L-9: Wavelength Measurement and Holograms

OBJECTIVES:

1. To measure the wavelength of laser light by using a steel rule as a diffraction grating; to observe and understand holograms.

APPARATUS:

He-Ne laser; steel rule; tripod stand & angle clamp; holograms; mercury arc; diverging lenses; 15 meter tape.

Part I - Wavelength Measurement

INTRODUCTION:

Specular reflection of the laser beam from a steel ruler gives a zero order image. In Fig. 1 the distance $h_{0}$ and $D$ determine $\theta_r$, the angle of reflection: $\theta_r=\tan^{-1}(D/h_{0}$) and hence also $\theta_i$, the angle of incidence.

The markings on the steel ruler form a coarse reflection grating which also yields higher order diffracted images. The angle of diffraction for the $n^{th}$ order is

$$\delta _{n} = \tan^{-1}(D/h_{n})   .$$

Figure 1: Schematic of diffraction from ruler markings.

Let $d$ be the separation of lines on the steel ruler. For the $n^{th}$ order the path difference, $\Delta$, between waves from successive ruler lines is $n \lambda$. From Fig. 2, $\Delta =\mbox{AC}-\mbox{BE}$. But $\mbox{AC} = d \cos (90^o-\theta_i)=d \sin \theta_i$ and $\mbox{BE}= d \cos (90^o-\delta_n)=d \sin \delta_n$.
Hence

$$\Delta = d (\sin \theta_i - \sin \delta _{n}) = n \lambda   .$$

Figure 2: Geometry required for constructive interference.

QUESTION: To minimize uncertainty in $\lambda$ should you measure a high or low order diffracted image?

EXPERIMENT:

Locate the zero order spectrum (specular reflection) by moving ruler sideways so beam misses the small (diffracting) divisions. Return ruler so one has the high order diffraction images. Make all the relevant measurements. Calculate $\lambda$ and compare with the accepted $\lambda =633$ nm. For the error estimate first find the uncertainty in i and $\delta _{n}$ (by estimating reasonable ranges of $D$, $h_0$ and $h_n$). Then find the corresponding uncertainties in sin i and in $\sin \delta_n$ (and hence in $\lambda$).

Part II - HOLOGRAMS

1. To observe a hologram: For this use a high intensity mercury arc with a green color filter. Have the room reasonably dark and stand about 3 meters in front of the mercury arc source. Look through the hologram but not directly at the source. The images are virtual and three-dimensional.

   Alternatively, diverge a laser beam to illuminate the hologram and view it as in Fig. 3.
   
   
   
   
   

   Figure 3:

   

   OPTIONAL: Simultaneously illuminate the hologram with a diverged red laser light and a green light from the Hg arc. Note the larger red image.

2. An explanation of the HOLOGRAM: To understand a hologram image, consider in Fig. 4 the formation for the simplest possible object, a small particle at O$_1$. The glass plate B partially reflects the plane waves from the laser and partially transmits them to illuminate the object.

   Figure 4:

   

   The reflected plane waves plus the spherical waves scattered from O$_1$ meet in the photograhic plate P and produce an interference pattern. The spacing d between regions of constructive interference is $d = \lambda /\sin \theta$ (see in Fig. 4 the enlargement of the region around I). This interference results in lines with spacing d in the photographic plate and constitute a diffraction grating of spacing d. Note that $\theta$ and $d$ vary with position on the plate. This diffraction grating of variable spacing constitutes the hologram for our small particle at O$_1$.

   If we illuminate this grating with plane waves traveling in the direction BD, the same direction as the reflected plane waves (from the beam splitter B) used to to form the hologram, we observe constructive interference (a first order spectrum) in the direction given by the regular grating equation $\sin \theta = \lambda /d$: namely the direction of the original wave from O$_1$ which formed the hologram. Because d varies from place to place on the plate, no matter where we look through the plate the light in the first order image comes from the direction O$_1$. Thus the spherical wave fronts originally from O$_1$ are reconstructed, and an observer at E sees a virtual image of O$_1$ at O$_1$.

   A similar treatment holds for a second scattering point O$_2$, and the brightness of the image point will depend on the contrast in the interference pattern which in turn depends on the brightness of the object point. The image of an extended object thus forms point by point, and the image is three dimensional because the virtual image points coincide with the corresponding object points in space. If one views the hologram with longer $\lambda$ light, all angles increase and a magnified image results.

   A first order diffraction beam also appears on the other side of the normal to the hologram, and the diffracted light makes the same angle with the normal to the hologram as the original waves from O$_1$. Thus a real image of O$_1$ forms at O$'_1$ (Fig. 5) and similarly for all other object points. This real image can be caught on a screen, one plane in focus at a time.

   Figure 5: