User:Tohline/Appendix/CGH/KAH2001 - Revision history

Tohline: /* From Our Separate Discussion of CGH */

2020-03-31T16:51:28Z

From Our Separate Discussion of CGH

← Older revision		Revision as of 16:51, 31 March 2020
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	dY \, .		dY \, .
	</math>		</math>
			</td>
			</tr>
			</table>

			Therefore, in order that the two expressions to match, we need to ignore the quadratic (Y<sup>2</sup>) term inside the last exponential, and make the association,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~V(Y)</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~ \biggl( \frac{2\pi i Z}{\lambda}\biggr) a_0(\Theta) e^{i\phi(\Theta)} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* From Our Separate Discussion of CGH */

2020-03-30T23:08:02Z

From Our Separate Discussion of CGH

← Older revision		Revision as of 23:08, 30 March 2020
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	<td align="left">		<td align="left">
	<math>~ (i k Z) \exp\biggl[i k \biggl( L-Z \biggr)\biggr]		<math>~ (i k Z) \exp\biggl[i k \biggl( L-Z \biggr)\biggr]
	\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY \, .		\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ (i k Z) \exp\biggl[\frac{i \pi y_1^2}{Z \lambda} \biggr]
			\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl\{ -i 2\pi Y \biggl(\frac{y_1 }{\lambda } \biggr) \frac{1}{Z}\biggl[1 - \frac{y_1^2}{2Z^2} \biggr] \biggr\} dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ \biggl( \frac{2\pi i Z}{\lambda}\biggr) \exp\biggl[\frac{i \pi y_1^2}{Z \lambda} \biggr]
			\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl\{ - \biggl(\frac{2\pi i y_1 Y}{Z \lambda } \biggr) \biggl[1 - \cancelto{0}{\frac{y_1^2}{2Z^2}} \biggr] \biggr\} dY \, ,
	</math>		</math>
	</td>		</td>
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	</table>		</table>

	~~Accounting~~ for the different variable notations that have been adopted in the two separate discussions, namely,		where we have used the approximate expressions,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~L - Z</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\frac{y_1^2}{2Z}</math>
			</td>
			<td align="center">      and
			<td align="right">
			<math>~\frac{1}{L}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\frac{1}{Z}\biggl[1 - \frac{y_1^2}{2Z^2} \biggr] \, .</math>
			</td>
			</tr>
			</table>

			Next, accounting for the different variable notations that have been adopted in the two separate discussions, namely,

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
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	\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{Z\lambda} \biggl[ - 2y_1Y + Y^2\biggr] \biggr\}		\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{Z\lambda} \biggl[ - 2y_1Y + Y^2\biggr] \biggr\}
	dY		dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \exp\biggl[ \frac{i \pi y_1^2}{Z\lambda} \biggr]
			\int_{-\infty}^{\infty} V(Y) \times
			\exp\biggl\{ - \frac{2\pi i y_1 Y}{Z\lambda} + \frac{i \pi Y^2}{Z\lambda} \biggr\}
			dY \, .
	</math>		</math>
	</td>		</td>

Tohline: /* From Our Separate Discussion of CGH */

2020-03-30T22:44:55Z

From Our Separate Discussion of CGH

← Older revision		Revision as of 22:44, 30 March 2020
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	</table>		</table>

	~~Let's demonstrate how similar this last integral expression is to~~ the ~~[[#Eq10KAH2001\|key integral from KAH2001~~ that ~~has~~ been ~~presented above]]~~, namely,		Accounting for the different variable notations that have been adopted in the two separate discussions, namely,

	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~I_\xi(x)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~ \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2}~~
	~~\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl[ \frac{i \pi \alpha^2}{2} \biggr] d\alpha \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
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	<tr>		<tr>
	<td align="center" colspan="1">KAH2001</td>		<td align="center" colspan="1">KAH2001</td>
	<td align="center" colspan="1">Our's</td>		<td align="center" colspan="1">Our</td>
	</tr>		</tr>

