>β</sup>*1) (32)

<sup>2</sup>*axay*(*<sup>x</sup>* <sup>+</sup> 1)(1 <sup>+</sup> *<sup>β</sup>*<sup>4</sup> / *<sup>β</sup>*3) (34)

<sup>2</sup> (33)

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 19

*xyyx* (35)

) ) (36)

) ) (37)

where *a <sup>x</sup>* and *a <sup>y</sup>* are respectively the amplitudes of the components in the *x*- and *y*-direc‐ tions, and *α* is their phase difference. These three parameters are generally functions of time.The referencelight is linearly polarized and its Jones vector is given by

$$\begin{array}{l}\stackrel{\text{\tiny \text{\\_}}}{e} = \begin{pmatrix} \cos \gamma\\ \sin \gamma \end{pmatrix} \\\\ \text{\\_} \tag{27}$$

where *γ* expresses the polarization direction. The polarization of the reference lightis inde‐ pendent of time.

Let us consider four different polarization orientations of the linearly polarized reference light beam, namely, *γ <sup>1</sup> =0*, *γ <sup>2</sup> =π/2, γ <sup>3</sup> =π/4*, and *γ <sup>4</sup> =π/4*.In the experiment, four photocur‐ rents due to the cross TPA between the signal light and these four linearly polarized referen‐ cebeams are measured by a lock-in amplifier. From Eq. (8), the cross-TPA probability, which is proportional to the measured photocurrent, is given by

$$
\beta\_1(\gamma = 0) \propto a\_\chi^2(\chi^{''''} \ll \chi^{''''}) + \chi^{''''} \llcorner \tag{28}
$$

$$
\beta\_2(\gamma = \pi \mid \mathfrak{D}) \propto a\_y^2 (\chi^{''}\llcorner + \chi^{''}\llcorner) + \chi^{''}\llcorner \tag{29}
$$

$$\begin{aligned} \beta\_3(\boldsymbol{\chi} = \boldsymbol{\pi} \mid \mathbf{4}) & \approx a\_{\mathbf{x}} a\_{\mathbf{y}} \cos \alpha \left( \boldsymbol{\chi} \nurarrow\_{\mathbf{x} \text{yy}} + \boldsymbol{\chi} \nurarrow\_{\mathbf{x} \text{yy}} \right) \\ + (\boldsymbol{\chi} \nurarrow\_{\mathbf{x} \text{yy}} + \mathbf{3} \nurarrow\_{\mathbf{x} \text{yy}}) / 2 \end{aligned} \tag{30}$$

$$\begin{aligned} \beta\_4(\boldsymbol{\gamma} = -\boldsymbol{\pi} \mid \boldsymbol{4}) & \propto -a\_{\boldsymbol{x}} a\_{\boldsymbol{y}} \cos \alpha \left( \boldsymbol{\chi''}\_{\ \boldsymbol{x} \boldsymbol{y} \boldsymbol{y}} + \boldsymbol{\chi''}\_{\ \boldsymbol{x} \boldsymbol{y} \boldsymbol{x}} \right) \\ + (\boldsymbol{\chi''}\_{\ \boldsymbol{x} \boldsymbol{y} \boldsymbol{y}} + \boldsymbol{3} \boldsymbol{\chi''}\_{\ \boldsymbol{x} \boldsymbol{y} \boldsymbol{y}}) / 2 \end{aligned} \tag{31}$$

Thus, the parameters of the measured light are given by

#### Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 19

$$a\_{\chi}2 = \frac{\chi + 2 - \beta\_2/\beta\_1}{(\chi + 1)(1 + \beta\_2/\beta\_1)}\tag{32}$$

$$a\_y \, ^2 = 1 - a\_x \, ^2$$

$$\cos \alpha = \frac{(\alpha + 3)(1 - \beta\_4/\beta\_3)}{2a\_x a\_y (\alpha + 1)(1 + \beta\_4/\beta\_3)} \tag{34}$$

$$\boldsymbol{\omega} \equiv \boldsymbol{\chi} \begin{array}{c} \text{\tiny \tiny \kern-5.526614pt}{ $\boldsymbol{\chi}$ } \text{\tiny \raisebox{0.526614pt}{ $\boldsymbol{\chi}$ }} \text{\tiny \raisebox{0.526614pt}{ $\boldsymbol{\chi}$ }} \text{\tiny \raisebox{0.526614pt}{ $\boldsymbol{\chi}$ }} \end{array} \tag{35}$$

