## Abstract

The nature of how a chirped Gaussian pulse is affected in the system with both polarization mode dispersion (PMD) and polarization-dependent loss (PDL) is analyzed. We develop a mathematical description and verify the results through numerical simulation. The delay of a chirped Gaussian pulse depends not only on the group delay characteristic of the transmission system but also on the chirp of the pulse itself. The delay also depends on the magnitude profile of the frequency response of the system. Therefore, the effective PMD for a chirped Gaussian pulse is related to both the PMD and PDL in the transmission system. The results show that the effective PMD may deteriorate when a PDL exists.

©2004 Optical Society of America

## 1. Introduction

Polarization mode dispersion (PMD) becomes a serious problem at bit-rates above 10 Gbps. The differential group delay (DGD) between two orthogonal states of polarization called the principal states of polarization (PSP’s) causes the PMD [1]. As a pulse propagates through a lightwave transmission system with a PMD, the pulse is split into a fast and slow one, and therefore becomes broadened. Moreover most optical components have a more or less polarization-dependent loss (PDL) or gain (PDG). When a PDL is present in the transmission system, the combined effects become more complicated. Therefore, a great interest has developed in the combined effects of PMD and PDL [2,3]. However, they have only been investigated for much simplified transmission systems. In this paper, we increase the level of complexity by investigating how a chirped Gaussian pulse is affected in a system with both PMD and PDL. Due to the difficulty of the mathematical analysis, we assume that the chirped pulse is Gaussian in form. We expect that our findings on a chirped pulse can be applied to other pulse shapes including a super-Gaussian shape. To analyze the effect of PDL and chirp, the concept of average pulse position is used, which defines the delay of a pulse and an analytic expression is derived for the pulse position of a chirped Gaussian pulse. From this concept, a time domain PMD can be defined as the difference between the maximum delay and the minimum delay [4]. As a result, for the first time, to our knowledge, we show that for a chirped Gaussian pulse, the PMD depends not only on the group delay properties of a transmission system but also on both the chirp and magnitude profiles of the frequency response of the system, a finding that is contrary to conventional expectations. In other words, the property of a transmission system alone does not define the delay of the system. The magnitude profile of the response results in the combined effect of PMD and PDL. This is an important result because the Jones matrix eigenanalysis (JME) method [5], which is generally used to determine the PMD of a system, has a limitation in the case of a chirped Gaussian input. We show that the PMD may deteriorate when a PDL exists and verify the result through a numerical simulation.

## 2. Theory

#### 2.1 Imaginary delay

Firstly, we consider a complex envelope of a 1-dimensional scalar pulse. We assume the transfer function of a system with a group delay as follows.

If the 1-dimensional transfer function has the magnitude that is exponentially dependent on the frequency, i.e., *H*_{m}
(*ω*)=exp(-*αω*)=exp[*j*(*jαω*)], we can interpret it in terms of an imaginary time delay and therefore the output pulse is *y*(*t*)=*x*(*t*-*t*
_{0}+*j*
*α*) for an input pulse of *x*(*t*). Although this odd treatment is unusual, it is useful in considering complex envelopes, as shown below.

#### 2.2 Delay due to chirp and exponential frequency response

We can obtain the output of a system with the exponential magnitude frequency response using the imaginary delay concept. When the system has an imaginary time delay and the input signal is a chirped Gaussian pulse, *A*(*t*)=*A*
_{0} exp[-(1+*jC*)(*t*/*T*
_{0})^{2}/2], where *C* is a chirp parameter and *T*
_{0} is the pulse width, the additional time delay occurs in the envelope of the pulse. If we neglect the phase term in the expression, the output pulse becomes as follows.

$$=\mathrm{exp}\left\{-\left[{\left(t-{t}_{0}\right)}^{2}-{\alpha}^{2}-2C\alpha \left(t-{t}_{0}\right)\right]\u20442{T}_{0}^{2}\right\}$$

$$=\mathrm{exp}\left\{-\left[{\left(t-{t}_{0}-\alpha C\right)}^{2}-{\alpha}^{2}-{\left(\alpha C\right)}^{2}\right]\u20442{T}_{0}^{2}\right\}.$$

