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let's consider a general waveguide with two ideal dielectrics inside it:

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Let's consider the interface condition about the tangential electric field:

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Generally E = E(x,y,z), where z is the axis of the waveguide. My book analyzes the special situation in which E may be expressed as the product of a z-dependent scalar Z(z) and of a x/y - dependent vector e(x,y):

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Then:

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My book says that it implies:

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Why is this true? I think it could be also, for instance:

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