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	<tr>		<tr>
	<td align="center" colspan="1"><math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} } - \sqrt{2d\lambda} ~\mu </math></td>		<td align="center" colspan="1"><math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} } - \sqrt{2d\lambda} ~\biggl(\frac{x}{d\lambda}\biggr) </math></td>
	<td align="center" colspan="1"><math>~\frac{\sqrt{2} Y}{ \sqrt{Z \lambda} } - \sqrt{2Z \lambda} ~\mu </math></td>		<td align="center" colspan="1"><math>~\frac{\sqrt{2} Y}{ \sqrt{Z \lambda} } - \sqrt{2Z \lambda} ~\biggl(\frac{y_1}{Z \lambda}\biggr) </math></td>
			</tr>
			</table>


			let's examine how similar this last integral expression is to the [[#Eq10KAH2001\|key integral from KAH2001 that has been presented above]]. We have,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~I_\xi(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2}
			\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl[ \frac{i \pi \alpha^2}{2} \biggr] d\alpha
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~ \biggl( \frac{Z \lambda}{2}\biggr)^{1 / 2}
			\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{2}
			\biggl[ \frac{\sqrt{2} Y}{ \sqrt{Z \lambda} } - \sqrt{2Z \lambda} ~\biggl(\frac{y_1}{Z \lambda}\biggr) \biggr]^2 \biggr\}
			\biggl[ \frac{2}{Z\lambda} \biggr]^{1 / 2}dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{2}
			\biggl[ \frac{2Y^2}{ Z \lambda } - \frac{2\sqrt{2} Y}{ \sqrt{Z \lambda} } \cdot \sqrt{2Z \lambda} ~\biggl(\frac{y_1}{Z \lambda}\biggr) + 2Z \lambda ~\biggl(\frac{y_1}{Z \lambda}\biggr)^2\biggr] \biggr\}
			dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{Z\lambda}
			\biggl[ Y^2 - 2y_1Y + y_1^2 \biggr] \biggr\}
			dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \exp\biggl[ \frac{i \pi y_1^2}{Z\lambda} \biggr]
			\int_{-\infty}^{\infty} V(Y) \times \exp\biggl\{ \frac{i \pi}{Z\lambda} \biggl[ - 2y_1Y + Y^2\biggr] \biggr\}
			dY
			</math>
			</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* From Our Separate Discussion of CGH */

2020-03-30T22:18:54Z

From Our Separate Discussion of CGH

← Older revision		Revision as of 22:18, 30 March 2020
Line 445:		Line 445:
	<tr>		<tr>
	<td align="center" colspan="1"><math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} } - \sqrt{2d\lambda} ~\mu </math></td>		<td align="center" colspan="1"><math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} } - \sqrt{2d\lambda} ~\mu </math></td>
	<td align="center" colspan="1">---</td>		<td align="center" colspan="1"><math>~\frac{\sqrt{2} Y}{ \sqrt{Z \lambda} } - \sqrt{2Z \lambda} ~\mu </math></td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Hologram Reconstruction Using a Digital Micromirror Device */

2020-03-30T16:48:33Z

Hologram Reconstruction Using a Digital Micromirror Device

← Older revision		Revision as of 16:48, 30 March 2020
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	</table>		</table>

	Let's demonstrate how similar this last integral expression is to the [[#Eq10KAH2001\|key integral from KAH2001 that has been ~~prsented,~~ above]].		Let's demonstrate how similar this last integral expression is to the [[#Eq10KAH2001\|key integral from KAH2001 that has been presented above]], namely,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~I_\xi(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2}
			\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl[ \frac{i \pi \alpha^2}{2} \biggr] d\alpha \, .
			</math>
			</td>
			</tr>
			</table>

			<table border="1" align="center" cellpadding="8">
			<tr>
			<td align="center" colspan="2">Notation</td>
			</tr>

			<tr>
			<td align="center" colspan="1">KAH2001</td>
			<td align="center" colspan="1">Our's</td>
			</tr>

			<tr>
			<td align="center" colspan="1"><math>~x</math></td>
			<td align="center" colspan="1"><math>~y_1</math></td>
			</tr>

			<tr>
			<td align="center" colspan="1"><math>~\xi</math></td>
			<td align="center" colspan="1"><math>~Y</math></td>
			</tr>

			<tr>
			<td align="center" colspan="1"><math>~d</math></td>
			<td align="center" colspan="1"><math>~Z</math></td>
			</tr>

			<tr>
			<td align="center" colspan="1"><math>~\lambda</math></td>
			<td align="center" colspan="1"><math>~\frac{2\pi}{k}</math></td>
			</tr>