The polarization can be determined from the ratios of the photocurrent *β <sup>2</sup> /β <sup>1</sup>* and *β <sup>4</sup> /β <sup>3</sup>*. *x* is the ratio of the two independent non-diagonal elements of the third-order nonlinear sus‐ ceptibility tensor; it was estimated to be 1.3 using values in Table 1. However, it was found that *a <sup>x</sup>*, *a <sup>y</sup>*, and *α*are quite insensitive to *x*.

Let us consider the case when the pulse width of the reference light is much shorter than that of the light to be measured. The measured photocurrent produced by APD due to cross-TPA samples the polarization of the light being measured during the reference light pulse. It is thus possible to measure polarization as a function of time by varying the timing of the short reference light pulse.

One problem with this measurement method is that the sign of sin*α* cannot be deter‐ mined as long as the reference light is linearly polarized. When it is important to deter‐ mine the sign ofsin*α*, it is necessary to compare the two photocurrents generated by cross-TPA for right and left circularly polarized localized lights; let us define these absorption coefficients as

$$
\beta\_5(\stackrel{\frown}{e} = \frac{1}{\sqrt{2}} \binom{1}{i}) \tag{36}
$$

and

tion is limited by the response speed of the PD and/or electrical devices employed. Measure‐ ments based on TPA can be employed to measure rapidly varying polarization without the need to use high-speed electronics. Since the reference beam can be short pulses, the tempo‐ ral polarization of a short-time period can be measured using this method. The temporal

The polarization of thelight to be measured can be generally described by the Jones vector

where *a <sup>x</sup>* and *a <sup>y</sup>* are respectively the amplitudes of the components in the *x*- and *y*-direc‐ tions, and *α* is their phase difference. These three parameters are generally functions of

where *γ* expresses the polarization direction. The polarization of the reference lightis inde‐

Let us consider four different polarization orientations of the linearly polarized reference light beam, namely, *γ <sup>1</sup> =0*, *γ <sup>2</sup> =π/2, γ <sup>3</sup> =π/4*, and *γ <sup>4</sup> =π/4*.In the experiment, four photocur‐ rents due to the cross TPA between the signal light and these four linearly polarized referen‐ cebeams are measured by a lock-in amplifier. From Eq. (8), the cross-TPA probability, which

*xxyy* + *χ* ″

*xxyy* + *χ* ″

*xyyx*) + *χ* ″

*xyyx*) + *χ* ″

*xxyy* + *χ* ″

*xxyy* + *χ* ″

*xyyx*)

*xyyx*)

*aye<sup>i</sup>α*) (26)

sin*γ*) (27)

*xyyx* (28)

*xyyx* (29)

(30)

(31)

*p* **^** =( *ax*

time.The referencelight is linearly polarized and its Jones vector is given by

*e* **^**=( cos*γ*

2(*χ* ″

*<sup>β</sup>*3(*<sup>γ</sup>* <sup>=</sup>*<sup>π</sup>* / 4)∝*axay*cos*<sup>α</sup>* (*<sup>χ</sup>* ″

*<sup>β</sup>*4(*<sup>γ</sup>* <sup>=</sup> <sup>−</sup>*<sup>π</sup>* / 4)<sup>∝</sup> <sup>−</sup>*axay*cos*<sup>α</sup>* (*<sup>χ</sup>* ″

*xyyx*) / 2

2(*χ* ″

*xyyx*) / 2

is proportional to the measured photocurrent, is given by

*β*1(*γ* =0)∝*ax*

*β*2(*γ* =*π* / 2)∝*ay*

*xxyy* + 3*χ* ″

*xxyy* + 3*χ* ″

+(*χ* ″

+(*χ* ″

Thus, the parameters of the measured light are given by

resolution is limited by only the pulse width of the reference light.

