Enhancement Factors for Prolate Spheroidal Geometry

For the purpose of calculating the DSP electromagnetic enhancement, we first consider the unaveraged normal enhancement at the tips of the prolate spheroid (tj = 1). Equation (22) expressed in terms of L ( l) becomes 

For the total enhancement, G , the effect of the metal particle on the emitting Raman radiation must also be included thus 

It should be remembered that the dielectric functions are dependent on the exciting frequency, wl. It is clear that the DSP resonance occurs when there is a zero in the denominator of Eq. (30), i.e., when 

As the aspect ratio a/b increases, the total enhancement at the tip increases, because the electric field at the tip is magnified. This is called the lightning rod effect and is given by the (Qi - oQl)/ratio in Eq. (32). The spheroidal coordinate fo can be approximated by 1 + (l/2)(f /a) for a/b 1, and as a/b increases, 1- This is the condition for a tip resonance. Equation (32) can then be explicitly expressed in terms of the aspect ratio a/b by using the approximations 

The real part of the dielectric function at resonance, Eq. (33), is negative and increases as a/b gets large, i.e., as the spheroid becomes more eccentric. Equation (34) is a good approximation for a/b 3/1. For a sphere a/b = 1), we have already calculated = 1.6 x 10 however, this occurred at = 382 nm. On the other hand, for an aspect ratio a/b = 2, = 6.90 x 10,  

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