Abstract:
In a projection display, input light is separated by a polarization beam splitter prism into first and second color components, and a third color component. The first and second color components, together with a light leakage component that is associated with the third color component, are directed by the polarization beam splitter prism to a color synthesizing prism or a color splitter prism set, which then separates the first and second color components and the light leakage component so as to travel in three different directions, respectively. The first and second color components from the color synthesizing prism or the color splitter prism set, and the third color component from the polarization beam splitter prism are processed by reflective first, second and third light modulators, respectively.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a projection display, more particularly a single-lens projection display that utilizes reflective light modulators. 
     2. Description of the Related Art 
     In a conventional projection display, an input light beam, which contains red, green and blue color components, is processed before providing the same to a projection lens so as to generate an output image. 
     Referring to FIG. 1, a conventional single-lens projection display  1  is shown to include a first light polarization selector  11 , a second light polarization selector  12 , a polarization beam splitter prism  13 , a dichroic beam splitter prism  14 , a first light modulator  15 , a second light modulator  16 , a third light modulator  17 , a polarizer  18 , and a projection lens  19 . An input light beam  10 , which is a white light beam, is separated into first, second and third color components  101 ,  102 ,  103 , such as red, green and blue color components, for processing by the first, second and third light modulators  15 ,  16 ,  17 , respectively. 
     Each of the first and second light polarization selectors  11 ,  12 , such as the ColorSelect™ filter products available from ColorLink Inc., serves to convert the polarization state of a predetermined color component, without altering the polarization state of the other color components. In the projection display  1  of FIG. 1, the polarization state of the third color component  103 , such as the green color component, is changed, whereas the polarization state of the first and second color components  101 ,  102  remains unaltered, after the color components  101 ,  102 ,  103  pass through the first and second light polarization selectors  11 ,  12 . 
     The polarization beam splitter prism  13  reflects light of a first polarization state, such as S-polarized light, in a transverse direction, and allows light of a second polarization state, such as P-polarized light, to pass therethrough. 
     The dichroic beam splitter prism  14  is used to separate two different color components. In the projection display  1  of FIG. 1, the dichroic beam splitter prism  14  allows the first color component  101  to pass directly therethrough, and reflects the second color component  102  in a transverse direction, thereby separating the first color component  101  from the second color component  102 . 
     Each of the first, second and third light modulators  15 ,  16 ,  17  is a reflective light valve, and is disposed adjacent to one of the dichroic beam splitter prism  14  and the polarization beam splitter prism  13  so as to receive a respective one of the first, second and third color components  101 ,  102 ,  103  therefrom. When activated, each of the first, second and third light modulators  15 ,  16 ,  17  modulates the respective one of the first, second and third color components  101 ,  102 ,  103 , and changes the polarization state of the respective one of the first, second and third color components  101 ,  102 ,  103 . The first, second and third light modulators  15 ,  16 ,  17  then reflect modulated first, second and third color components  101 ,  102 ,  103  back to the adjacent one of the dichroic beam splitter prism  14  and the polarization beam splitter prism  13 . 
     The polarizer  18  allows light of a predetermined polarization state to pass therethrough, and absorbs light of the other polarization state. In the projection display  1  of FIG. 1, the polarizer  18  allows P-polarized light to pass therethrough, and absorbs S-polarized light. 
     In operation, when the first light polarization selector  11  receives the S-polarized white input light beam  10 , the S-polarization state of the first and second color components  101 ,  102 , such as the red and blue color components, remains unaltered, whereas the S-polarization state of the third color component  103 , such as the green color component, is changed to the P-polarization state, after the first, second and third color components  101 ,  102 ,  103  pass through the first light polarization selector  11 . The polarization beam splitter prism  13  receives the S-polarized first and second color components  101 ,  102  and the P-polarized third color component  103  from the first light polarization selector  11  at a first side thereof, reflects the S-polarized first and second color components  101 ,  102  in a transverse direction such that the S-polarized first and second color components  101 ,  102  pass through a second side thereof, and permits the P-polarized third color component  103  to pass directly through a third side thereof opposite to the first side. 
