# Patent

## Title
Dynamic optical devices

## Abstract

The invention provides an optical device, including a light-transmissive substrate, and a pair of different, parallel gratings including a first grating and a second grating, located on the substrate at a constant distance from each other, each of the pair of parallel gratings including at least one sequence of a plurality of parallel lines, wherein the spacings between the lines gradually increase from one edge of the grating up to a maximum distance between the lines, and wherein the arrangement of lines in the second grating is in the same direction as that of the first grating. A system utilizing a plurality of such optical devices is also disclosed.

## Background
FIELD OF THE INVENTION
The present invention relates to diffractive optical devices (DOEs), and to devices which include a plurality of chirped diffractive optical elements carried by a common light-transmissive substrate.
The present invention is capable of being implemented in a large member of applications. For purposes of example only, implementations in division multiplexing/demultiplexing systems, compact optical switches and compact optical scanners are indicated herein.
BACKGROUND OF THE INVENTION
Recently, there have been significant advances in optical fiber technology for telecommunication systems. One of the proposed methods of exploiting the high potential bandwidth of optical fibers more efficiently, is by wavelength division multiplexing (WDM). With this technique, a large number of communication channels can be transmitted simultaneously over a single fiber. Various systems for implementing WDM have been proposed, including systems based on birefringent materials, surface relief gratings, Mach-Zender interferometry and waveguides. These proposed systems generally suffer from low efficiencies, or from a strict limitation on the number of possible channels.
Another proposed approach is to use a thick reflection hologram. However, the necessity to use a conventional aspheric lens for collimating and/or focusing the light waves makes such systems bully and space consuming. Furthermore, a single holographic element is very sensitive to the signal wavelength, which is usually strongly dependent on temperature.
In many optical systems, scanning of a plane wave over a wide field of view, or linear scanning of a focused beam on a plane, is required. A few examples are angular scanners for Laser-Radar, whereby the transmitted narrow beam is to cover a solid angle much wider than the angular divergence of the beam; aiming systems in which the central aiming point moves as a function of the target range and velocity; linear scanners for laser printers or plotters, and others. In the existing systems, beam steering is performed with conventional optical elements, such as a polygonal mirror or a pair of prisms. These systems suffer from various drawbacks: the scanning unit is relatively large and heavy, limiting the performance of systems in which compactness is a requirment; mass production is quite expensive; the scanning rate is severely limited by the mechanical system; rotating systems usually suffer from wobble which must be restrained in order to allow accurate scanning.
Several proposals have been made to perform beam steering by microlens array translation with either diffractive or refractive lenses. These approaches usually suffer from high aberrations at small f-numbers. In addition, they must rely on fairly complicated and costly equipment, which often limits the performance of the microlens arrays.
DISCLOSURE OF THE INVENTION
It is therefore a broad object of the present invention to provide a compact, relatively inexpensive, accurate and simple beam steering optical device having a high scanning rate.
It is a further object of the invention to provide a compact, relatively inexpensive, accurate and simple optical device for wavelength division multiplexion/demultiplexion having high spectral separation.
It is a still further object of the invention to provide an optical device having a substrate wherein a slight change in the refractive index of the substrate will cause an angular deviation in the output beam.
It is a still further object of the invention to provide an optical device providing a large deviation coefficient, so that with a minute refractive index change, significant deviation in the output beam is achieved.
In accordance with the present invention, there is therefore provided an optical device, comprising a light-transmissive substrate, and a pair of different, parallel gratings including a first grating and a second grating, located on said substrate at a constant distance from each other, each of said pair of parallel gratings comprising at least one sequence of a plurality of parallel lines, wherein the spacings between said lines gradually increase from one edge of the grating up to a maximum distance between said lines, and wherein the arrangement of lines in said second grating is in the same direction as that of said first grating.



BRIEF DESCRIPTION OF THE DRAWINGS
The invention will now be described in connection with certain preferred embodiments with reference to the following illustrative figures so that it may be more fully understood.
With specific reference now to the figures in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice.
In the drawings:
FIGS. 1a, 1b, 1c, 1d, 1e, 1f, 1g and 1h illustrate the geometry of some possible embodiments of a device according to the present invention;
FIGS. 2a and 2b schematically illustrate ray tracing output when passing through two gratings of the device according to the invention;
FIG. 3 schematically illustrates an iterative procedure according to the invention;
FIG. 4 schematically illustrates the conversion of a plane wave to a linear point scanner by means of a focusing lens;
FIG. 5 schematically illustrates the utilization of the double-grating configuration, constructed to provide a wavelength division demultiplexing system;
FIGS. 6a and 6b schematically illustrate the utilization of the double-grating configuration, constructed to be employed for light intensity attenuation or in light amplitude modulation;
FIG. 7 schematically illustrates an array of identical grating couples;
FIG. 8 schematically illustrates a further embodiment for reducing the thickness of the substrate in order to achieve a more compact device;
FIG. 9 is a graph illustrating the results of simulations which calculate the dispersion of a system as a function of the refractive index of the substrate, incorporating the device of the present invention;
FIG. 10 is a graph illustrating the results of simulations which calculate the output angle from the second grating as a function of the refractive index of the substrate, incorporating the device of the present invention;
FIG. 11 is a graph illustrating the results of simulations which calculate the wavelength, at an output angle of ρ=20° as a function of the refractive index of the substrate of the device according to the present invention;
FIGS. 12a and 12b are graphs illustrating the results of simulations which calculate the grating period (in line-pairs/mm) of gratings G1 and G2 as a function of x (FIG. 12a) and ξ (FIG. 12b), respectively;
FIG. 13 is a side view of a first stage of a switching system incorporating a device according to the present invention;
FIG. 14 is a side view of a second stage of the switching system of FIG. 13;
FIG. 15 is a top view of an optical switching system;
FIG. 16 is a schematic diagram illustrating S-polarization of an incoming beam, and
FIG. 17 is a schematic diagram illustrating S-polarization of an uniform and symmetrical incoming beam.



