Chirp Gaussian Pulse

[Paolo Carniello]

Hi all,
I have a doubt regarding the implementation of the “Optical Gaussian Pulse Generator” and the “Optical Fiber” components. I am using OptiSystem 14 (the version my university provides).

The first doubt is about the definition of “Chirp” in the “Analysis” part of the “Optical Time Domain Visualizer”. If we denote a generic complex signal as

s(t) = A * exp(+1j*phi),

where A>0 is the module and phi is the phase, it seems to me that by “Chirp” it is meant the quantity

d(phi)/dt

This is a common definition, but sometimes a sign ‘-‘ is used in front of the derivative. That is why I ask.
Anyway, doing some simulations with signals I generated (with the “Measured Optical Pulse” component), it seems to me that the chirp definition used in OptiSystem is d(phi)/dt.

Fixed that, I am wondering which is the definition of the Gaussian pulse implemented in the “Optical Gaussian Pulse Generator”. In particular, I am not sure about the “Chirp factor” which can be set for the component. Again, the Help of the component does not indicate a model for the Gaussian pulse where such “Chirp factor” is present. I thought that the standard expression for the chirped pulse had been used (as reported in eq5 of your tutorial <a href=”https://optiwave.com/resources/applications-resources/optical-system-effects-of-group-velocity-dispersion-gvd-on-gaussian-pulse-propagation/”>here</a&gt;). However, if eq5 was implemented, the chirp should be

-C*t/T0^2

ie, a decreasing quantity with time t. However, if one checks with a “Optical Time Domain Visualizer” the chirp of a gaussian pulse generated with “Optical Gaussian Pulse Generator” with chirp factor C=2, the chirp results to be increasing with time.
To me this indicates that a different definition (from eq5 of the above tutorial) has been used for the gaussian pulse. In particular, I would guess that a sign ‘-‘ has been used in front of the chirp factor ‘C’ in such equation.
Am I right? If not, why?

The second issue is with the propagation equation implemented by the “Optical fiber” component, assuming only group-velocity dispersion (GVD) is present, ie, attenuation, third-order dispersion, XPM and all other distorting effects are off. In the Help of the component is indicated that the equation implemented by the fiber is, in this case,

dE/dz = -j * beta2/2 * d2E/dt2, (model 1)

where beta2 is the second-order derivative of beta wrt frequency omega, and d2E/dt2 is the second-order derivative of the field amplitude E wrt time t.

However, I did a couple of simulations which brought me to believe that the actual equation implemented by the component is (in this case)

dE/dz = +j * beta2/2 * d2E/dt2 (model 2)

One of the simulations I did, consists in generating an unchirped gaussian pulse (ie, C=0), selecting beta2<0, and checking which is the chirp of the pulse after propagating in the fiber for a certain distance z. For a gaussian pulse the analytic expression for the output field when only GVD is present is known. The theory (ie, the mentioned analytic expression) tells us that, if (model 1) has been implemented, the chirp should be increasing with time. On the opposite, if (model 2) has been implemented, the chirp should be decreasing with time.
Doing such simulation with OptiSystem, I see from the “Optical Time Domain Visualizer” that the chirp is decreasing with time. Therefore, it seems to me that (model 2) is implemented by the “Optical fiber” component, instead of the (model 1) indicated in the Help.
Am I right? If not, why?

Thanks in advance to anyone who will answer!

[Ahmad Atieh]

Dear Paolo,
Sorry for late response. Please find answers below, which was sent to you by email as well.
[AA] we use +ve sign in the definition of chirp. The sign in the general filed equation is related to the directional of propagation of the optical field.
[AA] please note that the chirp definition is not exactly the derivative of the phase for the gaussian pulse as shown in the attached image and described in the component datasheet. It is derivative of the phase of the optical field E(t) which is related to the gaussian pulse p(t) according to the equation in the attachment.
[AA] as I’ve explained above, there is a difference between the gaussian pulses and the optical field used in the calculation of the propagated signals in the optical fiber.
[AA] please note that the equations in the tutorial on Optiwave website are related to the normalization process of the nonlinear Schrodinger equation. In equation 5, the initial pulse at z=0m, has a chirp parameter which is related to the phase. The field phase is Ct^2/2T^2. You can’t correlate this equation to the chirp definition show in the image for the Gaussian pulse.
[AA] if you derive the phase with respect to t, it gives Ct/T^2, which is linear varying function. The sign should be +ve. Typically, the positive or negative sign indicates the direction of the propagation of the field as I’ve mentioned above.
[AA] I guess the two equations 5 and the Gaussian pulse are different as described in the datasheet of the components.
[AA] the optical fiber components in OptiSystem support attenuation, GVD, Beta2, Beta 3, SPM, XPM, FWM, SBS, SRS, Self-steeping. etc.
You should be able to include all of these physical characteristic in your modeling. Please refer to the Example Library of OptiSystem at the location below for examples covering theses effect.
C:UsersUSER NAMEDocumentsOptiSystem 19.0 SamplesFiber analysis and design
[AA] OptiSystem solves the nonlinear Schrodinger equation using Split-step method. Please refer to the datasheet for the Optical Fiber and Bidirectional Optical Fiber components for more details. Please note that these models have been verified experimentally by many users in the industry and academia.
Regards,
Ahmad

[Ahmad Atieh]

Dear Paolo,
Sorry for late response. Please find answers below, which was sent to you by email as well.
[AA] we use +ve sign in the definition of chirp. The sign in the general filed equation is related to the directional of propagation of the optical field.
[AA] please note that the chirp definition is not exactly the derivative of the phase for the gaussian pulse as shown in the attached image and described in the component datasheet. It is derivative of the phase of the optical field E(t) which is related to the gaussian pulse p(t) according to the equation in the attachment.
[AA] as I’ve explained above, there is a difference between the gaussian pulses and the optical field used in the calculation of the propagated signals in the optical fiber.
[AA] please note that the equations in the tutorial on Optiwave website are related to the normalization process of the nonlinear Schrodinger equation. In equation 5, the initial pulse at z=0m, has a chirp parameter which is related to the phase. The field phase is Ct^2/2T^2. You can’t correlate this equation to the chirp definition show in the image for the Gaussian pulse.
[AA] if you derive the phase with respect to t, it gives Ct/T^2, which is linear varying function. The sign should be +ve. Typically, the positive or negative sign indicates the direction of the propagation of the field as I’ve mentioned above.
[AA] I guess the two equations 5 and the Gaussian pulse are different as described in the datasheet of the components.
[AA] the optical fiber components in OptiSystem support attenuation, GVD, Beta2, Beta 3, SPM, XPM, FWM, SBS, SRS, Self-steeping. etc.
You should be able to include all of these physical characteristic in your modeling. Please refer to the Example Library of OptiSystem at the location below for examples covering theses effect.
C:UsersUSER NAMEDocumentsOptiSystem 19.0 SamplesFiber analysis and design
[AA] OptiSystem solves the nonlinear Schrodinger equation using Split-step method. Please refer to the datasheet for the Optical Fiber and Bidirectional Optical Fiber components for more details. Please note that these models have been verified experimentally by many users in the industry and academia.
Regards,
Ahmad

[Author]

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