     The dichroic beam splitter prism  14 , which is disposed adjacent to the second side of the polarization beam splitter prism  13 , receives the S-polarized first and second color components  101 ,  102  therefrom. The first color component  101  passes directly through the dichroic beam splitter prism  14 , whereas the second color component  102  is reflected by the dichroic beam splitter prism  14  in a transverse direction. 
     The first and second light modulators  15 ,  16  are disposed adjacent to the dichroic beam splitter prism  14  so as to receive the S-polarized first and second color components  101 ,  102  respectively therefrom. The first and second light modulators  15 ,  16  modulate the respective one of the first and second color components  101 ,  102 , and change the polarization state of the respective first or second color component  101 ,  102  from the S-polarization state to the P-polarization state when the first and second light modulators  15 ,  16  are activated. The first and second light modulators  15 ,  16  reflect the corresponding modulated color component back to the dichroic beam splitter prism  14  for reception by the polarization beam splitter prism  13 . 
     The third light modulator  17  is disposed adjacent to the third side of the polarization beam splitter prism  13  so as to receive the P-polarized third color component  103  therefrom. The third light modulator  17  modulates the third color component  103 , and changes the polarization state of the third color component  103  from the P-polarization state to the S-polarization state when the third light modulator  17  is activated. The third light modulator  17  reflects the modulated S-polarized third color component  103  back to the polarization beam splitter prism  13 . 
     The modulated P-polarized first and second color components  101 ,  102  from the dichroic beam splitter prism  14  will be allowed by the polarization beam splitter prism  13  to pass directly through a fourth side thereof opposite to the second side for reception by the second light polarization selector  12 . The modulated S-polarized third color component  103  from the third light modulator  17  will be reflected by the polarization beam splitter prism  13  in a transverse direction so as to pass through the fourth side thereof for reception by the second light polarization selector  12 . 
     When the second light polarization selector  12  receives the modulated first, second and third color components  101 ,  102 ,  103  from the polarization beam splitter prism  13 , the polarization state of the modulated P-polarized first and second color components  101 ,  102  remains unaltered, whereas the polarization state of the modulated S-polarized third color component  103  will be changed to the P-polarization state. 
     The polarizer  18  permits only pure P-polarized color components to pass therethrough, and absorbs S-polarized color components. The modulated P-polarized first, second and third color components  101 ,  102 ,  103  from the polarizer  18  are recombined as they pass through the projection lens  19  for projecting a color image on a display screen (not shown). 
     In the conventional projection display  1 , the dichroic beam splitter prism  14  and the polarization beam splitter prism  13  cooperate with the first and second light polarization selectors  11 ,  12  to separate the white input light beam  10  into the three color components  101 ,  102 ,  103  that are modulated by the three light modulators  15 ,  16 ,  17 . However, in view of current manufacturing constraints, the polarization beam splitter  13  does not permit a very high transmission for P-polarized light. Thus, when the P-polarized third color component  103  is received by the polarization beam splitter prism  13 , a small portion of the P-polarized third color component  103  will be reflected by the polarization beam splitter prism  13  to pass through the second side thereof, i.e. toward the dichroic beam splitter prism  14 , thus resulting in a light leakage component  103 ′. This light leakage component  103 ′ will then be provided by the dichroic beam splitter prism  14  to the first or second light modulator  15 ,  16 . When the first or second light modulator  15 ,  16  is inactive, the light leakage component  103 ′ will be reflected back to the dichroic beam splitter prism  14 , and pass through the polarization beam splitter prism  13 , the second light polarization selector  12 , and the polarizer  18  to reach the projection lens  19 , thereby resulting in a shift in the gray scale coordinate of the projected image and in a reduction in the image contrast to adversely affect the output quality of the projection display  1 . 
     SUMMARY OF THE INVENTION 
     Therefore, the main object of the present invention is to provide a projection display that utilizes reflective light modulators and that is capable of overcoming the aforesaid light leakage drawback of the prior art to enhance both contrast and output image quality. 