**8.1. Principle of polarization measurement**

18 Photodiodes - From Fundamentals to Applications

pendent of time.

$$
\beta\_{\hat{e}}(\hat{e} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}) \tag{37}
$$

respectively. The sign of sin*α* is positive when *β <sup>6</sup> >β <sup>5</sup>*and vice versa.

#### **8.2. Measurement of stationary polarization**

Polarization measurements were performed using the same setup as that shown in Fig. 1. The reference light is linearly polarized and its polarization direction *γ* was varied in four ways by a half- wave plate. On the other hand, for the signal light, linear polarization was transferred to linear, elliptical, and circular polarization by a quarter wave plate. Because the transferred polarization is theoretically given by Eq.(13) (*ϕ*=π/2), it is possible to compare with the measured results.

polarization direction was tilted at an angle of 45° relative to the fast and slow axes of the fiber. The propagating optical pulse was separated by the birefringence of the polarizationmaintaining fiber since components polarized along the two axes have different the propa‐ gation velocities. Consequently, the polarization of the output optical pulse was made timedependent. A 20-m-long polarization-maintaining fiber imparted a propagation time

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 21

**Figure 10.** Jones vector of the output pulse from a polarization maintaining fiber. (a) amplitude along the x- and y-

Figure 10 shows the Jones vector of the output pulse of a polarization maintaining fiber. The x- and y-axes are parallel to the fastand slow axes, respectively. Figure 10(a) shows the measured amplitudes *a <sup>x</sup>* and *a y.*They vary due to the different group velocities of the polar‐ ized light along two axes. The head and tail of output pulse are polarized along the fast and slow axes, respectively. Figure 10(b) shows the measured phase difference *α*. It is deter‐ mined by the difference in the optical lengths for polarizations along the two axes. It varies

The measured phase difference *α* between the optical field oscillations along the x- and yaxes is affected by the wavelength chirping. This effect is exploited to measure the wave‐ length chirping. We consider the case when the linearly polarized signal light is injected to a wave plate whose principal axes are tilted relative the polarization directionof the incident light. The transit times through the wave plate differ by *ΔT* for components along the two

7

<sup>4</sup>*<sup>ν</sup>* (38)

*ΔT* = ±

with time due wavelength chirping and nonlinear phase shift in the fiber.

**8.4. Measurement of wavelength chirping**

major axes of the wave plate. For, a 7λ/4 wave plate

difference of about 30 ps between the two components.

axes. (b) phase difference.

Figure 9 shows the measured elements of the Jones vector of the light being measured. The circles and triangles represent the measured points, while the soid lines represent the theo‐ retical curves given by Eq. (13). Figure 9(a) shows the amplitudes of *a <sup>x</sup>* and *a <sup>y</sup>*. The meas‐ ured values agree reasonably well with the theoretical ones. Figure 9(b), on the other hand, shows the phase difference *α*. The measured*α* is slightly greater than the theoretical value for almost all values of *θ.* Small discrepancy between measured and theoretical phase differ‐ ence *α* is thought to be due to wavelength chirping of the measured light as will be dis‐ cussed insection 8.4.

**Figure 9.** Jones vector elements of light being measured with stationary polarization. (a) Amplitudes of the compo‐ nents in the *x*- and *y*-directions. (b) Phase difference between*x*- and *y*-directions. Circles and triangles are measured values. Solid lines are the theoretical values.

The light to be measured is circularly polarized (*ax* =*ay* =1 / 2, *δ* =*π* / 2) at *θ = π/ 4*, whereas, the lightis linearly polarized along the *x*-axis (i.e., *a <sup>x</sup> =1.0*and, *a <sup>y</sup> =0.0*)at *θ*= 0 and *π*/2.