DETAILED DESCRIPTION
In its simplest form, as shown in FIG. 1a, the optical device 2 of the present invention includes a light-transmissive substrate 4 having two facets or surfaces 6, 8. A plurality of parallel lines 10 are made on surface 6, constituting a first grating A. The spacings between the lines increase from one edge 12 of the surface to its other edge 14, according to mathematical formulae. The arrangement of lines 16 on surface 8 forms a second grating B. The spacings between the parallel lines 16 of second grating B increase in the same direction as those of grating A.
According to the embodiment of FIG. 1b, the surfaces 6 and 8, respectively, bear gratings C and D, each grating being formed of parallel lines, the spacings of which increase from one edge 18 and 20, respectively, of the surfaces to their centers, and then decrease towards the other respective edge 22, 24, in a symmetrical manner.
FIG. 1c depicts a modification of the arrangement of FIG. 1b, wherein the substrate 4 bears two gratings A, A′ and B, B′ according to the arrangement shown in FIG. 1a. Similarly, the substrate of FIG. 1d bears gratings C, C′ and D, D′ on its surfaces 6 and 8, as in the arrangement shown in FIG. 1b. 
The embodiment of FIG. 1e includes a substrate 4 having gratings A and B as shown in FIG. 1a, formed on a single surface 8. Similarly, FIG. 1f illustrates gratings C and D as in FIG. 1b, formed on surface 8; FIG. 1g illustrates the gratings A, A′, B, B′ of FIG. 1c , formed on a single surface 8, and FIG. 1h shows gratings C, C′, D, D′ formed on a single surface 8.
In the double grating system shown in FIGS. 2a and 2b, a monochromatic plane wave W is coupled inside a light-transmissive substrate 4 by a first grating G1 on surface 8 and then is coupled out by the second grating G2 formed on surface 6. The refractive index of the substrate can be dynamically controlled by external means, including, but not limited to, applying an electric field to the substrate or by illumination with a strong short-wavelength light source (not shown). There are many materials with which the electro-optic effect can be used to control the refractive index of the material. One such well-known material is Lithium-Niobate (LiNbO3), which is commercially available and which has a very fast time response, in the order of 10−9 second. However, many other materials, crystals and polymers, can just as well be used for the desired purpose.
The present invention is intended to provide an optical system wherein a change in the refractive index of the substrate yields an angular deviation of the output wave. That is, when the refractive index is ν1, the output wave Wo emerges from the second grating G2 formed on surface 6, at an angle ρ1, with respect to the substrate plane (FIG. 2a). However, when the refractive index is changed to ν2, the output wave Wo is deviated by an angle Δρ; that is, the output wave emerges from grating G2 at a different angle ρ2 to the substrate plane (FIG. 2b). Hence, a continuous change in the refractive index induces a continuous angular steering of the output wave. This angular steering can be converted into linear scanning of a focused beam by means of an appropriate converging lens.
It is assumed that the input wave Wi impinges on the first grating G1 at an angle βo to the normal of the substrate. The input and output rays remain in the same meridional plane without any loss of generality. Therefore, the two grating functions are invariant in y (the axis normal to the meridional plane), and depend only on x, the plane of grating G1. The distance d between G1 and G2 is normalized to 1. In the iterative procedure, shown in FIG. 3, an initial point x0 is chosen on grating G1. The incoming ray Wi of wavelength λ1 is traced (solid line) from x0 to a point ξ0 on grating G2 in a chosen direction β1(x0). The grating function of G1 at xo is




  
    
      
        
          
            
              φ
              1
            
            ⁡
            
              (
              
                x
                0
              
              )
            
          
          =
          
            
              
                2
                ⁢
                
                  
                
                ⁢
                π
              
              
                λ
                1
              
            
            [
            
              
                
                  v
                  1
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      o
                    
                    )
                  
                
              
              +
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  β
                  0
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




where both β0 and β1(x0) are defined to be positive in FIG. 3. Without any loss of generality, it is assumed that the output wave for the refractive index ν=ν1 emerges at an angle ρ1 to the substrate plane. Hence, the grating function of G2 on ξ0 is





  
    
      
        
          
            φ
            2
          
          ⁡
          
            (
            
              ξ
              0
            
            )
          
        
        =
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            
              λ
              1
            
          
          ⁢
          
            
              (
              
                
                  
                    v
                    1
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      β
                      1
                    
                    ⁡
                    
                      (
                      
                        x
                        o
                      
                      )
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        2
        )
      
    
  




The refractive index form is now changed from ν1 to ν2 and, as a result of the change in the refractive index, the output wave emerges from the substrate at an angle ρ2 to the substrate plane. The direction of the impinging ray at ξ0 is




  
    
      
        
          
            v
            2
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              o
            
            ⁡
            
              (
              
                ξ
                0
              
              )
            
          
        
        =
        
          
            
              
                
                  λ
                  1
                
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
              
              ⁢
              
                
                  φ
                  2
                
                ⁡
                
                  (
                  
                    ξ
                    0
                  
                  )
                
              
            
            -
            
              sin
              ⁢
              
                
              
              ⁢
              
                ρ
                2
              
            
          
          =
          
            
              
                v
                1
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              
                
                  β
                  1
                
                ⁡
                
                  (
                  
                    x
                    0
                  
                  )
                
              
            
            -
            
              sin
              ⁢
              
                
              
              ⁢
              
                ρ
                2
              
            
            +
            
              sin
              ⁢
              
                
              
              ⁢
              
                
                  ρ
                  1
                
                .
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




Tracing a ray 26 into G1 (dashed lines) calculates the grating function φ1 on x1 as




  
    
      
        
          
            
              
                
                  φ
                  1
                
                ⁡
                
                  (
                  
                    x
                    1
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  
                    λ
                    1
                  
                
                [
                
                  
                    
                      v
                      2
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        β
                        1
                        o
                      
                      ⁡
                      
                        (
                        
                          ξ
                          o
                        
                        )
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      0
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    
                      λ
                      1
                    
                  
                  [
                  
                    
                      
                        v
                        1
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          β
                          1
                        
                        ⁡
                        
                          (
                          
                            x
                            o
                          
                          )
                        
                      
                    
                    -
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      ρ
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        β
                        0
                      
                    
                  
                  ]
                
              
              ,
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




where the constant δρ is defined as δρ≡sin ρ2−sin ρ1. By switching back to ν1, the procedure can continue and the input angle at x1 is calculated to be





  
    
      
        
          
            
              v
              1
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                β
                1
              
              ⁡
              
                (
                
                  x
                  1
                
                )
              
            
          
          =
          
            
              
                
                  
                    λ
                    1
                  
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                
                ⁢
                
                  
                    φ
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      1
                    
                    )
                  
                
              
              -
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  β
                  0
                
              
            
            =
            
              
                
                  v
                  1
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      0
                    
                    )
                  
                
              
              -
              
                δ
                ⁢
                
                  
                
                ⁢
                ρ
              
            
          
        
        ,
      
    
    
      
        (
        5
        )
      
