User:Tohline/Appendix/CGH/Overview - Revision history

Tohline: /* CGH: Design Overview */

2017-12-30T01:55:00Z

CGH: Design Overview

← Older revision		Revision as of 01:55, 30 December 2017
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	=CGH: Design Overview=		=CGH: Design Overview=


			{{LSU_HBook_header}}


	Here we outline a research project that builds upon the extraordinary work of [http://www.phys.lsu.edu/~tohline/phys4412/howto/DynamicHolographic3dImageProjection.pdf Huebschman, Manjuluri, & Garner (2003)] titled, ''Dynamic holographic 3-D image projection''.		Here we outline a research project that builds upon the extraordinary work of [http://www.phys.lsu.edu/~tohline/phys4412/howto/DynamicHolographic3dImageProjection.pdf Huebschman, Manjuluri, & Garner (2003)] titled, ''Dynamic holographic 3-D image projection''.
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	</table>		</table>

	The objective is to create a fully three-dimensional (3D) ''rendering'' of the time-dependent surface of an object in 3-space, using efficiently calculated, two-dimensional digital holographic images. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the ~~separate partial~~		The objective is to create a fully three-dimensional (3D) ''rendering'' of the time-dependent surface of an object in 3-space, using efficiently calculated, two-dimensional digital holographic images. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small, rectangular mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the diffraction patterns that are generated separately, but simultaneously, by all of the small mirrors in the array.



	~~{{LSU_HBook_header}}~~


	==Slit Diffraction==		==Slit Diffraction==

Tohline: /* CGH: Design Overview */

2017-12-30T00:59:05Z

CGH: Design Overview

← Older revision		Revision as of 00:59, 30 December 2017
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	=CGH: Design Overview=		=CGH: Design Overview=

	Here we ~~provide an~~ outline of a research project that builds upon the extraordinary work of [http://www.phys.lsu.edu/~tohline/phys4412/howto/DynamicHolographic3dImageProjection.pdf Huebschman, Manjuluri, & Garner (2003)] titled, ''Dynamic holographic 3-D image projection''.		Here we outline a research project that builds upon the extraordinary work of [http://www.phys.lsu.edu/~tohline/phys4412/howto/DynamicHolographic3dImageProjection.pdf Huebschman, Manjuluri, & Garner (2003)] titled, ''Dynamic holographic 3-D image projection''.

	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
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	</table>		</table>

	~~Our~~ objective is to create a fully three-dimensional (3D) ''~~image~~'' of the surface of an object in 3-space. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the separate partial		The objective is to create a fully three-dimensional (3D) ''rendering'' of the time-dependent surface of an object in 3-space, using efficiently calculated, two-dimensional digital holographic images. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the separate partial

Tohline: /* CGH: Design Overview */

2017-12-29T23:05:44Z

CGH: Design Overview

← Older revision		Revision as of 23:05, 29 December 2017
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	=CGH: Design Overview=		=CGH: Design Overview=

			Here we provide an outline of a research project that builds upon the extraordinary work of [http://www.phys.lsu.edu/~tohline/phys4412/howto/DynamicHolographic3dImageProjection.pdf Huebschman, Manjuluri, & Garner (2003)] titled, ''Dynamic holographic 3-D image projection''.

			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="center">[[File:Output_seal_and_lsu_001.jpg\|300px\|hologram]]</td>
			<td align="center">[[File:Lsucghfocussed.jpg\|300px\|Photo focused on LSU logo]]</td>
			<td align="center">[[File:sealfocussed.jpg\|300px\|Photo focused on Louisiana seal]]</td>
			</tr>
			</table>

	Our objective is to create a fully three-dimensional (3D) ''image'' of the surface of an object in 3-space. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the separate partial		Our objective is to create a fully three-dimensional (3D) ''image'' of the surface of an object in 3-space. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the separate partial

Tohline: /* CGH: Philosophical Overview */

2017-12-29T18:57:38Z

CGH: Philosophical Overview

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	=CGH: ~~Philosophical~~ Overview=		=CGH: Design Overview=

			Our objective is to create a fully three-dimensional (3D) ''image'' of the surface of an object in 3-space. This will be accomplished by bouncing a monochromatic beam of light — the light source being, for example, an R, G, or B laser — off of a two-dimensional (2D) array of identical, small mirrors such as the mirror configuration that provides the central functional element of Texas Instrument's DLP chip. Each mirror will be uniformly illuminated, but the brightness of the light that is incident on each mirror will, in general, vary from mirror to mirror. Ultimately, the desired 3D image will be constructed by adding together the all of the separate partial



	{{LSU_HBook_header}}		{{LSU_HBook_header}}


	==Slit Diffraction==		==Slit Diffraction==

Tohline: /* Simplifications */

2017-12-29T04:34:50Z

Simplifications

← Older revision		Revision as of 04:34, 29 December 2017
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	</table>		</table>
	</div>		</div>
	where,		where (see the associated [[User:Tohline/Appendix/Ramblings/FourierSeries#Example_.232\|''Example #2'' discussion]]),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="left">		<td align="left">
	<math>~\frac{\beta}{2c} \int_{-c}^c \cos\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>		<math>~\frac{\beta}{2c} \int_{-c}^c \cos\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\beta \biggl[
			\frac{\sin\theta_n }{\theta_n}
			\biggr] \, ,
			</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>~-~\frac{\beta}{2c} \int_{-c}^c \sin\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>		<math>~-~\frac{\beta}{2c} \int_{-c}^c \sin\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0\, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\theta_n</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{2\pi n \Delta y_1 c}{\lambda L_n} \, .</math>
			</td>
			<td align="center">