#### **8.3. Measurement of time-dependent polarization**

The instantaneous polarization when the two light pulses overlap was measured for the cross-TPA. It is thus possible to measure the time-dependent polarization without using high-speed electronics using this method. An optical pulse compressed to 0.9 ps was used for the local oscillation *e* **^** in this measurement. The time resolution is equal to the width of this pulse. The timing of the short reference pulse was scanned over the signal light pulse to trace the variation of the polarization *p* **^**of the signal light pulse.

The polarization of the light being measured was varied with time using a polarizationmaintaining fiber. The output of the gain-switched LD was made linearly polarized and its polarization direction was tilted at an angle of 45° relative to the fast and slow axes of the fiber. The propagating optical pulse was separated by the birefringence of the polarizationmaintaining fiber since components polarized along the two axes have different the propa‐ gation velocities. Consequently, the polarization of the output optical pulse was made timedependent. A 20-m-long polarization-maintaining fiber imparted a propagation time difference of about 30 ps between the two components.

**Figure 10.** Jones vector of the output pulse from a polarization maintaining fiber. (a) amplitude along the x- and yaxes. (b) phase difference.

Figure 10 shows the Jones vector of the output pulse of a polarization maintaining fiber. The x- and y-axes are parallel to the fastand slow axes, respectively. Figure 10(a) shows the measured amplitudes *a <sup>x</sup>* and *a y.*They vary due to the different group velocities of the polar‐ ized light along two axes. The head and tail of output pulse are polarized along the fast and slow axes, respectively. Figure 10(b) shows the measured phase difference *α*. It is deter‐ mined by the difference in the optical lengths for polarizations along the two axes. It varies with time due wavelength chirping and nonlinear phase shift in the fiber.

#### **8.4. Measurement of wavelength chirping**

transferred to linear, elliptical, and circular polarization by a quarter wave plate. Because the transferred polarization is theoretically given by Eq.(13) (*ϕ*=π/2), it is possible to compare

Figure 9 shows the measured elements of the Jones vector of the light being measured. The circles and triangles represent the measured points, while the soid lines represent the theo‐ retical curves given by Eq. (13). Figure 9(a) shows the amplitudes of *a <sup>x</sup>* and *a <sup>y</sup>*. The meas‐ ured values agree reasonably well with the theoretical ones. Figure 9(b), on the other hand, shows the phase difference *α*. The measured*α* is slightly greater than the theoretical value for almost all values of *θ.* Small discrepancy between measured and theoretical phase differ‐ ence *α* is thought to be due to wavelength chirping of the measured light as will be dis‐

**Figure 9.** Jones vector elements of light being measured with stationary polarization. (a) Amplitudes of the compo‐ nents in the *x*- and *y*-directions. (b) Phase difference between*x*- and *y*-directions. Circles and triangles are measured

The light to be measured is circularly polarized (*ax* =*ay* =1 / 2, *δ* =*π* / 2) at *θ = π/ 4*, whereas,

The instantaneous polarization when the two light pulses overlap was measured for the cross-TPA. It is thus possible to measure the time-dependent polarization without using high-speed electronics using this method. An optical pulse compressed to 0.9 ps was used

this pulse. The timing of the short reference pulse was scanned over the signal light pulse to

The polarization of the light being measured was varied with time using a polarizationmaintaining fiber. The output of the gain-switched LD was made linearly polarized and its

**^**of the signal light pulse.

**^** in this measurement. The time resolution is equal to the width of

the lightis linearly polarized along the *x*-axis (i.e., *a <sup>x</sup> =1.0*and, *a <sup>y</sup> =0.0*)at *θ*= 0 and *π*/2.

with the measured results.