    
  




which yields

sin β1(x1)−sin β1(x0)=δρ/ν1  (6)

Now, the iterative procedure continues to calculate φ1 and φ2 at the points x1,x2,x3 . . . xi and ξ1,ξ2,ξ3 . . . ξi respectively. Following the same procedure as in Equations 1-6, it can be found that for each xi

sin β1(xi+1)−sin β1(x1)=δρ/ν1  (7)

With this iterative procedure, the desired grating functions φ1 and φ2 can be calculated only for a finite number of pairs. Knowing the discrete values of these functions, the values of the other points on the gratings can be calculated numerically by using interpolation methods, however, since the number of known values is comparatively small, the interpolation procedure may be complicated and time-consuming. Instead, analytic functions are presented herein, which are relatively easy to compute for an iterative procedure.
As shown in FIG. 3, the distance Δx1≡xi+1−x1 that the tracing procedure “moves” each step along the x axis is approximately




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              x
              i
            
          
          ≈
          
            
              Δ
              ⁢
              
                
              
              ⁢
              
                β
                i
              
            
            
              
                cos
                2
              
              ⁢
              
                
                  β
                  1
                
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




where Δβi is given by





  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              β
              i
            
          
          ≈
          
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      i
                    
                    )
                  
                
              
              -
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                    o
                  
                  ⁡
                  
                    (
                    
                      ξ
                      i
                    
                    )
                  
                
              
            
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  β
                  1
                
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                
                  (
                  
                    
                      
                        v
                        1
                      
                      
                        v
                        2
                      
                    
                    -
                    1
                  
                  )
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      i
                    
                    )
                  
                
              
              +
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  ρ
                
                
                  v
                  2
                
              
            
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  β
                  1
                
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        9
        )
      
    
  




Inserting Equation 9 into Equation 8 yields




  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            x
            i
          
        
        ≈
        
          
            
              
                
                  (
                  
                    
                      
                        v
                        1
                      
                      
                        v
                        2
                      
                    
                    -
                    1
                  
                  )
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    β
                    1
                  
                  ⁡
                  
                    (
                    
                      x
                      i
                    
                    )
                  
                
              
              +
              
                
                  
                    v
                    1
                  
                  
                    v
                    2
                  
                
                ⁢
                δ
                ⁢
                
                  
                
                ⁢
                ρ
              
            
            
              
                cos
                3
              
              ⁢
              
                
                  β
                  1
                
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        10
        )
      
    
  




Dividing Equation 10 by Equation 7 and generalizing the equation for each x on G1 yields




  
    
      
        
          
            
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        
                          x
                          +
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                        )
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                
              
              =
              
              ⁢
              
                
                  
                    v
                    1
                  
                  
                    v
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          (
                          
                            
                              v
                              1
                            
                            -
                            
                              v
                              2
                            
                          
                          )
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            β
                            1
                          
                          ⁡
                          
                            (
                            x
                            )
                          
                        
                      
                      
                        
                          cos
                          3
                        
                        ⁢
                        
                          
                            β
                            1
                          
                          ⁡
                          
                            (
                            x
                            )
                          
                        
                        ⁢
                        δ
                        ⁢
                        
                          
                        
                        ⁢
                        ρ
                      
                    
                    -
                    
                      1
                      
                        
                          cos
                          3
                        
                        ⁢
                        
                          
                            β
                            1
                          
                          ⁡
                          
                            (
                            x
                            )
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    a
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  
                    
                      cos
                      3
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                
                -
                
                  b
                  
                    
                      cos
                      3
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where the constants a and b are defined as a≡(ν1−ν2)ν1/(ν2δρ) and b≡ν1/ν2. Defining ƒ(x)≡sin β1(x) yields





  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            x
          
          
            
              f
              ⁡
              
                (
                
                  x
                  +
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                  
                
                )
              
            
            -
            
              f
              ⁡
              
                (
                x
                )
              
            
          
        
        =
        
          
            
              af
              ⁡
              
                (
                x
                )
              
            
            
              
                (
                
                  1
                  -
                  
                    
                      f
                      2
                    
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
                )
              
              
                3
                2
              
            
          
          -
          
            
              b
              
                
                  (
                  
                    1
                    -
                    
                      
                        f
                        2
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  )
                
                
                  3
                  2
                
              
            
            .
          
        
      
    
    
      
        (
        12
        )
      
    
  




For a system with a small change in the refractive index, it may be assumed that Δx<<1. Hence, the following approximation may be written:




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            x
          
          
            
              f
              ⁡
              
                (
                
                  x
                  +
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                  
                
                )
              
            
            -
            
              f
              ⁡
              
                (
                x
                )
              
            
          
        
        ≈
        
          
            
              d
              x
            
            
              d
              f
            
          
          .
        
      
    
    
      
        (
        13
        )
      
    
  




Inserting Equation 12 into Equation 13 yields the following differential equation:




  
    
      
        
          
            d
            x
          
          
            d
            f
          
        
        =
        
          
            
              af
              ⁡
              
                (
                x
                )
              
            
            
              
                (
                
                  1
                  -
                  
                    
                      f
                      2
                    
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
                )
              
              
                3
                2
              
            
          
          -
          
            
              b
              
                
                  (
                  
                    1
                    -
                    
                      
                        f
                        2
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  )
                
                
                  3
                  2
                
              
            
            .
          
        
      
    
    
      
        (
        14
        )
      
    
  




The solution of this equation is




  
    
      
        
          x
          =
          
            
              
                a
                -
                bf
              
              
                
                  1
                  -
                  
                    f
                    2
                  
                
              
            
            +
            
              c
              0
            
          
        
        ,
      
    
    
      
        (
        15
        )
      
    
  




where c0 is found from the boundary condition (ƒ(x=0)=0) to be

c0=−a.  (16)

Thus, the solution of Equation 15 is




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            f
            ⁡
            
              (
              x
              )
            
          
          =
          
            
              
                ab
                +
                
                  
                    
                      
                        (
                        ab
                        )
                      
                      2
                    
                    -
                    
                      
                        (
                        
                          
                            a
                            2
                          
                          -
                          
                            
                              (
                              
                                x
                                +
                                a
                              
                              )
                            
                            2
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            b
                            2
                          
                          +
                          
                            
                              (
                              
                                x
                                +
                                a
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  b
                  2
                
                +
                
                  
                    (
                    
                      x
                      +
                      a
                    
                    )
                  
                  2
                
              
            
            .
          