			<tr>
			<td align="center" colspan="1"><math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} } - \sqrt{2d\lambda} ~\mu </math></td>
			<td align="center" colspan="1">---</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* From Our Separate Discussion of CGH */

2020-03-30T16:34:45Z

From Our Separate Discussion of CGH

← Older revision		Revision as of 16:34, 30 March 2020
Line 388:		Line 388:
	<td align="left">		<td align="left">
	<math>~ (i k Z) \exp\biggl[i k \biggl( L-Z \biggr)\biggr]		<math>~ (i k Z) \exp\biggl[i k \biggl( L-Z \biggr)\biggr]
	\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY		\int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>

			Let's demonstrate how similar this last integral expression is to the [[#Eq10KAH2001\|key integral from KAH2001 that has been prsented, above]].

	=See Also=		=See Also=

Tohline: /* Optical Field in the Image Plane */

2020-03-30T16:32:33Z

Optical Field in the Image Plane

← Older revision		Revision as of 16:32, 30 March 2020
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	</table>		</table>

	the expression for <math>~I_\xi(x)</math> may be written as,		<span id="Eq10KAH2001">the expression for <math>~I_\xi(x)</math> may be written as,</span>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	The expression for <math>~I_\eta(y)</math> may be rewritten similarly.		The expression for <math>~I_\eta(y)</math> may be rewritten similarly.


	==SWMED==		==SWMED==

Tohline: /* SWMED */

2020-03-29T18:47:51Z

SWMED

← Older revision		Revision as of 18:47, 29 March 2020
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	In §4 of [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03] we find this general description:		In §4 of [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03] we find this general description:

	The following integral expression is "<font color="orange">… the mathematical transform containing the wave physics of monochromatic light emanating from each object point, passing through the optical system and being superimposed at each point in the holographic plane. It represents the integration over the object of spherical wave solutions of the Helmholtz form of the wave equation … with additional phase shifts due to a spherical converging lens in the light pathway.</font>"		The following integral expression is "<font color="orange">… the mathematical transform containing the wave physics of monochromatic light emanating from each object point, passing through the optical system and being superimposed at each point in the holographic plane. It represents the integration over the object of spherical wave solutions of the [https://en.wikipedia.org/wiki/Helmholtz_equation Helmholtz form of the wave equation] … with additional phase shifts due to a spherical converging lens in the light pathway.</font>"

	<table border="1" align="center" cellpadding="10" width="70%"><tr><td align="center">		<table border="1" align="center" cellpadding="10" width="70%"><tr><td align="center">
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	[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"		[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"

	"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> — see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, & R. Höfling (2001)], which is [[#Optical_Field_in_the_Image_Plane\|discussed above]]. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the Fresnel approximation in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"		"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> — see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, & R. Höfling (2001)], which is [[#Optical_Field_in_the_Image_Plane\|discussed above]]. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the [https://en.wikipedia.org/wiki/Fresnel_diffraction#The_Fresnel_approximation Fresnel approximation] in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"

	==From Our Separate Discussion of CGH==		==From Our Separate Discussion of CGH==

Tohline: /* SWMED */

2020-03-29T17:18:37Z

SWMED

← Older revision		Revision as of 17:18, 29 March 2020
Line 289:		Line 289:
	[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"		[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"

	"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> — see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, & R. Höfling (2001)], which is discussed above. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the Fresnel approximation in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"		"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> — see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, & R. Höfling (2001)], which is [[#Optical_Field_in_the_Image_Plane\|discussed above]]. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the Fresnel approximation in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"

	==From Our Separate Discussion of CGH==		==From Our Separate Discussion of CGH==

Tohline: /* SWMED */

2020-03-29T17:17:28Z

SWMED

← Older revision		Revision as of 17:17, 29 March 2020
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	[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"		[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.] As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">… <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> — i.e., immediately in front of the aperture — <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' … The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume. The wave number of the light is given by k and f is the focal length of the converging lens.</font>"

	"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity &~~hellip~~; The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the Fresnel approximation in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"		"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system … Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field. Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> — see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, & R. Höfling (2001)], which is discussed above. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point. We do not approximate this distance in our calculation, but we have incorporated the Fresnel approximation in the amplitude term in front of the exponential … Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"

	==From Our Separate Discussion of CGH==		==From Our Separate Discussion of CGH==