In class we will consider the sum of the electric field of two plane waves, Problem 1. u:(z1) and uz(z.1), both traveling in the positive z direction with slightly different frequencies and propagation constants and with equal amplitudes. The electric fields of both plane waves are oriented in the y direction, which means that they have the same "polarization". The time and space variation of the two electric fields are given by: u:(z) = cos[ot - kz) u:[2) = cos ([a + Ao]t - [k+ Ak]2) We will show that the sum of these two waves produce an intensity envelope in time and space that modulates the carrier frequency of+ Ao/2. The velocity of the envelope is called the group velocity. Assume Aeo is << than e. For this problem, consider the sum of two slightly different waves with similar electric fields: v:(z) = sin[ot - kz) v:(z) = sin[[@ + Ao]t - [k + Ak]z) Derive analytically an expression for a) the group and b) the phase velocity for the sum of the two fields. HINT: sinſa) + sin(B) = 2 sin[[a+B)/2]cos[[a-B)/2] Phase and Group Velocity E(t) = Es. cos(eat-kz) E(t) ! Vph = wk = c'n(),) Time E(t) = 2Ea. cos(Acot/2-Akz/2) sas((o+Aeit2-(k+Ak)z) E(t) I Vg = dea'dk = c/N,(0.)
In class we will consider the sum of the electric field of two plane waves, Problem 1. u:(z1) and uz(z.1), both traveling in the positive z direction with slightly different frequencies and propagation constants and with equal amplitudes. The electric fields of both plane waves are oriented in the y direction, which means that they have the same "polarization". The time and space variation of the two electric fields are given by: u:(z) = cos[ot - kz) u:[2) = cos ([a + Ao]t - [k+ Ak]2) We will show that the sum of these two waves produce an intensity envelope in time and space that modulates the carrier frequency of+ Ao/2. The velocity of the envelope is called the group velocity. Assume Aeo is << than e. For this problem, consider the sum of two slightly different waves with similar electric fields: v:(z) = sin[ot - kz) v:(z) = sin[[@ + Ao]t - [k + Ak]z) Derive analytically an expression for a) the group and b) the phase velocity for the sum of the two fields. HINT: sinſa) + sin(B) = 2 sin[[a+B)/2]cos[[a-B)/2] Phase and Group Velocity E(t) = Es. cos(eat-kz) E(t) ! Vph = wk = c'n(),) Time E(t) = 2Ea. cos(Acot/2-Akz/2) sas((o+Aeit2-(k+Ak)z) E(t) I Vg = dea'dk = c/N,(0.)
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![Problem 1. In class we will consider the sum of the electric field of two plane waves,
u:(z.t) and uz(z.t), both traveling in the positive z direction with slightly different
frequencies and propagation constants and with equal amplitudes. The electric fields of
both plane waves are oriented in the y direction, which means that they have the same
"polarization". The time and space variation of the two electric fields are given by:
u: (2.1) = cos(@t - kz)
uz(2.1) = cos ([o + Ao]t - [k + Ak]2)
We will show that the sum of these two waves produce an intensity envelope in time and
space that modulates the carrier frequency of m + Ao/2. The velocity of the envelope is
called the group velocity. Assume Ao is << than o.
For this problem, consider the sum of two slightly different waves with similar electric
fields:
v.(z) = sin(ot - kz)
vz(z1) = sin([o + Ao]t - [k + Ak]z)
Derive analytically an expression for a) the group and b) the phase velocity for the sum of
the two fields.
HINT: sinfa) + sin(B) = 2 sin[[a+B)/2]cos[[a-B)/2]
Phase and Group Velocity
E(t) = Eo. cos(ot-kz)
E(t) !
Vph = w/k - c/n().)
Time
E(t) = 2E0- cos(Aot/2-Akz/2)
•cos((o+Amt2-(k+Ak)z)
E(t) E
Vg = da'dk = c/Ng(2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8530e82-7d31-4442-ada5-dc030786f27d%2Fbb3ef0f2-27b5-47b9-8940-b1a62b3e19fd%2Fdfp0zvm_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. In class we will consider the sum of the electric field of two plane waves,
u:(z.t) and uz(z.t), both traveling in the positive z direction with slightly different
frequencies and propagation constants and with equal amplitudes. The electric fields of
both plane waves are oriented in the y direction, which means that they have the same
"polarization". The time and space variation of the two electric fields are given by:
u: (2.1) = cos(@t - kz)
uz(2.1) = cos ([o + Ao]t - [k + Ak]2)
We will show that the sum of these two waves produce an intensity envelope in time and
space that modulates the carrier frequency of m + Ao/2. The velocity of the envelope is
called the group velocity. Assume Ao is << than o.
For this problem, consider the sum of two slightly different waves with similar electric
fields:
v.(z) = sin(ot - kz)
vz(z1) = sin([o + Ao]t - [k + Ak]z)
Derive analytically an expression for a) the group and b) the phase velocity for the sum of
the two fields.
HINT: sinfa) + sin(B) = 2 sin[[a+B)/2]cos[[a-B)/2]
Phase and Group Velocity
E(t) = Eo. cos(ot-kz)
E(t) !
Vph = w/k - c/n().)
Time
E(t) = 2E0- cos(Aot/2-Akz/2)
•cos((o+Amt2-(k+Ak)z)
E(t) E
Vg = da'dk = c/Ng(2)
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