        
      
    
    
      
        (
        17
        )
      
    
  




Inserting Equation 18 into Equation 1 shows the grating function φ1(x) of grating G1 to be




  
    
      
        
          
            φ
            1
          
          ⁡
          
            (
            x
            )
          
        
        =
        
          
            
              
                2
                ⁢
                π
              
              
                λ
                1
              
            
            ⁡
            
              [
              
                
                  
                    v
                    1
                  
                  ⁢
                  
                    
                      ab
                      +
                      
                        
                          
                            
                              (
                              ab
                              )
                            
                            2
                          
                          -
                          
                            
                              (
                              
                                
                                  a
                                  2
                                
                                -
                                
                                  
                                    (
                                    
                                      x
                                      +
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                
                                  b
                                  2
                                
                                +
                                
                                  
                                    (
                                    
                                      x
                                      +
                                      a
                                    
                                    )
                                  
                                  2
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        b
                        2
                      
                      +
                      
                        
                          (
                          
                            x
                            +
                            a
                          
                          )
                        
                        2
                      
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    0
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        18
        )
      
    
  




The grating function φ2(ξ) of the second grating G2 can be calculated by a similar procedure to that described above with regard to Equations 1-18. The result of these calculations is




  
    
      
        
          
            
              φ
              2
            
            ⁡
            
              (
              ξ
              )
            
          
          =
          
            
              
                2
                ⁢
                π
              
              
                λ
                1
              
            
            ⁡
            
              [
              
                
                  
                    
                      a
                      ′
                    
                    ⁢
                    
                      b
                      ′
                    
                  
                  +
                  
                    
                      
                        
                          (
                          
                            
                              a
                              ′
                            
                            ⁢
                            
                              b
                              ′
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        
                          (
                          
                            
                              
                                a
                                ′
                              
                              2
                            
                            -
                            
                              
                                (
                                
                                  x
                                  +
                                  
                                    a
                                    ′
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              ′2
                            
                            +
                            
                              
                                (
                                
                                  x
                                  +
                                  
                                    a
                                    ′
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    b
                    ′2
                  
                  +
                  
                    
                      (
                      
                        x
                        +
                        
                          a
                          ′
                        
                      
                      )
                    
                    2
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        19
        )
      
    
  




where the constants a′ and b′ are now defined as a′≡(ν1−ν2)ν1/(ν2δρ) and b≡ν1/ν2−1.

It is important to note that the solution given in Equations 18 and 19 is not the most accurate analytical one, but rather, is an approximate solution, illustrating the capability of finding an easy and fast analytical solution for the aforementioned iterative process. However, for most cases, as will be described in further detail below, this solution is accurate enough.
In other cases, in which the system has a high numerical aperture, a diffraction-limited performance is required; hence, as the accuracy of the solution is crucial, a more accurate analytical solution must be found. This can be done by using a more accurate value for Δx in Equation 10, or by finding higher coefficients in the power series of Equation 13. Furthermore, after calculating the values of φ1 and φ2 on the discrete points x1,x2,x3 . . . xi and ξ1,ξ2,ξ3 . . . ξ4 respectively, as described earlier, a numerical or semi-numerical method can be employed to find the required accurate solution.
In addition, the solution, even the most accurate one, is computed for two discrete values of the refractive index, ν1 and ν2, but it can be assumed that for the dynamic range of ν={ν1−Δν/2, ν2+Δν/2}, where Δν≡ν2−ν1, a change of the refractive index to ν yields a deviation of the output wave to the direction

sin ρν=sin ρ1+δρ·g(ν),  (20)

where g(ν) is a monotonic function having the values of g(ν)=0,1 for ν=ν1,ν2, respectively. Since g(ν)=x is a continuous and monotonic function, the inverse function g−1(x)=ν can also be found.

The angular steering of the output wave can be translated into linear scanning. As shown in FIG. 4, there are provided two gratings, G1 and G2, parallel to each other and located at the surfaces of a light-transmissive substrate 4, where the refractive index of the substrate can be dynamically controlled. A focusing lens 28 is provided at a certain distance from grating G2, forming a focus at the imaging plane 30. The angular steering of the plane wave is converted by the focusing lens 28 into linear scanning of a point. Each plane wave, corresponding to a different refractive index, is focused by the focusing lens 28 onto the image plane 30, where the foci of the various plane waves are laterally displaced along a straight line.
The beam steering discussed above is performed only in the x axis. However, a two-dimensional scanner can easily be fabricated by combining two different parallel substrates, whereby the scanning direction of each substrate is normal to that of the other.
Another problem is that since the relation between the coordinates of the two gratings is x=ξ+tan β(ξ), the lateral dimension of the second grating is much smaller than that of the first grating in the x-axis, whereby the dimensions remain the same along the y-axis. That is, the output beam has a non-symmetrical form, whereby the lateral dimension along the y-axis is much larger than that along the x-axis. Apparently, this problem does not exist when the scanning system is composed of two substrates, as described above. But, even for a single substrate configuration, this problem can be solved by contracting the wider dimension, using optical means such as a folding prism.
Thus far, the exploitation of the double grating configuration embedded on a light-transmissive substrate, mainly for scanning purposes, has been described. However, the same configuration could be used also for other applications, including wavelength-division multiplexing, optical switching, light intensity attenuation, light amplitude-modulation and many others.
Considering now the above-described system of Equations 1-19, in the special case where β0=0 and the refractive index ν1 is fixed but the wavelength is changed from λ1 to λ, the direction of the output ray for each point x on the first grating G1 is




  
    
      
        
          
            v
            1
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              λ
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            
              λ
              2
            
            
              2
              ⁢
              π
            
          
          ⁢
          
            
              
                φ
                1
              
              ⁡
              
                (
                x
                )
              
            
            .
          
        
      
    
    
      
        (
        21
        )
      
    
  




Inserting Equation 1 into Equation 21 yields




  
    
      
        
          
            v
            1
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              λ
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          
            v
            1
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              
                β
                1
              
              ⁡
              
                (
                x
                )
              
            
            .
          
        
      
    
    
      
        (
        22
        )
      
    
  




Hence,




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              λ
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              
                β
                1
              
              ⁡
              
                (
                x
                )
              
            
            .
          
        
      
    
    
      
        (
        23
        )
      
    
  




However, in the former case, where the wavelength λ=λ1 is fixed and the variable is the refractive index ν, the direction of the output ray for each point x on the first grating G1 is




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              v
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            
              v
              1
            
            v
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              
                β
                1
              
              ⁡
              
                (
                x
                )
              
            
            .
          
