The standard parabolic wave equation can be derived by introducing a narrowangle approximation to a modal representation of the field in a waveguide. Let the modal solution be given by

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where the eigenfunctions Professional Australia Essay Writers | AustraliaEssayWriting.com satisfy the depth-separated wave equation

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Here, k0 is the reference wavenumber and Professional Australia Essay Writers | AustraliaEssayWriting.com the index of refraction. By assuming the modal eigenvalues to cluster around k0 (a narrow-angle approximation) and to be given in the form Professional Australia Essay Writers | AustraliaEssayWriting.com where Professional Australia Essay Writers | AustraliaEssayWriting.com is small compared to unity, show that to leading order in Professional Australia Essay Writers | AustraliaEssayWriting.com  the field solution can be written in the forProfessional Australia Essay Writers | AustraliaEssayWriting.com where the envelope function Professional Australia Essay Writers | AustraliaEssayWriting.com satisfies the standard parabolic equation (6.9).