Patent Publication Number: US-2004052449-A1

Title: Optical multiplexer/demultiplexer having decreased channel spacing

Description:
BACKGROUND OF THE INVENTION  
       [0001] With advances in technology, there is a continuous demand to increase data transmission rates and the volume of data transmission. Traditional communication lines, such as copper wires, have been used to meet this continuous demand. However, traditional communication lines are subject to many disadvantages including limited bandwidth and high signal attenuation, which imposes distance limitations. In addition, traditional communication lines are susceptible to interference during the transmission of data. An example of interference includes, but is not limited to, electromagnetic interference.  
       [0002] Optical fibers overcome many shortcomings of traditional communication lines. Communication via optical fibers is characterized by immunity to electromagnetic interference, long transmission range, and high bandwidth. In fact, telecommunication networks that use optical fibers typically have several Terahertz (THz) of bandwidth available for data transmission.  
       [0003] Until the late 1980s, communication via optical fibers was mainly confined to transmitting a single optical signal along an optical fiber using binary amplitude modulation. However, transmitting a single optical signal via an optical fiber rarely takes advantage of the potentially enormous bandwidth available in the optical fiber. Attenuation in the optical fiber requires that periodic regeneration be performed to maintain the intensity of the optical signal. Regeneration conventionally involves detection of the optical signal, electronic processing of the resulting electrical signal and retransmission of an optical generated in response to the processed electrical signal. Such electronic regeneration limits the data rate due to bandwidth limitations of the electronic processing.  
       [0004] To increase transmission capacity of telecommunication networks based on optical fibers, multiple optical signals, each having a different carrier wavelength, are transmitted simultaneously along a single optical fiber as a multi-wavelength optical signal. This technique is known as wavelength-division multiplexing (WDM). WDM systems in which the channel spacing between adjacent optical signals is reduced are known as dense wavelength division multiplexing (DWDM) systems. In this disclosure, dense wavelength division multiplexing (DWDM) will be regarded as encompassing wavelength-division multiplexing (WDM). In DWDM systems, multiple optical signals, each having a different carrier wavelength, are transmitted on a single optical fiber as a multi-wavelength optical signal.  
       [0005] In DWDM systems, each channel has one particular carrier wavelength assigned to it and carries an individual optical signal. Channel spacing is defined as the frequency difference between the center frequencies of two neighboring channels. A channel spacing of 50 GHz (corresponding to a carrier wavelength difference of approximately 0.4 nm at a wavelength of about 1.55 μm) is commonplace in modern DWDM systems. In future, even smaller channel spacings could be used.  
       [0006]FIG. 1 is a block diagram illustrating a typical dense wavelength division multiplexing (DWDM) communication system  102 . Communication system  102  might constitute part of a network, for example. The DWDM communication system transmits multiple information signals represented by a multi-wavelength optical signal carried on a single optical fiber. The multi-wavelength optical signal is composed of multiple single optical signals having different carrier wavelengths.  
       [0007] In a DWDM communication system, each channel carries a single optical signal having an assigned carrier wavelength. The single optical signal typically represents an individual information signal. The channels are separated by a channel spacing defined as described above. Minimizing the channel spacing maximizes the number of single optical signals that can be multiplexed within a given bandwidth. However, the channel spacing must be sufficient to accommodate the frequency sidebands generated by modulating the optical carrier with the information signal. Moreover, the channel spacing must be sufficient to accommodate tolerances in the carrier wavelengths. Finally, the channel spacing must be sufficient to enable the single optical signals to be combined or multiplexed to form the multi-wavelength optical signal and to enable the multi-wavelength optical signal to be separated or demultiplexed into its constituent single optical signals without crosstalk or interference among the single optical signals.  
       [0008] DWDM communication system  102  is composed of an optical transmitter  106 , an optical fiber  113  and an optical receiver  108 . Optical transmitter  106  receives electrical analog or digital signals and converts each electrical signal into a corresponding single optical signal. The single optical signal is in the form of an intensity-modulated beam of light. Any optical signal source, such as, but not limited to, one of the laser diodes  105  shown, may serve as the source of each single optical signal. Each single optical signal has a carrier wavelength (λ) different from the wavelengths of the other single optical signals with which it will be multiplexed. Single optical signals having carrier wavelengths λ 1 , λ 2  and λ 3 , respectively, are conveyed by input optical fibers  103 A,  103 B,  103 C, respectively, from their respective optical signal sources to a multiplexer  110  located in the optical transmitter  106 .  
       [0009] The multiplexer (mux)  110  combines the single optical signals, each having a different carrier wavelength, to form a multi-wavelength optical signal for transmission to optical receiver  108  via single optical fiber  113 . Specifically, multiplexer  110  combines the single optical signals carried by input optical fibers  103 A,  103 B and  103 C to form a multi-wavelength optical signal for transmission via single optical fiber  113 .  
       [0010] The multi-wavelength optical signal transmitted by optical transmitter  106  travels via optical fiber  113  to optical receiver  108 , which is typically located remotely from the optical transmitter. The optical receiver receives the multi-wavelength optical signal from the optical fiber. A demultiplexer (demux)  114  located in the optical receiver demultiplexes the multi-wavelength optical signal. The demultiplexing separates the multi-wavelength optical signal into its constituent single optical signals. The single optical signals are output via the output optical fibers  107 A,  107 B,  107 C. The information signals represented by the single optical signals obtained by demultiplexing the multi-wavelength optical signal may be detected using known optical signal detection techniques. For example, the photodiode receivers  109  may be used.  
       [0011] In DWDM communication system  102 , multiplexer  110  combines two or more single optical signals to form a multi-wavelength optical signal for transmission via optical fiber  113 . Demultiplexer  114  separates the multi-wavelength optical signal received from optical fiber  113  into its constituent single optical signals to enable the information signals represented by the single optical signals to be recovered, routed or otherwise processed.  
       [0012] Because of the reciprocal nature of optical devices, a device called an optical multiplexer/demultiplexer (mux/demux) can be used as multiplexer  110 . An optical mux/demux having the same structure as that used for multiplexer  110  can be used as demultiplexer  114  simply by reversing the direction of the optical signals. An optical mux/demux is referred to as having m channels, where m is as few as two and as many as forty or more. An m-channel optical mux/demux used as a multiplexer receives m single optical signals having different wavelengths as inputs and outputs an m-channel multi-wavelength optical signal. Used as a demultiplexer, the m-channel optical mux/demux receives an m-channel multi-wavelength optical signal as an input and outputs m single optical signals having different wavelengths.  