        
      
    
    
      
        (
        24
        )
      
    
  




Hence, the output direction for each point x on the grating G1 is equal in both cases, that is

sin β1ν(x)=sin β1λ(x)  (25)

in a condition that





  
    
      
        λ
        =
        
          
            
              v
              1
            
            v
          
          ⁢
          
            
              λ
              1
            
            .
          
        
      
    
    
      
        (
        26
        )
      
    
  




Consequently, each ray with a direction sin β1νimpinges on the second grating G2 at a point ξ where the grating function is




  
    
      
        
          
            φ
            2
          
          ⁡
          
            (
            ξ
            )
          
        
        =
        
          
            
              2
              ⁢
              π
            
            
              λ
              1
            
          
          ⁢
          
            
              (
              
                
                  v
                  ⁢
                  
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      β
                      1
                    
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    v
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        27
        )
      
    
  




The output direction for each point ξ on G2 is now




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              λ
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            
              λ
              
                2
                ⁢
                π
              
            
            ⁢
            
              
                φ
                2
              
              ⁡
              
                (
                ξ
                )
              
            
          
          -
          
            
              v
              1
            
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                
                  β
                  1
                  λ
                
                ⁡
                
                  (
                  x
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




Inserting Equations 25 and 27 into Equation 28 yields




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              λ
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            
              λ
              
                λ
                1
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                β
                1
              
              ⁡
              
                (
                x
                )
              
            
          
          +
          
            
              λ
              
                λ
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              ρ
              v
            
          
          -
          
            
              λ
              
                λ
                1
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                
                  β
                  1
                
                ⁡
                
                  (
                  x
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




Inserting Equations 20 and 26 into Equation 29 yields




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              λ
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            
              λ
              
                λ
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              ρ
              v
            
          
          =
          
            
              
                λ
                
                  λ
                  1
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ρ
                      1
                    
                  
                  +
                  
                    δρ
                    ·
                    
                      g
                      ⁡
                      
                        (
                        v
                        )
                      
                    
                  
                
                )
              
            
            =
            
              
                λ
                
                  λ
                  1
                
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        1
                      
                    
                    +
                    
                      δρ
                      ·
                      
                        g
                        ⁡
                        
                          (
                          
                            
                              
                                v
                                1
                              
                              ⁢
                              
                                λ
                                1
                              
                            
                            λ
                          
                          )
                        
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




Regarding now the more general case in which both the wavelength and the refractive index are variables, the direction of the output ray for each point x on the first grating G1 is




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              β
              1
              
                v
                ,
                λ
              
            
            ⁡
            
              (
              x
              )
            
          
        
        =
        
          
            
              v
              1
            
            v
          
          ⁢
          
            λ
            
              λ
              1
            
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              
                β
                1
              
              ⁡
              
                (
                x
                )
              
            
            .
          
        
      
    
    
      
        (
        31
        )
      
    
  




This case is equivalent to changing only the refractive index to νo, wherein




  
    
      
        
          v
          o
        
        =
        
          
            
              v
              ⁢
              
                
              
              ⁢
              
                λ
                1
              
            
            λ
          
          .
        
      
    
    
      
        (
        32
        )
      
    
  




Consequently, the output direction for each point ξ on G2 is




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              
                v
                ,
                λ
              
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              
                v
                o
              
            
            .
          
        
      
    
    
      
        (
        33
        )
      
    
  




Inserting Equation 20, where νo=, into Equation 33 yields




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              
                v
                ,
                λ
              
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                +
                
                  δρ
                  ·
                  
                    g
                    ⁡
                    
                      (
                      
                        v
                        o
                      
                      )
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        34
        )
      
    
  




Inserting Equation 32 into Equation 34 yields




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              ρ
              
                v
                ,
                λ
              
            
            ⁡
            
              (
              ξ
              )
            
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                +
                
                  δρ
                  ·
                  
                    g
                    ⁡
                    
                      (
                      
                        
                          v
                          ⁢
                          
                            
                          
                          ⁢
                          
                            λ
                            1
                          
                        
                        λ
                      
                      )
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        35
        )
      
    
  




Seemingly, sin ρν,λ(ξ)=sin ρν,λ is a constant over the entire area of the grating G2. As a consequence, the output beam is a plane wave. Therefore, the present invention can also be used as a wavelength-division multiplexing/demultiplexing device.
FIG. 5 illustrates the utilization of the double-grating configuration, constructed to provide a wavelength division demultiptexing system including an optical device 32 linking a single source fiber 34 and a plurality. of receiving fibers at receiving locations RL1, RL2 . . . RLn. The source fiber 34 contains n different communication channels, CC1 . . . CCn, with the wavelengths, λ1 . . . λn respectively. The first grating G1 couples, the corresponding incoming channels into the light transmissive substrate 4, and the second grating G2 couples them out and diffracts them into different directions. Each channel CCi is then focused by a focusing lens 36 onto its receiving fiber RL. The propagation direction of the waves can be inverted to provide a system which multiplexes a number of channels from their separated source fibers onto one receiving fiber. Since the light transmissive substrate can be located very close to the fibers, and the light waves are guided inside the substrate, the system can be compact and easy to use.
It is clear that the embodiment described above can also be used for WDM with a substrate composed of a material with a constant refractive index, that is, when ν=ν1=const. However, it is advantageous to use materials in which the refractive index can be controlled. One of the main drawbacks of using a simple grating as a WDM device is that it is very sensitive to the signal wavelength, which is usually strongly dependant on the temperature. In addition, since the diameter of the fiber core is smaller than 10 microns, the tolerances of the optical system should be very tight, so as to prevent degradation of the optical signal due to even a slight change in the environmental conditions. The necessity for tightening the optical and the mechanical tolerances makes the optical system very expensive and even impractical. In contrast to a simple grating, in the system according to the present invention, changes in environmental conditions can be compensated dynamically by changing the refractive index. For example, since g−1(x) is a continuous and monotonic function, it is possible to find a refractive index ν which fulfills the relation




  
    
      
        
          v
          =
          
            
              λ
              
                λ
                1
              
            
            ·
            
              
                g
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  
                    
                      
                        λ
                        1
                      
                      -
                      λ
                    
                    λ
                  
                  ·
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        1
                      
                    
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      ρ
                    
                  
                
                )
              
            
          
        
        ,
        
          

        
        ⁢
        or
      
    
    
      
        (
        36
        )
      
    
  
  
    
      
        
          
            g
            ⁡
            
              (
              
                
                  v
                  ⁢
                  
                    
                  