We can see that a time delay of *αC* is added to the conventional group delay *t*
_{0}, which is determined from the multiplication of two imaginary terms, *jC*(*t*-*t*
_{0}) and *jα*. This can be applied only when the system has a frequency response with an exponential magnitude. However, for a system with an arbitrary frequency response, we can infer from the relation between exp[*j*(*j*
*αω*)] and *α* that each frequency component is delayed by the derivative of the logarithm of the arbitrary frequency response magnitude multiplied by the minus of the chirp parameter *C*.

#### 2.3 Delay due to chirp and arbitrary frequency response

To apply our theory to a system with an arbitrary frequency response, we use the average pulse position concept. The average position of a pulse is defined as [4]

$$={\int}_{-\infty}^{\infty}{H}^{*}(\omega ){\Psi}_{\mathit{in}}^{*}\left(\omega \right)j\left[H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\right]\prime d\omega \u2044{\int}_{-\infty}^{\infty}{\mid H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}d\omega ,$$

where *ψ*_{out}
(*t*) is the output pulse of the transmission system, *H* is the transfer function of the system, and Ψ
_{in}
(*ω*) is the Fourier transform of the input pulse. If we consider a chirped Gaussian input and classify the transfer function into magnitude and phase terms, the expression will be

$$\times {\left\{{H}_{m}\left(\omega \right)\mathrm{exp}\left(j{\varphi}_{H}\right)\mathrm{exp}\left[-\left(1-jC\right){T}_{0}^{2}{\omega}^{2}\u20442\left(1+{C}^{2}\right)\right]\right\}}^{\prime}d\omega \u2044{\int}_{-\infty}^{\infty}{\mid H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}d\omega ,$$

where *H*_{m}
(*ω*) is the magnitude and *ϕ*_{H}
is the phase of the transfer function.

We can simplify the numerator to

Since <*t*> must be real-valued, this expression must also be real. Therefore, the imaginary terms in the expression are eliminated. If we eliminate all the imaginary terms, the expression becomes

Here, ${\int}_{-\infty}^{\infty}$
${H}_{m}^{\mathit{2}}$
(ω)exp[-${T}_{0}^{2}$ω^{2}/(1+*C*
^{2})][${\mathit{\text{CT}}}_{0}^{2}$
*ω*/(1+*C*
^{2})]*dω* can be calculated using integration by part as

$$=C{\int}_{-\infty}^{\infty}\mathrm{exp}\left[-{T}_{0}^{2}{\omega}^{2}\u2044\left(1+{C}^{2}\right)\right]{H}_{m}^{2}{\left\{\mathrm{ln}\left[{H}_{m}\left(\omega \right)\right]\right\}}^{\prime}d\omega .$$

Finally, we obtain

$$-C{\int}_{-\infty}^{\infty}\mathrm{exp}\left[-{T}_{0}^{2}{\omega}^{2}\u2044\left(1+{C}^{2}\right)\right]{H}_{m}^{2}{\left\{\mathrm{ln}\left[{H}_{m}\left(\omega \right)\right]\right\}}^{\prime}d\omega \}\u2044{\int}_{-\infty}^{\infty}{\mid H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}d\omega $$

as the total delay expression.

Here, the sign of the delay may be plus or minus according to the sign of the chirp parameter and the derivative of the logarithm of the magnitude of the frequency response. This includes the result reached in Section 2.2 and we can see that each frequency component is delayed by the conventional group delay -*ϕ′*_{H}
and the additional delay -*C*{ln[*H*_{m}
(*ω*)]}′. Each delay component is weighted by the proportion of the power of each output frequency component to the total output energy,

$$=\left\{{\mid H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}\right\}\u2044{\int}_{-\infty}^{\infty}{\mid H\left(\omega \right){\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}d\omega .$$

Of course, unlike an exponential response, an arbitrary frequency response does not lead to the exactly-delayed replica of the input pulse. Instead, due to the combined effect of delay and the distortion of the pulse shape, the average position of a pulse is changed.