       [0013] A prior art optical mux/demux such as the MICS™ mux/demux, manufactured by NetTest A/S of Kirkebjerg Allé 90, 2605 Brøndby, Denmark, can be used as multiplexer  110  or demultiplexer  114  in DWDM transmission system  102 . The MICS mux/demux is based on array waveguide grating (AWG) technology. Another example of an AWG-based prior art mux/demux is sold by NTT Electronics Corporation (NEL) of Shibuya Mark City, 1-12-1 Dogenzaka, Shibuya-ku, Tokyo 150-0043, Japan.  
       [0014] An AWG-based mux/demux has a typically Gaussian filter response. The characteristics of an AWG-based mux/demux can be better understood by referring to the graph of FIG. 2. The graph of FIG. 2 plots the insertion loss of a typical AWG-based mux/demux (y-axis) in three adjacent channels against optical frequency (x-axis). Each of the Gaussian filter shapes represents the fraction of the optical power coupled by the AWG-based mux/demux acting as a demultiplexer from the multi-wavelength optical signal to one of the three single optical signals illustrated.  
       [0015] A Gaussian filter has an amplitude response that is non-flat in the pass band and that has a relatively shallow slope in the stop band. The non-flat amplitude response in the pass band distorts the information signal with which the optical signal is modulated. Such distortion may result in an unacceptably-high bit error rate when the optical signal is detected to recover the information signal. The relatively shallow slope of the amplitude response in the stop band means that the frequency spacing between adjacent channels has to be made relatively large to obtain acceptable channel-to-channel crosstalk performance. The wide channel spacing results in is poor spectral efficiency. Spectral efficiency is a measure of how well available bandwidth is used to transmit required data. Specifically, spectral efficiency is the ratio of bit rate to channel spacing.  
       [0016] Finally, temperature sensitivity and mechanical tolerances make the design of conventional DWDM multiplexers and/or demultiplexers challenging, especially when the channel spacing is less than approximately 200 Gigahertz (GHz).  
       [0017] Thus, what is needed is an optical mux/demux having acceptable crosstalk performance at the channel spacing of modern DWDM optical transmission systems, e.g., 50 GHz, and additionally having a flat frequency response in the pass band to minimize distortion of the information signal and, hence, the bit error rate.  
       SUMMARY OF THE INVENTION  
       [0018] The invention provides an optical multiplexer/demultiplexer that comprises an input/output channel array, a diffractive element, an arraying device and a converging element. The input/output channel array is located adjacent an optical axis and includes input/output channels arrayed in a first direction, orthogonal to the optical axis, at a predetermined pitch. The diffractive element is arranged to receive light from the input/output channel array at a location separated from the input/output channel array along the optical axis. The diffractive element diffracts the light to array the light wavelength-dependently in a second direction, different from the first direction. The arraying device receives light diffracted by the diffractive element and arrays the light in the first direction at a pitch equivalent to the predetermined pitch. The converging element is located along the optical axis between the diffractive element and either or both the arraying device and the input/output channel array.  
       [0019] The invention additionally provides a method for demultiplexing a multi-wavelength optical signal. In the method, the multi-wavelength optical signal is received and is wavelength-dependently separated into single optical signals arrayed in a first direction. The single optical signals have different wavelengths. The single optical signals are wavelength-independently arrayed in a second direction, different from the first direction. The arraying in the first direction is wavelength-dependently reversed and the single optical signals arrayed in the second direction are individually output.  
       [0020] Finally, the invention provides a method for multiplexing single optical signals to form a multi-wavelength optical signal. In the method, the single optical signals are received arrayed in a first direction. The single optical signals have different wavelengths. The single optical signals are wavelength-dependently arrayed in a second direction, different from the first direction. The arraying in the first direction is wavelength-independently reversed. The arraying in the second direction is wavelength-dependently reversed to spatially overlap the single optical signals to form the single multi-wavelength optical signal and the multi-wavelength optical signal is output.  
       [0021] The optical multiplexer/demultiplexer, the optical demultiplexing method and the optical multiplexing method provided by the invention have a filter shape has a flat frequency response in the pass band, so that the single optical signals are not subject to distortion. Thus, the invention provides a low BER when an information signal is recovered from each single optical signal. In addition, the filter shape has a steep slope in the stop band. This allows the width of the spectral gaps between adjacent channels to be minimized without unacceptable cross talk between the channels. Hence, invention maximizes usage of the bandwidth of the optical communication system. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0022] The invention can be better understood with reference to the following drawings. The components of the drawings are not necessarily to scale. Emphasis is instead placed upon clearly illustrating the invention. Moreover, like reference numerals designate corresponding parts throughout the several views.  
     [0023]FIG. 1 is a block diagram illustrating a conventional optical communication system using wavelength division multiplexing.  
     [0024]FIG. 2 is a graph illustrating the Gaussian filter responses of an example of a conventional AWG-based optical mux/demux FIG. 3A is a schematic side view of a optical mux/demux according to the invention operating as a demultiplexer.  
     [0025]FIG. 3B is a schematic top view of the optical mux/demux shown in FIG. 3A.  
     [0026]FIG. 4A is a schematic side view of the optical mux/demux according to the invention operating as a multiplexer.  
     [0027]FIG. 4B is schematic top view of the optical mux/demux shown in FIG. 4A.  
     [0028]FIGS. 5A, 5B and  5 C are respectively a side view, a front view and a top view of a first embodiment of the roof prism array of the optical mux/demux shown in FIG. 3A.  
     [0029]FIGS. 5A, 5B and  5 C are respectively a side view, a front view and a top view of a second embodiment of the roof prism array of the optical mux/demux shown in FIG. 3A.  
     [0030]FIG. 6 is a schematic diagram illustrating spreading of the beam of one of the single optical signals in the respective roof prism of the optical mux/demux shown in FIG. 3A.  
     [0031]FIG. 7 is a graph illustrating the filter response of an example of the optical mux/demux shown in FIG. 3A.  
     [0032]FIG. 8A is a flowchart showing a first demultiplexing method according to the invention.  
     [0033]FIG. 8B is a flowchart showing a second demultiplexing method according to the invention.  
     [0034]FIG. 9A is a flowchart showing a first multiplexing method according to the invention.  
     [0035]FIG. 9B is a flowchart showing a second multiplexing method according to the invention.  
     [0036]FIGS. 10A and 10B are schematic diagrams illustrating examples of alternative arraying devices. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
     [0037] Referring now to the drawings, in which like reference numerals designate corresponding parts throughout the drawings, FIGS. 3A and 3B show an example  200  of an optical mux/demux according to the invention operating as a demultiplexer. Optical mux/demux  200  has a flat amplitude response in the filter pass band and a steeply-falling amplitude response in the stop band. The flat in-band response minimizes distortion of the optical signals while the steeply-falling response in the stop band provides low cross-talk and a high spectral efficiency, as will be described further below.  