                  ⁢
                  
                    λ
                    1
                  
                
                λ
              
              )
            
          
          =
          
            (
            
              
                
                  
                    λ
                    1
                  
                  -
                  λ
                
                λ
              
              ·
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  ρ
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        37
        )
      
    
  




which yields





  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            ρ
            1
          
        
        =
        
          
            λ
            
              λ
              1
            
          
          ⁢
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                +
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    ·
                    
                      g
                      ⁡
                      
                        (
                        
                          
                            v
                            ⁢
                            
                              
                            
                            ⁢
                            
                              λ
                              1
                            
                          
                          λ
                        
                        )
                      
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        38
        )
      
    
  




Substituting Equation 35 into Equation 38 yields

sin ρ1=sin ρν,λ.  (39)

That is, the output has the original direction as if the system has the values of ν1 and λ1. Consequently, compensation may be made for a change in the wavelength by changing the refractive index, and vice-versa. In addition, a taping element 35 attached to the receiving fibers and connected to a control unit 37 of the refractive index, can produce a closed-loop apparatus that will optimally control the performance of the optical system.

Another potential application of the present invention is as an optical switch. In the embodiment illustrated in FIG. 5, each one of the optical channels CC1 . . . CCn, having the wavelengths, λ1 . . . λn respectively, can be routed into one of the output locations RL1 . . . RLn. The routing of the channel CCi to the output location RL1 can be done by setting a refractive index νij that will solve the equation

sin ρν/,λ=sin ρνi,λj  (40)

Inserting Equation 35 into Equation 37 yields the equation




  
    
      
        
          
            
              
                λ
                i
              
              
                λ
                1
              
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                +
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    ·
                    g
                  
                  ⁢
                  
                    (
                    
                      
                        
                          v
                          i
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          λ
                          1
                        
                      
                      
                        λ
                        j
                      
                    
                    )
                  
                
              
              )
            
          
          =
          
            
              
                λ
                j
              
              
                λ
                1
              
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                +
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    ·
                    g
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            v
                            ⁢
                            
                              
                            
                          
                          1
                        
                        ⁢
                        
                          λ
                          1
                        
                      
                      
                        λ
                        j
                      
                    
                    )
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        41
        )
      
    
  




which has the solution





  
    
      
        
          
            v
            i
            j
          
          =
          
            
              λ
              
                λ
                1
              
            
            ·
            
              
                g
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  
                    
                      
                        
                          λ
                          1
                        
                        -
                        λ
                      
                      λ
                    
                    ·
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ρ
                          1
                        
                      
                      
                        δ
                        ⁢
                        
                          
                        
                        ⁢
                        ρ
                      
                    
                  
                  +
                  
                    
                      
                        λ
                        j
                      
                      
                        λ
                        i
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        
                          
                            
                              
                                v
                                ⁢
                                
                                  
                                
                              
                              1
                            
                            ⁢
                            
                              λ
                              1
                            
                          
                          
                            λ
                            j
                          
                        
                        )
                      
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        42
        )
      
    
  




whereby ν1 is the original refractive index designated to rout each channel CCk to its respective default output location RLk. Apparently, this embodiment can be generalized to produce an optical switch between n optical channels CC1 . . . CCn and m possible output locations RL1 . . . RLm, wherein n≠m .

Other potential applications of the present invention are in light intensity attenuation, or light amplitude modulation. In many fiber-optical applications, it is crucial to control the intensity of the light which is coupled into the fiber. The devices which are currently used for controlling coupled light intensity are mechanical attenuators in the main, which are fairly expensive and suffer from a slow time response. In the embodiment of FIGS. 6a and 6b, the exact direction of the optical wave Wo emerging from the substrate 4 can be controlled by changing the refractive index of the substrate. Angular steering of the wave is converted into linear shifting of the focused wave by means of converging lens 36. Hence, the exact portion of the focused wave to be coupled into the receiving fiber RL can be set, and the device can be operated as an optical intensity attenuator (FIG. 6b). In addition, since the time response of the change in the refractive index is usually very fast, an intensity modulation of the coupled wave can be performed with this device.
Another potential utilization of the present invention is as a monochromator. There are many applications in which it is desired to produce a monochromatic beam out of an optical wave having a much wider spectrum, where the exact selected wavelength of the output should be set dynamically during the operation of the device. As described above for optical switching, if the detector receives light only from a pre-determined direction, the desired wavelength can be set to emerge at that direction by setting the appropriate refractive index. In addition, this device can also be used as a spectrometer by measuring the intensity of the output wave as a function of the wavelength.
The embodiments described above are merely examples illustrating the implementation capabilities of the present invention. The invention can also be utilized in many other potential applications, including, but not limited to, angular drivers for laser range-finders, dynamic aiming systems, laser beam steerer for CD-ROM readers and others, where dynamic control of the direction of an optical wave is desired.
Another issue to be considered, and related principally to the utilization of the system for scanning purposes, is the dimensions of the light transmissive substrate. The relation between the apertures of the first grating G1 and the second grating G2 is given by




  
    
      
        
          
            x
            max
          
          =
          
            
              ξ
              max
            
            +
            
              tan
              ⁢
              
                
              
              ⁢
              
                
                  β
                  1
                  a
                
                ⁡
                
                  (
                  
                    ξ
                    max
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        43
        )
      
    
  




where xmax and ξmax are the apertures of the gratings G1 and G2, respectively; It can be seen that there are two contrary considerations for choosing ξmax. On the one hand, the size of the coupler aperture increases with ξmax; hence, for a given aperture, increasing ξmax decreases d, and thereby a more compact system is obtained. On the other hand, the aberrations of the output wave also increase with β1(ξmax), which increases monotonically with ξmax; hence, by increasing ξmax, the performance of the system is decreased. With these considerations, one might think that an optimal value of ξmax has to be chosen for each different design, according to the desired system size and optical performance. Unfortunately, this optimal value of ξmax usually does not exist. For example, a typical scanner aperture can be approximately a few tens of mm, while the thickness of the substrate cannot be more than 20-30 mm. On the other hand, the aberrations for ξmax>2·d are such that the optical performance is less than the diffraction limit. This limitation might be overcome by assembling an array of n identical gratings, given by





  
    
      
        
          
            G
            j
          
          =
          
            
              
                ∑
                
                  i
                  =
                  1
                
                n
              
              ⁢
              
                
              
              ⁢
              
                
                  G
                  j
                  i
                
                ⁢
                
                  
                
                ⁢
                j
              
            
            =
            1
          
        
        ,
        2
        ,
      
    
    
      
        (
        44
        )
      
    
  




where Gjk=Gji for 1≦i, k≦n. For all l≦i≦n the grating Gji is constructed in relation to the second grating G2i.

