- #1
test1234
- 13
- 2
Hi there, I'm kind of rusty on some stuff, so hope someone can help enlighten me.
I have an expression
[itex]E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z][/itex]
I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to a form with can be separated... I reckon the problem lies with my understanding of the 2nd derivative, especially with more variables coming into play.
From the Helmholtz equation,
[itex]\nabla^2 E+\epsilon (w) k_0^{\phantom{0}2} E=0[/itex]
Working out
[itex]\nabla^2 E =\nabla(\nabla E)[/itex]
[itex]=\nabla(A\exp[i\beta_0 z] \frac{\partial F}{\partial x}+A\exp[i\beta_0 z] \frac{\partial F}{\partial y}+ FA (i \beta_0 \exp[i\beta_0 z])+F \exp[i\beta_0 z] \frac{\partial A}{\partial z})
[/itex]
[itex]
=A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial x}+ \exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial x} [/itex]
[itex]
+ A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial y^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial y}+\exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}
[/itex]
[itex]
+\frac{\partial F}{\partial x} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ \frac{\partial F}{\partial y} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ F A (i \beta_0)^2 \exp[i\beta_0 z]
[/itex]
[itex]
+i \beta_0 F \exp[i\beta_0 z] \frac{\partial A}{\partial z}+F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]+F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}
[/itex]
Which gives
[itex]
=\exp[i\beta_0 z][A \frac{\partial^2 F}{\partial x^2}+2i \beta_0 A \frac{\partial F}{\partial x}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}+A \frac{\partial^2 F}{\partial y^2}+2i \beta_0 A \frac{\partial F}{\partial y}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}]
[/itex]
It seems like the terms [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}[/itex] and [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}[/itex] need to vanish...
------------------------------------------------------------------------------------------
Or have I done the derivative wrongly? Should it be the following instead?
[itex]
\nabla^2 E =\nabla_x ^{\phantom{0}2}E+\nabla_y ^{\phantom{0}2}E+\nabla_z ^{\phantom{0}2}E
[/itex]
where
[itex]
\nabla_x ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}
[/itex]
[itex]
\nabla_y ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^y}
[/itex]
[itex]
\nabla_z^{\phantom{0}2}E = \nabla_z [FA (i \beta_0) \exp[i\beta_0 z]+ F \exp[i\beta_0 z] \frac{\partial A}{\partial z}]
[/itex]
[itex]
=[F (i \beta_0) \exp[i\beta_0 z] \frac{\partial A}{\partial z}+FA (i \beta_0)^2 \exp[i\beta_0 z]+
F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}]+ F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]
[/itex]
Thanks in advance!
I have an expression
[itex]E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z][/itex]
I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to a form with can be separated... I reckon the problem lies with my understanding of the 2nd derivative, especially with more variables coming into play.
From the Helmholtz equation,
[itex]\nabla^2 E+\epsilon (w) k_0^{\phantom{0}2} E=0[/itex]
Working out
[itex]\nabla^2 E =\nabla(\nabla E)[/itex]
[itex]=\nabla(A\exp[i\beta_0 z] \frac{\partial F}{\partial x}+A\exp[i\beta_0 z] \frac{\partial F}{\partial y}+ FA (i \beta_0 \exp[i\beta_0 z])+F \exp[i\beta_0 z] \frac{\partial A}{\partial z})
[/itex]
[itex]
=A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial x}+ \exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial x} [/itex]
[itex]
+ A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial y^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial y}+\exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}
[/itex]
[itex]
+\frac{\partial F}{\partial x} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ \frac{\partial F}{\partial y} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ F A (i \beta_0)^2 \exp[i\beta_0 z]
[/itex]
[itex]
+i \beta_0 F \exp[i\beta_0 z] \frac{\partial A}{\partial z}+F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]+F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}
[/itex]
Which gives
[itex]
=\exp[i\beta_0 z][A \frac{\partial^2 F}{\partial x^2}+2i \beta_0 A \frac{\partial F}{\partial x}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}+A \frac{\partial^2 F}{\partial y^2}+2i \beta_0 A \frac{\partial F}{\partial y}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}]
[/itex]
It seems like the terms [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}[/itex] and [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}[/itex] need to vanish...
------------------------------------------------------------------------------------------
Or have I done the derivative wrongly? Should it be the following instead?
[itex]
\nabla^2 E =\nabla_x ^{\phantom{0}2}E+\nabla_y ^{\phantom{0}2}E+\nabla_z ^{\phantom{0}2}E
[/itex]
where
[itex]
\nabla_x ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}
[/itex]
[itex]
\nabla_y ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^y}
[/itex]
[itex]
\nabla_z^{\phantom{0}2}E = \nabla_z [FA (i \beta_0) \exp[i\beta_0 z]+ F \exp[i\beta_0 z] \frac{\partial A}{\partial z}]
[/itex]
[itex]
=[F (i \beta_0) \exp[i\beta_0 z] \frac{\partial A}{\partial z}+FA (i \beta_0)^2 \exp[i\beta_0 z]+
F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}]+ F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]
[/itex]
Thanks in advance!