     [0038] To simplify the drawings and related description, FIGS. 3A and 3B show a four-channel example of optical mux/demux  200 . This example is structured to multiplex or to demultiplex a multi-wavelength optical signal composed of no more than four single optical signals having a different wavelengths. Embodiments of optical mux/demux  200  having substantially more than four channels, as is typical in optical multiplexers/demultiplexers, will be described below. In this disclosure, m denotes the number of channels.  
     [0039] Turning first to FIG. 3A, the optical mux/demux  200  is composed of an I/O channel array  212 , a microlens array  202 , a lens  204 , a diffraction grating  206  and a roof prism array  208 . Also shown in FIG. 3A is an optional spatial filter  214 .  
     [0040] I/O channel array  212  is composed of m+1 I/O channels  211  arrayed in the y-direction. The I/O channel array is located offset in both the −y-direction and the −z-direction from an optical axis  180 . Optical fibers constitute the I/O channels in the example shown. Optical waveguides may alternatively be used as the I/O channels.  
     [0041] Of the (m+1) I/O channels  211  constituting I/O channel array  212 , the one I/O channel  190  differs in function from the m I/O channels  191 . The functions of I/O channels  190  and  191 , respectively, depend on whether the optical mux/demux  200  is operating as a multiplexer or as a demultiplexer. In the demultiplexer shown in FIG. 3A, I/O channel  190  is an input channel through which optical mux/demux  200  receives the multi-wavelength optical signal and each of I/O channels  191  is an output channel through which the optical mux/demux outputs one of the single optical signals demultiplexed from the multi-wavelength optical signal.  
     [0042] Microlens array  202  is composed of (m+1) microlenses  201  arrayed in the y-direction at a pitch corresponding to that of I/O channels  211  constituting I/O channel array  212 . Each microlens re-images the light output by a respective one of I/O channels  211  on first focal plane  194  of lens  204  or alternatively re-images light focused in first focal plane  194  by lens  204  on a respective one of I/O channels  211 .  
     [0043] Microlens array  202  matches the divergence of the beams input to and output by the channels constituting I/O array  212  to the numerical aperture of lens  204 . Specifically, the microlens array allows the numerical aperture to be reduced (larger f-number), which simplifies the design of the lens. Since the beams input to and output by the I/O array have a fixed divergence, a telescope using conventional spherical optics can alternatively be used to double the beam spot size from 5 μm to 10 μm to reduce the beam divergence. However, microlens array  202  allows the channel spacing of the I/O channels  211  and the spot size to be varied independently.  
     [0044] Of the (m+1) microlenses constituting microlens array  212 , microlens  192  performs a different one of the functions just described from the m microlenses  193 , depending on whether the optical mux/demux  200  is operating as a multiplexer or as a demultiplexer. In the demultiplexer shown in FIG. 3A, microlens  192  re-images the multi-wavelength optical signal received via I/O channel  190  on first focal plane  194  of lens  204  and each of the m microlenses  193  re-images one of the single optical signals focused in first focal plane  194  by lens  204  on a respective one of I/O channels  191 .  
     [0045] Diffraction grating  206  is located on optical axis  180  in a second focal plane  195  of lens  204 . Second focal plane  195  is on the opposite side of the lens from first focal plane  194 . The lines of diffraction grating are oriented in the y-direction.  
     [0046] Roof prism array  208  is located near first focal plane  194  of lens  204  and is offset in both the +y-direction and the +z-direction from optical axis  180 . Specifically, roof prism array  208  is located and the roof prism array is configured so that the focal point of a parallel beam converged by lens  204  is located mid-way along the optical path through each of the m roof prisms  209  that constitute the roof prism array. In roof prism array  208 , roof prisms  209  are arrayed in the z-direction. The roof prisms are arranged to receive light from lens  204  and are structured to return such light to the lens. The roof prism array returns the light to the lens displaced in the +y-direction relative to the light received from the lens. The roof prism array displaces the light received from the lens by a different distance in the +y-direction. The difference in the y-direction displacement between adjacent ones of the of the light corresponds to the pitch of I/O channel array  212 , i.e., the distance in the y-direction between adjacent ones of I/O channels  211  constituting the I/O channel array.  
     [0047] The number of roof prisms  209  in roof prism array  208  is equal to the number of channels that the optical mux/demux  200  is designed to multiplex or demultiplex. However, roof prism array  208  may be composed of more roof prisms than the number of channels. In the four-channel example shown in FIGS. 3A and 3B, roof prism array  208  is an array of four individual roof prisms  209 . Roof prism array  208  will be described in further detail below with reference to FIGS.  5 A- 5 G.  
     [0048] Optical mux/demux  200  is shown in FIG. 3A as being additionally composed of a spatial filter  214  located between lens  204  and roof prism array  208 . In the example shown, the spatial filter is composed of slits  215  (FIG. 3B) arrayed in the z-direction and located in the path of light passing from lens  204  to roof prism array  208 . Thus, in the demultiplexer shown in FIG. 3A, exemplary beam  207  passes through one of the slits constituting the spatial filter.  
     [0049] Slits  215  constituting spatial filter  214  have a width, i.e., dimension in the z-direction, less than the width, i.e., dimension in the z-direction, of roof prisms  209  constituting roof prism array  208 . The relationship between the width of roof prisms  209 , the width of slits  215  and the spot size of the single optical signals at roof prism array  208  will be described in detail below.  
     [0050] In the example shown in FIG. 3A, spatial filter  214  is shown as an element independent of roof prism array  208 . The spatial filter is preferably located as close as possible to the roof prism array, so the spatial filter may be supported by the roof prism array or, as in the examples to be described below with reference to FIGS.  5 A- 5 G, by the individual roof prisms  209 .  
     [0051] Spatial filter  214  reduces cross-talk between the single optical signals constituting the multi-wavelength optical signal. Cross-talk becomes problematical when the position of a given single optical signals deviates significantly in the z-direction from the widthwise center of respective roof prism  209  constituting roof prism array  208 . Manufacturing tolerances, variations due to environmental changes and/or static or dynamic differences in the wavelengths of the single optical signals from their respective channel center wavelengths can cause such a deviation. A widthwise positional deviation can cause at least part of the single optical signal to enter the adjacent roof prism  209 , and consequently be displaced by the wrong distance in the y-direction. This causes the part of the single optical signal enter the wrong I/O channel of I/O channel array  212 . The single optical signal entering the wrong I/O channel causes crosstalk.  
     [0052] In embodiments and applications of optical mux/demux  200  in which the single optical signals do not deviate significantly in the z-direction from the widthwise centers of the respective roof prisms  209 , acceptable cross-talk performance can be obtained with spatial filter  214  omitted.  