Since the first grating G1 is constructed of a large number of facets, the diffraction efficiencies of these facets should be properly set so as to achieve uniform illumination on the second grating G. In addition, special care should be taken to avoid spaces between the elements G1i in order to avoid loss of energy and to ensure an output wave with uniform intensity. To guarantee this, the coordinates xmaxi of the facet G1i should be identical to the coordinate xi+1=0 of the adjacent facet G1i+1. The total area of the grating G1 is given by




  
    
      
        
          R
          ⁢
          
            
          
          ⁢
          
            L
            ⁡
            
              (
              
                G
                1
              
              )
            
          
        
        =
        
          
            n
            ·
            
              x
              max
            
          
          =
          
            n
            ·
            
              
                (
                
                  
                    ξ
                    max
                  
                  +
                  
                    tan
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        β
                        1
                      
                      ⁡
                      
                        (
                        
                          ξ
                          max
                        
                        )
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




For a given system, the parameter ξmax is calculated according to the desired performance and RL(G1) is set according to the system aperture. Hence, it is possible to calculate the required number of facets n, from Equation 45.
An alternative manner of reducing the thickness of the substrate in order to achieve a more compact system is illustrated in FIG. 8. The wave W1, coupled inside the substrate by means of grating G1, does not proceed directly to the second grating G2, but is first reflected a few times off the surfaces 38, 40 of the substrate 4. This reflectance can be induced by a reflective coating on the surfaces of the substrate or, if the coupling angles are high enough, by total internal reflection. In any case, the actual thickness of the substrate is dact=d/(n+1), where n is the number of reflections of the substrate surfaces. Care should be taken to avoid overlap between the coupled wave after the first reflection and the active area of the gratings. In the original embodiment, which is described above, the light proceeds directly from G1 to G2, and hence the gratings are located on opposite sides of the substrate. There are however, configurations with an odd number of reflections from the substrate surfaces, in which the input and the image waves could be located on the same side of the substrate.
An example of the device of FIG. 8 has the following parameters:

δρ/(ν2−ν1)=5; ν1=1.5; λ1=1.5 μm ; d=30 mm; ρ1=20°; β0=0  (46)

The maximum dispersion as a function of the refractive index for a double grating designed according to Equations 18 and 19 can be calculated. The dispersion in the band is ν1±Δν/2, where Δν≡0.02 ν1 and where xmax is d/3=10 mm, hence, ξmax is set to be  ̃1 mm. As a result, the diffraction limit of the system is approximately 1.5 milliradians. The beams are reflected twice off the surfaces of the substrate, hence, the actual thickness of substrate 4 is 10 mm.
FIG. 9 shows the angular dispersion as a function of the refractive index (normalized to ν1). It is evident from this Figure that the maximal dispersion of the double grating is only 0.4 milliradians, which is actually a diffraction limited performance for the system defined in Equation 46. It is even possible to increase the system aperture by a factor of three and still achieve a nearly diffraction limited performance of 0.5 milliradians.
FIG. 10 shows the output angle from the second grating as a function of the refractive index normalized to ν1 for the system defined in Equation 46. The system, has a scanning range of 100 milliradians ( ̃6°) for a refractive index with a dynamic range of Δν≡0.02ν1.
FIG. 11 illustrates the utilization of the present invention as an optical switch and/or as a monochromator. The figure depicts the wavelength, at an output angle of ρ=20°, as a function of the refractive index. By combining the results of FIGS. 10 and 11, it is clear to see that a wavelength change of Δλ=1.5 nm yields an angular deviation of 5 milliradians, which illustrates the performance of the device as a WDM. An even better spectral sensitivity can be achieved by selecting higher deviation ratio (δρ/(ν2−ν1)) for the system.
FIGS. 12a and 12b show the grating period (in line-pairs/mm) of the gratings G1 and G2 as a function of x (FIG. 12a) and ξ (FIG. 12b), respectively. Both functions monotonically increase with ξ and x, and the fabrication process for both gratings should be fairly simple. Furthermore, other gratings with constant grating periods, which can be easily fabricated by conventional techniques such as holographic recording or photo-lithography, may be added to the surfaces of the substrate. As a result, the maximal grating period values of the gratings G1 and G2 can be less than 250 line-pairs/mm.
A combination of the embodiments described above can be utilized to materialize an all-optical switching system, having as an input n communication channels and n different wavelengths λ1 . . . λn, and as an output m, receiving channels RC1 . . . RCm (not shown).
FIG. 13 illustrates a side view (a projection on the ξ-ζ plan) of the first stage of the switching system. The n input channels 42 are coupled by a diffraction grating G1 into the substrate 4, at a location where the substrate is constructed of a passive material. The pair of gratings G1, G2 is used to perform the wavelength-division-demultiplexing, using the method described above with reference to FIG. 5. The second grating traps the output waves inside the substrate by total internal reflection.
FIG. 13 illustrates how a lateral separation between two different channels 44, 46, having the wavelengths λi and λj, respectively, can be obtained. After a total lateral separation between the channels is achieved, an array of gratings 481 to 48n is used to rotate the trapped waves in such a way that the propagation direction of the separated waves is along the η axis, which is normal to the figure plan.
FIG. 14 depicts a side view (a projection on the η-ζ plan) of the second stage of the switching system. After rotation by the gratings 48, the trapped waves impinge on an array of gratings 50, where the substrate at this location is constructed of a dynamic material having a refractive index that can be controlled by an external voltage. Though it cannot be seen in the Figure, it is noted that the different channels are laterally separated along the ξ axis, where each channel 44, 46 has its respective pair of gratings 50, 52 and a separate part of the substrate where the refractive index of each part can be separately controlled. Each pair of gratings 50, 52 is used to control the output angles of the channels 44, 46, using the method described above with reference to FIG. 3. The gratings array 54 couples the trapped waves from the substrate onto a coupling optics 56, 58 to focus the waves into their respective receivers 60, 62.
For a multi-stage optical switching system, an optional array of wavelength converters 64, 66 can be inserted into the optical pass of the waves, in order to convert the wavelength of the wave, which is routed into a receiver RCk into the wavelength λk.
Another subject which should be addressed is the polarization of the incoming light. It is well known that it is simpler to design and fabricate diffracting gratings or elements using the electro-optic effect for S-polarized light than it is for non-polarized or P-polarized light. There are cases where light sources like VCSELs (Vertical Catity Surface Emitting Lasers) are linearly polarized. However, there are many cases, especially those associated with fiber-optical communication, where the polarization of the incoming beam is unknown.
This problem can be solved by utilizing a half-wavelength plate and a polarizing beam-splitter. As illustrated in FIG. 16, light beam 64, having undefined polarization, emerges from a light source 66 and impinges on a polarizing beam-splitter 68. The part of the light beam 70 having P-polarization continues in the same direction, while the S-polarized light beam 72 is reflected and impinges on a folding mirror 74 which reflects the light again to its original direction. By using a half-wavelength plate 76, it is possible to rotate the polarization of the P-polarized light such that it is S-polarized in relation to the grating plane. In such a way, the grating G1 is illuminated with S-polarized light. This solution suffers, however, from two main problems: First, the cross-section of the incoming beam is not symmetrical any more. That is, the lateral dimension of the beam in the ξ-axis is twice as much as the lateral dimension in the η-axis. In addition, since the polarization of the incoming beam is unknown and can be in any orientation, there is uncertainty regarding the energy distribution of the incoming beam along the ξ-axis. This undesgined energy distribution presents a drawback for optical systems where a diffraction-limited performance is required.
FIG. 17 illustrates a modified version of FIG. 16, which solves these two problems. After crossing the half-wavelength plate 76, the light beam 70 is rotated by the folding mirror 78 and then combined with the light beam 72, using a 50% beam-splitter 80. The grating G1 is illuminated now by an S-polarized, uniform and symmetrical light beam 82. Half of the incoming energy 84 is lost during this process, but usually the energy of the incoming light beam is high enough to stand this loss.
Not only S-polarized incoming beams, but also any other linearly polarized light beams, can be created with the system described above.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrated embodiments and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