20 Photodiodes - From Fundamentals to Applications

cussed insection 8.4.

values. Solid lines are the theoretical values.

for the local oscillation *e*

trace the variation of the polarization *p*

**8.3. Measurement of time-dependent polarization**

The measured phase difference *α* between the optical field oscillations along the x- and yaxes is affected by the wavelength chirping. This effect is exploited to measure the wave‐ length chirping. We consider the case when the linearly polarized signal light is injected to a wave plate whose principal axes are tilted relative the polarization directionof the incident light. The transit times through the wave plate differ by *ΔT* for components along the two major axes of the wave plate. For, a 7λ/4 wave plate

$$
\Delta T = \pm \frac{\nabla}{4\nu} \tag{38}
$$

where *ν* is the optical frequency. The linearly polarized light is converted circularly polar‐ ized light because the phase shift between polarizations along the two principal axes is7*π* / 2 which is equivalent to−*π* / 2 *.*

**Figure 11.** Measurement of wavelength chirping of optical pulse from a gain switched LD. The left vertical axis is the phase difference between polarization components along the two principal axes of the wave plate. The right vertical axis is the estimated wavelength chirping gradient.

As the optical frequency is shifted by the wavelength chirping during the time period of *ΔT*, the optical frequencies of components polarized along the two principal axes after the pulse passes through the wave plate differ by

$$
\Delta \nu = \frac{d\nu}{dt} \Delta T \tag{39}
$$

The light is, therefore, converted into elliptically polarized light.

plate to make the phase shift sufficiently large to detect.

the output pulse from a SOA varies with time.

show the measured absolute value of phase difference.

Because *α* can be measured from the TPA of the Si-APD, the wavelength chirping gradient *dν* / *dt* can be determined. A 7λ/4 wave plate was used instead of a conventional λ/4 wave

Two-Photon Absorption in Photodiodes http://dx.doi.org/10.5772/50491 23

Figure 11 shows the measured wavelength chirping of an optical pulse from a gain-switch‐ ed LD. The linearly polarization is tilted at 45° relative to the principal axis of the 7λ/4 wave plate. The optical pulse passes through the wave plate and propagates in 40-cm of free space. The chirping gradient |*dν* / *dt* |is shown by the left vertical axis in Fig. 11. The measured value is consistent with the wavelength broadening observed by an optical spectrum ana‐ lyzer. The chirping gradient is large at the head of pulse due to the asymmetry pulse shape.

Semiconductor optical amplifiers (SOAs) generally exhibit birefringence due to the real and/or imaginary parts of the optical gain having different values for transverse electric (TE) and the transverse magnetic (TM) polarizations. The real and imaginary parts of the SOA gain are nonlinear for intense propagating light and induce dynamic birefringence [15,16]. Intense optical pulse affects the polarization of the pulse itself. Consequently, polarization of

**Figure 12.** Polarization of output pulse from an SOA when the polarization of the input pulse is tilted by 45 ° degree against the x- and y-axes. The waveform of the output pulse is also shown. (a) The closed circles and triangles show the measured polarization directions. The open circles show the measured output waveform. (b) The closed circles

A linearly polarized signal light was injected into a SOA witha polarization direction tilted at 45 ° against TE and TM modes. Time dependent Jones vector components of the output pulse from the SOA are measured by the cross-TPA with a reference light pulse with a pulse width of 0.9 ps. The results are shown in Figs. 12(a) and (b). The closed circles and triangles

**8.5. Measurement of dynamic birefringence of a semiconductor optical amplifier**

where*dν* / *dt* is the wavelength chirping gradient. The output pulse propagates in free space for a length of *L* reaching the PD. During the propagation time, polarization components along the two principal axis of the wave plate have different oscillation frequencies. Thus, the optical phase difference *α* between the two polarization components accumulates dur‐ ing the time period *L/c*, where *c* is the speed of light. The phase difference at the position of PD is

$$
\alpha = -\frac{\pi}{2} \pm 2\pi\Delta\nu \frac{L}{c} \tag{40}
$$

The light is, therefore, converted into elliptically polarized light.

where *ν* is the optical frequency. The linearly polarized light is converted circularly polar‐ ized light because the phase shift between polarizations along the two principal axes is7*π* / 2