     [0053]FIGS. 5A, 5B and  5 C are enlarged views of a first embodiment of roof prism array  208 . This embodiment is also shown in FIG. 3A. Each of the single optical signals is received from lens  204  by a different one of roof prisms  209  constituting roof prism array  208  and is reflected back towards the lens displaced by a different distance in the y-direction. In this embodiment, the roof prism array is structured so that the surface  220  that constitutes the hypotenuse of each roof prism  209  has a different length. Surface  220  will be called the hypotenuse surface of the roof prism. The dimension of the hypotenuse surface in the y-direction will be called the hypotenuse length of the roof prism. The hypotenuse length of a given roof prism determines the distance in the y-direction by which the roof prism displaces the respective single optical signal. The different hypotenuse lengths of roof prisms  209  cause the roof prism array to spatially separate the single optical signals by a predetermined distance in the y-direction, as shown in FIG. 5A, while maintaining the spatial separation of the single optical signals in the z-direction, as shown in FIG. 5C. Thus, after passing through the roof prism array, the single optical signals are arrayed both in the y-direction and in the z-direction. The roof prism array is structured such that the component in the y-direction of the separation between the single optical signals is equal to the pitch of I/O channel array  212 , i.e., the separation in the y-direction between I/O channels  211  constituting the I/O array.  
     [0054]FIG. 5B shows the locations of the spots  216  into which the single optical signals are focused on the hypotenuse surfaces  220  of the roof prisms  209  in relation to the widths of the roof prisms and slits  215  constituting spatial filter  214 . The spot sizes of the spots are enlarged to enable them to be seen in the drawing. FIG. 5B also shows the locations of the spots  217  corresponding to the single optical signals exiting the hypotenuse surfaces of the roof prisms  209 . The spot sizes of the spots  217  are also enlarged to enable them to be seen in the drawing.  
     [0055]FIGS. 5B and 5C show an embodiment of spatial filter  214  in which the spatial filter is divided into elements each of which is supported by a corresponding one of roof prisms  209 . In such an embodiment, the spatial filter element is composed of a patterned layer of metal deposited on part of the hypotenuse surface  220  of the roof prism. In the example shown, the metal pattern supported by hypotenuse surface  220  of largest roof prism  218  is composed of metal portions  222  and  223  that collectively define slit  215  of the spatial filter element. The slit is centered in the z-direction on the width of hypotenuse surface  220 . Spatial filter  214  does not extend in the y-direction to the part of hypotenuse surface  220  from which the single optical signals exit roof prism  209 . The spatial filter elements supported by the other roof prisms shown are similarly structured. Opaque materials capable of deposition on a surface of a roof prism may be used instead of metal to define the elements of the spatial filter.  
     [0056] In embodiments of optical mux/demux  200  specifically configured for operation as a multiplexer, the elements of spatial filter  214  may be located in the alternative positions shown in FIG. 5D.  
     [0057] FIGS.  5 E- 5 G show a second embodiment of roof prism array  208  in which roof prisms  219  all have the same hypotenuse length and are aligned in the roof prism array progressively offset from one another in the y-direction by a distance equal to one-half of the channel spacing. An offset of p between one roof prism and another in the y-direction causes the displacement in the y-direction between the single optical signals reflected by the roof prisms to differ by 2p. Thus, an offset of 125 μm between adjacent roof prisms results in a difference in beam displacement of 250 μm. In an exemplary embodiment, each of the roof prisms  209  had a hypotenuse length of 10.125 mm and a width of 250 μm.  
     [0058] The embodiment of roof prism array  208  shown in FIGS.  5 E- 5 G may be fabricated by cutting a large roof prism of the requisite dimensions into slices 250 μm thick. The slices are then attached to one another offset from one another by an offset of 125 μm in the y-direction, as described above.  
     [0059] Operation of optical mux/demux  200  as a demultiplexer will now be described initially with reference to FIG. 3A. A multi-wavelength optical signal is received via I/O channel  190 , which constitutes part of I/O channel array  212 . The multi-wavelength optical signal passes to micro lens  192  associated with I/O channel  190 . Micro lens  192  is one of the microlenses  201  constituting micro lens array  202 .  
     [0060] Micro lens  192  re-images the multi-wavelength optical signal received via I/O channel  190  on first focal plane  194  of lens  204 . Lens  204  collimates light traveling towards diffraction grating  206 , and focuses light received from diffraction grating  206 , as will be described below. Although lens  204  is shown as a single, bi-convex lens, a multi-element converging lens is used as lens  204  in practical embodiments to reduce common optical aberrations. Such multi-element lens may include one or more diverging elements to reduce the overall length of the optical mux/demux  200 .  
     [0061] Lens  204  collimates the multi-wavelength optical signal diverging from first focal plane  194  on which it has been focused by micro lens  192 . The multi-wavelength optical signal collimated by the lens passes to diffraction grating  206 . The collimated illuminates a substantial area of diffraction grating  206 , which enables the diffraction grating to have a high diffraction efficiency.  
     [0062] Diffraction grating  206  diffracts the multi-wavelength optical signal. This angularly separates the multi-wavelength optical signal into its constituent single optical signals. The diffraction grating  206  diffracts each single optical signal at a diffraction angle that depends on the wavelength of the single optical signal. Thus, since the single optical signals have different wavelengths, each single optical signal is diffracted at a different diffraction angle. Diffraction grating  206  is reflective, so that the single optical signals diffracted at different diffraction angles by the diffraction grating return to and pass through lens  204 . After the single optical signals diffracted by the diffraction grating pass through the lens, they become spatially arrayed in a first direction, namely, in the z-direction.  
     [0063]FIG. 3A is a side view of optical mux/demux  200 , so it appears as if one single optical signal  207  returns to and passes through lens  204  after diffraction by diffraction grating  206 . However, in the example in which the multi-wavelength optical signal is composed of four single optical signals, diffraction grating  206  diffracts the four single optical signals at different diffraction angles. The four single optical signals all lie in the x-z plane after diffraction, so that the single optical signal  207  hides the remaining three single optical signals in FIG. 3A.  
     [0064]FIG. 3B is a top view of optical mux/demux  200  that better illustrates the four single optical signals resulting from diffraction grating  206  diffracting the multi-wavelength optical signal. These single optical signals are indicated by open arrows. These single optical signals are in fact hidden by the single optical signals, indicated by closed arrows, returning from roof prism array  208  to the diffraction grating. Passing through lens  204  arrays in the z-direction the single optical signals angularly separated by the diffraction grating. Single optical signal  207  is identified in FIG. 3B for the purpose of comparison to FIG. 3A.  