     According to one aspect of the present invention, a projection display is adapted to process an input light beam that includes a first color component, a second color component and a third color component, and comprises: 
     a polarization beam splitter prism having a first side adapted to receive the input light beam, a second side, a third side opposite to the first side, and a fourth side opposite to the second side, the polarization beam splitter prism being adapted to separate the input light beam into the first and second color components, which pass through the second side thereof, and the third color component, a large portion of which passes through the third side thereof, the polarization beam splitter prism further allowing a small portion of the third color component, which serves as a light leakage component, to pass through the second side thereof; 
     a color synthesizing prism disposed adjacent to the second side of the polarization beam splitter prism, and adapted to separate the first and second color components and the light leakage component from the polarization beam splitter prism so as to travel in three different directions, respectively; 
     reflective first and second light modulators disposed adjacent to the color synthesizing prism so as to receive the first and second color components respectively therefrom, the first and second light modulators modulating the respective one of the first and second color components and changing polarization state of the respective one of the first and second color components when activated, the first and second light modulators reflecting modulated first and second color components back to the color synthesizing prism; and 
     a reflective third light modulator disposed adjacent to the third side of the polarization beam splitter prism so as to receive the third color component therefrom, the third light modulator modulating the third color component and changing polarization state of the third color component when activated, the third light modulator reflecting a modulated third color component back to the polarization beam splitter prism. 
     According to another aspect of the present invention, a projection display is adapted to process an input light beam that includes a first color component, a second color component and a third color component, and comprises: 
     a polarization beam splitter prism having a first side adapted to receive the input light beam, a second side, a third side opposite to the first side, and a fourth side opposite to the second side, the polarization beam splitter prism being adapted to separate the input light beam into the first and second color components, which pass through the second side thereof, and the third color component, a large portion of which passes through the third side thereof, the polarization beam splitter prism further allowing a small portion of the third color component, which serves as a light leakage component, to pass through the second side thereof; 
     a color splitter prism set disposed adjacent to the second side of the polarization beam splitter prism, and adapted to separate the first and second color components and the light leakage component from the polarization beam splitter prism so as to travel in three different directions, respectively; 
     reflective first and second light modulators disposed adjacent to the color splitter prism set so as to receive the first and second color components respectively therefrom, the first and second light modulators modulating the respective one of the first and second color components and changing polarization state of the respective one of the first and second color components when activated, the first and second light modulators reflecting modulated first and second color components back to the color splitter prism set; and 
     a reflective third light modulator disposed adjacent to the third side of the polarization beam splitter prism so as to receive the third color component therefrom, the third light modulator modulating the third color component and changing polarization state of the third color component when activated, the third light modulator reflecting a modulated third color component back to the polarization beam splitter prism. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other features and advantages of the present invention will become apparent in the following detailed description of the preferred embodiments with reference to the accompanying drawings, of which: 
     FIG. 1 is a schematic view showing a conventional single-lens projection display; 
     FIG. 2 is a schematic view showing the first preferred embodiment of a projection display according to the present invention; 
     FIG. 3 is a schematic view showing the second preferred embodiment of a projection display according to the present invention; 
     FIG. 4 is a schematic view showing the third preferred embodiment of a projection display according to the present invention; 
     FIG. 5 is a schematic view showing a first light polarization selector for the projection display of this invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring to FIG. 2, the first preferred embodiment of a projection display  2  according to the present invention is shown to include a first light polarization selector  21 , a second light polarization selector  22 , a polarization beam splitter prism  23 , a polarizer  24 , a color synthesizing prism  25 , a first light modulator  26 , a second light modulator  27 , a third light modulator  28 , and a projection lens (not shown). A white input light beam  3  contains S-polarized first, second and third color components  31 S,  32 S,  33 S, such as red, green and blue color components, that are to be modulated by the first, second and third light modulators  26 ,  27 ,  28 , respectively. 
     Each of the first and second light polarization selectors  21 ,  22 , such as the ColorSelect™ filter products available from ColorLink Inc., serves to convert the polarization state of a predetermined color component, without altering the polarization state of the other color components. In the projection display of this embodiment, the polarization state of the third color component  33 , such as the green color component, is changed, whereas the polarization state of the first and second color components  31 ,  32 , such as the red and blue color components, remains unaltered, after the color components  31 ,  32 ,  33  pass through the first and second light polarization selectors  21 ,  22 . 