**Figure 11.** Measurement of wavelength chirping of optical pulse from a gain switched LD. The left vertical axis is the phase difference between polarization components along the two principal axes of the wave plate. The right vertical

As the optical frequency is shifted by the wavelength chirping during the time period of *ΔT*, the optical frequencies of components polarized along the two principal axes after the pulse

where*dν* / *dt* is the wavelength chirping gradient. The output pulse propagates in free space for a length of *L* reaching the PD. During the propagation time, polarization components along the two principal axis of the wave plate have different oscillation frequencies. Thus, the optical phase difference *α* between the two polarization components accumulates dur‐ ing the time period *L/c*, where *c* is the speed of light. The phase difference at the position

*dt ΔT* (39)

*<sup>c</sup>* (40)

*Δν* <sup>=</sup> *<sup>d</sup><sup>ν</sup>*

*<sup>α</sup>* <sup>=</sup> <sup>−</sup> *<sup>π</sup>*

<sup>2</sup> <sup>±</sup> <sup>2</sup>*πΔν*

*L*

which is equivalent to−*π* / 2 *.*

22 Photodiodes - From Fundamentals to Applications

axis is the estimated wavelength chirping gradient.

passes through the wave plate differ by

of PD is

Because *α* can be measured from the TPA of the Si-APD, the wavelength chirping gradient *dν* / *dt* can be determined. A 7λ/4 wave plate was used instead of a conventional λ/4 wave plate to make the phase shift sufficiently large to detect.

Figure 11 shows the measured wavelength chirping of an optical pulse from a gain-switch‐ ed LD. The linearly polarization is tilted at 45° relative to the principal axis of the 7λ/4 wave plate. The optical pulse passes through the wave plate and propagates in 40-cm of free space.

The chirping gradient |*dν* / *dt* |is shown by the left vertical axis in Fig. 11. The measured value is consistent with the wavelength broadening observed by an optical spectrum ana‐ lyzer. The chirping gradient is large at the head of pulse due to the asymmetry pulse shape.

#### **8.5. Measurement of dynamic birefringence of a semiconductor optical amplifier**

Semiconductor optical amplifiers (SOAs) generally exhibit birefringence due to the real and/or imaginary parts of the optical gain having different values for transverse electric (TE) and the transverse magnetic (TM) polarizations. The real and imaginary parts of the SOA gain are nonlinear for intense propagating light and induce dynamic birefringence [15,16]. Intense optical pulse affects the polarization of the pulse itself. Consequently, polarization of the output pulse from a SOA varies with time.

**Figure 12.** Polarization of output pulse from an SOA when the polarization of the input pulse is tilted by 45 ° degree against the x- and y-axes. The waveform of the output pulse is also shown. (a) The closed circles and triangles show the measured polarization directions. The open circles show the measured output waveform. (b) The closed circles show the measured absolute value of phase difference.

A linearly polarized signal light was injected into a SOA witha polarization direction tilted at 45 ° against TE and TM modes. Time dependent Jones vector components of the output pulse from the SOA are measured by the cross-TPA with a reference light pulse with a pulse width of 0.9 ps. The results are shown in Figs. 12(a) and (b). The closed circles and triangles in Fig. 12(a) show the measured amplitudes *a <sup>e</sup> <sup>2</sup>* and *a <sup>m</sup> <sup>2</sup>* , respectively. *a <sup>e</sup>* and *a <sup>m</sup>*are the amplitudes of the Jones vectors for TE and TM polarization. The open circles and dashed line show the measured output pulse shape. The polarization at the head of the pulse is al‐ most the same as that of the injected light pulse. However, the carrier density modulation in the SOA rotates the polarization because the gains for the polarizations of the TE and TM modes have different carrier density dependences. Figure 12(b) shows the measured phase difference *α*. The phase difference varies dynamically due to self-phase modulation in the SOA as a result of the carrier density modulation and spectrum hole burning.

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