     [0065] Lens  204  focuses each of the single optical signals received from diffraction grating  206  onto a different one of roof prisms  209  constituting roof prism array  208 . While the single optical signals are collimated and are partially overlaid entering lens  204 , at roof prism array  208 , the single optical signals have been focused by lens  204  to small spots having a spot size of about 20 μm. As will be described in detail below, the spot size of the spots is a fraction of the width of the roof prisms and the width of the slits  215  constituting spatial filter  214 . The spots are arrayed in the z-direction at the roof prism array, as shown in FIG. 3B, are spatially separated from one another in the z-direction by a distance equal to the width of roof prisms  209  and each is nominally centered widthwise on the respective one of the roof prisms. As a result, each of the single optical signals passes through a different one of roof prisms  209 .  
     [0066] Returning now to FIG. 3A, the single optical signals exit from roof prism array  208  arrayed in the y-direction in addition to being arrayed in the z-direction. A more detailed description of the arraying provided by roof prism array  208  is set forth below. The single optical signals exiting roof prism array  208  travel in the +x-direction towards lens  204 . The direction of travel of the single optical signals is parallel to, opposite to and offset in the y-direction from, their direction prior to arraying by the roof prism array. Lens  204  collimates the diverging beams of the single optical signals received from the roof prism array  208 .  
     [0067] The single optical signals then pass from lens  204  to diffraction grating  206 . The diffraction grating diffracts the single optical signals a second time and reflects the single optical signals back toward lens  204 . The second diffraction by diffraction grating  206  reverses the angular dispersal that resulted from the first diffraction and, hence, reverses the arraying of the single optical signals in the z-direction. However, the single optical signals remain arrayed in the y-direction.  
     [0068] Diffraction grating  206  reflects the single optical signals arrayed only in the y-direction towards lens  204 . Lens  204  focuses each of the single optical signals received from the diffracted grating  206  in first focal plane  194  of the lens and at a point corresponding to a different one of the m micro lenses  193  that constitute part of micro lens array  202 . Micro lens  192  does not receive any of the single optical signals from lens  204 . Each of micro lenses  193  re-images one of the single optical signals received from lens  204  on a respective one of the m I/O channels  191  constituting part of I/O channel array  212 . Each single optical signal is re-imaged in a manner that efficiently couples the single optical signal into the respective I/O channel  211 .  
     [0069] The single output signals are then individually output via respective ones of I/O channels  191 .  
     [0070]FIGS. 4A and 4B show an embodiment of optical mux/demux  200  operating as an optical multiplexer. It should be noted that the structure of optical mux/demux of FIGS. 4A and 4B is the same as that of optical mux/demux  200  operating as a demultiplexer shown in FIGS. 3A and 3B. However, the functions of I/O channels  190  and  191  and of microlenses  192  and  193  differ from the functions of these elements in the demultiplexer. Moreover, the position of spatial filter  214  may be changed from the position shown, as described above with reference to FIG. 5D.  
     [0071] Single optical signals are received via m respective I/O channels  191  that constitute part of I/O channel array  212 . The I/O channel array arrays the single optical signals in the y-direction. The spacing in the y-direction between the single optical signals is defined by the pitch of the I/O channel array. The single optical signals pass to respective ones of micro lenses  193  that constitute part of micro lens array  202 . The micro lenses focus the single optical signals in first focal plane  194  of lens  204 . Lens  204  collimates the diverging beams of the single optical signals, and the collimated single optical signals pass to diffraction grating  206 .  
     [0072] Diffraction grating  206  diffracts the single optical signals through angles dependent on their wavelengths and reflects the single optical signals back towards lens  204 . Passing through lens  204  arrays the single optical signals in the z-direction in addition to their arraying in the y-direction. The lens additionally focuses the single optical signals on respective ones of the roof prisms  209  constituting roof prism array  208 . The single optical signals pass through roof prism array  208 . The difference in y-direction displacement between adjacent ones of the roof prisms of the roof prism array corresponds to the pitch of I/O array  212 . As a result, all of the single optical signals exit from the roof prism in the same x-z plane. Thus, the roof prism array has wavelength-independently reversed the arraying of the single optical signals in the y-direction. The single optical signals exit the roof prism array arrayed only in the z-direction and traveling in the +x-direction towards lens  204 .  
     [0073] Lens  204  collimates the single optical signals received from roof prism array  208 . The single optical signals then pass from lens  204  to diffraction grating  206 . The diffraction grating diffracts the single optical signals a second time and reflects the single optical signals back toward lens  204 . The second diffraction by diffraction grating  206  reverses the angular dispersal that resulted from the first diffraction and, hence, reverses the arraying of the single optical signals in the z-direction. Reversing the arraying of the single optical signals in the z-direction spatially overlays the single optical signals to form a multi-wavelength optical signal. Lens  204  then focuses the multi-wavelength optical signal at a point in first focal plane  194  aligned with microlens  192 , an element of micro lens array  202 . Micro lens  192  re-images the multi-wavelength optical signal on I/O channel  190 , an element of I/O channel array  212 .  
     [0074] The multi-wavelength optical signal is then output via I/O channel  190 .  
     [0075] In this disclosure, the term spot size is used to denote the radius of a single optical signal and is defined as the radius of a monochromatic single optical signal at which the power density falls to 1/e 2  of the peak power density. The term spot waist size denotes the minimum spot size to which the single optical signal is focused as it enters the roof prism array.  
     [0076] The extent to which optical mux/demux  200  can be miniaturized is limited in one aspect by diffraction that causes lateral spreading of the single optical signals as they pass through roof prism array  208 . Lateral spreading of the single optical signals defines a minimum width of roof prisms  209  constituting the roof prism array. Proper selection of the spot waist size of the single optical signals at the roof prism array minimizes beam diffraction. As a single optical signal passes through its corresponding roof prism, it should spread laterally to a width less than the width of the roof prism. Otherwise, the roof prism will clip the single optical signal. Clipping is undesirable as it reduces the intensity of the single optical signal and can cause cross talk.  
     [0077] Clipping is avoided by an appropriate choice of the spot waist size of the single optical signal in relation to the dimensions of the roof prism. An example of a process for determining the minimum spot size selection will now be described with reference to an 80-channel embodiment of optical mux/demux  200 . In such embodiment, I/O channel array  212  is composed of 80 I/O channels  191 . Of the 80 I/O channels, 40 are arrayed in the +y direction and 40 are arrayed in the −y-direction relative to I/O channel  190 . Additionally, roof prism array  208  is composed of at least 80 roof prisms  209 . It should be noted that the following is an example and that other methods and systems for selecting the spot waist size may be used.  
     [0078] The filter shape of optical mux/demux  200  is determined by three parameters, namely, the spot waist size of the single optical signals at the roof prism array  208 , the width of slits  215  and the channel spacing. The channel spacing is the spacing in the z-direction between the single optical signals at the roof prism array. The channel spacing corresponds to the width, i.e., the dimension in the z-direction, of the roof prisms  209 . Also, the example shown has a magnification of one, so the channel spacing is also equal to the pitch in the y-direction of I/O channel array  212 . At other magnifications, the channel spacing corresponds to the pitch in the y-direction of the I/O channel array.  