			</td>
			<td align="left">

	</td>		</td>
	</tr>		</tr>

Tohline: /* Simplifications */

2017-12-28T23:29:09Z

Simplifications

← Older revision		Revision as of 23:29, 28 December 2017
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	If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math>; and if we assume that, for all <math>~j</math>,		If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math> and if, furthermore, we assume that, for all <math>~j</math>,
	<div align="center">		<div align="center">
	<math>~\biggl\| \frac{Y_j}{L}\biggr\| \ll 1</math>		<math>~\biggl\| \frac{Y_j}{L}\biggr\| \ll 1 \, ,</math>
	</div>		</div>
	we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and [[User:Tohline/Appendix/CGH/ParallelApertures#Simplify\|deduce that]],		we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and [[User:Tohline/Appendix/CGH/ParallelApertures#Simplify\|deduce that]],
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	\, .		\, .
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>

			Let's consider the case of a (single) aperture of width, <math>~w = 2c</math>, that is uniformly illuminated — that is, <math>~\beta</math> is independent of location across the aperture. If we allow the summation index, <math>~j</math>, to go to infinity, this (simplified) amplitude expression can be integrated to give, at various discrete <math>~(n)</math> locations along the image screen,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(n \Delta y_1)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			e^{i2\pi L_n/\lambda} \biggl[ a_n + i b_n\biggr] \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\beta}{2c} \int_{-c}^c \cos\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~b_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-~\frac{\beta}{2c} \int_{-c}^c \sin\biggl( \frac{2\pi n \Delta y_1 x}{\lambda L_n} \biggr) dx</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Single Aperture */

2017-12-28T22:19:47Z

Single Aperture

← Older revision		Revision as of 22:19, 28 December 2017
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	</table>		</table>
	</td></tr></table>		</td></tr></table>
	As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#One-Dimensional_Aperture\|accompanying discussion]], we ~~consider~~, first, the amplitude (and phase) of light ~~that is incident at a location~~ <math>~~~y_1~~</math> on an image screen that is located a distance <math>~Z</math> from ~~a slit of width~~ <math>~~~w = (Y_1 - Y_2) = 2c~~</math>~~. The amplitude~~ is given by the expression,		As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#One-Dimensional_Aperture\|accompanying discussion]], we examine, first, the amplitude (and phase) of light which, after passing through an aperture of width <math>~w = (Y_1 - Y_2) = 2c</math>, is incident on an image screen that is parallel to, and located a distance <math>~Z</math> from, the aperture. The amplitude at any location, <math>~y_1</math>, along the image screen is given by the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~A(~~n\Delta~~ y_1)</math>		<math>~A(y_1)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~\sum_j		<math>~\sum_j
	~~a_j~~ \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) + i \sin\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) \biggr]		\beta_j \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) + i \sin\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) \biggr]
	\, ,		\, ,
	</math>		</math>
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	</table>		</table>
	</div>		</div>
	where, <math>~~~a_j~~ = a(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,		where, <math>~\beta_j = \beta(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,
	<div align="center" id="Distance">		<div align="center" id="Distance">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we also will find that the various natural lengths of this problem ~~are~~ can be related via the expression,
			The green solid curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2\|Figures 2 & 3 of our accompanying, more detailed discussion]] display — for a fixed set of values of the parameter-pair, <math>~(\lambda, w)</math> — how the amplitude of the incident light, <math>~(A\cdot A^*)^{1 / 2}</math>, varies across an image screen that is placed at two different distances from the aperture.

			Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As [[#Simplifications\|discussed below]], our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. (The red dotted curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2\|Figures 2 & 3 of our accompanying discussion]] illustrate how the image-screen diffraction pattern is altered as a result of this simplification.) Via a related simplification, we also will find that the various natural lengths of this problem can be related to one another via the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.		CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.

			===Simplifications===


			If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math>; and if we assume that, for all <math>~j</math>,
			<div align="center">
			<math>~\biggl\| \frac{Y_j}{L}\biggr\| \ll 1</math>
			</div>
			we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and [[User:Tohline/Appendix/CGH/ParallelApertures#Simplify\|deduce that]],
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~e^{i2\pi L/\lambda} \sum_j
			\beta_j \biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr]
			\, .
			</math>
			</td>
			</tr>
			</table>
			</div>

	=See Also=		=See Also=

Tohline: /* Single Aperture */

2017-12-28T05:03:14Z

Single Aperture

← Older revision		Revision as of 05:03, 28 December 2017
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	</table>		</table>
	</div>		</div>
	where,		where, <math>~a_j = a(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,
	<div align="center" id="Distance">		<div align="center" id="Distance">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Single Aperture */

2017-12-28T04:55:47Z

Single Aperture

← Older revision		Revision as of 04:55, 28 December 2017
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~A(y_1)</math>		<math>~A(n\Delta y_1)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>
	</div>		</div>
	Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,		Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we also will find that the various natural lengths of this problem are can be related via the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~y_1\biggr\|_{1^\mathrm{st} \mathrm{fringe}}</math>		<math>~y_1\biggr\|_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\lambda Z}{w} \, .</math>		<math>~\frac{\lambda Z}{2c} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>

			CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.

	=See Also=		=See Also=

Tohline: /* Single Aperture */

2017-12-28T04:47:44Z

Single Aperture

← Older revision		Revision as of 04:47, 28 December 2017
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	[Z^2 + y_1^2 ]^{1 / 2} \, .		[Z^2 + y_1^2 ]^{1 / 2} \, .
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>
			Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~y_1\biggr\|_{1^\mathrm{st} \mathrm{fringe}}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\frac{\lambda Z}{w} \, .</math>
	</td>		</td>
	</tr>		</tr>