#### Document Type

Article

#### Publication Date

6-1992

#### Abstract

For an interface between two isotropic media the power reflectance *R _{v}* (ø) is an even function,

*R*(ø) =

_{v}*R*(--), of the angle of incidence ø; hence all the odd derivatives,

_{v}*R*

_{v}^{(n)}=

*d*ø

^{n}R_{v}/d^{n}(

*n*odd), are identically 0 at ø = 0, independent of the incident polarization

*v*. When the incident light is unpolarized (

*v = u*), the second derivative,

*R*

_{u}^{(2)}, is also 0 at ø = 0, so that the flatness of the

*R*-versus-ø curve over an initial range of ø starting from ø = 0 is determined by the fourth derivative,

_{u}*R*

_{u}^{(4)}. The condition that

*R*

_{u}^{(4)}= 0 at ø = 0 gives the maximally flat response and leads to a specific constraint on the complex relative refractive index

*N*, namely, that Re[(

*N*- 1)

^{2}/

*N*

^{3}] = 2/

*NN**. The corresponding complex plane contour is the limaçon of Pascal, η = 2 cos θ ± √3 in polar form, where

*N*= η exp(

*j*θ). The two branches of this contour constitute the boundary lines that separate the region of the complex plane in which the-function

*R*(ø) is monotonic from that in which the function exhibits a minimum at oblique incidence. Families of curves that illustrate the maximally flat response in external and internal reflection are presented. New equations that determine the angle of incidence of minimum unpolarized-light reflectance of a dielectric-dielectric or a dielectric-conductor interface are derived.

_{u}#### Journal Name

Journal of the Optical Society of America A

#### Recommended Citation

R. M. A. Azzam, "Limaçon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light," J. Opt. Soc. Am. A 9, 957-563 (1992)

## Comments

This paper was published in

Journal of the Optical Society of America Aand is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-9-6-957. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.