     [0079] Setting the ratios between spot size, slit width and channel spacing to 1:5:8 provides the optical mux/demux  200  with a filter response having stop band slope sufficiently steep for operation with a channel spacing of 50 GHz and additionally provides a flat-topped response in the filter pass band.  
     [0080] The design of roof prism array  208  will now be described with reference to the simplified example shown in FIGS.  5 A- 5 C. In roof prism array, largest roof prism  218  has the largest hypotenuse length and the largest length of the portion of the optical path disposed in the y-direction and smallest roof prism  219  has the smallest hypotenuse length and the smallest length of the portion of the optical path disposed in the y-direction. The remaining roof prisms have hypotenuse lengths and optical path lengths intermediate between those of the largest and smallest roof prisms. In some embodiments of the roof prism array  208 , the roof prisms have equal hypotenuse lengths, as will be described below with reference to FIGS.  5 E- 5 G.  
     [0081] Using the 1:5:8 ratios described above, in an 80-channel embodiment of optical mux/demux  200 , the maximum length of the optical path disposed in the y-direction in largest roof prism  218  is 320 times (8×40) the spot size. In an embodiment in which the channel spacing is equal to the pitch of I/O array  212 , this corresponds to 40 times the pitch of the I/O array.  
     [0082] Calculation of the minimum size of largest roof prism  218  will now be described with reference to FIG. 6. FIG. 6 illustrates the way in which the single optical signal spreads due to diffraction as it passes through largest roof prism  218 . The single optical signals passing through the other roof prisms of roof prism array  208  spread similarly, but less. The optical path through roof prism  218  is shown unfolded to simplify the drawing. Slit  215  is located immediately in front of the largest roof prism. For the purpose of this calculation, the single optical signal is composed of monochromatic light having a wavelength corresponding to the center frequency of the longest-wavelength channel, i.e., the channel corresponding to largest roof prism  218 .  
     [0083] For the purpose of simplifying the following calculation, the single optical signal shown is focused by lens  204  to a minimum spot size at slit  215 . The minimum spot size is the spot waist size and is denoted by the symbol ω o . The slit has a width of 5 ω o  and is centered in the z-direction relative to the width of largest roof prism  218 . The width of each of roof prisms  209  is 8 ω o . In a practical embodiment, lens  204  would focus the single optical signal at a point half-way along the optical path shown.  
     [0084] The length of the part of the optical path disposed in the y-direction in largest roof prism  218  is equal to 320 times the spot waist size ω o , as noted above. This is equal to the separation in the y-direction between the single optical signal returned to lens  204  by largest roof prism  218  and the single optical signal returned to the lens by smallest roof prism  219 . Clipping of the single optical signal is avoided when the center of the single optical signal is located a distance ω o , from one edge of slit  215  and the edge of the single optical signal does not intersect the nearest edge of the roof prism after the single optical signal has propagated a distance of 320 ω o  in the y-direction through the roof prism. To account for the portions of the optical path in the x-direction, assumed each to have a length of 2 ω o  a propagation distance of 324 ω o  will be used in the following calculations.  
     [0085] Referring additionally to FIG. 3A, dependence of the spot size ω(y) on defocus y is given by equation (1):  
               ω        (   y   )       =       ω   0              1   +     (     y     z   r       )         2               (   1   )                       
 
     [0086] where z r  is the Rayleigh range, defined by equation (2).  
               z   r     =       π                 n                   ω   0   2       λ             (   2   )                       
 
     [0087] where n is the refractive index of largest roof prism  218  and λ is the wavelength of the monochromatic single optical signal in vacuo. After propagating a distance of 324 times the spot waist size through largest roof prism  218 , the spot size should not increase to more than 2.5 ω o  if clipping is to be avoided. Setting y=324 ω o  and ω(y)=2.5 ω o  in equation (1) provides equation (3):  
     324 ω o =229 z   r   (3)  
     [0088] Substituting equation (2) into equation (3), the spot waist size in the above-described exemplary optical mux/demux is given by equation (4).  
               ω   o     =       141.5                 λ       π                 n               (   4   )                       
 
     [0089] There is a substantial benefit to using a high refractive index (n) material for each roof prism  209 . As an example, in an embodiment in which the roof prism is made of silicon, which has a refractive index (n) of 3.5, the spot waist size would be 19.9 μm. With a spot waist size of 19.9 μm, the minimum width of each of the roof prisms is 8×19.9 μm, which is approximately 160 μm. This is substantially less than 250 μm, which is the pitch of commercially-available fiber optic arrays suitable for use as I/O array  212 .  
     [0090] An 80-channel embodiment of optical mux/demux  200  structured to incorporate a conventional, commercially-available fiber optic array as I/O array  212  will be described next. Such fiber optic array has a pitch of 250 μm, which, with a magnification of one, sets the channel spacing to 250 μm. With a channel spacing of 250 μm, the spot size is 31.25 μm and the slit width is 156.25 μm.  
     [0091] The angular dispersion of diffraction grating  206  is given by equation (5):  
                    β          λ       =       2                 tan                 β     λ             (   5   )                       
 
     [0092] where β is the angle between the normal to diffraction grating  206  and the incident multi-wavelength optical signal. The relationship between the wavelength λ of the single optical signal, the angle β and the pitch d of diffraction grating  206  is defined by equation (6).  
     λ=2 d  sin β  (6)  
     [0093] Lens  204  transforms the angular dispersion of the single optical signals to spatial arraying in the z-direction in accordance with the relationship shown by equation (7).  
                    z          λ       =     f                        β          λ                 (   7   )                       
 
     [0094] where f is the focal length of lens  204 . Substituting equation (6) into equation (5), and incorporating the result into equation (7) yields equation (8), which defines the focal length f of the lens  204 :  
             f   =         δ                 z       δ                 λ          d                 cos                 β             (   8   )                       
 
     [0095] where δz is the channel spacing at roof prism array  208  and δλ is the channel wavelength spacing, e.g., about 0.4 nm in a DWDM system with a 50 GHz channel spacing.  
     [0096] Making the focal length f of lens  204  as short as possible minimizes the size of optical mux/demux  200 . To minimize the focal length f the minimum pitch d consistent with low loss and low polarization dependence is selected. For example, a 600 lines/mm grating having a pitch d of 1.667 μm will give good results when used as diffraction grating  206 . Diffraction gratings with pitches different from that exemplified may be used with a proportional change in the focal length of the lens  204 . The value of β calculated from equation (6) is 27.71 degrees for a center wavelength of 1.55 μm. Substituting the above results into equation (8) gives a focal length f of lens  204  of 922 mm.  