#### 2.4 Expanding to a polarized pulse

Because the Jones formalism for polarization is 2-dimensional, the above results cannot be applied directly. However, this analysis can be applied to output components in arbitrary two orthogonal states. In each orthogonal state, a 1-dimensional frequency response can be defined. The component of a polarized chirped Gaussian pulse in each orthogonal state experiences a delay by the conventional group delay, the chirp and the magnitude profile of the 1-dimensional frequency response individually. In addition, the output pulse will be the sum of the two components that experienced their corresponding delays. Because the power of the polarized output pulse is the sum of the power of the output pulses in arbitrary two orthogonal states, the delay of the power of the polarized pulse is the weighted average of the delays of the components in each orthogonal state. The weight is the proportion of the energy of each component to the total output energy. This arises from the fact that the average pulse position is calculated by considering the normalized power profile of the polarized pulse as a probability density function.

Therefore, we can define the magnitude of 1-dimensional transfer functions using the projection of the output state on two arbitrary unit orthogonal states *ρ*
_{1}, *ρ*
_{2} as follows.

where *T*(*ω*) is the Jones matrix of the transmission system and in *$\widehat{\phi}$ _{in}* is the input state of polarization (SOP). Therefore, if we define the power of the 2-dimensional frequency response,

*H*

_{m}, as

then the relation *${\mathrm{H}}_{1m}^{2}$*+${H}_{2\mathrm{m}}^{2}$=${H}_{m}^{2}$
holds.

The power of the frequency component of the output pulse can be expressed as

We can consider the delay due to chirp only without any loss of generality, and therefore each frequency component experiences the weighted delay in the amount of

$$=-C\left({H}_{1m}{H}_{1m}^{\prime}+{H}_{2m}{H}_{2m}^{\prime}\right)[{\mid {\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}\u2044{\int}_{-\infty}^{\infty}{P}_{\mathit{out}}\left(\omega \right)d\omega ].$$

Since the relation of *H*
_{1m}
*H*′_{1m}+*H*
_{2m}
*H*′_{2}
_{m}
=*H*_{m}*H*′
_{m}
from ${H}_{1m}^{2}$+*${\mathrm{H}}_{2m}^{2}$*=${\mathrm{H}}_{m}^{2}$
is valid, we can obtain the delay expression for a polarized chirped Gaussian pulse as

$$=-C{H}_{m}{H}_{m}^{\prime}[{\mid {\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}\u2044{\int}_{-\infty}^{\infty}{P}_{\mathit{out}}\left(\omega \right)d\omega ]$$

$$=-C{\left\{\mathrm{ln}\left[{H}_{m}\left(\omega \right)\right]\right\}}^{\prime}[{H}_{m}^{2}{\mid {\Psi}_{\mathit{in}}\left(\omega \right)\mid}^{2}\u2044{\int}_{-\infty}^{\infty}{P}_{\mathit{out}}\left(\omega \right)d\omega ]\phantom{\rule{.2em}{0ex}}.$$

Therefore, the weighted delay experienced by each frequency component of the pulse is

We find that the magnitude of 2-dimensional frequency response determines the amount of delay due to chirp. For example, if *H′*_{m}
is zero or an even function around the center frequency, the integral of the delay of each frequency component is zero. We note that the magnitude of the delay is independent of the multiplication of the frequency response by a real number, since the expression is normalized by ${\int}_{-\infty}^{\infty}$
*P*_{out}
(*ω*)*dω* and the factor is proportional to {ln[*H*_{m}
(*ω*)]}′. In other words, amplification does not affect the delay due to chirp. If the magnitude of the frequency response *H*_{m}
(*ω*) is a function of both the input SOP and the frequency, the time domain PMD, which is defined as the difference between the maximum delay and the minimum delay [4], may change. This is because a chirped Gaussian pulse may have different values of additional delay according to the input SOP. The combined effect between the PMD and PDL may cause the magnitude of the frequency response to change with frequency. In such a case, even if the PMD is completely compensated, delay due to chirp is still changed by the input SOP and the residual PMD remains uncompensated. If the PMD property varies dynamically with time, since the delay due to chirp is changed with time, the change in the average position of the pulse may also affect the pulse as jitter.

