Both Polarizer and Analyzer Along the y Axis

Both Polarizer and Analyzer Along the y Axis Подпись: (3.51)

In this case the general expression of the transmitted light becomes:

0 0 1 і m и Ш12 1 – i 0 0 0

0 1 і 1 m 21 Ш22 – i 1 0 1 Ay

from which:

Ay a Ay(ш 11 + Ш22)


(ш 11 + m 22) = (1 + ei A) = 0

verified for:

^A = -1


In the case of the analyzer along y axis, the destructive light interference gives (with k integer):

Подпись: (3.54) (3.55) A 2k + 1

= n



N = = 0.5, 1.5, 2.5,…


Alternatively, the light intensity expression is, Eq. 3.31:

I a | Ay |2(Ші1 + Ш22)(Ш11 + Ш22)

Both Polarizer and Analyzer Along the y Axis

or, with the same zeros of Eq. 3.53:

3.9 Isochromatics

As it was previously shown with Eq. 3.38, the difference of the principal stresses in a point of a plane model is directly proportional to the photoelastic constant, to the parameter N and inversely to the thickness d at that point.


Both Polarizer and Analyzer Along the y Axis

Fig. 3.12 Isochromatics on a black and bright background of an annular disc loaded at three points

If the observation is carried out in monochromatic light with wavelength X, the loci of points of light extinction with crossed filters (or the maximum transmission with parallel filters), are interference fringes that are called isochromatics. So such fringes appear black on a black background with crossed filters and bright on a bright background with parallel filters (Fig. 3.12). Details of the contact points are shown in Fig. 3.13.

Isochromatics join the points having the same value of the difference o1 – o2, namely the same diameter as the Mohr’s circle, i. e. the same maximum shear stress;

Both Polarizer and Analyzer Along the y Axis

Fig. 3.13 Details of isochromatics in contact points without (up) and with a light friction (bottom)

Both Polarizer and Analyzer Along the y Axis

Table 3.1 NX values up to the first order with crossed polarizers in white light, modified from [13]

N X (nm)







Iron gray


Lavender gray


Gray blue




Green white




Yellow white


Pale straw yellow


Straw yellow


Bright yellow



Brilliant yellow (corr. to N = 1 with sodium light)


Brown yellow








Deep red




Brilliant yellow

Indigo-violet (passage from warm to cold colors)



Sky blue


Green blue





Bright green



♦Correspondent to order 1 with the use of Sodium vapor light

in other words they are the contour lines of the difference of the principal N is called Isochromatic order.

If observation is carried out in white light, isochromatics appear colored by the complementary colors of the extinguished ones,[31] whose wavelength X is linked to the principal stresses difference by the relationship, Eq. 3.37:

B ■ (ai — 02)d = NX (3.58)

with N given respectively by Eqs. 3.41 and 3.55 for crossed and parallel polarizers. In this way, each value of d(n2 — n1) i. e. (a1 – a2) is distinguishable for a characteristic color of transmitted light, Table3.1.


Подпись: Absorbed colorsПодпись: Transmitted colorsПодпись:Подпись: First Order Second Order Third Order К — »K Подпись:Both Polarizer and Analyzer Along the y AxisLet us now continuously increase the difference (o – a2), starting from zero value. If the principal stresses difference is zero, the transmitted light intensity is zero and a black fringe appears. The light of all wavelengths is absorbed and order N = 0 occurs.

For higher stress differences, (keeping N = 1), extinction happens for wave­lengths smaller than the minimum of the visible spectrum (violet) and the transmitted light varies from black, through shades of gray, up to white at about 260 nm. The first colors clearly defined appear due to the extinction of a wavelength of approximately 390 nm (violet) (NX = 1 ■ 390 nm), and the transmitted light color, complementary to the absorbed one is yellow. The extinction occurs for the following wavelengths of the visible spectrum in this order: blue, green, yellow, orange and red. The colors are then transmitted respectively in the sequence: yellow, orange, red, purple, blue, green, Fig.3.14.[32]

At about 577 nm extinction of the first order of colors N = 1 is complete. If the difference (o – a2) increases, extinctions begin for twice the wavelengths of the visible spectrum (N = 2) and the colors are extinguished a second time. If the difference becomes higher, the number of wavelengths cancelled at the same time tends to increase. The second order is complete at about 1100 nm. From the third order the transmitted colors are the sum of several complementary colors and appear faded, tending to pink, pale green, and finally, to white. Theoretically, the estimation
of the color of the transmitted light could offer a way to calculate the difference of the principal stresses. Since the isochromatic of integer order corresponds to a precise multiple of the wavelength of the monochromatic light used, the estimation of an intermediate color in white light would make it possible to evaluate the frac­tional order of isochromatic. However, estimation is uncertain and it is preferable to use white light only for identifying the isochromatic of zero order, which appears always black.

Conversely, the accurate counting of of isochromatic orders is done in mono­chromatic light, because they appear sharp up to very high (10 and more) orders. If sodium vapor lamps are used (I = 576.6nm), the first order corresponds to the indigo-violet color observed in white light.