     [0097] The spot size ω at lens  204 , which is approximately the same as the spot size at diffraction grating  206 , can be calculated from equation (1) to be 7.27 mm. To avoid power loss in optical mux/demux  200 , the diameter of lens  204  should be greater than three times the spot size at the lens, i.e., greater than 21.8 mm. Diffraction grating  206  is approximately the same size as the lens  204  when corrected for the angle of incidence on the diffraction grating. In practice, to fulfill telecentricity requirements, the lens will be substantially larger than the minimum size just stated. The fiber optic array used as I/O array  212  has an overall length of about 20 mm in the y-direction, so that the lens diameter will be about 50 mm in practice.  
     [0098] In the embodiment of roof prism array  208  shown in FIGS.  5 A- 5 C, the different optical path lengths in the y-direction are achieved by each roof prism  209  having a different hypotenuse length. The individual roof prisms are assembled to form the roof prism array with one of their apices adjacent their hypotenuse surfaces located in a common x-z plane and their hypotenuse surfaces staggered in the x-direction, as will be described below. In an example, the largest roof prism has a hypotenuse length of 10.125 mm, which corresponds to 324 ω 0 , and the smallest roof prism has a hypotenuse length of 0.375 mm, which corresponds to 12 ω 0 . All of the prisms have a width of 250 μm.  
     [0099] The roof prisms  209  are assembled to form roof prism array  208  with one of their apices adjacent their hypotenuse surfaces aligned in a common x-z plane and their hypotenuse surfaces offset from one another in the x-direction to make the lengths of the optical paths through all the roof prisms equal. This keeps all of the single optical signals in focus at I/O channel array  212 . The offset a in the x-direction of smallest roof prism  219  relative to largest roof prism  218  is:  
             a   =       39        (     320                   ω   0       )         2      n               (   9   )                       
 
     [0100] in which 39 is the number of I/O channels  191  arrayed on one side of the I/O channel  190 , minus one. 320 ω 0  is the largest optical path length in the y-direction, and is approximately equal to 10 mm. For silicon prisms and with the other parameters as defined above, the x-direction offset a of smallest roof prism  219  is 1.39 mm. The x-direction offsets of the remaining roof prisms of roof prism array  208  are distributed in equal increments between zero and 1.39 mm.  
     [0101]FIG. 7 is a graph showing an example of the spectral efficiency provided by optical mux/demux  200 . In FIG. 7, the optical frequency of three of the single optical signals is plotted on the x-axis and the insertion loss of optical mux/demux  200  is plotted on the y-axis. The filter shape has a flat frequency response in the pass band, so that the single optical signal is not subject to distortion by passing through optical mux/demux  200 . This provides a low BER when the information signal is recovered from the single optical signal. In addition, the filter has a steep slope in the stop band. This allows the width of the spectral gaps between adjacent channels to be minimized without unacceptable cross talk. Hence, optical mux/demux  200  maximizes usage of the bandwidth of the optical fiber.  
     [0102] The optical mux/demux  200  has been described above with reference to examples in which lenses are used to re-image, collimate and focus the various optical signals, in which diffraction grating  206  is reflective, and in which roof prism array  208  is used to array the single optical signals in the y-direction or to reverse such arraying. Elements other than lenses, such as mirrors and diffractive optical elements, may be used instead of the lenses shown to re-image, collimate and focus the optical signals. Also, lens  204 , shown as a single-element lens, may be composed of more than one element. Moreover, a transmissive diffraction grating may be used as diffraction grating  206 . Finally, optical elements other than the roof prisms shown may be used to array the single optical signals in the y-direction and to reverse such arraying, as will be described below with reference to FIGS.  10 A- 10 C. Light may exit such alternative optical elements in the same direction as that in which the light entered instead of being reflected back towards lens  204  as in the examples shown.  
     [0103] To reduce the size of optical mux/demux  200 , the long air paths on opposite sides of lens  204  may be folded using reflective elements such as mirrors. Alternatively, a beam-expanding telescope with a focal length equivalent to that of lens  204  may used instead of lens  204 .  
     [0104]FIG. 8A is a flowchart illustrating a first method  300  in accordance with the invention for demultiplexing a multi-wavelength signal. With regard to the flowchart of FIG. 8A, and in the other flowcharts depicted in this disclosure, it should also be noted that in some implementations, the functions performed in the blocks may occur out of the order shown. For example, two blocks shown in succession may in fact be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order.  
     [0105] In block  302 , a multi-wavelength optical signal is received. In block  306 , the multi-wavelength optical signal is wavelength-dependently separated into its constituent single optical signals arrayed in a first direction. Each of the single optical signals has a different wavelength. In block  310 , the single optical signals are wavelength-independently arrayed in a second direction, different from the first direction. In block  314 , the arraying of the single optical signals in the first direction is wavelength-dependently reversed, leaving the single optical signals arrayed in the second direction. In block  318 , the single optical signals arrayed only in the second direction are individually output.  
     [0106]FIG. 8B is a flowchart illustrating a second method  320  in accordance with the invention for demultiplexing a multi-wavelength signal. In block  302 , a multi-wavelength optical signal is received. In block  304 , the multi-wavelength optical signal is collimated. In block  306 , the multi-wavelength optical signal is wavelength-dependently separated into its constituent single optical signals arrayed in a first direction. Each of the single optical signals has a different wavelength. In block  308 , the single optical signals arrayed in the first direction are focused. In block  310 , the single optical signals are wavelength-independently arrayed in a second direction, different from the first direction.  
     [0107] In block  312 , the single optical signals, arrayed in both the first direction and the second direction, are collimated. In block  314 , the arraying of the single optical signals in the first direction is wavelength-dependently reversed, leaving the single optical signal arrayed in the second direction. In block  316 , the single optical signals arrayed in the second direction are focused. In block  318 , the optical signals arrayed only in the second direction are individually output.  
     [0108] The multi-wavelength optical signal received in block  302  may be re-imaged before being collimated in block  304 . Moreover, the single optical signals arrayed in the second direction may be re-imaged before being individually output in block  318 . The single optical signals arrayed in the first direction may be spatially filtered before or after being wavelength-independently arrayed in block  310 . In the focusing performed in block  308 , the spot size of the single optical signals is changed to a spot waist size at which each of the single optical signals is individually wavelength-independently arrayed. This prevents clipping of and cross talk between the single optical signals.  
     [0109] The methods  300  and  320  described in FIGS. 8A and 8B may be performed by the optical mux/demux  200  shown operating as a demultiplexer in FIGS. 3A and 3B.  
     [0110]FIG. 9A is a flowchart illustrating a first method  400  in accordance with the invention for multiplexing single optical signals to form a multi-wavelength signal.  