     The polarization beam splitter prism  23  reflects light of a first polarization state, such as S-polarized light, in a transverse direction, and allows light of a second polarization state, such as P-polarized light, to pass therethrough. In the embodiment of FIG. 2, the polarization beam splitter prism  23  has a first side disposed adjacent to the first light polarization selector  21 , a second side disposed adjacent to the color synthesizing prism  25 , a third side opposite to the first side and disposed adjacent to the third light modulator  28 , and a fourth side opposite to the second side and disposed adjacent to the second light polarization selector  22 . In view of the characteristics of the polarization beam splitter prism  23 , when the P-polarized third color component  33 P is received by the polarization beam splitter prism  23  from the first light polarization selector  21 , a small portion of the P-polarized third color component  33 P will be reflected by the polarization beam splitter prism  23  to pass through the second side thereof, i.e. toward the color synthesizing prism  25 , thus resulting in a light leakage component  33 P′. 
     The polarizer  24 , which is disposed adjacent to the second light polarization selector  22 , allows light of a predetermined polarization state to pass therethrough, and absorbs light of the other polarization state. In the present embodiment, the polarizer  24  allows P-polarized light to pass therethrough, and absorbs S-polarized light. 
     The color synthesizing prism  25  receives the S-polarized first and second color components  31 S,  32 S and the light leakage component  33 P′ from the polarization beam splitter prism  23 . In the present embodiment, the color synthesizing prism  25  reflects the S-polarized first and second color components  31 S,  32 S in opposing transverse directions, and allows the light leakage component  33 P′ to pass directly therethrough. 
     Each of the first, second and third light modulators  26 ,  27 ,  28  is a reflective light valve, and is disposed adjacent to one of the color synthesizing prism  25  and the polarization beam splitter prism  23  so as to receive a respective one of the first, second and third color components  31 ,  32 ,  33  therefrom. In this embodiment, the first and second light modulators  26 ,  27  are disposed adjacent to opposite sides of the color synthesizing prism  25 , whereas the third light modulator  28  is disposed adjacent to the third side of the polarization beam splitter prism  23 . When activated, each of the first, second and third light modulators  26 ,  27 ,  28  modulates the respective one of the first, second and third color components  31 ,  32 ,  33 , and changes the polarization state of the respective one of the first, second and third color components  31 ,  32 ,  33 . The first, second and third light modulators  26 ,  27 ,  28  then reflect modulated first, second and third color components  31 ,  32 ,  33  back to the adjacent one of the color synthesizing prism  25  and the polarization beam splitter prism  23 . 
     In operation, when the first light polarization selector  21  receives the S-polarized white input light beam  3 , the S-polarization state of the first and second color components  31 S,  32 S remains unaltered, whereas the polarization state of the third color component  33 S is changed to the P-polarization state, after the first, second and third color components  31 S,  32 S,  33 S pass through the first light polarization selector  21 . The polarization beam splitter prism  23  receives the S-polarized first and second color components  31 S,  32 S and the P-polarized third color component  33 P from the first light polarization selector  21  at the first side thereof, reflects the S-polarized first and second color components  31 S,  32 S in a transverse direction such that the S-polarized first and second color components  31 S,  32 S pass through the second side thereof toward the color synthesizing prism  25 , permits a large portion of the P-polarized third color component  33 P to pass directly through the third side thereof toward the third light modulator  28 , and reflects a small portion of the P-polarized third color component  33 P, i.e. the light leakage component  33 P′, in the transverse direction so as to pass through the second side thereof. 
     The color synthesizing prism  25  receives the S-polarized first and second color components  31 S,  32 S and the light leakage component  33 P′ from the polarization beam splitter prism  23 . The color synthesizing prism  25  reflects the S-polarized first color component  31 S in a first transverse direction for reception by the first light modulator  26 , and further reflects the S-polarized second color component  32 S in a second transverse direction for reception by the second light modulator  27 . The color synthesizing prism  25  permits the light leakage component  33 P′ to pass directly therethrough, thereby preventing the light leakage component  33 P′ from reaching the projection lens (not shown). A shift in the gray scale coordinate of the projected image can thus be avoided to enhance both the image contrast and the output quality of the projection display  2 . 
     The first and second light modulators  26 ,  27  receive the S-polarized first and second color components  31 S,  32 S from the color synthesizing prism  25 , modulate the respective one of the S-polarized first and second color components  31 S,  32 S, and change the polarization state of the respective one of the first and second color components  31 S,  32 S from the S-polarization state to the P-polarization state when the first and second light modulators  26 ,  27  are activated. The first and second light modulators  26 ,  27  then reflect the corresponding modulated P-polarized color component  31 P,  32 P back to the color synthesizing prism  25  for reception by the polarization beam splitter prism  23 . 
