Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38

, o f C Z t c   f

, o f C Z t c   2 r

C Z t c  

,

,

(4)

f

r

t

, o t

t

where c f,r,t

are the so-called stress-optical constants defined as:

ν

η

2 o



 

 

c

c

k c

c c

c

,

,

(5)

f

r

f

t

r

f

E

with k >0 being obtained by the interferometric method and η o ℑ the imaginary part, while prime indicates first derivative. In turn, zeroing of the Jacobian of the transformation of Eqs. (2), yields the radii of the so-called initial curves, i.e., the loci of points on the disc faces providing upon illumination the caustics, by solving the equation: 4 1 , , , , , ( ) f r t m f r t ζ C Φ λ    (6) where double prime denotes the second derivative. From Eq.(6) it is clear that there are three kinds of initial curves, one on the front face of the disc, providing the reflected caustics from the front face of the disc, and two on the rear face of the disc providing the reflected and transmitted caustics from the rear face of the disc. Substituting in Eq.(6) for Φ ( ζ ) from Eqs.(1), and using the transformations i       1 2 i 1 2 e , e θ θ ζ r ζ r , θ , θ 1 , θ 2 ϵ [0, –π] (Fig.1), the radii r o,f,r,t of the initial curves, the one on the front ( r o,f ) and the two ( r o,r and r o,t ) on the rear face of the disc, are obtained as: denoting the air-refraction index, equaling approximately zero. Finally, ℜ is the real and

 4 3

   

C

 ρ

1

(7)

, , f r t



2 sin 2

  







r

θ

θ

cos2

2



, , , o f r t

λ

KR

, , , m f r t

In Eq.(7), the absolute value of the ratio C ,f,r,t / λ m,f,r,t of Eq.(6); that absolute value suffices, also, the demand r 1

has been considered to avoid the double sign ± on the right-hand side be real positive numbers as well as the fact that and r 2

 4 3

   

C

ρ

1

, , f r t

  KR will be a real positive number, too. However, r o,f,r,t , given by Eq.(7), may turn out to be a complex . In this context, the double sign before the internal square root, in complete correspondence with the superscript ± on the left-hand side of Eq.(7), ensures that all possible real values for r o,f,r,t will be taken into account.   , , , m f r t 2 λ number, something that should be excluded from the solution, keeping only the real values for r o,f,r,t

 4 3

   4 3

Namely, if θ ϵ [0, –π],   

C

   

C



 ρ

1

C

ρ

1

ρ

1

, , f r t

, , f r t



, , f r t

2

1

≥sin 2 2 θ , or simpler if

, then

 

2 sin 2

2

θ

2

 

λ

KR

  

λ

KR

 

λ

KR

, , , m f r t

, , , m f r t

, , , m f r t

is a real positive quantity θ ϵ [0, –π] and the value , , , o f r t r  

for the plus sign before the internal square root in Eq.(7) should be

chosen for r o,f,r,t is lying com pletely in the imaginary plane). That is the classic case of a single initial (and in turn caustic) curve, sketched in Fig.3c (for the front initial curve). In Fig.3a, the variation of cos2 θ = 2 1 sin 2 θ  is also shown as the lower accepted limiting value , rejecting the value , , , o f r t r  as a purely imaginary one (see Fig.4b, where it is seen that , , , o f r t r 

   4 3

   4 3

   

being a real quantity; beyond that limit, 

C

C

ρ

ρ

1

1

, , f r t

, , f r t

and r o,f,r,t

for

partly

2 sin 2

2 sin 2

  

θ

θ

2

2

  

  

λ

KR

λ

KR

, , , m f r t

, , , m f r t

“pass” in the imaginary plane (Fig.4) and they must be accordingly rejected as it is shown next for a double initial curve.

 4 3

   4 3

   

does not exceed sin 2 2 θ , θ ϵ [0, –π], then   

C

C

ρ

1

ρ

1

, , f r t

, , f r t

Namely, if

is found to attain

 

2 sin 2

2

θ

2

 

  

λ

KR

 

λ

KR

, , , m f r t

, , , m f r t

454