If the variation of delay due to chirp with respect to SOP is much larger than the conventional PMD, the PMD would be expected to increase. Since {ln[*H*_{m}
(*ω*)+*a*]}′ is *H′*_{m}
(*ω*)/[*H*_{m}
(*ω*)+*a*], where *a* is a constant, even if the derivative of the output response *H*_{m}
(*ω*) is equal, the system with a lower *H*_{m}
(*ω*) is further affected by chirp.

#### 2.5 The calculation of the time domain PMD

Lu *et al*. proposed a method by which the time domain PMD of a system with both PMD and PDL can be easily calculated [3]. Because we use this method to verify our theory, a brief outline is provided here. For a given transmission system, the output frequency response of the system can be expressed as Ψ
_{out}
(*ω*)=*T*(*ω*)Ψ
_{in}
(*ω*)*$\widehat{\phi}$*
_{in}
. The average position of a pulse can be calculated as follows.

where *ψ*_{out}
(*t*) is the output electrical field vector as a function of time *t*, *E*_{out}
is the output energy of the input pulse, and

is a 2×2 Hermitian matrix.

Although less intuitive, the delay expression in the above section can also be derived from Eq. (16). The time domain PMD is calculated from max(<*t*>)-min(<*t*>). We regard two states, which lead to the maximum and the minimum <*t*>, as PSP’s in the extended concept. A superposition of the PSP’s causes pulse broadening. The corresponding output pulse is not the exact sum of the maximum and the minimum delay pulses because of interference between the two PSP’s. However the tails of the output pulses, which do not suffer extensive interference, can be attributed to broadening.

## 3. Simple examples

Since we know from our theory that chirp leads to the additional delay, and this additional delay may lead to a deterioration in the PMD, we calculate the time domain PMD to verify our theory. A simplified model is assumed here. In this calculation, we consider a concatenation of a uniform PMD element with a PDL element. We assume that the elements have linear birefringence and PDL, respectively. We designate the PMD of the first element section as PMD1 and the PDL of the second PDL element section as PDL_{2} hereafter. The first element has only PMD and the second only PDL. Therefore, if we change the frequency of an input sinusoidal lightwave, the polarization of the output of the first PMD element changes with frequency. Due to the PDL of the second section, the magnitude of the output wave of the concatenated system, *H*_{m}
(*ω*), then also changes with frequency. The input pulses are assumed to be chirped Gaussian. We assume that the chirp parameters of the input pulses are 0, 2, 4, 7, 8 and 10. The time domain PMD of the concatenated system, which includes the effects of all frequency components is then calculated while changing the angle between the PMD axis and the PDL axis from 0° to 180° in Stokes space. To change the angle, we rotate the PDL element about the PMD element from 0° to 90° since the axes lie in the linear polarization plane. The relative angle is the rotation angle of the PDL element, which is half of the angle between the PMD and the PDL axis in Stokes space.

Figure 1 shows the effect of chirp on PMD. The PMD initially increases with chirp parameter while the rate of increase is reduced as the chirp parameter increases. Finally, the PMD decreases with chirp from the maximum value. The bandwidth of a chirped Gaussian pulse increases with chirp parameter. Because it is generally known that in the case of a chirp-free pulse, as the bandwidth of the pulse becomes broader, the effective PMD becomes lower [3], this effect cannot be explained by conventional PMD theory. As shown in the figure, a pulse with a width of 50^{1/2} ps, which has the same bandwidth as a pulse with a chirp parameter of 7, has a very low PMD. The increase in PMD is explained as a PMD increase phenomenon due to chirp. The combined effect between chirp and the magnitude of the frequency response leads to an increase in PMD, as shown in the above theory.