     The third light modulator  28  receives the P-polarized third color component  33 P from the polarization beam splitter prism  23 , modulates the P-polarized third color component  33 P, and changes the polarization state of the third color component  33 P from the P-polarization state to the S-polarization state when the third light modulator  28  is activated. The third light modulator  28  reflects the modulated S-polarized third color component  33 S back to the polarization beam splitter prism  23 . 
     The modulated P-polarized first and second color components  31 P,  32 P from the color synthesizing prism  25  will be allowed by the polarization beam splitter prism  23  to pass directly through the fourth side thereof for reception by the second light polarization selector  22 . The modulated S-polarized third color component  33 S from the third light modulator  28  will be reflected by the polarization beam splitter prism  23  in a transverse direction so as to pass through the fourth side thereof for reception by the second light polarization selector  22 . 
     When the second light polarization selector  22  receives the modulated first, second and third color components  31 P,  32 P,  33 S from the polarization beam splitter prism  23 , the polarization state of the modulated P-polarized first and second color components  31 P,  32 P remains unaltered, whereas the polarization state of the modulated S-polarized third color component  33 S will be changed to the P-polarization state. 
     The polarizer  24  permits only pure P-polarized color components to pass therethrough, and absorbs S-polarized color components. The modulated P-polarized first, second and third color components  31 P,  32 P,  33 P from the polarizer  24  are recombined as they pass through the projection lens (not shown) for projecting a color image on a display screen (not shown). 
     Referring to FIG. 3, the second preferred embodiment of a projection display  4  according to the present invention is shown to include a first light polarization selector  41 , a second light polarization selector  42 , a polarization beam splitter prism  43 , a polarizer  44 , a first light modulator  46 , a second light modulator  47 , a third light modulator  48 , and a projection lens (not shown). However, unlike the first preferred embodiment, which uses a color synthesizing prism  25 , the projection display  4  of this embodiment additionally includes a color splitter prism set formed from first and second color splitter prisms  451 ,  452 . The first color splitter prism  451  has a first side formed as a total reflection interface  456 , a second side formed as a light splitting interface  457  and disposed adjacent to the first light modulator  46 , and a third side. The second color splitter prism  452  has a first side formed as a total reflection interface  454  and disposed adjacent to the second side of the polarization beam splitter prism  43 , a second side formed as a light splitting interface  455  and disposed adjacent to the total reflection interface  456  of the first color splitter prism  451 , and a third side disposed adjacent to the second light modulator  47 . The light splitting interface  455  of the second color splitter prism  452  cooperates with the total reflection interface  456  of the first color splitter prism  451  to form a clearance  453  therebetween. 
     In use, when the second color splitter prism  452  receives the second color component  52  from the polarization beam splitter prism  43  at the total reflection interface  454  thereof, the second color component  52  is subsequently reflected entirely back to the total reflection interface  454  by the second light splitting interface  455  so as to pass through the third side of the second color splitter prism  452  for reception by the second light modulator  47 . When activated, the second light modulator  47  modulates the second color component  52 , and changes the polarization state of the second color component  52 . The second light modulator  47  then reflects the modulated second color component  52  back to the second color splitter prism  452  for reception by the polarization beam splitter prism  43 . 
     On the other hand, when the second color splitter prism  452  receives the first color component  51  from the polarization beam splitter prism  43  at the total reflection interface  454  thereof, the first color component  51  will be allowed to pass directly through the light splitting interface  455  for reception by the first color splitter prism  451  at the total reflection interface  456  of the latter. The first color splitter prism  451  allows the first color component  51  to pass directly through the light splitting interface  457  for reception by the first light modulator  46 . When activated, the first light modulator  46  modulates the first color component  51 , and changes the polarization state of the first color component  51 . The first light modulator  46  then reflects the modulated first color component  51  back to the first color splitter prism  451  for passage through the second color splitter prism  452  prior to reception by the polarization beam splitter prism  43 . 