     [0111] In block  402 , single optical signals arrayed in a first direction are received. Each of the single optical signals has a different wavelength. In block  406 , the single optical signals are wavelength-dependently arrayed in a second direction, different from the first direction. In block  410 , the arraying of the single optical signals in the first direction is wavelength-independently reversed. In block  414 , the arraying of the single optical signals in the second direction is wavelength-dependently reversed. This spatially overlaps the single optical signals to form a multi-wavelength optical signal. In block  418 , the multi-wavelength optical signal is output.  
     [0112]FIG. 9B is a flowchart illustrating a second method  420  in accordance with the invention for multiplexing single optical signals to form a multi-wavelength signal. In block  402 , the single optical signals arrayed in a first direction are received. Each of the single optical signals has a different wavelength. In block  404 , the single optical signals are collimated. In block  406 , the single optical signals are wavelength-dependently arrayed in a second direction. In block  408 , the single optical signals arrayed in both the first direction and the second direction are focused. In block  410 , the arraying of the single optical signals in the first direction is wavelength-independently reversed, leaving the single optical signals arrayed in the second direction.  
     [0113] In block  412 , the single optical signals arrayed in the second direction are collimated. In block  414 , the arraying of the single optical signals in the second direction is wavelength-dependently reversed. This spatially overlaps the single optical signals to form a multi-wavelength optical signal. In block  416 , the multi-wavelength optical signal is focused. In block  418 , the multi-wavelength optical signal is output.  
     [0114] The single optical signals received in block  402  may be re-imaged before being collimated in block  404 . Moreover, the multi-wavelength optical signal focused in block  416  may be re-imaged before being output in block  418 . The single optical signals arrayed in both the first direction and the second direction may be spatially filtered before their arraying in the first direction is reversed in block  410 . Additionally or alternatively, the single optical signals arrayed in the second direction may be spatially filtered before being collimated in block  412 . In the focusing performed in block  408 , the spot size of the single optical signals is changed to a spot size at which the arraying of each of the single optical signals is individually reversed. This prevents clipping of and cross talk between the single optical signals.  
     [0115] The methods  400  and  420  described in FIGS. 9A and 9B may be performed by the optical mux/demux  200  shown operating as a multiplexer in FIGS. 4A and 4B.  
     [0116] In the optical mux/demux  200  described above, roof prism array  208  performs the function of receiving light diffracted by the diffractive element and arraying the light in the first direction at a pitch equivalent to the pitch of the input/output channel array. However, elements other than a roof prism array may be used to perform this function. For example, each roof prism of the roof prism array may be replaced by a pair of reflective surfaces located in positions equivalent to those of the reflective surfaces of the roof prism. A medium having a refractive index greater than unity may be located in the optical path between the reflective surfaces to reduce diffraction effects, as described above.  
     [0117]FIGS. 10A and 10B show examples of other arraying devices that may be used to perform the function of receiving light diffracted by the diffractive element and arraying the light in the first direction at a pitch equivalent to the pitch of the input/output channel array. These examples will now be described with reference to FIGS. 10A and 10B and with additional reference to FIG. 3A.  
     [0118]FIG. 10A shows arraying device  500  composed of prism array  502  and mirror  504 . Prism array  502  is composed of one prism for each channel of the optical mux/demux. Each prism of the prism array has a thickness in the z-direction equal to the channel spacing, i.e., the spacing in the z-direction between the single optical signals. Each prism has a different prism angle α. The prisms of two exemplary channels are shown at  506  (solid line) and  508  (broken line).  
     [0119] The paths through arraying device  500  of the single optical signals  510  and  512  respectively aligned with prisms  506  and  508  are shown by a solid line and a broken line, respectively. The single optical signals are received from lens  204  arrayed in the z-direction. Single optical signal  512  is hidden by single optical signal  510 . Single optical signal  510  passes through prism  506  and is refracted towards the optical axis  180  through an angle that depends on the prism angle α 1 . Single optical signal  512  passes through prism  508  and is refracted towards the optical axis  180  through an angle that depends on the prism angle α 2 , greater than prism angle α 1 . Single optical signals  510  and  512  pass to mirror  504 . Single optical signal  512  impinges on the mirror at a point offset in the y-direction from single optical signal  510  and with a greater angle of incidence. The mirror reflects the single optical signals back towards prisms  506  and  508 , respectively. The single optical signals impinge on the prisms at points offset from one another in the y-direction. The prisms refract the respective single optical signals through angles that depend on their respective prisms. The single optical signals emerge from the respective prisms in a direction parallel to the axis  180  and arrayed in the y-direction.  
     [0120] The prism angles required to provide a given offset between the single optical signals for a given size and refractive index of the prisms of prism array  502  and a given distance of the prism array from mirror  504  can be calculated using simple optical calculations.  
     [0121]FIG. 10B shows an example  520  of an arraying device composed of a cylindrical converging element and a mirror array. In the arraying device  520 , a cylindrical lens  522  with its axis disposed in the y-direction is used as the cylindrical converging element. Mirror array  524  is located in the focal plane of the cylindrical lens. Mirror array  524  is composed of one reflective surface for each channel of the optical mux/demux. The reflective surfaces of two exemplary channels are shown at  526  (solid line) and  528  (broken line). Each reflective surface is disposed at a different angle relative to the y-z plane. In the example shown, reflective surface  526  is disposed parallel to the y-z plane whereas reflective surface  526  is disposed at a non-zero angle relative to the y-z plane. Each reflective surface has a width, i.e., dimension in the z-direction, equal to the channel spacing.  
     [0122] The paths through arraying device  520  of the single optical signals  510  and  512  respectively aligned with reflective surfaces  526  and  528  of mirror array  524  are shown by a solid line and a broken line, respectively. The single optical signals are received from lens  204  arrayed in the z-direction. Single optical signal  512  is hidden by single optical signal  510 . The single optical signals pass through cylindrical lens  522  and are refracted towards the optical axis  180 . Single optical signals  510  and  512  pass to reflective surfaces  526  and  528 , respectively, and are reflected by the reflective surfaces back towards cylindrical lens  522 . Due to the different angles of the reflective surfaces, the single optical signals impinge on the cylindrical lens at points offset from one another in the y-direction. The cylindrical lens refracts the respective single optical signals through angles at which the single optical signals emerge from the lens in a direction parallel to the optical axis  180  and to one another, and arrayed in the y-direction.  
     [0123] The angles of the refractive surfaces required to provide a given offset between adjacent ones of the single optical signals for a given focal length of cylindrical lens  522  can be calculated using simple optical calculations.  
     [0124] This disclosure describes the invention in detail using illustrative embodiments. However, it is to be understood that the invention defined by the following claims is not limited to the precise embodiments described.