The decrease in the rate of increase can be explained as follows. Figure 2 shows the magnitude of the frequency responses and output spectra corresponding to the maximum and the minimum delay for a chirp parameter of 7 and a relative angle of 45°. As shown in Fig. 2, the magnitude of the frequency response at the center frequency can be approximated as an exponential function. However, since the bandwidth increases with chirp, the frequency response profile deviates from the exponential function. Therefore, while the delay that each frequency component experiences is increased by chirp, the region of frequency components having the lower |{ln[*H*_{m}
(*ω*)]}′| value becomes broader. This is the reason for why the rate of increase of PMD is reduced. When the relative angle is 0° and 90°, since the output power does not vary with frequency, PMD is unaffected by chirp.

Figure 3 shows pulses corresponding to the maximum delay and minimum delay for different chirp parameters. An increase in PMD with an additional delay can be seen. As shown in Fig. 2 (a), the magnitude of the frequency response may represent gain. The reason for this is that the PDL matrix is assumed to be the form of [$\left[\begin{array}{cc}\mathrm{exp}(+\alpha \u20442)& 0\\ 0& \mathrm{exp}(-\alpha \u20442)\end{array}\right]$], where *α* is the PDL parameter. From the discussion in the section 2.4, it can be seen that the amplification has no effect on delay due to chirp, and, thus, this assumption is appropriate. Therefore, both PDL and PDG can be considered simultaneously. The additional delay for the maximum and minimum delayed pulse is about ±20 ps. In Fig. 3, we can see that the corresponding -*C*[ln*H*_{m}
(*ω*)]′ value is also almost ±20 ps while the whole PMD predicted by the JME method is only 10 ps and is independent of chirp. This shows that the JME method has limitations in the case of a system with both PMD and PDL when the input is a chirped Gaussian pulse, while our theory accounts for this well. Because of the form of the PDL matrix mentioned above, the amplitude of pulses which experience PDL may be amplified.

In the case of a chirp-free and PDL-free pulse the input SOP’s which represent the maximum and the minimum delay are almost the same as the input PSP’s in the frequency domain. However, we saw that the SOP’s which represent the maximum and minimum delay for a chirped Gaussian pulse change significantly compared with those for a chirp-free pulse. Figure 4 shows this tendency. If the input SOP deviates from either of the input PSP’s, the group delay decreases from the maximum delay or increases from the minimum delay due to mixing of the two PSP’s. Instead, the profile of the magnitude of the frequency response is also changed. Therefore, since the effect of chirp may increase, there are two input SOP’s which maximize the sum of the two effects. The SOP’s lead to a maximum PMD. If the variation in delay due to chirp is dominant over the conventional PMD, the effective PMD is largely determined by chirp and the magnitude of the frequency response.

Although we mainly deal with the combined effects of PMD, PDL, and chirp in this paper, whenever the magnitude of the response of a transmission system varies with both frequency and SOP in any way, the PMD of a chirped-Gaussian pulse may be affected. This means that if the frequency-dependent PDL is not compensated, even if the PMD is compensated, the variance of the average pulse position may increase. If a Gaussian pulse experiences a chirping effect during propagation, the elimination of chirp before PMD compensation is preferable.

## 4. Conclusion

In conclusion, an analysis of the combined effect between the chirp of a Gaussian pulse and the frequency response of a system indicates that an additional delay can occur. For a chirped Gaussian pulse, the system characteristics are not the only factors that determine pulse delay. We derived a delay expression which represents the additional delay due to chirp and the magnitude of the frequency response. This result was then expanded to a 2-dimensional analysis to include PMD phenomenon. We showed that the delay of the pulse is determined not only by the phase shift characteristic of the transmission system but also by both the chirp of the input pulse and the magnitude profile of the frequency response of the system. The combined effect of PMD and PDL determines the magnitude profile. This is readily explained by our theory that the effective PMD is actually changed by chirp and PDL (or PDG) and therefore the PMD may deteriorate. This is an important result because the JME method considers only the phase derivative of the frequency response, which cannot explain the above effect. We further showed, using a numerical simulation, that the time domain PMD may increase with chirp parameter and explained this result using our theory.

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