     Like the previous embodiment, the polarization beam splitter prism  43  permits a large portion of the third color component  53  to pass directly through the third side thereof toward the third light modulator  48  for processing in the manner described beforehand. A small portion of the third color component  53 , i.e. the light leakage component  53 ′, is reflected by the polarization beam splitter prism  43  in the transverse direction so as to be received by the second color splitter prism  452  at the total reflection interface  454  of the latter. The second color splitter prism  452  permits the light leakage component  53 ′ to pass directly through the color splitting interface  455  thereof for reception by the first color splitter prism  451  at the total reflection interface  456 . The light leakage component  53 ′ is reflected entirely by the color splitting interface  457  back to the total reflection interface  456 , which then reflects the same to pass through the third side of the first color splitter prism  451 , thereby preventing the light leakage component  53 ′ from reaching the projection lens (not shown). Stability in the gray scale coordinate of the projected image can thus be ensured to enhance both the image contrast and the output quality of the projection display  4 . 
     The modulated first, second and third color components  51 ,  52 ,  53  will be directed by the polarization beam splitter prism  43  to pass in sequence through the second light polarization selector  42 , the polarizer  44 , and the projection lens (not shown) for projecting a color image on a display screen (not shown). 
     FIG. 4 illustrates the third preferred embodiment of a projection display  6  according to the present invention, which is a modification of the second preferred embodiment. Like the embodiment of FIG. 3, a color splitter prism set includes first and second color splitter prisms  63 ,  64  that have a clearance  62  formed therebetween. However, unlike the previous embodiment, the first color splitter prism  63  has a first side formed as a total reflection interface  631  adjacent to the second color splitter prism  64 , a second side formed as a light splitting interface  632 , and a third side disposed adjacent to the first light modulator  65 . 
     When the first color component  71  passes through the second color splitter prism  64  for reception by the first color splitter prism  63  at the total reflection interface  631  thereof, the first color component  71  will be reflected entirely by the color splitting interface  632  back to the total reflection interface  631 , which then reflects the same to pass through the third side of the first color splitter prism  63  for processing by the first light modulator  65 . On the other hand, the small portion of the third color component  73 , i.e. the light leakage component  73 ′, that is reflected by the polarization beam splitter prism  61  and that passes through the second color splitter prism  64 , will be allowed by the first color splitter prism  63  to pass through the total reflection interface  631  and the light splitting interface  632 , thereby preventing the light leakage component  73 ′ from reaching the projection lens (not shown) of the projection display  6 . 
     It has thus been shown that, with the inclusion of a color synthesizing prism or a color splitter prism set, the light leakage component that is attributed to the third color component can be prevented from reaching the projection lens so as to ensure stability in the gray scale coordinate of the projected image and to thereby enhance both the image contrast and the output quality of the projection display. 
     FIG. 5 illustrates a first light polarization selector that is suitable for use in any one of the projection displays of FIGS. 2,  3  and  4 . As shown, a first dichroic mirror  91  receives S-polarization first, second and third color components Rs, Bs, Gs. The first dichroic mirror  91  allows the first and second color components Rs, Bs to pass therethrough in a first direction, and reflects the third color component Gs such that the third color components Gs travels in a second direction transverse to the first direction. A first reflective mirror  92  reflects the first and second color components Rs, Bs from the first dichroic mirror  91  such that the first and second color components Rs, Bs travel in a third direction transverse to the first direction and parallel to the second direction. A second reflective mirror  93  reflects the third color component Gs from the first dichroic mirror  91  such that the third color component Gs travels in a fourth direction transverse to the second direction and parallel to the first direction. A second dichroic mirror  94  receives the first and second color components Rs, Bs from the first reflective mirror  92 . A half-wavelength plate  95  is disposed between the second reflective mirror  93  and the second dichroic mirror  94  and converts the polarization state of the third color component from S-polarization to P-polarization before the third color component reaches the second dichroic mirror  94 . As such, the input light beam can be obtained by the polarization beam splitter prism of the projection display from the second dichroic mirror  94  of the first light polarization selector, and includes the S-polarization first and second color components Rs, Bs and the P-polarization third color component Gp. 
     While the present invention has been described in connection with what is considered the most practical and preferred embodiments, it is understood that this invention is not limited to the disclosed embodiments but is intended to cover various arrangements included within the spirit and scope of the broadest interpretation so as to encompass all such modifications and equivalent arrangements.