## Claims

1. An optical device comprising:
a light transmissive substrate having at least two major surfaces, a first surface and a second surface, and edges;
a first and a second grating, each having an X-axis and a Y-axis, located on at least one of said substrate surfaces, at a constant distance from each other;
each grating having a lateral dimension including at least one sequence of a plurality of parallel lines, wherein the spacing between said lines gradually increases in a direction away from an edge of the grating up to a maximum distance between said lines, and wherein the spacing between the parallel lines of the second grating increases in the same direction as the spacing of the first grating;
the dimension of the second grating along the X-axis being smaller than the dimension of said first grating in the X-axis, while the dimensions of the gratings along the Y-axis are substantially the same;
a light source coupling a light beam into said substrate by said first grating in a first direction, and the light beam being coupled out from the substrate in a second direction by said second grating;
said second direction of the light beam being controlled by changing a refractive index of the substrate or by changing a wavelength of the light source;
wherein the refractive index of said light-transmissive substrate between said gratings is dynamically, externally controlled.

2. The device according to claim 1, wherein the refractive index of said light-transmissive substrate is dynamically, externally controlled by using electro-optical effect.

3. The device according to claim 1, wherein the refractive index of the substrate is dynamically controlled by applying an electric field to the substrate.

4. The device according to claim 1, wherein the refractive index of the substrate is dynamically controlled by illuminating the substrate with a strong, short-wavelength light source.

5. The device according to claim 1, wherein the change in the refractive index of the substrate causes an angular steering of an output beam passing through the two parallel gratings.

6. The device according to claim 5, further comprising a focusing lens located in front of one of said pair of parallel optical gratings, producing a point scan at imaging plane defined by the focus of said focusing lens.

7. The device according to claim 5, comprising two different parallel optical devices combined to form two dimensional beam steering, wherein the steering direction of each optical device is different than that of the other.

8. The device according to claim 7, wherein the steering direction of each optical device is normal to that of the other.

9. The device according to claim 1, wherein said gratings are located on opposite sides of said substrate.

10. The device according to claim 1, wherein said gratings are located on the same side of said substrate.

11. The device according to claim 1, wherein more than one pair of said parallel gratings is embedded on said light transmissive substrate.

12. An optical system, comprising:
a pair of gratings according to claim 1, wherein light constituting an output from one pair of gratings is utilized as an input to another pair of gratings.

13. The device according to claim 1, further comprising a light source for transmitting light onto said first grating, wherein said first grating diffracts the light onto said second grating.

14. The device according to claim 1, wherein the refractive index of said light-transmissive substrate is dynamically, externally controlled by using electro-optical effect without changing grating functions.

15. The device according to claim 14, wherein the refractive index of the substrate is dynamically controlled by applying an electric field to the substrate.

16. The device according to claim 14, wherein the refractive index of the substrate is dynamically controlled by illuminating the substrate with a strong, short-wavelength light source.

17. The device according to claim 14, wherein the change in the refractive index of the substrate causes an angular deviation of an output beam passing through the two parallel gratings.

18. The device according to claim 17, further comprising a focusing lens located in front of one of said pair of parallel optical gratings, producing a point scan at an imaging plane defined by the focus of said focusing lens.

19. The device according to claim 17, comprising two different parallel optical devices combined to form two-dimensional steering, wherein the steering direction of each optical device is different than that of the other.

20. The device according to claim 19, wherein the steering direction of each optical device is normal to that of the other.

21. The device according to claim 1, wherein said gratings are located on opposite sides of said substrate.

22. The device according to claim 1, wherein said gratings are located on the same side of said substrate.

23. The device according to claim 1, wherein more than one pair of said parallel gratings is embedded on said light transmissive substrate.

24. An optical device comprising:
a light transmissive substrate having at least two major surfaces, a first surface and a second surface, and edges;
a first and a second grating, each having an X-axis and a Y-axis, located on at least one of said substrate surfaces, at a constant distance from each other;
each grating having a lateral dimension, including at least one sequence of a plurality of parallel lines, wherein the spacing between said lines gradually increases in a direction away from an edge of the grating up to a maximum distance between said lines, and wherein the spacing between the parallel lines of the second grating increase in the same direction as the spacing of the first grating;
a light source coupling a light beam into said substrate by said first grating in a first direction, and the light beam being coupled out from the substrate in a second direction by said second grating;
said second direction of the light beam controlled by changing a refractive index of the substrate wherein the refractive index of said light-transmissive substrate between said gratings is dynamically, externally controlled, without changing the grating functions.

25. The optical device according to claim 24 wherein the dimension of the second grating along the X-axis is smaller than the dimension of said first grating in the X-axis, while the dimensions of the gratings along the Y-axis are substantially the same.